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The Signal Toolkit contains signal processing tools, including filtering, windowing and display functions.
The Signal Toolkit must be installed and then loaded to be used.
It can be installed in GNU Octave directly from octave-forge,
If running in Windows, the package may already be installed, to check run:
pkg list signal
With an internet connection available, the Signal package can be installed from octave-forge using the following command within GNU Octave:
pkg install -forge signal
The latest released version of the toolkit will be downloaded and installed.
Otherwise, if the package file has already been downloaded it can be installed using the follwoing command in GNU Octave:
pkg install signal-1.4.6.tar.gz
Regardless of the method of installing the toolkit, in order to use its functions, the toolkit must be loaded using the pkg load command:
pkg load signal
The toolkit must be loaded on each GNU Octave session.
y = buffer (x, n, p, opt) ¶[y, z, opt] = buffer (…) ¶Buffer a signal into a data frame. The arguments to buffer are
The data to be buffered.
The number of rows in the produced data buffer. This is an positive integer value and must be supplied.
An integer less than n that specifies the under- or overlap between column in the data frame. The default value of p is 0.
In the case of an overlap, opt can be either a vector of length p or the string ’nodelay’. If opt is a vector, then the first p entries in y will be filled with these values. If opt is the string ’nodelay’, then the first value of y corresponds to the first value of x.
In the can of an underlap, opt must be an integer between 0 and
-p. The represents the initial underlap of the first
column of y.
The default value for opt the vector zeros (1, p)
in the case of an overlap, or 0 otherwise.
In the case of a single output argument, y will be padded with zeros to fill the missing values in the data frame. With two output arguments z is the remaining data that has not been used in the current data frame.
Likewise, the output opt is the overlap, or underlap that might
be used for a future call to code to allow continuous buffering.
(t) ¶(t, f0) ¶(t, f0, t1) ¶(t, f0, t1, f1) ¶(t, f0, t1, f1, shape) ¶(t, f0, t1, f1, shape, phase) ¶Evaluate a chirp signal at time t. A chirp signal is a frequency swept cosine wave.
vector of times to evaluate the chirp signal
frequency at time t=0 [ 0 Hz ]
time t1 [ 1 sec ]
frequency at time t=t1 [ 100 Hz ]
shape of frequency sweep ’linear’ f(t) = (f1-f0)*(t/t1) + f0 ’quadratic’ f(t) = (f1-f0)*(t/t1)^2 + f0 ’logarithmic’ f(t) = (f1/f0)^(t/t1) * f0
phase shift at t=0
For example:
specgram (chirp ([0:0.001:5])); # default linear chirp of 0-100Hz in 1 sec specgram (chirp ([-2:0.001:15], 400, 10, 100, "quadratic")); soundsc (chirp ([0:1/8000:5], 200, 2, 500, "logarithmic"), 8000);
If you want a different sweep shape f(t), use the following:
y = cos (2 * pi * integral (f(t)) + phase);
[psi, x] = cmorwavf (lb, ub, n, fb, fc) ¶Compute the Complex Morlet wavelet.
y = diric (x,n) ¶Compute the dirichlet function.
See also: sinc, gauspuls, sawtooth.
y = pulstran (t, d, func, …) ¶y = pulstran (t, d, p) ¶y = pulstran (t, d, p, Fs) ¶y = pulstran (t, d, p, Fs, method) ¶Generate the signal y=sum(func(t+d,...)) for each d. If d is a matrix of two columns, the first column is the delay d and the second column is the amplitude a, and y=sum(a*func(t+d)) for each d,a. Clearly, func must be a function which accepts a vector of times. Any extra arguments needed for the function must be tagged on the end.
Example:
fs = 11025; # arbitrary sample rate f0 = 100; # pulse train sample rate w = 0.001; # pulse width of 1 millisecond auplot (pulstran (0:1/fs:0.1, 0:1/f0:0.1, "rectpuls", w), fs);
If instead of a function name you supply a pulse shape sampled at frequency Fs (default 1 Hz), an interpolated version of the pulse is added at each delay d. The interpolation stays within the the time range of the delayed pulse. The interpolation method defaults to linear, but it can be any interpolation method accepted by the function interp1.
Example:
fs = 11025; # arbitrary sample rate f0 = 100; # pulse train sample rate w = boxcar(10); # pulse width of 1 millisecond at 10 kHz auplot (pulstran (0:1/fs:0.1, 0:1/f0:0.1, w, 10000), fs);
y = rectpuls (t) ¶y = rectpuls (t, w) ¶Generate a rectangular pulse over the interval [-w/2,w/2),
sampled at times t. This is useful with the function pulstran
for generating a series of pulses.
Example:
fs = 11025; # arbitrary sample rate f0 = 100; # pulse train sample rate w = 0.3/f0; # pulse width 3/10th the distance between pulses plot (pulstran (0:1/fs:4/f0, 0:1/f0:4/f0, "rectpuls", w));
See also: gauspuls, pulstran, tripuls.
y = sawtooth (t) ¶y = sawtooth (t, width) ¶Generates a sawtooth wave of period 2 * pi with limits +1/-1
for the elements of t.
width is a real number between 0 and 1 which specifies
the point between 0 and 2 * pi where the maximum is. The
function increases linearly from -1 to 1 in [0, 2 *
pi * width] interval, and decreases linearly from 1 to
-1 in the interval [2 * pi * width, 2 * pi].
If width is 0.5, the function generates a standard triangular wave.
If width is not specified, it takes a value of 1, which is a standard sawtooth function.
[psi, x] = shanwavf (lb, ub, n, fb, fc) ¶Compute the Complex Shannon wavelet.
[y s] = sigmoid_train (t, ranges, rc) ¶Evaluate a train of sigmoid functions at t.
The number and duration of each sigmoid is determined from ranges.
Each row of ranges represents a real interval, e.g. if sigmoid
i starts at t=0.1 and ends at t=0.5, then
ranges(i,:) = [0.1 0.5].
The input rc is an array that defines the rising and falling time
constants of each sigmoid. Its size must equal the size of ranges.
The individual sigmoids are returned in s. The combined sigmoid train
is returned in the vector y of length equal to t, and such that
Y = max (S).
Run demo sigmoid_train to some examples of the use of this function.
(x) ¶(x, n) ¶(x, n, Fs) ¶(x, n, Fs, window) ¶(x, n, Fs, window, overlap) ¶[S, f, t] = specgram (…) ¶Generate a spectrogram for the signal x. The signal is chopped into
overlapping segments of length n, and each segment is windowed and
transformed into the frequency domain using the FFT. The default segment
size is 256. If fs is given, it specifies the sampling rate of the
input signal. The argument window specifies an alternate window to
apply rather than the default of hanning (n). The argument
overlap specifies the number of samples overlap between successive
segments of the input signal. The default overlap is
length (window)/2.
If no output arguments are given, the spectrogram is displayed. Otherwise, S is the complex output of the FFT, one row per slice, f is the frequency indices corresponding to the rows of S, and t is the time indices corresponding to the columns of S.
Example:
x = chirp([0:0.001:2],0,2,500); # freq. sweep from 0-500 over 2 sec.
Fs=1000; # sampled every 0.001 sec so rate is 1 kHz
step=ceil(20*Fs/1000); # one spectral slice every 20 ms
window=ceil(100*Fs/1000); # 100 ms data window
specgram(x, 2^nextpow2(window), Fs, window, window-step);
## Speech spectrogram
[x, Fs] = auload(file_in_loadpath("sample.wav")); # audio file
step = fix(5*Fs/1000); # one spectral slice every 5 ms
window = fix(40*Fs/1000); # 40 ms data window
fftn = 2^nextpow2(window); # next highest power of 2
[S, f, t] = specgram(x, fftn, Fs, window, window-step);
S = abs(S(2:fftn*4000/Fs,:)); # magnitude in range 0<f<=4000 Hz.
S = S/max(S(:)); # normalize magnitude so that max is 0 dB.
S = max(S, 10^(-40/10)); # clip below -40 dB.
S = min(S, 10^(-3/10)); # clip above -3 dB.
imagesc (t, f, log(S)); # display in log scale
set (gca, "ydir", "normal"); # put the 'y' direction in the correct direction
The choice of window defines the time-frequency resolution. In speech for example, a wide window shows more harmonic detail while a narrow window averages over the harmonic detail and shows more formant structure. The shape of the window is not so critical so long as it goes gradually to zero on the ends.
Step size (which is window length minus overlap) controls the horizontal scale of the spectrogram. Decrease it to stretch, or increase it to compress. Increasing step size will reduce time resolution, but decreasing it will not improve it much beyond the limits imposed by the window size (you do gain a little bit, depending on the shape of your window, as the peak of the window slides over peaks in the signal energy). The range 1-5 msec is good for speech.
FFT length controls the vertical scale. Selecting an FFT length greater than the window length does not add any information to the spectrum, but it is a good way to interpolate between frequency points which can make for prettier spectrograms.
After you have generated the spectral slices, there are a number of decisions for displaying them. First the phase information is discarded and the energy normalized:
S = abs(S); S = S/max(S(:));
Then the dynamic range of the signal is chosen. Since information in speech is well above the noise floor, it makes sense to eliminate any dynamic range at the bottom end. This is done by taking the max of the magnitude and some minimum energy such as minE=-40dB. Similarly, there is not much information in the very top of the range, so clipping to a maximum energy such as maxE=-3dB makes sense:
S = max(S, 10^(minE/10)); S = min(S, 10^(maxE/10));
The frequency range of the FFT is from 0 to the Nyquist frequency of one half the sampling rate. If the signal of interest is band limited, you do not need to display the entire frequency range. In speech for example, most of the signal is below 4 kHz, so there is no reason to display up to the Nyquist frequency of 10 kHz for a 20 kHz sampling rate. In this case you will want to keep only the first 40% of the rows of the returned S and f. More generally, to display the frequency range [minF, maxF], you could use the following row index:
idx = (f >= minF & f <= maxF);
Then there is the choice of colormap. A brightness varying colormap such as copper or bone gives good shape to the ridges and valleys. A hue varying colormap such as jet or hsv gives an indication of the steepness of the slopes. The final spectrogram is displayed in log energy scale and by convention has low frequencies on the bottom of the image:
imagesc(t, f, flipud(log(S(idx,:))));
s = square (t, duty) ¶s = square (t) ¶Generate a square wave of period 2 pi with limits +1/-1.
If duty is specified, it is the percentage of time the square wave is "on". The square wave is +1 for that portion of the time.
on time * 100
duty cycle = ------------------
on time + off time
See also: cos, sawtooth, sin, tripuls.
y = tripuls (t) ¶y = tripuls (t, w) ¶y = tripuls (t, w, skew) ¶Generate a triangular pulse over the interval [-w/2,w/2),
sampled at times t. This is useful with the function pulstran
for generating a series of pulses.
skew is a value between -1 and 1, indicating the relative placement of the peak within the width. -1 indicates that the peak should be at -w/2, and 1 indicates that the peak should be at w/2. The default value is 0.
Example:
fs = 11025; # arbitrary sample rate f0 = 100; # pulse train sample rate w = 0.3/f0; # pulse width 3/10th the distance between pulses plot (pulstran (0:1/fs:4/f0, 0:1/f0:4/f0, "tripuls", w));
See also: gauspuls, pulstran, rectpuls.
[out] = unshiftdata (in, perm, shifts) ¶Reverse what is done by shiftdata.
See also: shiftdata.
y = vco (x, fc, fs) ¶y = vco (x, [fmin, fmax], fs) ¶Creates a signal that oscillates at a frequency determined by input x with a sampling frequency fs.
Inputs:
Outputs:
[pks, loc, extra] = findpeaks (data) ¶… = findpeaks (…, property, value) ¶… = findpeaks (…, "DoubleSided") ¶Finds peaks on data.
Peaks of a positive array of data are defined as local maxima. For double-sided data, they are maxima of the positive part and minima of the negative part. data is expected to be a single column vector.
The function returns the value of data at the peaks in pks. The index indicating their position is returned in loc.
The third output argument is a structure with additional information:
A structure containing the parabola fitted to each returned peak. The structure has two fields, "x" and "pp". The field "pp" contains the coefficients of the 2nd degree polynomial and "x" the extrema of the interval where it was fitted.
The estimated height of the returned peaks (in units of data).
The height at which the roots of the returned peaks were calculated (in units of data).
The abscissa values (in index units) at which the parabola fitted to each of the returned peaks realizes its width as defined below.
This function accepts property-value pair given in the list below:
Minimum peak height (non-negative scalar). Only peaks that exceed this
value will be returned. For data taking positive and negative values
use the option "DoubleSided". Default value eps.
Minimum separation between (positive integer). Peaks separated by less than this distance are considered a single peak. This distance is also used to fit a second order polynomial to the peaks to estimate their width, therefore it acts as a smoothing parameter. The neighborhood size is equal to the value of "MinPeakDistance". Default value 1.
Minimum width of peaks (positive integer). The width of the peaks is estimated using a parabola fitted to the neighborhood of each peak. The width is caulculated with the formula
a * (width - x0)^2 = 1
where a is the the concavity of the parabola and x0 its vertex. Default value 1.
Maximum width of peaks (positive integer).
Default value Inf.
Tells the function that data takes positive and negative values. The base-line for the peaks is taken as the mean value of the function. This is equivalent as passing the absolute value of the data after removing the mean.
Run demo findpeaks to see some examples.
y = peak2peak (x) ¶y = peak2peak (x, dim) ¶Compute the difference between the maximum and minimum values in the vector x.
If x is a matrix, compute the difference for each column and return them in a row vector.
If the optional argument dim is given, operate along this dimension.
See also: max, min, peak2rms, rms, rssq.
y = peak2rms (x) ¶y = peak2rms (x, dim) ¶Compute the ratio of the largest absolute value to the root-mean-square (RMS) value of the vector x.
If x is a matrix, compute the peak-magnitude-to-RMS ratio for each column and return them in a row vector.
If the optional argument dim is given, operate along this dimension.
See also: max, min, peak2peak, rms, rssq.
y = rms (x) ¶y = rms (x, dim) ¶Compute the root-mean-square (RMS) of the vector x.
The root-mean-square is defined as
rms (x) = SQRT (1/N SUM_i x(i)^2)
If x is a matrix, compute the root-mean-square for each column and return them in a row vector.
If the optional argument dim is given, operate along this dimension.
See also: mean, meansq, peak2rms, rssq, sumsq.
y = rssq (x) ¶y = rssq (x, dim) ¶Compute the root-sum-of-squares (RSS) of the vector x.
The root-sum-of-squares is defined as
rssq (x) = SQRT (SUM_i x(i)^2)
If x is a matrix, compute the root-sum-of-squares for each column and return them in a row vector.
If the optional argument dim is given, operate along this dimension.
See also: mean, meansq, sumsq, rms.
c = cconv (a, b, n) ¶c = cconv (a, b) ¶Compute the modulo-N circular convolution.
a and b are input vectors and c is the modolo-n
convolution of a and b. If n is not provided,
its assumed default value is length(a) + length(b) - 1,
which provides the same result as a linear convolution.
Examples:
cconv (1:2, 1:4)
⇒ 1 4 7 10 8
cconv (1:2, 1:4, 2)
⇒ 16 14
cconv (1:2, 1:4, 4)
⇒ 9 4 7 10
See also: conv, circshift.
(a, n) ¶If a is a column vector and x is a column vector of length n, then
convmtx(a, n) * x
gives the convolution of of a and x and is the
same as conv(a, x). The difference is if
many vectors are to be convolved with the same vector, then
this technique is possibly faster.
Similarly, if a is a row vector and x is a row vector of length n, then
x * convmtx(a, n)
is the same as conv(x, a).
See also: conv.
[R, lag] = xcorr ( X ) ¶… = xcorr ( X, Y ) ¶… = xcorr ( …, maxlag) ¶… = xcorr ( …, scale) ¶Estimates the cross-correlation.
Estimate the cross correlation R_xy(k) of vector arguments X and Y or, if Y is omitted, estimate autocorrelation R_xx(k) of vector X, for a range of lags k specified by argument "maxlag". If X is a matrix, each column of X is correlated with itself and every other column.
The cross-correlation estimate between vectors "x" and "y" (of length N) for lag "k" is given by
N
R_xy(k) = sum x_{i+k} conj(y_i),
i=1
where data not provided (for example x(-1), y(N+1)) is zero. Note the
definition of cross-correlation given above. To compute a
cross-correlation consistent with the field of statistics, see xcov.
ARGUMENTS
[non-empty; real or complex; vector or matrix] data
[real or complex vector] data
If X is a matrix (not a vector), Y must be omitted. Y may be omitted if X is a vector; in this case xcorr estimates the autocorrelation of X.
[integer scalar] maximum correlation lag If omitted, the default value is N-1, where N is the greater of the lengths of X and Y or, if X is a matrix, the number of rows in X.
[character string] specifies the type of scaling applied to the correlation vector (or matrix). is one of:
return the unscaled correlation, R,
return the biased average, R/N,
return the unbiased average, R(k)/(N-|k|),
return the correlation coefficient, R/(rms(x).rms(y)), where "k" is the lag, and "N" is the length of X. If omitted, the default value is "none". If Y is supplied but does not have the same length as X, scale must be "none".
