pspectrum
Analyze signals in the frequency and timefrequency domains
Syntax
Description
returns the power spectrum of p
= pspectrum(x
)x
.
If
x
is a vector or a timetable with a vector of data, then it is treated as a single channel.If
x
is a matrix, a timetable with a matrix variable, or a timetable with multiple vector variables, then the spectrum is computed independently for each channel and stored in a separate column ofp
.
specifies additional options using namevalue pair arguments. Options include
the frequency resolution bandwidth and the percent overlap between adjoining
segments.p
= pspectrum(___,Name,Value
)
Examples
Power Spectra of Sinusoids
Generate 128 samples of a twochannel complex sinusoid.
The first channel has unit amplitude and a normalized sinusoid frequency of $\pi /4$ rad/sample
The second channel has an amplitude of $1/\sqrt{2}\text{\hspace{0.17em}}$and a normalized frequency of $\pi /2$ rad/sample.
Compute the power spectrum of each channel and plot its absolute value. Zoom in on the frequency range from $0.15\pi $ rad/sample to $0.6\pi $ rad/sample. pspectrum
scales the spectrum so that, if the frequency content of a signal falls exactly within a bin, its amplitude in that bin is the true average power of the signal. For a complex exponential, the average power is the square of the amplitude. Verify by computing the discrete Fourier transform of the signal. For more details, see Measure Power of Deterministic Periodic Signals.
N = 128; x = [1 1/sqrt(2)].*exp(1j*pi./[4;2]*(0:N1)).'; [p,f] = pspectrum(x); plot(f/pi,abs(p)) hold on stem(0:2/N:21/N,abs(fft(x)/N).^2) hold off axis([0.15 0.6 0 1.1]) legend("Channel 1, pspectrum","Channel 2, pspectrum", ... "Channel 1, fft","Channel 2, fft") grid
Generate a sinusoidal signal sampled at 1 kHz for 296 milliseconds and embedded in white Gaussian noise. Specify a sinusoid frequency of 200 Hz and a noise variance of 0.1². Store the signal and its time information in a MATLAB® timetable.
Fs = 1000; t = (0:1/Fs:0.296)'; x = cos(2*pi*t*200)+0.1*randn(size(t)); xTable = timetable(seconds(t),x);
Compute the power spectrum of the signal. Express the spectrum in decibels and plot it.
[pxx,f] = pspectrum(xTable); plot(f,pow2db(pxx)) grid on xlabel('Frequency (Hz)') ylabel('Power Spectrum (dB)') title('Default Frequency Resolution')
Recompute the power spectrum of the sinusoid, but now use a coarser frequency resolution of 25 Hz. Plot the spectrum using the pspectrum
function with no output arguments.
pspectrum(xTable,'FrequencyResolution',25)
TwoSided Spectra
Generate a signal sampled at 3 kHz for 1 second. The signal is a convex quadratic chirp whose frequency increases from 300 Hz to 1300 Hz during the measurement. The chirp is embedded in white Gaussian noise.
fs = 3000; t = 0:1/fs:11/fs; x1 = chirp(t,300,t(end),1300,'quadratic',0,'convex') + ... randn(size(t))/100;
Compute and plot the twosided power spectrum of the signal using a rectangular window. For real signals, pspectrum
plots a onesided spectrum by default. To plot a twosided spectrum, set TwoSided
to true.
pspectrum(x1,fs,'Leakage',1,'TwoSided',true)
Generate a complexvalued signal with the same duration and sample rate. The signal is a chirp with sinusoidally varying frequency content and embedded in white noise. Compute the spectrogram of the signal and display it as a waterfall plot. For complexvalued signals, the spectrogram is twosided by default.
x2 = exp(2j*pi*100*cos(2*pi*2*t)) + randn(size(t))/100; [p,f,t] = pspectrum(x2,fs,'spectrogram'); waterfall(f,t,p') xlabel('Frequency (Hz)') ylabel('Time (seconds)') wtf = gca; wtf.XDir = 'reverse'; view([30 45])
Window Leakage and Tone Resolution
Generate a twochannel signal sampled at 100 Hz for 2 seconds.
The first channel consists of a 20 Hz tone and a 21 Hz tone. Both tones have unit amplitude.