RETURNED VARIABLES
array of correlation estimates
row vector of correlation lags [-maxlag:maxlag]
The array of correlation estimates has one of the following forms: (1) Cross-correlation estimate if X and Y are vectors.
(2) Autocorrelation estimate if is a vector and Y is omitted.
(3) If X is a matrix, R is an matrix containing the cross-correlation estimate of each column with every other column. Lag varies with the first index so that R has 2*maxlag+1 rows and P^2 columns where P is the number of columns in X.
If Rij(k) is the correlation between columns i and j of X
R(k+maxlag+1,P*(i-1)+j) == Rij(k)
for lag k in [-maxlag:maxlag], or
R(:,P*(i-1)+j) == xcorr(X(:,i),X(:,j)).
reshape(R(k,:),P,P) is the cross-correlation matrix for X(k,:).
See also: xcov.
(a) ¶(a, b) ¶(…, scale) ¶Compute the 2D cross-correlation of matrices a and b.
If b is not specified, computes autocorrelation of a, i.e.,
same as xcorr (a, a).
The optional argument scale, defines the type of scaling applied to the cross-correlation matrix. Possible values are:
No scaling.
Scales the raw cross-correlation by the maximum number of elements of a and b involved in the generation of any element of c.
Scales the raw correlation by dividing each element in the cross-correlation matrix by the number of products a and b used to generate that element.
Scales the normalized cross-correlation on the range of [0 1] so that a value of 1 corresponds to a correlation coefficient of 1.
See also: conv2, corr2, xcorr.
[R, lag] = xcov ( X ) ¶… = xcov ( X, Y ) ¶… = xcov ( …, maxlag) ¶… = xcov ( …, scale) ¶Compute covariance at various lags [=correlation(x-mean(x),y-mean(y))].
input vector
if specified, compute cross-covariance between X and Y, otherwise compute autocovariance of X.
is specified, use lag range [-maxlag:maxlag], otherwise use range [-n+1:n-1].
for covariance=raw/N,
for covariance=raw/(N-|lag|),
for covariance=raw/(covariance at lag 0),
for covariance=raw
is the default.
Returns the covariance for each lag in the range, plus an optional vector of lags.
See also: xcorr.
y = filtfilt (b, a, x) ¶Forward and reverse filter the signal. This corrects for phase distortion introduced by a one-pass filter, though it does square the magnitude response in the process. That’s the theory at least. In practice the phase correction is not perfect, and magnitude response is distorted, particularly in the stop band.
Example
[b, a]=butter(3, 0.1); # 5 Hz low-pass filter t = 0:0.01:1.0; # 1 second sample x=sin(2*pi*t*2.3)+0.25*randn(size(t)); # 2.3 Hz sinusoid+noise y = filtfilt(b,a,x); z = filter(b,a,x); # apply filter plot(t,x,';data;',t,y,';filtfilt;',t,z,';filter;')
zf = filtic (b, a, y) ¶zf = filtic (b, a, y, x) ¶Set initial condition vector for filter function The vector zf has the same values that would be obtained from function filter given past inputs x and outputs y
The vectors x and y contain the most recent inputs and outputs respectively, with the newest values first:
x = [x(-1) x(-2) ... x(-nb)], nb = length(b)-1 y = [y(-1) y(-2) ... y(-na)], na = length(a)-a
If length(x)<nb then it is zero padded If length(y)<na then it is zero padded
zf = filtic(b, a, y) Initial conditions for filter with coefficients a and b and output vector y, assuming input vector x is zero
zf = filtic(b, a, y, x) Initial conditions for filter with coefficients a and b input vector x and output vector y
y = medfilt1 (x, n) ¶y = medfilt1 (x, n, [], dim) ¶y = medfilt1 (..., NaN_flag, padding) ¶Apply a one dimensional median filter with a window size of n to the data x, which must be real, double and full. For n = 2m+1, y(i) is the median of x(i-m:i+m). For n = 2m, y(i) is the median of x(i-m:i+m-1).
The calculation is performed over the first non-singleton dimension, or over dimension dim if that is specified as the fourth argument. (The third argument is ignored; Matlab used to use it to tune its algorithm.)
NaN_flag may be omitnan or includenan (the default).
If it is omitnan then any NaN values are removed from the window
before the median is taken.
Otherwise, any window containing an NaN returns a median of NaN.
padding determines how the partial windows at the start and end of
x are treated.
It may be truncate or zeropad (the default).
If it is truncate then the window for y(i) is
the intersection of the window stated above with 1:length(x).
If it is zeropad, then partial windows have additional zeros
to bring them up to size n.
See also: filter, medfilt2.
[rmsx,w] = movingrms (x,w,rc,Fs=1) ¶Calculate moving RMS value of the signal in x.
The signal is convoluted against a sigmoid window of width w and risetime rc. The units of these parameters are relative to the value of the sampling frequency given in Fs (Default value = 1).
Run demo movingrms to see an example.
See also: sigmoid_train.
y = sgolayfilt (x) ¶y = sgolayfilt (x, p) ¶y = sgolayfilt (x, p, n) ¶y = sgolayfilt (x, p, n, m) ¶y = sgolayfilt (x, p, n, m, ts) ¶y = sgolayfilt (x, p, n, m, ts) ¶y = sgolayfilt (x, f) ¶Smooth the data in x with a Savitsky-Golay smoothing filter of polynomial order p and length n, n odd, n > p. By default, p=3 and n=p+2 or n=p+3 if p is even.
If f is given as a matrix, it is expected to be a filter as
computed by sgolay.
These filters are particularly good at preserving lineshape while removing high frequency squiggles. Particularly, compare a 5 sample averager, an order 5 butterworth lowpass filter (cutoff 1/3) and sgolayfilt(x, 3, 5), the best cubic estimated from 5 points:
[b, a] = butter (5, 1/3);
x = [zeros(1,15), 10*ones(1,10), zeros(1,15)];
plot (sgolayfilt (x), "r;sgolayfilt;", ...
filtfilt (ones (1,5)/5, 1, x), "g;5 sample average;", ...
filtfilt (b, a, x), "c;order 5 butterworth;", ...
x, "+b;original data;");
See also: sgolay.
y = sosfilt (sos, x) ¶Second order section IIR filtering of x. The second order section filter is described by the matrix sos with:
| [ B1 A1 ] | |
| sos = | [ … ], |
| [ BN AN ] |
where B1 = [b0 b1 b2] and A1 = [1 a1 a2] for
section 1, etc. The b0 entry must be nonzero for each section.
L = filternorm (b, a) ¶L = filternorm (b, a, pnorm) ¶L = filternorm (b, a, 2, tol) ¶Compute the 2-norm of a digital filter defined by the numerator coefficients, b, and the denominator coefficients, a. It is also possible to compute the infinity-norm by passing inf in the pnorm parameter. pnorm only accepts 2 or inf.
Example:
[b, a] = butter (8, 0.5); filternorm (b, a)
n = filtord (b, a) ¶n = filtord (sos) ¶Returns the filter order n for a filter defined by the numerator coefficients, b, and the denominator coefficients, a. It also accepts a filter defined by a matrix of second-order sections, sos.
Example:
[b, a] = butter (8, 0.5); filtord (b, a)
h = freqs (b, a, w) ¶(b, a, w) ¶Compute the s-plane frequency response of the IIR filter B(s)/A(s) as H = polyval(B,j*W)./polyval(A,j*W). If called with no output argument, a plot of magnitude and phase are displayed.
Example:
b = [1 2]; a = [1 1]; w = linspace (0, 4, 128); freqs (b, a, w);
f = fwhm (y) ¶f = fwhm (x, y) ¶f = fwhm (…, "zero") ¶f = fwhm (…, "min") ¶f = fwhm (…, "alevel", level) ¶f = fwhm (…, "rlevel", level) ¶Compute peak full-width at half maximum (FWHM) or at another level of peak maximum for vector or matrix data y, optionally sampled as y(x). If y is a matrix, return FWHM for each column as a row vector.
The default option "zero" computes fwhm at half maximum, i.e. 0.5*max(y). The option "min" computes fwhm at the middle curve, i.e. 0.5*(min(y)+max(y)).
The option "rlevel" computes full-width at the given relative level of peak
profile, i.e. at rlevel*max(y) or rlevel*(min(y)+max(y)),
respectively. For example, fwhm (…, "rlevel", 0.1) computes
full width at 10 % of peak maximum with respect to zero or minimum; FWHM is
equivalent to fwhm(…, "rlevel", 0.5).
The option "alevel" computes full-width at the given absolute level of y.
Return 0 if FWHM does not exist (e.g. monotonous function or the function does not cut horizontal line at rlevel*max(y) or rlevel*(max(y)+min(y)) or alevel, respectively).
[g, w] = grpdelay (b) ¶[g, w] = grpdelay (b, a) ¶[g, w] = grpdelay (…, n) ¶[g, w] = grpdelay (…, n, "whole") ¶[g, f] = grpdelay (…, n, Fs) ¶[g, f] = grpdelay (…, n, "whole", Fs) ¶[g, w] = grpdelay (…, w) ¶[g, f] = grpdelay (…, f, Fs) ¶(…) ¶Compute the group delay of a filter.
[g, w] = grpdelay(b) returns the group delay g of the FIR filter with coefficients b. The response is evaluated at 512 angular frequencies between 0 and pi. w is a vector containing the 512 frequencies. The group delay is in units of samples. It can be converted to seconds by multiplying by the sampling period (or dividing by the sampling rate fs).
[g, w] = grpdelay(b,a) returns the group delay of the rational IIR filter whose numerator has coefficients b and denominator coefficients a.
[g, w] = grpdelay(b,a,n) returns the group delay evaluated at n angular frequencies. For fastest computation n should factor into a small number of small primes.
[g, w] = grpdelay(b,a,n,’whole’) evaluates the group delay at n frequencies between 0 and 2*pi.
[g, f] = grpdelay(b,a,n,Fs) evaluates the group delay at n frequencies between 0 and Fs/2.
[g, f] = grpdelay(b,a,n,’whole’,Fs) evaluates the group delay at n frequencies between 0 and Fs.
[g, w] = grpdelay(b,a,w) evaluates the group delay at frequencies w (radians per sample).
[g, f] = grpdelay(b,a,f,Fs) evaluates the group delay at frequencies f (in Hz).
grpdelay(...) plots the group delay vs. frequency.
If the denominator of the computation becomes too small, the group delay is set to zero. (The group delay approaches infinity when there are poles or zeros very close to the unit circle in the z plane.)
Theory: group delay, g(w) = -d/dw [arg{H(e^jw)}], is the rate of change of phase with respect to frequency. It can be computed as:
d/dw H(e^-jw)
g(w) = -------------
H(e^-jw)
where
H(z) = B(z)/A(z) = sum(b_k z^k)/sum(a_k z^k).
By the quotient rule,
A(z) d/dw B(z) - B(z) d/dw A(z)
d/dw H(z) = -------------------------------
A(z) A(z)
Substituting into the expression above yields:
A dB - B dA
g(w) = ----------- = dB/B - dA/A
A B
Note that,
d/dw B(e^-jw) = sum(k b_k e^-jwk)
d/dw A(e^-jw) = sum(k a_k e^-jwk)
which is just the FFT of the coefficients multiplied by a ramp.
As a further optimization when nfft>>length(a), the IIR filter (b,a) is converted to the FIR filter conv(b,fliplr(conj(a))). For further details, see http://ccrma.stanford.edu/~jos/filters/Numerical_Computation_Group_Delay.html
[x, t] = impz (b) ¶[x, t] = impz (b, a) ¶[x, t] = impz (b, a, n) ¶[x, t] = impz (b, a, n, fs) ¶(…) ¶Generate impulse-response characteristics of the filter. The filter coefficients correspond to the the z-plane rational function with numerator b and denominator a. If a is not specified, it defaults to 1. If n is not specified, or specified as [], it will be chosen such that the signal has a chance to die down to -120dB, or to not explode beyond 120dB, or to show five periods if there is no significant damping. If no return arguments are requested, plot the results.
See also: freqz, zplane.
L = isallpass (b, a) ¶L = isallpass (sos) ¶Determine whether a digital filter is allpass. The filter might be defined by the numerator coefficients, b, and the denominator coefficients, a, or, alternatively, by a matrix of second-order sections, sos.
Example:
a = [1 2 3]; b = [3 2 1]; isallpass (b, a)
Ref [1] Shyu, Jong-Jy, & Pei, Soo-Chang, A new approach to the design of complex all-pass IIR digital filters, Signal Processing, 40(2–3), 207–215, 1994. https://doi.org/10.1016/0165-1684(94)90068-x
Ref [2] Vaidyanathan, P. P. Multirate Systems and Filter Banks. 1st edition, Pearson College Div, 1992.
L = ismaxphase (b, a) ¶L = ismaxphase (sos) ¶L = ismaxphase (…, tol) ¶Determine whether a digital filter is maximum phase (maximum energy-delay). The filter might be defined by the numerator coefficients, b, and the denominator coefficients, a, or, alternatively, by a matrix of second-order sections, sos. A tolerance tol might be given to define when two numbers are close enough to be considered equal.
Example:
b = [1 2 4 4 2 1]; zplane (b); ismaxphase (b)
Ref [1] Oppenheim, Alan, and Ronald Schafer. Discrete-Time Signal Processing. 3rd edition, Pearson, 2009.
L = isminphase (b, a) ¶L = isminphase (sos) ¶L = isminphase (…, tol) ¶Determine whether a digital filter is minimum phase. The filter might be defined by the numerator coefficients, b, and the denominator coefficients, a, or, alternatively, by a matrix of second-order sections, sos. A toleranve tol might be given to define when two numbers are close enough to be considered equal.
Example:
a = [1 0.5]; b = [3 1]; isminphase (b, a)
Ref [1] Oppenheim, Alan, and Ronald Schafer. Discrete-Time Signal Processing. 3rd edition, Pearson, 2009.
FLAG = isstable (B, A) ¶Returns a logical output equal to TRUE, if the filter is stable. This can be done with coeffients of the filer B and A. Alternatively by using a second order sections matrix (SOS).
Inputs:
Output:
Examples:
b = [1 2 3 4 5 5 1 2]; a = [4 5 6 7 9 10 4 6]; flag = isstable (b, a) flag = 0
Using SOS
[z, p, k] = butter (6, 0.7, 'high'); sos = zp2sos (z, p, k); flag = isstable (sos) flag = 1
[phi, w] = phasez (b, a, n) ¶[phi, w] = phasez (b, a) ¶[phi, w] = phasez (sos, n) ¶[phi, w] = phasez (sos) ¶[phi, w] = phasez (…, n, "whole") ¶[phi, w] = phasez (…, n, Fs) ¶(…) ¶Compute the phase response of digital filter defined either by its coefficients (b and a are the numerator and denominator coefficients respectively) or by its second-order sections representation, given by the matrix sos. The output phi is the phase response computed in a vector the vector of frequencies w.
The phase response is evaluated at n angular frequencies between 0 and pi.
If a is omitted, the denominator is assumed to be 1 (this corresponds to a simple FIR filter).
If n is omitted, a value of 512 is assumed.
If the third/forth argument, "whole", is given, the response is
evaluated at n angular frequencies between 0 and
2*pi.
It is possible also to pass the value "half", which will lead to
the default behaviour.
Example:
[b, a] = butter (2, [.15,.3]); phasez (b, a);
Ref [1] Oppenheim, Alan, and Ronald Schafer. Discrete-Time Signal Processing. 3rd edition, Pearson, 2009.
See also: freqz, phasedelay.
(z, p) ¶(b, a) ¶Plot the poles and zeros on a complex plane. If the arguments are column vectors z and p, the complex zeros z and poles p are displayed. If the arguments are row vectors b and a, the zeros and poles of the transfer function represented by these filter coefficients are displayed.
If z and p are matrices, the columns are distinct sets of zeros and poles and are displayed together in distinct colors.
Note that due to the nature of the roots function, poles and zeros
may be displayed as occurring around a circle rather than at a single
point.
The transfer function is
B(z) b0 + b1 z^(-1) + b2 z^(-2) + ... + bM z^(-M)
H(z) = ---- = --------------------------------------------
A(z) a0 + a1 z^(-1) + a2 z^(-2) + ... + aN z^(-N)
b0 (z - z1) (z - z2) ... (z - zM)
= -- z^(-M+N) ------------------------------
a0 (z - p1) (z - p2) ... (z - pN)
If called with only one argument, the poles p defaults to an empty vector, and the denominator coefficient vector a defaults to 1.
[r, p, f, m] = residued (b, a) ¶Compute the partial fraction expansion (PFE) of filter
H(z) = B(z)/A(z). In the usual PFE function residuez, the
IIR part (poles p and residues r) is driven in parallel
with the FIR part (f). In this variant, the IIR part is driven by
the output of the FIR part. This structure can be more accurate in
signal modeling applications.
INPUTS:
b and a are vectors specifying the digital filter
H(z) = B(z)/A(z). See help filter for documentation of the
b and a filter coefficients.
RETURNED:
EXAMPLES:
See test residued verbose to see a number of examples.
For the theory of operation, see
‘http://ccrma.stanford.edu/~jos/filters/residued.html’
See also: residue, residued.
[r, p, f, m] = residuez (b, a) ¶Compute the partial fraction expansion of filter H(z) = B(z)/A(z).
INPUTS:
b and a are vectors specifying the digital filter
H(z) = B(z)/A(z). See help filter for documentation of the
b and a filter coefficients.
RETURNED:
EXAMPLES:
See test residuez verbose to see a number of examples.
For the theory of operation, see
‘http://ccrma.stanford.edu/~jos/filters/residuez.html’
See also: residue, residued.
[a, b, c, d] = sos2ss (sos) ¶Convert series second-order sections to state-space.
See also: sos2ss, ss2tf.
[b, a] = sos2tf (sos) ¶[b, a] = sos2tf (sos, g) ¶Convert series second-order sections to transfer function.
INPUTS:
sos = [B1.' A1.'; ...; BN.' AN.']
where
B1.' = [b0 b1 b2] and A1.' = [a0 a1 a2] for
section 1, etc.
a0 is usually equal to 1 because all 2nd order transfer functions
can be scaled so that a0 = 1.
However, this is not mandatory for this implementation, which supports
all kinds of transfer functions, including first order transfer functions.
See filter for documentation of the second-order direct-form filter
coefficients Bi and Ai.
RETURNED:
b and a are vectors specifying the analog or digital filter
H(s) = B(s)/A(s) or H(z) = B(z)/A(z).
See filter for further details.
See also: tf2sos, zp2sos, sos2pz, zp2tf, tf2zp.
[z, p, k] = sos2zp (sos) ¶[z, p, k] = sos2zp (sos, g) ¶Convert series second-order sections to zeros, poles, and gains (pole residues).
INPUTS:
sos = [B1.' A1.'; ...; BN.' AN.']
where
B1.' = [b0 b1 b2] and A1.' = [a0 a1 a2] for
section 1, etc.
a0 is usually equal to 1 because all 2nd order transfer functions can
be scaled so that a0 = 1.
However, this is not mandatory for this implementation, which supports
all kinds of transfer functions, including first order transfer functions.
See filter for documentation of the second-order direct-form filter
coefficients Bi and Ai.
RETURNED:
EXAMPLE:
[z, p, k] = sos2zp ([1 0 1, 1 0 -0.81; 1 0 0, 1 0 0.49])
⇒ z =
0 + 1i
0 - 1i
0 + 0i
0 + 0i
⇒ p =
-0.9000 + 0i
0.9000 + 0i
0 + 0.7000i
0 - 0.7000i
⇒ k = 1
See also: zp2sos, sos2tf, tf2sos, zp2tf, tf2zp.
[num, den] = ss2tf (a, b, c, d) ¶Conversion from state-space to transfer function representation. The state space system:
.
x = Ax + Bu
y = Cx + Du
is converted to a transfer function:
num(s)
G(s)=-------
den(s)
[z, p, k] = ss2zp (a, b, c, d) ¶Converts a state space representation to a set of poles and zeros; k is a gain associated with the zeros.
[sos, g] = tf2sos (b, a) ¶sos = tf2sos (b, a) ¶Convert direct-form filter coefficients to series second-order sections.
INPUTS:
b and a are vectors specifying the digital filter
H(z) = B(z)/A(z). See filter for documentation of the b
and a filter coefficients.
RETURNED:
sos = [b1.' a1.'; ...; bn.' an.']
where
B1.' = [b0 b1 b2] and A1.' = [1 a1 a2] for
section 1, etc. The b0 entry must be nonzero for each section (zeros at
infinity not supported).
If called with only one output argument, the overall filter gain is applied to the first second-order section in the matrix sos.
EXAMPLE:
B = [1 0 0 0 0 1];
A = [1 0 0 0 0 .9];
[sos, g] = tf2sos (B, A)
sos =
1.00000 0.61803 1.00000 1.00000 0.60515 0.95873
1.00000 -1.61803 1.00000 1.00000 -1.58430 0.95873
1.00000 1.00000 -0.00000 1.00000 0.97915 -0.00000
g = 1
See also: sos2tf, zp2sos, sos2pz, zp2tf, tf2zp.
[a, b, c, d] = tf2ss (num, den) ¶Conversion from transfer function to state-space. The state space system:
.
x = Ax + Bu
y = Cx + Du
is obtained from a transfer function:
num(s)
G(s)=-------
den(s)
The state space system matrices obtained from this function will be in observable companion form as Wolovich’s Observable Structure Theorem is used.
[z, p, k] = tf2zp (num, den) ¶Convert transfer functions to poles-and-zero representations.
Returns the zeros and poles of the system defined by num/den. k is a gain associated with the system zeros.
[sos, g] = zp2sos (z) ¶[sos, g] = zp2sos (z, p) ¶[sos, g] = zp2sos (z, p, k) ¶sos = zp2sos (…) ¶Convert filter poles and zeros to second-order sections.
INPUTS:
RETURNED:
sos = [B1.' A1.'; ...; BN.' AN.']
where
B1.' = [b0 b1 b2] and A1.' = [a0 a1 a2] for
section 1, etc.
See filter for documentation of the second-order direct-form filter
coefficients Bi and %Ai, i=1:N.
If called with only one output argument, the overall filter gain is applied to the first second-order section in the matrix sos.
EXAMPLE:
[z, p, k] = tf2zp ([1 0 0 0 0 1], [1 0 0 0 0 .9]);
[sos, g] = zp2sos (z, p, k)
sos =
1.0000 0.6180 1.0000 1.0000 0.6051 0.9587
1.0000 -1.6180 1.0000 1.0000 -1.5843 0.9587
1.0000 1.0000 0 1.0000 0.9791 0
g =
1
See also: sos2zp, sos2tf, tf2sos, zp2tf, tf2zp.
[a, b, c, d] = zp2ss (z, p, k) ¶Conversion from zero / pole to state space.
Inputs
Vectors of (possibly) complex poles and zeros of a transfer function. Complex values must come in conjugate pairs (i.e., x+jy in z means that x-jy is also in z).
Real scalar (leading coefficient).
Outputs
The state space system, in the form:
.
x = Ax + Bu
y = Cx + Du
[zero, pole, gain] = besselap (n) ¶Return bessel analog filter prototype.
References:
http://en.wikipedia.org/wiki/Bessel_polynomials
[b, a] = besself (n, w) ¶[b, a] = besself (n, w, "high") ¶[z, p, g] = besself (…) ¶[a, b, c, d] = besself (…) ¶[…] = besself (…, "z") ¶Generate a Bessel filter. Default is a Laplace space (s) filter.
[b,a] = besself(n, Wc) low pass filter with cutoff pi*Wc radians
[b,a] = besself(n, Wc, ’high’) high pass filter with cutoff pi*Wc radians
[z,p,g] = besself(...) return filter as zero-pole-gain rather than coefficients of the numerator and denominator polynomials.
[...] = besself(...,’z’) return a discrete space (Z) filter, W must be less than 1.
[a,b,c,d] = besself(...) return state-space matrices
References:
Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company.
[Zb, Za] = bilinear (Sb, Sa, T) ¶[Zb, Za] = bilinear (Sz, Sp, Sg, T) ¶[Zz, Zp, Zg] = bilinear (…) ¶Transform a s-plane filter specification into a z-plane specification. Filters can be specified in either zero-pole-gain or transfer function form. The input form does not have to match the output form. 1/T is the sampling frequency represented in the z plane.
Note: this differs from the bilinear function in the signal processing toolbox, which uses 1/T rather than T.
Theory: Given a piecewise flat filter design, you can transform it from the s-plane to the z-plane while maintaining the band edges by means of the bilinear transform. This maps the left hand side of the s-plane into the interior of the unit circle. The mapping is highly non-linear, so you must design your filter with band edges in the s-plane positioned at 2/T tan(w*T/2) so that they will be positioned at w after the bilinear transform is complete.
The following table summarizes the transformation:
+---------------+-----------------------+----------------------+ | Transform | Zero at x | Pole at x | | H(S) | H(S) = S-x | H(S)=1/(S-x) | +---------------+-----------------------+----------------------+ | 2 z-1 | zero: (2+xT)/(2-xT) | zero: -1 | | S -> - --- | pole: -1 | pole: (2+xT)/(2-xT) | | T z+1 | gain: (2-xT)/T | gain: (2-xT)/T | +---------------+-----------------------+----------------------+
With tedious algebra, you can derive the above formulae yourself by substituting the transform for S into H(S)=S-x for a zero at x or H(S)=1/(S-x) for a pole at x, and converting the result into the form:
H(Z)=g prod(Z-Xi)/prod(Z-Xj)
Please note that a pole and a zero at the same place exactly cancel. This is significant since the bilinear transform creates numerous extra poles and zeros, most of which cancel. Those which do not cancel have a "fill-in" effect, extending the shorter of the sets to have the same number of as the longer of the sets of poles and zeros (or at least split the difference in the case of the band pass filter). There may be other opportunistic cancellations but I will not check for them.
Also note that any pole on the unit circle or beyond will result in an unstable filter. Because of cancellation, this will only happen if the number of poles is smaller than the number of zeros. The analytic design methods all yield more poles than zeros, so this will not be a problem.
References:
Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company.
[z, p, g] = buttap (n) ¶Design lowpass analog Butterworth filter.
This function exists for MATLAB compatibility only, and is equivalent
to butter (n, 1, "s").
See also: butter.
[b, a] = butter (n, wc) ¶[b, a] = butter (n, wc, filter_type) ¶[z, p, g] = butter (…) ¶[a, b, c, d] = butter (…) ¶[…] = butter (…, "s") ¶Generate a Butterworth filter. Default is a discrete space (Z) filter.
The cutoff frequency, wc should be specified in radians for
analog filters. For digital filters, it must be a value between zero
and one. For bandpass filters, wc is a two-element vector
with w(1) < w(2).
The filter type must be one of "low", "high",
"bandpass", or "stop". The default is "low"
if wc is a scalar and "bandpass" if wc is a
two-element vector.
If the final input argument is "s" design an analog Laplace
space filter.
Low pass filter with cutoff pi*Wc radians:
[b, a] = butter (n, Wc)
High pass filter with cutoff pi*Wc radians:
[b, a] = butter (n, Wc, "high")
Band pass filter with edges pi*Wl and pi*Wh radians:
[b, a] = butter (n, [Wl, Wh])
Band reject filter with edges pi*Wl and pi*Wh radians:
[b, a] = butter (n, [Wl, Wh], "stop")
Return filter as zero-pole-gain rather than coefficients of the numerator and denominator polynomials:
[z, p, g] = butter (...)
Return a Laplace space filter, Wc can be larger than 1:
[...] = butter (..., "s")
Return state-space matrices:
[a, b, c, d] = butter (...)
References:
Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company.
n = buttord (wp, ws, rp, rs) ¶n = buttord ([wp1, wp2], [ws1, ws2], rp, rs) ¶n = buttord ([wp1, wp2], [ws1, ws2], rp, rs, "s") ¶[n, wc_p] = buttord (…) ¶[n, wc_p, wc_s] = buttord (…) ¶Compute the minimum filter order of a Butterworth filter with the desired response characteristics. The filter frequency band edges are specified by the passband frequency wp and stopband frequency ws. Frequencies are normalized to the Nyquist frequency in the range [0,1]. rp is the allowable passband ripple measured in decibels, and rs is the minimum attenuation in the stop band, also in decibels.
The output arguments n and wc_p (or n and wc_n) can
be given as inputs to butter.
Using wc_p makes the filter characteristic touch at least one pass band
corner and using wc_s makes the characteristic touch at least one
stop band corner.
If wp and ws are scalars, then wp is the passband cutoff frequency and ws is the stopband edge frequency. If ws is greater than wp, the filter is a low-pass filter. If wp is greater than ws, the filter is a high-pass filter.
If wp and ws are vectors of length 2, then wp defines the passband interval and ws defines the stopband interval. If wp is contained within ws (ws1 < wp1 < wp2 < ws2), the filter is a band-pass filter. If ws is contained within wp (wp1 < ws1 < ws2 < wp2), the filter is a band-stop or band-reject filter.
If the optional argument "s" is given, the minimum order for an analog
elliptic filter is computed. All frequencies wp and ws are
specified in radians per second.
Theory: For Low pass filters, |H(W)|^2 = 1/[1+(W/Wc)^(2N)] = 10^(-R/10). With some algebra, you can solve simultaneously for Wc and N given Ws,Rs and Wp,Rp. Rounding N to the next greater integer, one can recalculate the allowable range for Wc (filter caracteristic touching the pass band edge or the stop band edge).
For other types of filter, before making the above calculation, the requirements must be transformed to LP requirements. After calculation, Wc must be transformed back to original filter type.
See also: butter, cheb1ord, cheb2ord, ellipord.
(n, x) ¶Returns the value of the nth-order Chebyshev polynomial calculated at the point x. The Chebyshev polynomials are defined by the equations:
/ cos(n acos(x), |x| <= 1
Tn(x) = |
\ cosh(n acosh(x), |x| > 1
If x is a vector, the output is a vector of the same size, where each element is calculated as y(i) = Tn(x(i)).
[z, p, g] = cheb1ap (n, Rp) ¶Design lowpass analog Chebyshev type I filter.
This function exists for MATLAB compatibility only, and is equivalent
to cheby1 (n, Rp, 1, "s").
Input:
Output:
Example
[z, p, g] = cheb1ap (2, 1) z = [](0x1) p = -0.54887 - 0.89513i -0.54887 + 0.89513i g = 0.98261
See also: buttap, cheby1, cheb2ap, ellipap.
n = cheb1ord (wp, ws, rp, rs) ¶n = cheb1ord ([wp1, wp2], [ws1, ws2], rp, rs) ¶n = cheb1ord ([wp1, wp2], [ws1, ws2], rp, rs, "s") ¶[n, wc] = cheb1ord (…) ¶[n, wc_p, wc_s] = cheb1ord (…) ¶Compute the minimum filter order of a Chebyshev type I filter with the desired response characteristics. The filter frequency band edges are specified by the passband frequency wp and stopband frequency ws. Frequencies are normalized to the Nyquist frequency in the range [0,1]. rp is the allowable passband ripple measured in decibels, and rs is the minimum attenuation in the stop band, also in decibels.
The output arguments n and wc_p (or n and wc_s) can
be given as inputs to cheby1.
Using wc_p makes the filter characteristic touch at least one pass band
corner and using wc_s makes the characteristic touch at least one
stop band corner.
If wp and ws are scalars, then wp is the passband cutoff frequency and ws is the stopband edge frequency. If ws is greater than wp, the filter is a low-pass filter. If wp is greater than ws, the filter is a high-pass filter.
If wp and ws are vectors of length 2, then wp defines the passband interval and ws defines the stopband interval. If wp is contained within ws (ws1 < wp1 < wp2 < ws2), the filter is a band-pass filter. If ws is contained within wp (wp1 < ws1 < ws2 < wp2), the filter is a band-stop or band-reject filter.
If the optional argument "s" is given, the minimum order for an analog
elliptic filter is computed. All frequencies wp and ws are
specified in radians per second.
See also: buttord, cheby1, cheb2ord, ellipord.
[z, p, g] = cheb2ap (n, Rs) ¶Design lowpass analog Chebyshev type II filter.
This function exists for MATLAB compatibility only, and is equivalent
to cheby2 (n, Rs, 1, "s").
Demo
demo cheb2ap
See also: cheby2.
n = cheb2ord (wp, ws, rp, rs) ¶n = cheb2ord ([wp1, wp2], [ws1, ws2], rp, rs) ¶n = cheb2ord ([wp1, wp2], [ws1, ws2], rp, rs, "s") ¶[n, wc_s] = cheb2ord (…) ¶[n, wc_s, wc_p] = cheb2ord (…) ¶Compute the minimum filter order of a Chebyshev type II filter with the desired response characteristics. The filter frequency band edges are specified by the passband frequency wp and stopband frequency ws. Frequencies are normalized to the Nyquist frequency in the range [0,1]. rp is the allowable passband ripple measured in decibels, and rs is the minimum attenuation in the stop band, also in decibels.
The output arguments n and wc_p (or n and wc_s) can
be given as inputs to cheby2.
Using wc_p makes the filter characteristic touch at least one pass band
corner and using wc_s makes the characteristic touch at least one
stop band corner.
If wp and ws are scalars, then wp is the passband cutoff frequency and ws is the stopband edge frequency. If ws is greater than wp, the filter is a low-pass filter. If wp is greater than ws, the filter is a high-pass filter.
If wp and ws are vectors of length 2, then wp defines the passband interval and ws defines the stopband interval. If wp is contained within ws (ws1 < wp1 < wp2 < ws2), the filter is a band-pass filter. If ws is contained within wp (wp1 < ws1 < ws2 < wp2), the filter is a band-stop or band-reject filter.