The second channel also has two tones. One tone has unit amplitude and a frequency of 20 Hz. The other tone has an amplitude of 1/100 and a frequency of 30 Hz.
fs = 100; t = (0:1/fs:21/fs)'; x = sin(2*pi*[20 20].*t) + [1 1/100].*sin(2*pi*[21 30].*t);
Embed the signal in white noise. Specify a signaltonoise ratio of 40 dB. Plot the signals.
x = x + randn(size(x)).*std(x)/db2mag(40); plot(t,x)
Compute the spectra of the two channels and display them.
pspectrum(x,t)
The default value for the spectral leakage, 0.5, corresponds to a resolution bandwidth of about 1.29 Hz. The two tones in the first channel are not resolved. The 30 Hz tone in the second channel is visible, despite being much weaker than the other one.
Increase the leakage to 0.85, equivalent to a resolution of about 0.74 Hz. The weak tone in the second channel is clearly visible.
pspectrum(x,t,'Leakage',0.85)
Increase the leakage to the maximum value. The resolution bandwidth is approximately 0.5 Hz. The two tones in the first channel are resolved. The weak tone in the second channel is masked by the large window sidelobes.
pspectrum(x,t,'Leakage',1)
Compare spectrogram
and pspectrum
Functions
Generate a signal that consists of a voltagecontrolled oscillator and three Gaussian atoms. The signal is sampled at ${\mathit{f}}_{\mathit{s}}=2$ kHz for 1 second.
fs = 2000; tx = (0:1/fs:2); gaussFun = @(A,x,mu,f) exp((xmu).^2/(2*0.03^2)).*sin(2*pi*f.*x)*A'; s = gaussFun([1 1 1],tx',[0.1 0.65 1],[2 6 2]*100)*1.5; x = vco(chirp(tx+.1,0,tx(end),3).*exp(2*(tx1).^2),[0.1 0.4]*fs,fs); x = s+x';
ShortTime Fourier Transforms
Use the pspectrum
function to compute the STFT.
Divide the ${\mathit{N}}_{\mathit{x}}$sample signal into segments of length $\mathit{M}=80$ samples, corresponding to a time resolution of $80/2000=40$ milliseconds.
Specify $\mathit{L}=16$ samples or 20% of overlap between adjoining segments.
Window each segment with a Kaiser window and specify a leakage $\ell =0.7$.
M = 80; L = 16; lk = 0.7; [S,F,T] = pspectrum(x,fs,"spectrogram", ... TimeResolution=M/fs,OverlapPercent=L/M*100, ... Leakage=lk);
Compare to the result obtained with the spectrogram
function.
Specify the window length and overlap directly in samples.
pspectrum
always uses a Kaiser window as $\mathit{g}\left(\mathit{n}\right)$. The leakage $\ell $ and the shape factor $\beta $ of the window are related by $\beta =40\times \left(1\ell \right)$.pspectrum
always uses ${\mathit{N}}_{\mathrm{DFT}}=1024$ points when computing the discrete Fourier transform. You can specify this number if you want to compute the transform over a twosided or centered frequency range. However, for onesided transforms, which are the default for real signals,spectrogram
uses $1024/2+1=513$ points. Alternatively, you can specify the vector of frequencies at which you want to compute the transform, as in this example.If a signal cannot be divided exactly into $\mathit{k}=\lfloor \frac{{\mathit{N}}_{\mathit{x}}\mathit{L}}{\mathit{M}\mathit{L}}\rfloor $ segments,
spectrogram
truncates the signal whereaspspectrum
pads the signal with zeros to create an extra segment. To make the outputs equivalent, remove the final segment and the final element of the time vector.spectrogram
returns the STFT, whose magnitude squared is the spectrogram.pspectrum
returns the segmentbysegment power spectrum, which is already squared but is divided by a factor of ${\sum}_{\mathit{n}}\mathit{g}\left(\mathit{n}\right)$ before squaring.For onesided transforms,
pspectrum
adds an extra factor of 2 to the spectrogram.
g = kaiser(M,40*(1lk)); k = (length(x)L)/(ML); if k~=floor(k) S = S(:,1:floor(k)); T = T(1:floor(k)); end [s,f,t] = spectrogram(x/sum(g)*sqrt(2),g,L,F,fs);
Use the waterplot
function to display the spectrograms computed by the two functions.
subplot(2,1,1) waterplot(sqrt(S),F,T) title("pspectrum") subplot(2,1,2) waterplot(s,f,t) title("spectrogram")
maxd = max(max(abs(abs(s).^2S)))
maxd = 2.4419e08
Power Spectra and Convenience Plots
The spectrogram
function has a fourth argument that corresponds to the segmentbysegment power spectrum or power spectral density. Similar to the output of pspectrum
, the ps
argument is already squared and includes the normalization factor ${\sum}_{\mathit{n}}\mathit{g}\left(\mathit{n}\right)$. For onesided spectrograms of real signals, you still have to include the extra factor of 2. Set the scaling argument of the function to "power"
.