If the optional argument "s" is given, the minimum order for an analog
elliptic filter is computed. All frequencies wp and ws are
specified in radians per second.
See also: buttord, cheb1ord, cheby2, ellipord.
[b, a] = cheby1 (n, rp, w) ¶[b, a] = cheby1 (n, rp, w, "high") ¶[b, a] = cheby1 (n, rp, [wl, wh]) ¶[b, a] = cheby1 (n, rp, [wl, wh], "stop") ¶[z, p, g] = cheby1 (…) ¶[a, b, c, d] = cheby1 (…) ¶[…] = cheby1 (…, "s") ¶Generate a Chebyshev type I filter with rp dB of passband ripple.
[b, a] = cheby1(n, Rp, Wc) low pass filter with cutoff pi*Wc radians
[b, a] = cheby1(n, Rp, Wc, ’high’) high pass filter with cutoff pi*Wc radians
[b, a] = cheby1(n, Rp, [Wl, Wh]) band pass filter with edges pi*Wl and pi*Wh radians
[b, a] = cheby1(n, Rp, [Wl, Wh], ’stop’) band reject filter with edges pi*Wl and pi*Wh radians
[z, p, g] = cheby1(...) return filter as zero-pole-gain rather than coefficients of the numerator and denominator polynomials.
[...] = cheby1(...,’s’) return a Laplace space filter, W can be larger than 1.
[a,b,c,d] = cheby1(...) return state-space matrices
References:
Parks & Burrus (1987). Digital Filter Design. New York: John Wiley & Sons, Inc.
[b, a] = cheby2 (n, rs, wc) ¶[b, a] = cheby2 (n, rs, wc, "high") ¶[b, a] = cheby2 (n, rs, [wl, wh]) ¶[b, a] = cheby2 (n, rs, [wl, wh], "stop") ¶[z, p, g] = cheby2 (…) ¶[a, b, c, d] = cheby2 (…) ¶[…] = cheby2 (…, "s") ¶Generate a Chebyshev type II filter with rs dB of stopband attenuation.
[b, a] = cheby2(n, Rs, Wc) low pass filter with cutoff pi*Wc radians
[b, a] = cheby2(n, Rs, Wc, ’high’) high pass filter with cutoff pi*Wc radians
[b, a] = cheby2(n, Rs, [Wl, Wh]) band pass filter with edges pi*Wl and pi*Wh radians
[b, a] = cheby2(n, Rs, [Wl, Wh], ’stop’) band reject filter with edges pi*Wl and pi*Wh radians
[z, p, g] = cheby2(...) return filter as zero-pole-gain rather than coefficients of the numerator and denominator polynomials.
[...] = cheby2(...,’s’) return a Laplace space filter, W can be larger than 1.
[a,b,c,d] = cheby2(...) return state-space matrices
References:
Parks & Burrus (1987). Digital Filter Design. New York: John Wiley & Sons, Inc.
[b, a] = ellip (n, rp, rs, wp) ¶[b, a] = ellip (n, rp, rs, wp, "high") ¶[b, a] = ellip (n, rp, rs, [wl, wh]) ¶[b, a] = ellip (n, rp, rs, [wl, wh], "stop") ¶[z, p, g] = ellip (…) ¶[a, b, c, d] = ellip (…) ¶[…] = ellip (…, "s") ¶Generate an elliptic or Cauer filter with rp dB of passband ripple and rs dB of stopband attenuation.
[b,a] = ellip(n, Rp, Rs, Wp) low pass filter with order n, cutoff pi*Wp radians, Rp decibels of ripple in the passband and a stopband Rs decibels down.
[b,a] = ellip(n, Rp, Rs, Wp, ’high’) high pass filter with cutoff pi*Wp...
[b,a] = ellip(n, Rp, Rs, [Wl, Wh]) band pass filter with band pass edges pi*Wl and pi*Wh ...
[b,a] = ellip(n, Rp, Rs, [Wl, Wh], ’stop’) band reject filter with edges pi*Wl and pi*Wh, ...
[z,p,g] = ellip(...) return filter as zero-pole-gain.
[...] = ellip(...,’s’) return a Laplace space filter, W can be larger than 1.
[a,b,c,d] = ellip(...) return state-space matrices
References:
- Oppenheim, Alan V., Discrete Time Signal Processing, Hardcover, 1999. - Parente Ribeiro, E., Notas de aula da disciplina TE498 - Processamento Digital de Sinais, UFPR, 2001/2002.
[z, p, g] = ellipap (n, Rp, Rs) ¶Design lowpass analog elliptic filter.
This function exists for MATLAB compatibility only, and is equivalent
to ellip (n, Rp, Rs, 1, "s").
See also: ellip.
n = ellipord (wp, ws, rp, rs) ¶n = ellipord ([wp1, wp2], [ws1, ws2], rp, rs) ¶n = ellipord ([wp1, wp2], [ws1, ws2], rp, rs, "s") ¶[n, wc] = ellipord (…) ¶Compute the minimum filter order of an elliptic filter with the desired
response characteristics. The filter frequency band edges are specified
by the passband frequency wp and stopband frequency ws.
Frequencies are normalized to the Nyquist frequency in the range [0,1].
rp is the allowable passband ripple measured in decibels, and rs
is the minimum attenuation in the stop band, also in decibels. The output
arguments n and wc can be given as inputs to ellip.
If wp and ws are scalars, then wp is the passband cutoff frequency and ws is the stopband edge frequency. If ws is greater than wp, the filter is a low-pass filter. If wp is greater than ws, the filter is a high-pass filter.
If wp and ws are vectors of length 2, then wp defines the passband interval and ws defines the stopband interval. If wp is contained within ws (ws1 < wp1 < wp2 < ws2), the filter is a band-pass filter. If ws is contained within wp (wp1 < ws1 < ws2 < wp2), the filter is a band-stop or band-reject filter.
If the optional argument "s" is given, the minimum order for an analog
elliptic filter is computed. All frequencies wp and ws are
specified in radians per second.
Reference: Lamar, Marcus Vinicius, Notas de aula da disciplina TE 456 - Circuitos Analogicos II, UFPR, 2001/2002.
See also: buttord, cheb1ord, cheb2ord, ellip.
b = firpm (n, f, a) ¶b = firpm (n, f, @respFn) ¶b = firpm (n, f, {@respFn, …}) ¶b = firpm (…, w) ¶b = firpm (…, class) ¶b = firpm (…, {accuracy, …}) ¶[b, minimax] = firpm (…) ¶[b, minimax, res] = firpm (…) ¶Designs a linear-phase FIR filter according to given specifications and the ‘minimax’ criterion. The method (per McClellan et al.1) uses successive approximation to minimize the maximum weighted error between the desired and actual frequency response of the filter. Such filters are variably described as being ‘minimax’, ‘equiripple’, or ‘optimal (in the Chebyshev sense)’.
Where shown as the first argument to firpm, indicates that any
previously-indicated list of arguments may substitute for the ellipsis.
A positive integer giving the filter order.
A vector of real-numbers, increasing in the range [0,1], giving the frequencies of the left and right edges of each band for which a specific amplitude response is desired: [l1 r1 l2 r2 …]. 1 represents the Nyquist-frequency. Transition-bands are defined implicitly as the regions between or outside the given bands.
A vector of real-numbers giving the desired amplitude response. An amplitude value is given either for each band edge: [a(l1) a(r1) a(l2) a(r2) …], or for each band: [a1 a2 …]. In the former case, in-band amplitude is determined by linear interpolation between the given band-edge values. 1 represents unity-gain, 0 represents infinite attenuation, and −1 represents a phase change of pi radians.
Note that amplitude response is necessarily zero at f=0 for type III and IV filters, and at f=1 for type II and III filters.
A handle to a ‘response function’ that supplies the desired amplitude response
and error-weighting. This, unlike a above, allows the response to be
arbitrary (subject to the note above). firpm invokes the response
function according to the following syntax:
ag =respFn(n,f,g,w, ...) [ag wg] =respFn(n,f,g,w, ...) symmetry =respFn("defaults", {n,f,g,w, ...})
where:
firpm.
firpm, or ones if not given.
"even" or "odd"; this provides an
alternative to using the class values "symmetric"
and "antisymmetric".
When used in conjunction with a, w is a vector of positive real-numbers giving error-weightings to be applied at each given band-edge [w(l1) w(r1) w(l2) w(r2) …], or for each band [w1 w2 …]. In the former case, in-band weighting is determined by linear interpolation between the given band-edge values. A higher relative error weighting yields a lower relative error.
When used in conjunction with @respFn, w is a vector (constrained as above) that is passed through to respFn.
A string, which may be abbreviated, giving the filter-class:
"symmetric" (the default) for type I or II filters,
"antisymmetric" (or "hilbert") for standard type III or IV
filters,
"differentiator" for type III or IV filters with inverted phase and
with error-weighting (further to w) of 2/f applied in the pass-band(s).
Up to three properties contained within a cell-array: accuracy, persistence, robustness, that respectively control how close the computed filter will be to the ideal minimax solution, the number of computation iterations over which the required accuracy will be sought, and the precision of certain internal processing. Each can each be set to a small positive number (typically ≤3), to increase the relevant item; this may increase computation time, but the need to do so should be rare. A value of 0 can be used to leave an item unchanged.
Alternatively, setting accuracy ≥16 emulates MATLAB’s lgrid argument.
If a problem occurs during the computation, a diagnostic message will normally be displayed. If this happens, adjusting accuracy, persistence, or robustness may provide the solution. Some filters however, may not be realizable due to machine-precision limitations. If a filter can be computed, returned values are as follows:
A length N+1 row-vector containing the computed filter coefficients.
The absolute value of the minimized, maximum weighted error, or this number negated if the required accuracy could not be achieved.
A structure of data relating to the filter computation and a partial response-analysis of the resultant filter; fields are vectors:
fgridAnalysis frequencies per f. desDesired amplitude response. wtError weighting. HComplex frequency response. errorDesired minus actual amplitude response. iextrIndices of local peaks in error.fextrFrequencies of local peaks in error.
Using res is not recommended because it can be slow to compute and, since
the analysis excludes transition-bands, any ‘anomalies’2 therein are not easy to
discern. In general, freqz suffices to check that the response of the
computed filter is satisfactory.
# Low-pass with frequencies in Hz: Fs = 96000; Fn = Fs/2; # Sampling & Nyquist frequencies. b = firpm (50, [0 20000 28000 48000] / Fn, [1 0]);
# Type IV high-pass: b = firpm (31, [0 0.5 0.7 1], [0 1], "antisym");
# Inverse-sinc (arbitrary response): b = firpm (20, [0 0.5 0.9 1], @(n,f,g) ... deal ((g<=f(2))./sinc (g), (g>=f(3))*9+1));
# Band-pass with filter-response check: freqz (firpm (40, [0 3 4 6 8 10]/10, [0 1 0]))
Further examples can be found in the firpm and firpmord
demonstration scripts.
Given invalid filter specifications, Octave emits an error and does not produce a filter; MATLAB in such circumstances may still produce filter coefficients.
Unlike with MATLAB, with Octave minimax can be negative; for compatibility, take the absolute value.
See also: firpmord.
[n, Fout, a, w] = firpmord (f, a, d) ¶[n, Fout, a, w] = firpmord (f, a, d, fs) ¶c = firpmord (f, a, d, "cell") ¶c = firpmord (f, a, d, fs, "cell") ¶Estimate the filter-order needed for firpm to design a type-I or
type-II linear-phase FIR filter according to the given specifications.
A vector of real-numbers, increasing in the range (0, fs/2), giving the frequencies of the left and right edges of each band for which a specific amplitude response is desired (omitting 0 and fs/2, which are implied): [r1 l2 r2 …]. Transition-bands are defined implicitly as the regions between the given bands.
A vector of real-numbers giving the ideal amplitude response. An amplitude value is given for each band specified by f: [a1 a2 …]. 1 represents unity-gain, 0 represents infinite attenuation, and −1 represents a phase change of pi radians.
A vector of positive real-numbers giving the maximum allowable linear
deviation from the amplitudes given in a, thus constraining the actual
amplitude response (where defined by f) to be within a +/−
d. Note that, though related, d does not equate to
firpm’s w argument.
The sampling-frequency, which defaults to 2.
The function returns the estimated filter-order, together with the other
design specification values, in one of two forms suitable for use with
firpm. By default, multiple return values are used; alternatively, by
giving "cell" (or "c") as the last argument to firpmord,
the returned values are contained within a cell-array that can, if desired,
be passed directly to firpm.
The following examples illustrate the use of both mechanisms, as well as
aspects of firpmord usage in general:
# Low-pass; frequencies in kHz: [n f a w] = firpmord ([2.5 3], [1 0], [0.01 db2mag(-60)], 8); b = firpm (n, f, a, w);
# Band-pass:
c = firpmord ([3 4 8 9], [0 1 0], [1e-3 1e-2 1e-3], 20, "cell");
b = firpm (c{:});
# High-pass:
b = firpm (firpmord ([6.4 8]/16, [0 1], [1e-4 0.01], "c"){:});
In cases where elements of d follow a repeating pattern (e.g. all the elements are equal, or elements corresponding to pass-bands are equal and elements corresponding to stop-bands are equal), only as many elements as are needed to establish the pattern need be given.
For example, the following firpmord invocation pairs are equivalent:
# Low-pass: firpmord ([0.4 0.5], [0 1], [db2mag(-72) db2mag(-72)]); firpmord ([0.4 0.5], [0 1], [db2mag(-72)]);
# Multi-band-pass: ds = db2mag(-80); dp = 0.01; firpmord ([1 2 3 4 5 6 7 8]/10, [0 1 0 1 0], [ds dp ds dp ds]); firpmord ([1 2 3 4 5 6 7 8]/10, [0 1 0 1 0], [ds dp]);
The estimation algorithm used is per Ichige et al.3 Accuracy tends to decrease as
the number of bands increases. Even with two bands (i.e. high-pass or
low-pass), the algorithm may under- or over-estimate. See the
firpmord demonstrations for some examples.
In order to precisely determine the minimum order needed for a particular
design, firpmord could be used to seed an algorithm iterating
invocations of firpm (as exemplified in demonstration number five).
See also: firpm, kaiserord.
[b_out, a_out] = impinvar (b, a, fs, tol) ¶[b_out, a_out] = impinvar (b, a, fs) ¶[b_out, a_out] = impinvar (b, a) ¶Converts analog filter with coefficients b and a to digital, conserving impulse response.
If fs is not specified, or is an empty vector, it defaults to 1Hz.
If tol is not specified, it defaults to 0.0001 (0.1%) This function does the inverse of impinvar so that the following example should restore the original values of a and b.
invimpinvar implements the reverse of this function.
[b, a] = impinvar (b, a); [b, a] = invimpinvar (b, a);
Reference: Thomas J. Cavicchi (1996) “Impulse invariance and multiple-order poles”. IEEE transactions on signal processing, Vol 44 (9): 2344–2347
See also: bilinear, invimpinvar.
[b_out, a_out] = invimpinvar (b, a, fs, tol) ¶[b_out, a_out] = invimpinvar (b, a, fs) ¶[b_out, a_out] = invimpinvar (b, a) ¶Converts digital filter with coefficients b and a to analog, conserving impulse response.
This function does the inverse of impinvar so that the following example should restore the original values of a and b.
[b, a] = impinvar (b, a); [b, a] = invimpinvar (b, a);
If fs is not specified, or is an empty vector, it defaults to 1Hz.
If tol is not specified, it defaults to 0.0001 (0.1%)
Reference: Thomas J. Cavicchi (1996) “Impulse invariance and multiple-order poles”. IEEE transactions on signal processing, Vol 40 (9): 2344–2347
See also: bilinear, impinvar.
[z, p, g] = cauer(Rp, Rs, n) ¶Analog prototype for Cauer filter.
Passband ripple
Stopband ripple
Desired order
complex vector of zeros for the model.
complex vector of poles for the model.
gain value.
References:
- Serra, Celso Penteado, Teoria e Projeto de Filtros, Campinas: CARTGRAF, 1983.
- Lamar, Marcus Vinicius, Notas de aula da disciplina TE 456 - Circuitos Analogicos II, UFPR, 2001/2002.
[b, a] = pei_tseng_notch (frequencies, bandwidths) ¶Return coefficients for an IIR notch-filter with one or more filter frequencies and according (very narrow) bandwidths
to be used with filter or filtfilt.
The filter construction is based on an allpass which performs a reversal of phase at the filter frequencies.
Thus, the mean of the phase-distorted and the original signal has the respective frequencies removed.
See the demo for an illustration.
Original source: Pei, Soo-Chang, and Chien-Cheng Tseng "IIR Multiple Notch Filter Design Based on Allpass Filter" 1996 IEEE Tencon doi: 10.1109/TENCON.1996.608814)
[Sz, Sp, Sg] = sftrans (Sz, Sp, Sg, W, stop) ¶Transform band edges of a generic lowpass filter (cutoff at W=1) represented in splane zero-pole-gain form. W is the edge of the target filter (or edges if band pass or band stop). Stop is true for high pass and band stop filters or false for low pass and band pass filters. Filter edges are specified in radians, from 0 to pi (the nyquist frequency).
Theory: Given a low pass filter represented by poles and zeros in the splane, you can convert it to a low pass, high pass, band pass or band stop by transforming each of the poles and zeros individually. The following table summarizes the transformation:
Transform Zero at x Pole at x
---------------- ------------------------- ------------------------
Low Pass zero: Fc x/C pole: Fc x/C
S -> C S/Fc gain: C/Fc gain: Fc/C
---------------- ------------------------- ------------------------
High Pass zero: Fc C/x pole: Fc C/x
S -> C Fc/S pole: 0 zero: 0
gain: -x gain: -1/x
---------------- ------------------------- ------------------------
Band Pass zero: b +- sqrt(b^2-FhFl) pole: b +- sqrt(b^2-FhFl)
S^2+FhFl pole: 0 zero: 0
S -> C -------- gain: C/(Fh-Fl) gain: (Fh-Fl)/C
S(Fh-Fl) b=x/C (Fh-Fl)/2 b=x/C (Fh-Fl)/2
---------------- ------------------------- ------------------------
Band Stop zero: b +- sqrt(b^2-FhFl) pole: b +- sqrt(b^2-FhFl)
S(Fh-Fl) pole: +-sqrt(-FhFl) zero: +-sqrt(-FhFl)
S -> C -------- gain: -x gain: -1/x
S^2+FhFl b=C/x (Fh-Fl)/2 b=C/x (Fh-Fl)/2
---------------- ------------------------- ------------------------
Bilinear zero: (2+xT)/(2-xT) pole: (2+xT)/(2-xT)
2 z-1 pole: -1 zero: -1
S -> - --- gain: (2-xT)/T gain: (2-xT)/T
T z+1
---------------- ------------------------- ------------------------
where C is the cutoff frequency of the initial lowpass filter, Fc is the edge of the target low/high pass filter and [Fl,Fh] are the edges of the target band pass/stop filter. With abundant tedious algebra, you can derive the above formulae yourself by substituting the transform for S into H(S)=S-x for a zero at x or H(S)=1/(S-x) for a pole at x, and converting the result into the form:
H(S)=g prod(S-Xi)/prod(S-Xj)
The transforms are from the references. The actual pole-zero-gain changes I derived myself.
Please note that a pole and a zero at the same place exactly cancel. This is significant for High Pass, Band Pass and Band Stop filters which create numerous extra poles and zeros, most of which cancel. Those which do not cancel have a "fill-in" effect, extending the shorter of the sets to have the same number of as the longer of the sets of poles and zeros (or at least split the difference in the case of the band pass filter). There may be other opportunistic cancellations but I will not check for them.
Also note that any pole on the unit circle or beyond will result in an unstable filter. Because of cancellation, this will only happen if the number of poles is smaller than the number of zeros and the filter is high pass or band pass. The analytic design methods all yield more poles than zeros, so this will not be a problem.
References:
Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company.
h = cl2bp (m, w1, w2, up, lo) ¶h = cl2bp (m, w1, w2, up, lo, gridsize) ¶Constrained L2 bandpass FIR filter design. This is a fast implementation of the algorithm cited below. Compared to remez, it offers implicit specification of transition bands, a higher likelihood of convergence, and an error criterion combining features of both L2 and Chebyshev approaches.
Inputs:
degree of cosine polynomial, i.e. the number of output coefficients will be m*2+1
bandpass filter cutoffs in the range 0 <= w1 < w2 <= pi, where pi is the Nyquist frequency
vector of 3 upper bounds for [stopband1, passband, stopband2]
vector of 3 lower bounds for [stopband1, passband, stopband2]
search grid size; larger values may improve accuracy, but greatly increase calculation time. Default value is 2048, max value is 1e6.
Output:
A vector of m*2+1 FIR coefficients, or an empty value if the solver failed to converge.
Example:
h = cl2bp(30, 0.3*pi, 0.6*pi, [0.02, 1.02, 0.02], [-0.02, 0.98, -0.02], 2^11);
Original Paper: I. W. Selesnick, M. Lang, and C. S. Burrus. A modified algorithm for constrained least square design of multiband FIR filters without specified transition bands. IEEE Trans. on Signal Processing, 46(2):497-501, February 1998.
See also: remez.
b = fir1 (n, w) ¶b = fir1 (n, w, type) ¶b = fir1 (n, w, type, window) ¶b = fir1 (n, w, type, window, noscale) ¶Produce an order n FIR filter with the given frequency cutoff w, returning the n+1 filter coefficients in b. If w is a scalar, it specifies the frequency cutoff for a lowpass or highpass filter. If w is a two-element vector, the two values specify the edges of a bandpass or bandstop filter. If w is an N-element vector, each value specifies a band edge of a multiband pass/stop filter.
The filter type can be specified with one of the following strings: "low", "high", "stop", "pass", "bandpass", "DC-0", or "DC-1". The default is "low" is w is a scalar, "pass" if w is a pair, or "DC-0" if w is a vector with more than 2 elements.
An optional shaping window can be given as a vector with length n+1. If not specified, a Hamming window of length n+1 is used.
With the option "noscale", the filter coefficients are not normalized. The default is to normalize the filter such that the magnitude response of the center of the first passband is 1.
To apply the filter, use the return vector b with the filter
function, for example y = filter (b, 1, x).
Examples:
freqz (fir1 (40, 0.3)); freqz (fir1 (15, [0.2, 0.5], "stop")); # note the zero-crossing at 0.1 freqz (fir1 (15, [0.2, 0.5], "stop", "noscale"));
See also: filter, fir2.
b = fir2 (n, f, m) ¶b = fir2 (n, f, m, grid_n) ¶b = fir2 (n, f, m, grid_n, ramp_n) ¶b = fir2 (n, f, m, grid_n, ramp_n, window) ¶Produce an order n FIR filter with arbitrary frequency response m over frequency bands f, returning the n+1 filter coefficients in b. The vector f specifies the frequency band edges of the filter response and m specifies the magnitude response at each frequency.
The vector f must be nondecreasing over the range [0,1], and the first and last elements must be 0 and 1, respectively. A discontinuous jump in the frequency response can be specified by duplicating a band edge in f with different values in m.
The resolution over which the frequency response is evaluated can be controlled with the grid_n argument. The default is 512 or the next larger power of 2 greater than the filter length.
The band transition width for discontinuities can be controlled with the ramp_n argument. The default is grid_n/25. Larger values will result in wider band transitions but better stopband rejection.
An optional shaping window can be given as a vector with length n+1. If not specified, a Hamming window of length n+1 is used.
To apply the filter, use the return vector b with the filter
function, for example y = filter (b, 1, x).
Example:
f = [0, 0.3, 0.3, 0.6, 0.6, 1]; m = [0, 0, 1, 1/2, 0, 0]; [h, w] = freqz (fir2 (100, f, m)); plot (f, m, ";target response;", w/pi, abs (h), ";filter response;");
See also: filter, fir1.
b = firls (n, f, a) ¶b = firls (n, f, a, w) ¶FIR filter design using least squares method. Returns a length n+1 linear phase filter such that the integral of the weighted mean squared error in the specified bands is minimized.
The vector f specifies the frequencies of the band edges, normalized so that half the sample frequency is equal to 1. Each band is specified by two frequencies, to the vector must have an even length.
The vector a specifies the amplitude of the desired response at each band edge.
The optional argument w is a weighting function that contains one value for each band that weights the mean squared error in that band.
a must be the same length as f, and w must be half the
length of f. n must be even. If given an odd value,
firls increments it by 1.
The least squares optimization algorithm for computing FIR filter coefficients is derived in detail in:
I. Selesnick, "Linear-Phase FIR Filter Design by Least Squares," http://cnx.org/content/m10577
[n, Wn, beta, ftype] = kaiserord (f, m, dev) ¶[…] = kaiserord (f, m, dev, fs) ¶Return the parameters needed to produce a filter of the desired specification from a Kaiser window. The vector f contains pairs of frequency band edges in the range [0,1]. The vector m specifies the magnitude response for each band. The values of m must be zero for all stop bands and must have the same magnitude for all pass bands. The deviation of the filter dev can be specified as a scalar or a vector of the same length as m. The optional sampling rate fs can be used to indicate that f is in Hz in the range [0,fs/2].
The returned value n is the required order of the filter (the length
of the filter minus 1). The vector Wn contains the band edges of
the filter suitable for passing to fir1. The value beta is
the parameter of the Kaiser window of length n+1 to shape the filter.
The string ftype contains the type of filter to specify to
fir1.
The Kaiser window parameters n and beta are computed from the relation between ripple (A=-20*log10(dev)) and transition width (dw in radians) discovered empirically by Kaiser:
/ 0.1102(A-8.7) A > 50
beta = | 0.5842(A-21)^0.4 + 0.07886(A-21) 21 <= A <= 50
\ 0.0 A < 21
n = (A-8)/(2.285 dw)
Example:
[n, w, beta, ftype] = kaiserord ([1000, 1200], [1, 0], [0.05, 0.05], 11025); b = fir1 (n, w, kaiser (n+1, beta), ftype, "noscale"); freqz (b, 1, [], 11025);
See also: fir1, kaiser.
(nb, at) ¶(nb, at, linear) ¶Computes a finite impulse response (FIR) filter for use with a quasi-perfect reconstruction polyphase-network filter bank. This version utilizes a Kaiser window to shape the frequency response of the designed filter. Tha number nb of bands and the desired attenuation at in the stop-band are given as parameters.
The Kaiser window is multiplied by the ideal impulse response h(n)=a.sinc(a.n) and converted to its minimum-phase version by means of a Hilbert transform.
By using a third non-null argument, the minimum-phase calculation is omitted at all.
b = remez (n, f, a) ¶b = remez (n, f, a, w) ¶b = remez (n, f, a, w, ftype) ¶b = remez (n, f, a, w, ftype, griddensity) ¶Parks-McClellan optimal FIR filter design.
gives the filter order, where the generated filter length taps is n+1
gives frequency at the band edges [b1 e1 b2 e2 b3 e3 …]
gives amplitude at the band edges [a(b1) a(e1) a(b2) a(e2) …]
gives weighting applied to each band
is "bandpass", "hilbert" or "differentiator"
determines how accurately the filter will be constructed. The minimum value is 16, but higher numbers are slower to compute.
Frequency is in the range (0, 1), with 1 being the Nyquist frequency.
f = sgolay (p, n) ¶f = sgolay (p, n, m) ¶f = sgolay (p, n, m, ts) ¶Computes the filter coefficients for all Savitzsky-Golay smoothing filters of order p for length n (odd). m can be used in order to get directly the mth derivative. In this case, ts is a scaling factor.
The early rows of F smooth based on future values and later rows smooth based on past values, with the middle row using half future and half past. In particular, you can use row i to estimate x(k) based on the i-1 preceding values and the n-i following values of x values as y(k) = F(i,:) * x(k-i+1:k+n-i).
Normally, you would apply the first (n-1)/2 rows to the first k points of the vector, the last k rows to the last k points of the vector and middle row to the remainder, but for example if you were running on a realtime system where you wanted to smooth based on the all the data collected up to the current time, with a lag of five samples, you could apply just the filter on row n-5 to your window of length n each time you added a new sample.
Reference: Numerical recipes in C. p 650
See also: sgolayfilt.
[zc, zr] = cplxreal (z) ¶[zc, zr] = cplxreal (z, tol) ¶[zc, zr] = cplxreal (z, tol, dim) ¶Sort the numbers z into complex-conjugate-valued and real-valued elements. The positive imaginary complex numbers of each complex conjugate pair are returned in zc and the real numbers are returned in zr.
tol is a weighting factor in the range [0, 1) which determines the
tolerance of the matching. The default value is 100 * eps and the
resulting tolerance for a given complex pair is
tol * abs (z(i))).
By default the complex pairs are sorted along the first non-singleton dimension of z. If dim is specified, then the complex pairs are sorted along this dimension.
Signal an error if some complex numbers could not be paired. Signal an error if all complex numbers are not exact conjugates (to within tol). Note that there is no defined order for pairs with identical real parts but differing imaginary parts.
See also: cplxpair.
(x) ¶(x, m) ¶(x, m, w) ¶(x, m, w, a) ¶Chirp z-transform. Compute the frequency response starting at a and stepping by w for m steps. a is a point in the complex plane, and w is the ratio between points in each step (i.e., radius increases exponentially, and angle increases linearly).
To evaluate the frequency response for the range f1 to f2 in a signal with sampling frequency Fs, use the following:
m = 32; ## number of points desired w = exp(-j*2*pi*(f2-f1)/((m-1)*Fs)); ## freq. step of f2-f1/m a = exp(j*2*pi*f1/Fs); ## starting at frequency f1 y = czt(x, m, w, a);
If you don’t specify them, then the parameters default to a Fourier transform: m=length(x), w=exp(-j*2*pi/m), a=1
If x is a matrix, the transform will be performed column-by-column.
(x) ¶(x, n) ¶Compute the discrete cosine transform of x. If n is given, then x is padded or trimmed to length n before computing the transform. If x is a matrix, compute the transform along the columns of the the matrix. The transform is faster if x is real-valued and has even length.
The discrete cosine transform x can be defined as follows:
N-1
X[k] = w(k) sum x[n] cos (pi (2n+1) k / 2N ), k = 0, ..., N-1
n=0
with w(0) = sqrt(1/N) and w(k) = sqrt(2/N), k = 1, ..., N-1. There are other definitions with different scaling of X[k], but this form is common in image processing.
See also: idct, dct2, idct2, dctmtx.
(n) ¶Return the DCT transformation matrix of size n-by-n.
If A is an n-by-n matrix, then the following are true:
T*A == dct(A), T'*A == idct(A)
T*A*T' == dct2(A), T'*A*T == idct2(A)
A DCT transformation matrix is useful for doing things like jpeg image compression, in which an 8x8 DCT matrix is applied to non-overlapping blocks throughout an image and only a subblock on the top left of each block is kept. During restoration, the remainder of the block is filled with zeros and the inverse transform is applied to the block.
See also: dct, idct, dct2, idct2.
d = dftmtx (n) ¶Compute the n-by-n Fourier transformation matrix. This is
the matrix d such that the Fourier transform of a column vector of
length n is given by dftmtx(n) * x and the
inverse Fourier transform is given by inv(dftmtx(n)) * x.
In general this is less efficient than calling the fft and
ifft functions directly.
See also: fft, ifft.
y = digitrevorder (x, r) ¶[y, i] = digitrevorder (x, r) ¶Reorder the elements of the vector x in digit-reversed order. The elements of x are converted to radix r and reversed. The reordered indices of the elements of x are returned in i.
See also: bitrevorder, fft, ifft.
y = dst (x) ¶y = dst (x, n) ¶Computes the type I discrete sine transform of x. If n is given, then x is padded or trimmed to length n before computing the transform. If x is a matrix, compute the transform along the columns of the the matrix.
The discrete sine transform X of x can be defined as follows:
N
X[k] = sum x[n] sin (pi n k / (N+1) ), k = 1, ..., N
n=1
See also: idst.
[u, v] = dwt (x, wname) ¶[u, v] = dwt (x, Hp, Gp) ¶[u, v] = dwt (x, Hp, Gp, …) ¶Discrete wavelet transform (1D).
Inputs
Signal vector.
Wavelet name.
Coefficients of low-pass decomposition FIR filter.
Coefficients of high-pass decomposition FIR filter.
Outputs
Signal vector of average, approximation.
Signal vector of difference, detail.
m = fht (d) ¶m = fht (d, n) ¶m = fht (d, n, dim) ¶Calculate the Fast Hartley Transform of real input d. If d is a matrix, the Hartley transform is calculated along the columns by default. The options n and dim are similar to the options of FFT function.
The forward and inverse Hartley transforms are the same (except for a scale factor of 1/N for the inverse Hartley transform), but implemented using different functions.
The definition of the forward hartley transform for vector d, m[K] = \sum_{i=0}^{N-1} d[i]*(cos[K*2*pi*i/N] + sin[K*2*pi*i/N]), for 0 <= K < N. m[K] = \sum_{i=0}^{N-1} d[i]*CAS[K*i], for 0 <= K < N.
fht(1:4)
See also: ifht, fft.
(x) ¶(x, n) ¶(x, n, order) ¶Compute the Walsh-Hadamard transform of x using the Fast Walsh-Hadamard Transform (FWHT) algorithm. If the input is a matrix, the FWHT is calculated along the columns of x.
The number of elements of x must be a power of 2; if not, the input will be extended and filled with zeros. If a second argument is given, the input is truncated or extended to have length n.
The third argument specifies the order in which the returned Walsh-Hadamard transform coefficients should be arranged. The order may be any of the following strings:
The coefficients are returned in sequency order. This is the default if order is not given.
The coefficients are returned in Hadamard order.
The coefficients are returned in Gray code order.