[~,~,~,ps] = spectrogram(x*sqrt(2),g,L,F,fs,"power");
max(abs(S(:)ps(:)))
ans = 2.4419e08
When called with no output arguments, both pspectrum
and spectrogram
plot the spectrogram of the signal in decibels. Include the factor of 2 for onesided spectrograms. Set the colormaps to be the same for both plots. Set the xlimits to the same values to make visible the extra segment at the end of the pspectrum
plot. In the spectrogram
plot, display the frequency on the yaxis.
subplot(2,1,1) pspectrum(x,fs,"spectrogram", ... TimeResolution=M/fs,OverlapPercent=L/M*100, ... Leakage=lk) title("pspectrum") cc = clim; xl = xlim; subplot(2,1,2) spectrogram(x*sqrt(2),g,L,F,fs,"power","yaxis") title("spectrogram") clim(cc) xlim(xl)
function waterplot(s,f,t) % Waterfall plot of spectrogram waterfall(f,t,abs(s)'.^2) set(gca,XDir="reverse",View=[30 50]) xlabel("Frequency (Hz)") ylabel("Time (s)") end
Persistence Spectrum of Transient Signal
Visualize an interference narrowband signal embedded within a broadband signal.
Generate a chirp sampled at 1 kHz for 500 seconds. The frequency of the chirp increases from 180 Hz to 220 Hz during the measurement.
fs = 1000; t = (0:1/fs:500)'; x = chirp(t,180,t(end),220) + 0.15*randn(size(t));
The signal also contains a 210 Hz sinusoid. The sinusoid has an amplitude of 0.05 and is present only for 1/6 of the total signal duration.
idx = floor(length(x)/6); x(1:idx) = x(1:idx) + 0.05*cos(2*pi*t(1:idx)*210);
Compute the spectrogram of the signal. Restrict the frequency range from 100 Hz to 290 Hz. Specify a time resolution of 1 second. Both signal components are visible.
pspectrum(x,fs,'spectrogram', ... 'FrequencyLimits',[100 290],'TimeResolution',1)
Compute the power spectrum of the signal. The weak sinusoid is obscured by the chirp.
pspectrum(x,fs,'FrequencyLimits',[100 290])
Compute the persistence spectrum of the signal. Now both signal components are clearly visible.
pspectrum(x,fs,'persistence', ... 'FrequencyLimits',[100 290],'TimeResolution',1)
Spectrogram and Reassigned Spectrogram of Chirp
Generate a quadratic chirp sampled at 1 kHz for 2 seconds. The chirp has an initial frequency of 100 Hz that increases to 200 Hz at t = 1 second. Compute the spectrogram using the default settings of the pspectrum
function. Use the waterfall
function to plot the spectrogram.
fs = 1e3; t = 0:1/fs:2; y = chirp(t,100,1,200,"quadratic"); [sp,fp,tp] = pspectrum(y,fs,"spectrogram"); waterfall(fp,tp,sp') set(gca,XDir="reverse",View=[60 60]) ylabel("Time (s)") xlabel("Frequency (Hz)")
Compute and display the reassigned spectrogram.
[sr,fr,tr] = pspectrum(y,fs,"spectrogram",Reassign=true); waterfall(fr,tr,sr') set(gca,XDir="reverse",View=[60 60]) ylabel("Time (s)") xlabel("Frequency (Hz)")
Recompute the spectrogram using a time resolution of 0.2 second. Visualize the result using the pspectrum
function with no output arguments.
pspectrum(y,fs,"spectrogram",TimeResolution=0.2)
Compute the reassigned spectrogram using the same time resolution.
pspectrum(y,fs,"spectrogram",TimeResolution=0.2,Reassign=true)
Spectrogram of Dial Tone Signal
Create a signal, sampled at 4 kHz, that resembles pressing all the keys of a digital telephone. Save the signal as a MATLAB® timetable.
fs = 4e3; t = 0:1/fs:0.51/fs; ver = [697 770 852 941]; hor = [1209 1336 1477]; tones = []; for k = 1:length(ver) for l = 1:length(hor) tone = sum(sin(2*pi*[ver(k);hor(l)].*t))'; tones = [tones;tone;zeros(size(tone))]; end end % To hear, type soundsc(tones,fs) S = timetable(seconds(0:length(tones)1)'/fs,tones);
Compute the spectrogram of the signal. Specify a time resolution of 0.5 second and zero overlap between adjoining segments. Specify the leakage as 0.85, which is approximately equivalent to windowing the data with a Hann window.
pspectrum(S,'spectrogram', ... 'TimeResolution',0.5,'OverlapPercent',0,'Leakage',0.85)
The spectrogram shows that each key is pressed for half a second, with halfsecond silent pauses between keys. The first tone has a frequency content concentrated around 697 Hz and 1209 Hz, corresponding to the digit '1'
in the DTMF standard.