See also: ifwht.
h = hilbert (f, N, dim) ¶Analytic extension of real valued signal.
h = hilbert (f) computes the extension of the real
valued signal f to an analytic signal. If f is a matrix,
the transformation is applied to each column. For N-D arrays,
the transformation is applied to the first non-singleton dimension.
real (h) contains the original signal f.
imag (h) contains the Hilbert transform of f.
hilbert (f, N) does the same using a length N
Hilbert transform. The result will also have length N.
hilbert (f, [], dim) or
hilbert (f, N, dim) does the same along
dimension dim.
y = idct (x) ¶y = idct (x, n) ¶Compute the inverse discrete cosine transform of x. If n is given, then x is padded or trimmed to length n before computing the transform. If x is a matrix, compute the transform along the columns of the the matrix. The transform is faster if x is real-valued and even length.
The inverse discrete cosine transform x can be defined as follows:
N-1
x[n] = sum w(k) X[k] cos (pi (2n+1) k / 2N ), n = 0, ..., N-1
k=0
with w(0) = sqrt(1/N) and w(k) = sqrt(2/N), k = 1, ..., N-1
See also: dct, dct2, idct2, dctmtx.
m = ifht (d, n, dim) ¶Calculate the inverse Fast Hartley Transform of real input d. If d is a matrix, the inverse Hartley transform is calculated along the columns by default. The options n and dim are similar to the options of FFT function.
The forward and inverse Hartley transforms are the same (except for a scale factor of 1/N for the inverse hartley transform), but implemented using different functions.
The definition of the forward hartley transform for vector d, m[K] = 1/N \sum_{i=0}^{N-1} d[i]*(cos[K*2*pi*i/N] + sin[K*2*pi*i/N]), for 0 <= K < N. m[K] = 1/N \sum_{i=0}^{N-1} d[i]*CAS[K*i], for 0 <= K < N.
ifht(1:4)
See also: fht, fft.
(x) ¶(x, n) ¶(x, n, order) ¶Compute the inverse Walsh-Hadamard transform of x using the Fast Walsh-Hadamard Transform (FWHT) algorithm. If the input is a matrix, the inverse FWHT is calculated along the columns of x.
The number of elements of x must be a power of 2; if not, the input will be extended and filled with zeros. If a second argument is given, the input is truncated or extended to have length n.
The third argument specifies the order in which the returned inverse Walsh-Hadamard transform coefficients should be arranged. The order may be any of the following strings:
The coefficients are returned in sequency order. This is the default if order is not given.
The coefficients are returned in Hadamard order.
The coefficients are returned in Gray code order.
See also: fwht.
[y, ym] = rceps (x) ¶Return the cepstrum of the signal x.
If x is a matrix, return the cepstrum of each column.
If called with two output arguments, the minimum phase reconstruction of the signal x is returned in ym.
For example:
f0 = 70; Fs = 10000; # 100 Hz fundamental, 10kHz sampling rate a = poly (0.985 * exp (1i * pi * [0.1, -0.1, 0.3, -0.3])); # two formants s = 0.005 * randn (1024, 1); # Noise excitation signal s(1:Fs/f0:length(s)) = 1; # Impulse glottal wave x = filter (1, a, s); # Speech signal [y, ym] = rceps (x .* hanning (1024));
Reference: Programs for Digital Signal Processing, IEEE Press, John Wiley & Sons, New York, 1979.
[P, w] = __power (b, a) ¶[…] = __power (b, a, nfft) ¶[…] = __power (b, a, nfft, Fs) ¶[…] = __power (b, a, nfft, Fs, range) ¶[…] = __power (b, a, nfft, Fs, range, units) ¶(…) ¶Plot the power spectrum of the given ARMA model.
b, a: filter coefficients (b=numerator, a=denominator) nfft is number of points at which to sample the power spectrum Fs is the sampling frequency of x range is ’half’ (default) or ’whole’ units is ’squared’ or ’db’ (default) range and units may be specified any time after the filter, in either order
Returns P, the magnitude vector, and w, the frequencies at which it is sampled. If there are no return values requested, then plot the power spectrum and don’t return anything.
(a, v) ¶(a, v, freq) ¶(a, v, freq, Fs) ¶(…, range) ¶(…, method) ¶(…, plottype) ¶[psd, f_out] = ar_psd (…) ¶Calculate the power spectrum of the autoregressive model
M
x(n) = sqrt(v).e(n) + SUM a(k).x(n-k)
k=1
where x(n) is the output of the model and e(n) is white noise.
This function is intended for use with
[a, v, k] = arburg (x, poles, criterion)
which use the Burg (1968) method to calculate a "maximum entropy"
autoregressive model of x.
If the freq argument is a vector (of frequencies) the spectrum is calculated using the polynomial method and the method argument is ignored. For scalar freq, an integer power of 2, or method = "FFT", causes the spectrum to be calculated by FFT. Otherwise, the spectrum is calculated as a polynomial. It may be computationally more efficient to use the FFT method if length of the model is not much smaller than the number of frequency values. The spectrum is scaled so that spectral energy (area under spectrum) is the same as the time-domain energy (mean square of the signal).
ARGUMENTS: All but the first two arguments are optional and may be empty.
CONTROL-STRING ARGUMENTS – each of these arguments is a character string. Control-string arguments can be in any order after the other arguments.
Range:
’half’, ’onesided’ : frequency range of the spectrum is from zero up to but not including sample_f/2. Power from negative frequencies is added to the positive side of the spectrum. ’whole’, ’twosided’ : frequency range of the spectrum is -sample_f/2 to sample_f/2, with negative frequencies stored in "wrap around" order after the positive frequencies; e.g. frequencies for a 10-point ’twosided’ spectrum are 0 0.1 0.2 0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1 ’shift’, ’centerdc’ : same as ’whole’ but with the first half of the spectrum swapped with second half to put the zero-frequency value in the middle. (See "help fftshift". If "freq" is vector, ’shift’ is ignored. If model coefficients "ar_coeffs" are real, the default range is ’half’, otherwise default range is ’whole’.
Method:
’fft’: use FFT to calculate power spectrum. ’poly’: calculate power spectrum as a polynomial of 1/z N.B. this argument is ignored if the "freq" argument is a vector. The default is ’poly’ unless the "freq" argument is an integer power of 2.
Plot type:
’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’: specifies the type of plot. The default is ’plot’, which means linear-linear axes. ’squared’ is the same as ’plot’. ’dB’ plots "10*log10(psd)". This argument is ignored and a spectrum is not plotted if the caller requires a returned value.
RETURNED VALUES: If returned values are not required by the caller, the spectrum is plotted and nothing is returned.
REFERENCE [1] Equation 2.28 from Steven M. Kay and Stanley Lawrence Marple Jr.: "Spectrum analysis – a modern perspective", Proceedings of the IEEE, Vol 69, pp 1380-1419, Nov., 1981
[Pxx, freq] = cohere(x,y,Nfft,Fs,window,overlap,range,plot_type,detrend) ¶Estimate (mean square) coherence of signals "x" and "y".
Use the Welch (1967) periodogram/FFT method.
Compatible with Matlab R11 cohere and earlier.
See "help pwelch" for description of arguments, hints and references — especially hint (7) for Matlab R11 defaults.
[Pxx, freq] = cpsd (x, y) ¶[…] = cpsd (x, y, window) ¶[…] = cpsd (x, y, window, overlap) ¶[…] = cpsd (x, y, window, overlap, Nfft) ¶[…] = cpsd (x, y, window, overlap, Nfft, Fs) ¶[…] = cpsd (x, y, window, overlap, Nfft, Fs, range) ¶(…) ¶Estimate cross power spectrum of data x and y by the Welch (1967) periodogram/FFT method.
See also: pwelch.
[Pxx,freq] = csd(x, y, Nfft, Fs, window, overlap, range, plot_type, detrend) ¶Estimate cross power spectrum of data "x" and "y" by the Welch (1967) periodogram/FFT method.
Compatible with Matlab R11 csd and earlier.
See "help pwelch" for description of arguments, hints and references — especially hint (7) for Matlab R11 defaults.
(x) ¶Convert decibels (dB) to power.
The power of x is defined as
p = 10 ^ (x/10).
If x is a vector, matrix, or N-dimensional array, the power is computed over the elements of x.
Example:
db2pow ([-10, 0, 10]) ⇒ 0.1000 1.0000 10.0000
See also: pow2db.
[Pxx, freq] = mscohere (x, y) ¶[…] = mscohere (x, y, window) ¶[…] = mscohere (x, y, window, overlap) ¶[…] = mscohere (x, y, window, overlap, Nfft) ¶[…] = mscohere (x, y, window, overlap, Nfft, Fs) ¶[…] = mscohere (x, y, window, overlap, Nfft, Fs, range) ¶(…) ¶Estimate (mean square) coherence of signals x and y. Use the Welch (1967) periodogram/FFT method.
See also: pwelch.
[psd,f_out] = pburg(x, poles, freq, Fs, range, method, plot_type, criterion) ¶Calculate Burg maximum-entropy power spectral density.
The functions "arburg" and "ar_psd" do all the work. See "help arburg" and "help ar_psd" for further details.
ARGUMENTS:
All but the first two arguments are optional and may be empty.
[vector] sampled data
[integer scalar] required number of poles of the AR model
[real vector] frequencies at which power spectral density is calculated.
[integer scalar] number of uniformly distributed frequency values at which spectral density is calculated. [default=256]
[real scalar] sampling frequency (Hertz) [default=1]
CONTROL-STRING ARGUMENTS – each of these arguments is a character string.
Control-string arguments can be in any order after the other arguments.
’half’, ’onesided’ : frequency range of the spectrum is from zero up to but not including sample_f/2. Power from negative frequencies is added to the positive side of the spectrum.
’whole’, ’twosided’ : frequency range of the spectrum is -sample_f/2 to sample_f/2, with negative frequencies stored in "wrap around" order after the positive frequencies; e.g. frequencies for a 10-point ’twosided’ spectrum are 0 0.1 0.2 0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1
’shift’, ’centerdc’ : same as ’whole’ but with the first half of the spectrum swapped with second half to put the zero-frequency value in the middle. (See "help fftshift". If "freq" is vector, ’shift’ is ignored. If model coefficients "ar_coeffs" are real, the default range is ’half’, otherwise default range is ’whole’.
’fft’: use FFT to calculate power spectral density.
’poly’: calculate spectral density as a polynomial of 1/z N.B. this argument is ignored if the "freq" argument is a vector. The default is ’poly’ unless the "freq" argument is an integer power of 2.
’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’: specifies the type of plot. The default is ’plot’, which means linear-linear axes. ’squared’ is the same as ’plot’. ’dB’ plots "10*log10(psd)". This argument is ignored and a spectrum is not plotted if the caller requires a returned value.
[optional string arg] model-selection criterion. Limits the number of poles so that spurious poles are not added when the whitened data has no more information in it (see Kay & Marple, 1981). Recognized values are
’AKICc’ – approximate corrected Kullback information criterion (recommended),
’KIC’ – Kullback information criterion
’AICc’ – corrected Akaike information criterion
’AIC’ – Akaike information criterion
’FPE’ – final prediction error" criterion
The default is to NOT use a model-selection criterion
RETURNED VALUES:
If return values are not required by the caller, the spectrum is plotted and nothing is returned.
[real vector] power-spectral density estimate
[real vector] frequency values
HINTS
This function is a wrapper for arburg and ar_psd.
See "help arburg", "help ar_psd".
(x) ¶Convert power to decibels (dB).
The decibel value of x is defined as
d = 10 * log10 (x).
If x is a vector, matrix, or N-dimensional array, the decibel value is computed over the elements of x.
Examples:
pow2db ([0, 10, 100]) ⇒ -Inf 10 20
See also: db2pow.
[spectra,freq] = pwelch(x, window, overlap, Nfft, Fs, range, plot_type, detrend, sloppy) ¶Estimate power spectral density of data "x" by the Welch (1967) periodogram/FFT method.
All arguments except "x" are optional.
The data is divided into segments. If "window" is a vector, each segment has the same length as "window" and is multiplied by "window" before (optional) zero-padding and calculation of its periodogram. If "window" is a scalar, each segment has a length of "window" and a Hamming window is used.
The spectral density is the mean of the periodograms, scaled so that area under the spectrum is the same as the mean square of the data. This equivalence is supposed to be exact, but in practice there is a mismatch of up to 0.5% when comparing area under a periodogram with the mean square of the data.
[spectra,freq] = pwelch(x,y,window,overlap,Nfft,Fs, range,plot_type,detrend,sloppy,results)
Two-channel spectrum analyser. Estimate power spectral density, cross- spectral density, transfer function and/or coherence functions of time- series input data "x" and output data "y" by the Welch (1967) periodogram/FFT method.
pwelch treats the second argument as "y" if there is a control-string argument "cross", "trans", "coher" or "ypower"; "power" does not force the 2nd argument to be treated as "y". All other arguments are optional. All spectra are returned in matrix "spectra".
[spectra,Pxx_ci,freq] = pwelch(x,window,overlap,Nfft,Fs,conf, range,plot_type,detrend,sloppy)
[spectra,Pxx_ci,freq] = pwelch(x,y,window,overlap,Nfft,Fs,conf, range,plot_type,detrend,sloppy,results)
Estimates confidence intervals for the spectral density.
See Hint (7) below for compatibility options.
Confidence level "conf" is the 6th or 7th numeric argument. If "results" control-string arguments are used, one of them must be "power" when the "conf" argument is present; pwelch can estimate confidence intervals only for the power spectrum of the "x" data. It does not know how to estimate confidence intervals of the cross-power spectrum, transfer function or coherence; if you can suggest a good method, please send a bug report.
ARGUMENTS
All but the first argument are optional and may be empty, except that the "results" argument may require the second argument to be "y".
[non-empty vector] system-input time-series data
[non-empty vector] system-output time-series data
[real vector] of window-function values; the data segment has the same length as the window. Default window shape is Hamming.
[integer scalar] length of each data segment. The default value is window=sqrt(length(x)) rounded up to the nearest integer power of 2; see ’sloppy’ argument.
[real scalar] segment overlap expressed as a multiple of window or segment length. 0 <= overlap < 1, The default is overlap=0.5 .
[integer scalar] Length of FFT. The default is the length of the "window" vector or has the same value as the scalar "window" argument. If Nfft is larger than the segment length, "seg_len", the data segment is padded with "Nfft-seg_len" zeros. The default is no padding. Nfft values smaller than the length of the data segment (or window) are ignored silently.
[real scalar] sampling frequency (Hertz); default=1.0
[real scalar] confidence level between 0 and 1. Confidence intervals of the spectral density are estimated from scatter in the periodograms and are returned as Pxx_ci. Pxx_ci(:,1) is the lower bound of the confidence interval and Pxx_ci(:,2) is the upper bound. If there are three return values, or conf is an empty matrix, confidence intervals are calculated for conf=0.95 . If conf is zero or is not given, confidence intervals are not calculated. Confidence intervals can be obtained only for the power spectral density of x; nothing else.
CONTROL-STRING ARGUMENTS – each of these arguments is a character string. Control-string arguments must be after the other arguments but can be in any order.
’half’, ’onesided’ : frequency range of the spectrum is zero up to but not including Fs/2. Power from negative frequencies is added to the positive side of the spectrum, but not at zero or Nyquist (Fs/2) frequencies. This keeps power equal in time and spectral domains. See reference [2].
’whole’, ’twosided’ : frequency range of the spectrum is -Fs/2 to Fs/2, with negative frequencies stored in "wrap around" order after the positive frequencies; e.g. frequencies for a 10-point ’twosided’ spectrum are 0 0.1 0.2 0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1
’shift’, ’centerdc’ : same as ’whole’ but with the first half of the spectrum swapped with second half to put the zero-frequency value in the middle. (See "help fftshift".
If data (x and y) are real, the default range is ’half’, otherwise default range is ’whole’.
’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’: specifies the type of plot. The default is ’plot’, which means linear-linear axes. ’squared’ is the same as ’plot’. ’dB’ plots "10*log10(psd)". This argument is ignored and a spectrum is not plotted if the caller requires a returned value.
’no-strip’, ’none’ – do NOT remove mean value from the data
’short’, ’mean’ – remove the mean value of each segment from each segment of the data.
’linear’, – remove linear trend from each segment of the data.
’long-mean’ – remove the mean value from the data before splitting it into segments. This is the default.
’sloppy’: FFT length is rounded up to the nearest integer power of 2 by zero padding. FFT length is adjusted after addition of padding by explicit Nfft argument. The default is to use exactly the FFT and window/ segment lengths specified in argument list.
specifies what results to return (in the order specified and as many as desired).
’power’ calculate power spectral density of "x"
’cross’ calculate cross spectral density of "x" and "y"
’trans’ calculate transfer function of a system with input "x" and output "y"
’coher’ calculate coherence function of "x" and "y"
’ypower’ calculate power spectral density of "y"
The default is ’power’, with argument "y" omitted.
RETURNED VALUES:
If return values are not required by the caller, the results are plotted and nothing is returned.
[real-or-complex matrix] columns of the matrix contain results in the same order as specified by "results" arguments. Each column contains one of the result vectors.
[real matrix] estimate of confidence interval for power spectral density of x. First column is the lower bound. Second column is the upper bound.
[real column vector] frequency values
HINTS
1) overlap is expressed as a multiple of window length — effect of overlap scales with window length
2) default values of length(window), Nfft and Fs are more sensible, and
3) Goertzel algorithm is not available so Nfft cannot be an array of frequencies as in Matlab 2006b.
Pwelch has four persistent Matlab-compatibility levels. Calling pwelch with an empty first argument sets the order of arguments and defaults specified above in the USAGE and ARGUMENTS section of this documentation.
prev_compat=pwelch([]);
[Pxx,f]=pwelch(x,window,overlap,Nfft,Fs,conf,...);
Calling pwelch with a single string argument (as described below) gives compatibility with Matlab R11 or R12, or the R14 spectrum.welch defaults. The returned value is the PREVIOUS compatibility string.
Matlab R11: For compatibility with the Matlab R11 pwelch:
prev_compat=pwelch('R11-');
[Pxx,f]=pwelch(x,Nfft,Fs,window,overlap,conf,range,units);
%% units of overlap are "number of samples"
%% defaults: Nfft=min(length(x),256), Fs=2*pi, length(window)=Nfft,
%% window=Hanning, do not detrend,
%% N.B. "Sloppy" is not available.
Matlab R12: For compatibility with Matlab R12 to 2006a pwelch:
prev_compat=pwelch('R12+');
[Pxx,f]=pwelch(x,window,overlap,nfft,Fs,...);
%% units of overlap are "number of samples"
%% defaults: length(window)==length(x)/8, window=Hamming,
%% Nfft=max(256,NextPow2), Fs=2*pi, do not detrend
%% NextPow2 is the next power of 2 greater than or equal to the
%% window length. "Sloppy", "conf" are not available. Default
%% window length gives very poor amplitude resolution.
To adopt defaults of the Matlab R14 "spectrum.welch" spectrum object associated "psd" method.
prev_compat=pwelch('psd');
[Pxx,f] = pwelch(x,window,overlap,Nfft,Fs,conf,...);
%% overlap is expressed as a percentage of window length,
%% defaults: length(window)==64, Nfft=max(256,NextPow2), Fs=2*pi
%% do not detrend
%% NextPow2 is the next power of 2 greater than or equal to the
%% window length. "Sloppy" is not available.
%% Default window length gives coarse frequency resolution.
REFERENCES
[1] Peter D. Welch (June 1967): "The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms." IEEE Transactions on Audio Electroacoustics, Vol AU-15(6), pp 70-73
[2] William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery", "Numerical recipes in C, The art of scientific computing", 2nd edition, Cambridge University Press, 2002 — Section 13.7.
[psd,f_out] = pyulear(x,poles,freq,Fs,range,method,plot_type) ¶Calculates a Yule-Walker autoregressive (all-pole) model of the data "x" and computes the power spectrum of the model.
This is a wrapper for functions "aryule" and "ar_psd" which perform the argument checking.
See "help aryule" and "help ar_psd" for further details.
ARGUMENTS:
All but the first two arguments are optional and may be empty.
[vector] sampled data
[integer scalar] required number of poles of the AR model
[real vector] frequencies at which power spectral density is calculated
[integer scalar] number of uniformly distributed frequency values at which spectral density is calculated. [default=256]
[real scalar] sampling frequency (Hertz) [default=1]
CONTROL-STRING ARGUMENTS – each of these arguments is a character string.
Control-string arguments can be in any order after the other arguments.
’half’, ’onesided’ : frequency range of the spectrum is from zero up to but not including sample_f/2. Power from negative frequencies is added to the positive side of the spectrum.
’whole’, ’twosided’ : frequency range of the spectrum is -sample_f/2 to sample_f/2, with negative frequencies stored in "wrap around" order after the positive frequencies; e.g. frequencies for a 10-point ’twosided’ spectrum are 0 0.1 0.2 0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1
’shift’, ’centerdc’ : same as ’whole’ but with the first half of the spectrum swapped with second half to put the zero-frequency value in the middle. (See "help fftshift". If "freq" is vector, ’shift’ is ignored.
If model coefficients "ar_coeffs" are real, the default range is ’half’, otherwise default range is ’whole’.
’fft’: use FFT to calculate power spectrum.
’poly’: calculate power spectrum as a polynomial of 1/z N.B. this argument is ignored if the "freq" argument is a vector. The default is ’poly’ unless the "freq" argument is an integer power of 2.
’plot’, ’semilogx’, ’semilogy’, ’loglog’, ’squared’ or ’db’: specifies the type of plot. The default is ’plot’, which means linear-linear axes. ’squared’ is the same as ’plot’. ’dB’ plots "10*log10(psd)". This argument is ignored and a spectrum is not plotted if the caller requires a returned value.
RETURNED VALUES:
If return values are not required by the caller, the spectrum is plotted and nothing is returned.
[real vector] power-spectrum estimate
[real vector] frequency values
HINTS
This function is a wrapper for aryule and ar_psd.
See "help aryule", "help ar_psd".
[Pxx,freq] = tfe(x,y,Nfft,Fs,window,overlap,range,plot_type,detrend) ¶Estimate transfer function of system with input "x" and output "y".
Use the Welch (1967) periodogram/FFT method.
Compatible with Matlab R11 tfe and earlier.
See "help pwelch" for description of arguments, hints and references — especially hint (7) for Matlab R11 defaults.
(x, y) ¶(x, y, window) ¶(x, y, window, overlap) ¶(x, y, window, overlap, Nfft) ¶(x, y, window, overlap, Nfft, Fs) ¶(x, y, window, overlap, Nfft, Fs, range) ¶[Pxx, freq] = tfestimate (…) ¶Estimate transfer function of system with input x and output y. Use the Welch (1967) periodogram/FFT method.
See also: pwelch.
(m) ¶Return the filter coefficients of a modified Bartlett-Hann window of length m.
See also: rectwin, bartlett.
(m) ¶(m, "periodic") ¶(m, "symmetric") ¶Return the filter coefficients of a Blackman-Harris window of length m.
If the optional argument "periodic" is given, the periodic form
of the window is returned. This is equivalent to the window of length
m+1 with the last coefficient removed. The optional argument
"symmetric" is equivalent to not specifying a second argument.
See also: rectwin, bartlett.
(m) ¶(m, "periodic") ¶(m, "symmetric") ¶Return the filter coefficients of a Blackman-Nuttall window of length m.
If the optional argument "periodic" is given, the periodic form
of the window is returned. This is equivalent to the window of length
m+1 with the last coefficient removed. The optional argument
"symmetric" is equivalent to not specifying a second argument.
See also: nuttallwin, kaiser.
(m) ¶Return the filter coefficients of a Bohman window of length m.
See also: rectwin, bartlett.
(m) ¶Return the filter coefficients of a rectangular window of length m.
(m) ¶(m, at) ¶Return the filter coefficients of a Dolph-Chebyshev window of length m. The Fourier transform of the window has a stop-band attenuation of at dB. The default attenuation value is 100 dB.
For the definition of the Chebyshev window, see
* Peter Lynch, "The Dolph-Chebyshev Window: A Simple Optimal Filter", Monthly Weather Review, Vol. 125, pp. 655-660, April 1997. (http://www.maths.tcd.ie/~plynch/Publications/Dolph.pdf)
* C. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level", Proc. IEEE, 34, pp. 335-348.
The window is described in frequency domain by the expression:
Cheb(m-1, beta * cos(pi * k/m))
W(k) = -------------------------------
Cheb(m-1, beta)
with
beta = cosh(1/(m-1) * acosh(10^(at/20))
and Cheb(m,x) denoting the m-th order Chebyshev polynomial calculated at the point x.
Note that the denominator in W(k) above is not computed, and after the inverse Fourier transform the window is scaled by making its maximum value unitary.
See also: kaiser.
expwin (m, alpha) ¶expwin (m, sll) ¶expwin (…, "canonical") ¶Return the coefficients of an exponential window4 of length m.
The window’s adjustable parameter can be set directly as alpha (≥0) or indirectly with sll (<0), the latter being the desired side-lobe level in dBc.
By default, the returned window is both symmetric and periodic. With the
optional argument "canonical", the canonical form of the window is
returned, which is symmetric but not periodic.
See also: kaiser, poisswin.
(m) ¶(m, "periodic") ¶(m, "symmetric") ¶Return the filter coefficients of a Flat Top window of length m. The Flat Top window is defined by the function f(w):
f(w) = 1 - 1.93 cos(2 pi w) + 1.29 cos(4 pi w)
- 0.388 cos(6 pi w) + 0.0322cos(8 pi w)
where w = i/(m-1) for i=0:m-1 for a symmetric window, or w = i/m for i=0:m-1 for a periodic window. The default is symmetric. The returned window is normalized to a peak of 1 at w = 0.5.
This window has low pass-band ripple, but high bandwidth.
According to [1]:
The main use for the Flat Top window is for calibration, due to its negligible amplitude errors.
[1] Gade, S; Herlufsen, H; (1987) "Use of weighting functions in DFT/FFT analysis (Part I)", Bruel & Kjaer Technical Review No.3.
(m) ¶(m, a) ¶Return a Gaussian convolution window of length m. The width of the window is inversely proportional to the parameter a. Use larger a for a narrower window. Use larger m for longer tails.
w = exp ( -(a*x)^2/2 )
for x = linspace ( -(m-1)/2, (m-1)/2, m ).
Width a is measured in frequency units (sample rate/num samples). It should be f when multiplying in the time domain, but 1/f when multiplying in the frequency domain (for use in convolutions).
(m) ¶(m, a) ¶Return the filter coefficients of a Gaussian window of length m. The width of the window is inversely proportional to the parameter a. Use larger a for a narrow window. Use larger m for a smoother curve.
w = exp ( -(a*x)^2/2 )
for x = linspace(-(m-1)/m, (m-1)/m, m)
(m) ¶(m, "periodic") ¶(m, "symmetric") ¶Return the filter coefficients of a Hanning window of length m.
If the optional argument "periodic" is given, the periodic form
of the window is returned. This is equivalent to the window of length
m+1 with the last coefficient removed. The optional argument
"symmetric" is equivalent to not specifying a second argument.
This function exists for MATLAB compatibility only, and is equivalent
to hanning (m).
See also: hanning.
(m) ¶(m, beta) ¶Return the filter coefficients of a Kaiser window of length m. The Fourier transform of the window has a stop-band attenuation that is derived from the parameter beta.
For the definition of the Kaiser window, see A. V. Oppenheim & R. W. Schafer, "Discrete-Time Signal Processing".
The continuous version of width m centered about x=0 is:
besseli(0, beta * sqrt(1-(2*x/m).^2))
k(x) = -------------------------------------, m/2 <= x <= m/2
besseli(0, beta)
See also: kaiserord.
(m) ¶(m, "periodic") ¶(m, "symmetric") ¶Return the filter coefficients of a Blackman-Harris window defined by Nuttall of length m.
If the optional argument "periodic" is given, the periodic form
of the window is returned. This is equivalent to the window of length
m+1 with the last coefficient removed. The optional argument
"symmetric" is equivalent to not specifying a second argument.
See also: blackman, blackmanharris.
(m) ¶Return the filter coefficients of a Parzen window of length m.
See also: rectwin, bartlett.
(m) ¶Return the filter coefficients of a rectangular window of length m.
See also: boxcar, hamming, hanning.
taylorwin (m) ¶taylorwin (m, nbar) ¶taylorwin (m, nbar, sll) ¶taylorwin (m, nbar, sll, plots) ¶Return the coefficients of a Taylor window of length m, whose Fourier transform has nbar (default 4) quasi-equiripple side-lobes adjacent to the main-lobe, at a nominal level of sll (default −30) dBc.
If plots is non-zero then time and frequency domains plots of the resultant window are shown.
Reference: Doerry, ‘Catalog of Window Taper Functions for Sidelobe Control’, 2017.
See also: chebwin.
(m) ¶Return the filter coefficients of a triangular window of length m.
Unlike the Bartlett window, triang does not go to zero at the edges
of the window. For odd m, triang (m) is equal to
bartlett (m + 2) except for the zeros at the edges of the
window.
See also: bartlett.
(m) ¶(m, r) ¶Return the filter coefficients of a Tukey window (also known as the cosine-tapered window) of length m. r defines the ratio between the constant section and and the cosine section. It has to be between 0 and 1. The function returns a Hanning window for r equal to 1 and a rectangular window for r equal to 0. The default value of r is 1/2.
For a definition of the Tukey window, see e.g. Fredric J. Harris, "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proceedings of the IEEE", Vol. 66, No. 1, January 1978, Page 67, Equation 38.
See also: hanning.
[w, xmu] = ultrwin (m, mu, beta) ¶[w, xmu] = ultrwin (m, mu, att, "att") ¶[w, xmu] = ultrwin (m, mu, latt, "latt") ¶w = ultrwin (m, mu, xmu, "xmu") ¶Return the coefficients of an Ultraspherical window of length m. The parameter mu controls the window’s Fourier transform’s side-lobe to side-lobe ratio, and the third given parameter controls the transform’s main-lobe width/side-lobe-ratio; normalize w such that the central coefficient(s) value is unitary.
By default, the third parameter is beta, which sets the main lobe width to beta times that of a rectangular window. Alternatively, giving att or latt sets the ripple ratio at the first or last side-lobe respectively, or giving xmu sets the (un-normalized) window’s Fourier transform according to its canonical definition:
(MU)
W(k) = C [ XMU cos(pi k/M) ], k = 0, 1, ..., M-1,
M-1
where C is the Ultraspherical (a.k.a. Gegenbauer) polynomial, which can be defined using the recurrence relationship:
(l) 1 (l) (l)
C (x) = - [ 2x(m + l - 1) C (x) - (m + 2l - 2) C (x) ]
m m m-1 m-2
(l) (l)
for m an integer > 1, and C (x) = 1, C (x) = 2lx.
0 1
For given beta, att, or latt, the corresponding (determined) value of xmu is also returned.
The Dolph-Chebyshev and Saramaki windows are special cases of the Ultraspherical window, with mu set to 0 and 1 respectively. Note that when not giving xmu, stability issues may occur with mu <= -1.5. For further information about the window, see
See also: chebwin, kaiser.
(m) ¶(m, "periodic") ¶(m, "symmetric") ¶Return the filter coefficients of a Welch window of length m. The Welch window is given by w(n)=1-(n/N-1)^2, n=[0,1, ... m-1]. The optional argument specifies a "symmetric" window (the default) or a "periodic" window.
A symmetric window has zero at each end and maximum in the middle, and the
length must be an integer greater than 2. The variable N in the
formula above is (m-1)/2.
A periodic window wraps around the cyclic interval [0,1, ... m-1],
and is intended for use with the DFT. The length must be an integer
greater than 1. The variable N in the formula above is
m/2.
See also: blackman, kaiser.
[a, v, k] = arburg (x, poles) ¶[a, v, k] = arburg (x, poles, criterion) ¶Calculate coefficients of an autoregressive (AR) model of complex data
x using the whitening lattice-filter method of Burg (1968). The
inverse of the model is a moving-average filter which reduces x to
white noise. The power spectrum of the AR model is an estimate of the
maximum entropy power spectrum of the data. The function ar_psd
calculates the power spectrum of the AR model.
ARGUMENTS:
RETURNED VALUES:
P+1
x(n) = sqrt(v).e(n) + SUM a(k).x(n-k)
k=1
v mean square of residual noise from the whitening operation of the Burg lattice filter.
HINTS:
(1) arburg does not remove the mean from the data. You should remove the mean from the data if you want a power spectrum. A non-zero mean can produce large errors in a power-spectrum estimate. See "help detrend". (2) If you don’t know what the value of "poles" should be, choose the largest (reasonable) value you could want and use the recommended value, criterion=’AKICc’, so that arburg can find it. E.g. arburg(x,64,’AKICc’) The AKICc has the least bias and best resolution of the available model-selection criteria. (3) Autoregressive and moving-average filters are stored as polynomials which, in matlab, are row vectors.
NOTE ON SELECTION CRITERION:
AIC, AICc, KIC and AKICc are based on information theory. They attempt to balance the complexity (or length) of the model against how well the model fits the data. AIC and KIC are biased estimates of the asymmetric and the symmetric Kullback-Leibler divergence respectively. AICc and AKICc attempt to correct the bias. See reference [4].
REFERENCES:
[1] John Parker Burg (1968) "A new analysis technique for time series data", NATO advanced study Institute on Signal Processing with Emphasis on Underwater Acoustics, Enschede, Netherlands, Aug. 12-23, 1968.
[2] Steven M. Kay and Stanley Lawrence Marple Jr.: "Spectrum analysis – a modern perspective", Proceedings of the IEEE, Vol 69, pp 1380-1419, Nov., 1981
[3] William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery "Numerical recipes in C, The art of scientific computing", 2nd edition, Cambridge University Press, 2002 — Section 13.7.
[4] Abd-Krim Seghouane and Maiza Bekara "A small sample model selection criterion based on Kullback’s symmetric divergence", IEEE Transactions on Signal Processing, Vol. 52(12), pp 3314-3323, Dec. 2004
See also: ar_psd.
a = aryule (x, p) ¶[a, v, k] = aryule (x, p) ¶Fit an AR (p)-model with Yule-Walker estimates.
data vector to estimate
AR coefficients
variance of white noise
reflection coefficients for use in lattice filter
The power spectrum of the resulting filter can be plotted with pyulear(x, p), or you can plot it directly with ar_psd(a,v,...).
See also: pyulear, power, freqz, impz – for observing characteristics of the model arburg – for alternative spectral estimators
Example: Use example from arburg, but substitute aryule for arburg.
Note: Orphanidis ’85 claims lattice filters are more tolerant of truncation errors, which is why you might want to use them. However, lacking a lattice filter processor, I haven’t tested that the lattice filter coefficients are reasonable.
[B,A] = invfreq(H,F,nB,nA) ¶[B,A] = invfreq(H,F,nB,nA,W) ¶[B,A] = invfreq(H,F,nB,nA,W,[],[],plane) ¶[B,A] = invfreq(H,F,nB,nA,W,iter,tol,plane) ¶Fit filter B(z)/A(z) or B(s)/A(s) to complex frequency response at frequency points F.
A and B are real polynomial coefficients of order nA and nB respectively. Optionally, the fit-errors can be weighted vs frequency according to the weights W. Also, the transform plane can be specified as either ’s’ for continuous time or ’z’ for discrete time. ’z’ is chosen by default. Eventually, Steiglitz-McBride iterations will be specified by iter and tol.
H: desired complex frequency response It is assumed that A and B are real polynomials, hence H is one-sided.
F: vector of frequency samples in radians
nA: order of denominator polynomial A
nB: order of numerator polynomial B
plane=’z’: F on unit circle (discrete-time spectra, z-plane design)
plane=’s’: F on jw axis (continuous-time spectra, s-plane design)
H(k) = spectral samples of filter frequency response at points zk, where zk=exp(sqrt(-1)*F(k)) when plane=’z’ (F(k) in [0,.5]) and zk=(sqrt(-1)*F(k)) when plane=’s’ (F(k) nonnegative)
Example:
[B,A] = butter(12,1/4);
[H,w] = freqz(B,A,128);
[Bh,Ah] = invfreq(H,F,4,4);
Hh = freqz(Bh,Ah);
disp(sprintf('||frequency response error|| = %f',norm(H-Hh)));
References:
J. O. Smith, "Techniques for Digital Filter Design and System Identification with Application to the Violin, Ph.D. Dissertation, Elec. Eng. Dept., Stanford University, June 1983, page 50; or,
http://ccrma.stanford.edu/~jos/filters/FFT_Based_Equation_Error_Method.html
[B,A] = invfreqs(H,F,nB,nA) ¶[B,A] = invfreqs(H,F,nB,nA,W) ¶[B,A] = invfreqs(H,F,nB,nA,W,iter,tol,'trace') ¶Fit filter B(s)/A(s)to the complex frequency response H at frequency points F.
A and B are real polynomial coefficients of order nA and nB.
Optionally, the fit-errors can be weighted vs frequency according to the weights W.
Note: all the guts are in invfreq.m
H: desired complex frequency response
F: frequency (must be same length as H)
nA: order of the denominator polynomial A
nB: order of the numerator polynomial B
W: vector of weights (must be same length as F)
Example:
B = [1/2 1];
A = [1 1];
w = linspace(0,4,128);
H = freqs(B,A,w);
[Bh,Ah] = invfreqs(H,w,1,1);
Hh = freqs(Bh,Ah,w);
plot(w,[abs(H);abs(Hh)])
legend('Original','Measured');
err = norm(H-Hh);
disp(sprintf('L2 norm of frequency response error = %f',err));
[B,A] = invfreqz(H,F,nB,nA) ¶[B,A] = invfreqz(H,F,nB,nA,W) ¶[B,A] = invfreqz(H,F,nB,nA,W,iter,tol,'trace') ¶Fit filter B(z)/A(z)to the complex frequency response H at frequency points F.
A and B are real polynomial coefficients of order nA and nB. Optionally, the fit-errors can be weighted vs frequency according to the weights W.
Note: all the guts are in invfreq.m
H: desired complex frequency response
F: normalized frequency (0 to pi) (must be same length as H)
nA: order of the denominator polynomial A
nB: order of the numerator polynomial B
W: vector of weights (must be same length as F)
Example:
[B,A] = butter(4,1/4);
[H,F] = freqz(B,A);
[Bh,Ah] = invfreq(H,F,4,4);
Hh = freqz(Bh,Ah);
disp(sprintf('||frequency response error|| = %f',norm(H-Hh)));
[a, v, ref] = levinson (acf) ¶[…] = levinson (acf, p) ¶Use the Durbin-Levinson algorithm to solve: toeplitz(acf(1:p)) * x = -acf(2:p+1). The solution [1, x’] is the denominator of an all pole filter approximation to the signal x which generated the autocorrelation function acf.
acf is the autocorrelation function for lags 0 to p. p defaults to length(acf)-1. Returns a=[1, x’] the denominator filter coefficients. v= variance of the white noise = square of the numerator constant ref = reflection coefficients = coefficients of the lattice implementation of the filter Use freqz(sqrt(v),a) to plot the power spectrum.
REFERENCE [1] Steven M. Kay and Stanley Lawrence Marple Jr.: "Spectrum analysis – a modern perspective", Proceedings of the IEEE, Vol 69, pp 1380-1419, Nov., 1981
a = lpc (x) ¶a = lpc (x, p) ¶[a, g] = lpc (…) ¶[a, g] = lpc (x, p) ¶Determines the forward linear predictor by minimizing the prediction error in the least squares sense. Use the Durbin-Levinson algorithm to solve the Yule-Walker equations obtained by the autocorrelation of the input signal.
x is a data vector used to estimate the lpc model of p-th order,
given by the prediction polynomial a = [1 a(2) …
a(p+1)]. If p is not provided, length(p) - 1
is used as default.
x might also be a matrix, in which case each column is regarded as a
separate signal. lpc will return a model estimate for each column of
x.
g is the variance (power) of the prediction error for each signal in x.
See also: aryule,levinson.
[fhandle, fullname] = data2fun (ti, yi) ¶[…] = data2fun (…, property, value) ¶Create a vectorized function based on data samples using interpolation.
The values given in yi (N-by-k matrix) correspond to evaluations of the function y(t) at the points ti (N-by-1 matrix). The data is interpolated and the function handle to the generated interpolant is returned.
The function accepts property-value pairs described below.
Code is generated and .m file is created. The value contains the name
of the function. The returned function handle is a handle to that file. If
value is empty, then a name is automatically generated using
tempname and the file is created in the current directory. value
must not have an extension, since .m will be appended.
Numerical values used in the function are stored in a .mat file with the same
name as the function.
Type of interpolation. See interp1.
See also: interp1.
y = decimate (x, q) ¶y = decimate (x, q, n) ¶y = decimate (…, "fir") ¶Downsample the signal x by a reduction factor of q. A lowpass antialiasing filter is applied to the signal prior to reducing the input sequence. By default, an order n Chebyshev type I filter is used. If n is not specified, the default is 8.
If the optional argument "fir" is given, an order n FIR filter
is used, with a default order of 30 if n is not given.
Note that q must be an integer for this rate change method.
Example:
## Generate a signal that starts away from zero, is slowly varying ## at the start and quickly varying at the end, decimate and plot. ## Since it starts away from zero, you will see the boundary ## effects of the antialiasing filter clearly. Next you will see ## how it follows the curve nicely in the slowly varying early ## part of the signal, but averages the curve in the quickly ## varying late part of the signal. t = 0:0.01:2; x = chirp (t, 2, .5, 10, "quadratic") + sin (2*pi*t*0.4); y = decimate (x, 4); stem (t(1:121) * 1000, x(1:121), "-g;Original;"); hold on; # original stem (t(1:4:121) * 1000, y(1:31), "-r;Decimated;"); hold off; # decimated
y = downsample (x, n) ¶y = downsample (x, n, offset) ¶Downsample the signal, selecting every nth element. If x is a matrix, downsample every column.
For most signals you will want to use decimate instead since
it prefilters the high frequency components of the signal and
avoids aliasing effects.
If offset is defined, select every nth element starting at sample offset.
See also: decimate, interp, resample, upfirdn, upsample.
y = interp (x, q) ¶y = interp (x, q, n) ¶y = interp (x, q, n, Wc) ¶Upsample the signal x by a factor of q, using an order 2*q*n+1 FIR filter. Note that q must be an integer for this rate change method. n defaults to 4 and Wc defaults to 0.5.
Example
# Generate a signal. t=0:0.01:2; x=chirp(t,2,.5,10,'quadratic')+sin(2*pi*t*0.4); y = interp(x(1:4:length(x)),4,4,1); # interpolate a sub-sample stem(t(1:121)*1000,x(1:121),"-g;Original;"); hold on; stem(t(1:121)*1000,y(1:121),"-r;Interpolated;"); stem(t(1:4:121)*1000,x(1:4:121),"-b;Subsampled;"); hold off;
See also: decimate, resample.
[y, h] = resample (x, p, q) ¶y = resample (x, p, q, h) ¶Change the sample rate of x by a factor of p/q. This is performed using a polyphase algorithm. The impulse response h of the antialiasing filter is either specified or either designed with a Kaiser-windowed sinecard.
Ref [1] J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th ed., Prentice Hall, 2007. Chap. 6
Ref [2] A. V. Oppenheim, R. W. Schafer and J. R. Buck, Discrete-time signal processing, Signal processing series, Prentice-Hall, 1999
y = upfirdn (x, h, p, q) ¶Upsample, FIR filtering, and downsample.
y = upsample (x, n) ¶y = upsample (x, n, offset) ¶Upsample the signal, inserting n-1 zeros between every element.
If x is a matrix, upsample every column.
If offset is specified, control the position of the inserted sample in the block of n zeros.
See also: decimate, downsample, interp, resample, upfirdn.
levels = statelevels (A) ¶levels = statelevels (A, nbins) ¶levels = statelevels (A, nbins, method) ¶levels = statelevels (A, nbins, method, bounds) ¶[levels, histograms] = statelevels (…) ¶[levels, histograms, binlevels] = statelevels (…) ¶(…) ¶Estimate state-level for bilevel waveform A using histogram method
INPUTS:
Bylevel waveform
Number of histogram bins (100 default)
State-level estimation method ’mode’ (default) or ’mean’.
2 element vector for histogram lower and upper bounds. Values outside of this will be ignored.
OUTPUTS:
Levels of high and low states
Histogram counts
Histogram bincenters
If no outputs are provided, the signal and histogram will be plotted, and display the levels.
y = buffer (x, n, p, opt) ¶[y, z, opt] = buffer (…) ¶Buffer a signal into a data frame. The arguments to buffer are
The data to be buffered.
The number of rows in the produced data buffer. This is an positive integer value and must be supplied.
An integer less than n that specifies the under- or overlap between column in the data frame. The default value of p is 0.
In the case of an overlap, opt can be either a vector of length p or the string ’nodelay’. If opt is a vector, then the first p entries in y will be filled with these values. If opt is the string ’nodelay’, then the first value of y corresponds to the first value of x.
In the can of an underlap, opt must be an integer between 0 and
-p. The represents the initial underlap of the first
column of y.
The default value for opt the vector zeros (1, p)
in the case of an overlap, or 0 otherwise.
In the case of a single output argument, y will be padded with zeros to fill the missing values in the data frame. With two output arguments z is the remaining data that has not been used in the current data frame.
Likewise, the output opt is the overlap, or underlap that might
be used for a future call to code to allow continuous buffering.
clusteridx = clustersegment (unos) ¶Calculate boundary indexes of clusters of 1’s.
The function calculates the initial index and end index of the sequences of 1’s in the rows of unos. The clusters are sought in the rows of the array unos.
The result is returned in a cell array of size 1-by-Np, where Np is the number of rows in unos. Each element of the cell has two rows. The first row is the initial index of a sequence of 1’s and the second row is the end index of that sequence.
If Np == 1 the output is a matrix with two rows.
The function works by finding the indexes of jumps between consecutive values in the rows of unos.
q = marcumq (a, b) ¶q = marcumq (a, b, m) ¶q = marcumq (a, b, m, tol) ¶Compute the generalized Marcum Q function of order m with
noncentrality parameter a and argument b. If the order
m is omitted it defaults to 1. An optional relative tolerance
tol may be included, the default is eps.
If the input arguments are commensurate vectors, this function will produce a table of values.
This function computes Marcum’s Q function using the infinite Bessel series, truncated when the relative error is less than the specified tolerance. The accuracy is limited by that of the Bessel functions, so reducing the tolerance is probably not useful.
Reference: Marcum, "Tables of Q Functions", Rand Corporation.
Reference: R.T. Short, "Computation of Noncentral Chi-squared and Rice Random Variables", www.phaselockedsystems.com/publications
F = primitive (f, t, F0) ¶Calculate the primitive of a function.
The function approximates the primitive (indefinite integral) of the univariate function handle f with constant of integration F0. The output is the primitive evaluated at the points t. The vector t must be ordered and ascending.
This function is a fancy way of calculating the cumulative sum,
F = cumsum (f(t).*diff (t)([1 1:end])).
Example:
f = @(t) sin (2 * pi * 3 * t); t = [0; sort(rand (100, 1))]; F = primitive (f, t, 0); t_true = linspace (0, 1, 1e3).'; F_true = (1 - cos (2 * pi * 3 * t_true)) / (2 * pi * 3); plot (t, F, 'o', t_true, F_true);
See also: quadgk, cumsum.
xt = sampled2continuous (xn, T, t) ¶Calculate the x(t) reconstructed from samples x[n] sampled at a rate 1/T samples per unit time.
t is all the instants of time when you need x(t) from x[n]; this time is relative to x[0] and not an absolute time.
This function can be used to calculate sampling rate effects on aliasing, actual signal reconstruction from discrete samples.
schtrig (x,lvl,rst=1) ¶schtrig (…) ¶Implements a multisignal Schmitt trigger with levels lvl.
The triger works along the first dimension of the 2-dimensional array x.
It compares each column in x to the levels in lvl, when the
value is higher max (lvl) the output v is high (i.e. 1); when the
value is below min (lvl) the output is low (i.e. 0); and when
the value is between the two levels the output retains its value.
The threshold levels are passed in the array lvl. If this is a scalar,
the thresholds are symmetric around 0, i.e. [-lvl lvl].
The second output argument stores the ranges in which the output is high, so
the indexes rng(1,i):rng(2,i) point to the i-th segment of 1s in v.
See clustersegment for a detailed explanation.
The function conserves the state of the trigger across calls (persistent variable).
If the reset flag is active, i.e. rst== true, then the state of
the trigger for all signals is set to the low state (i.e. 0).
Example:
x = [0 0.5 1 1.5 2 1.5 1.5 1.2 1 0 0].'; y = schtrig (x, [1.3 1.6]); disp ([x y]); 0.0 0 0.5 0 1.0 0 1.5 0 2.0 1 1.5 1 1.5 1 1.2 0 1.0 0 0.0 0 0.0 0
Run demo schtrig to see further examples.
See also: clustersegment.
y = upsamplefill (x, v) ¶y = upsamplefill (…, copy) ¶Upsamples a vector interleaving given values or copies of the vector elements.
The values in the vector v are placed between the elements of x.
If the optional argument copy is true then v should be a scalar and each value in x are repeat v times.
Example:
upsamplefill (eye (2), 2, true)
⇒ 1 0
1 0
1 0
0 1
0 1
0 1
upsamplefill (eye (2), [-1 -1 -1])
⇒ 1 0
-1 -1
-1 -1
-1 -1
0 1
-1 -1
-1 -1
-1 -1
See also: upsample.
J. H. McClellan, T. W. Parks and L. R. Rabiner, ‘A Computer Program for Designing Optimum FIR Linear Phase Digital Filters’, IEEE Trans. Audio Electroacoust., vol. AU-21, 1973, pp. 506–525.
Tapio Saramäki, ‘Finite impulse response filter design’, Chapter 4 in ‘Handbook for Digital Signal Processing’, edited by S. K. Mitra and J. F. Kaiser, John Wiley and Sons, New York, 1993, pp. 155–277. (https://homepages.tuni.fi/tapio.saramaki/Mitra_Kaiser.pdf)
K. Ichige, M. Iwaki, algorithm and R. Ishii, ‘Accurate Estimation of Minimum Filter Length for Optimum FIR Digital Filters’, IEEE Transactions on Circuits and Systems, Vol. 47, No. 10, 2000, pp. 1008–1017
K. Avci & A. Nacaroglu, ‘Exponential Window Family’, SIPIJ Vol. 4 No. 4, August 2013.
S. Gade & H. Herlufsen, ‘Windows to FFT analysis (Part I)’, Technical Review 3, Bruel & Kjaer, 1987