Input Arguments
x
— Input signal
vector  matrix  timetable
Input signal, specified as a vector, a matrix, or a MATLAB^{®}
timetable
.
If
x
is a timetable, then it must contain increasing finite row times.Note
If a timetable has missing or duplicate time points, you can fix it using the tips in Clean Timetable with Missing, Duplicate, or Nonuniform Times.
If
x
is a timetable representing a multichannel signal, then it must have either a single variable containing a matrix or multiple variables consisting of vectors.
If x
is nonuniformly sampled, then
pspectrum
interpolates the signal to a uniform grid
to compute spectral estimates. The function uses linear interpolation and
assumes a sample time equal to the median of the differences between
adjacent time points. For a nonuniformly sampled signal to be supported, the
median time interval and the mean time interval must obey
$$\frac{1}{100}<\frac{\text{Mediantimeinterval}}{\text{Meantimeinterval}}100.$$
Example: cos(pi./[4;2]*(0:159))'+randn(160,2)
is a
twochannel signal consisting of sinusoids embedded in white
noise.
Example: timetable(seconds(0:4)',rand(5,2))
specifies a
twochannel random variable sampled at 1 Hz for 4 seconds.
Example: timetable(seconds(0:4)',rand(5,1),rand(5,1))
specifies a twochannel random variable sampled at 1 Hz for 4
seconds.
Data Types: single
 double
Complex Number Support: Yes
fs
— Sample rate
2π (default)  positive numeric scalar
Sample rate, specified as a positive numeric scalar.
t
— Time values
vector  datetime
array  duration
array  duration
scalar
Time values, specified as a vector, a datetime
or duration
array, or a
duration
scalar representing the time interval
between samples.
Example: seconds(0:1/100:1)
is a
array representing
1 second of sampling at 100 Hz.duration
Example: seconds(1)
is a
scalar representing
a 1second time difference between consecutive signal
samples.duration
type
— Type of spectrum to compute
'power'
(default)  'spectrogram'
 'persistence'
Type of spectrum to compute, specified as 'power'
,
'spectrogram'
, or 'persistence'
:
'power'
— Compute the power spectrum of the input. Use this option to analyze the frequency content of a stationary signal. For more information, Spectrum Computation.'spectrogram'
— Compute the spectrogram of the input. Use this option to analyze how the frequency content of a signal changes over time. For more information, see Spectrogram Computation.'persistence'
— Compute the persistence power spectrum of the input. Use this option to visualize the fraction of time that a particular frequency component is present in a signal. For more information, see Persistence Spectrum Computation.
Note
The 'spectrogram'
and
'persistence'
options do not support multichannel
input.
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'Leakage',1,'Reassign',true,'MinThreshold',35
windows
the data using a rectangular window, computes a reassigned spectrum estimate, and
sets all values smaller than –35 dB to zero.
FrequencyLimits
— Frequency band limits
[0 fs/2]
(default)  twoelement numeric vector
Frequency band limits, specified as the commaseparated pair
consisting of 'FrequencyLimits'
and a twoelement
numeric vector:
If the input contains time information, then the frequency band is expressed in Hz.
If the input does not contain time information, then the frequency band is expressed in normalized units of rad/sample.
By default, pspectrum
computes the spectrum over
the whole Nyquist range:
If the specified frequency band contains a region that falls outside the Nyquist range, then
pspectrum
truncates the frequency band.If the specified frequency band lies completely outside of the Nyquist range, then
pspectrum
throws an error.
See Spectrum Computation for more information about the Nyquist range.
If x
is nonuniformly sampled, then
pspectrum
linearly interpolates the signal to a
uniform grid and defines an effective sample rate equal to the inverse
of the median of the differences between adjacent time points. Express
'FrequencyLimits'
in terms of the effective
sample rate.
Example: [0.2*pi 0.7*pi]
computes the spectrum of a
signal with no time information from
0.2π
to
0.7π
rad/sample.
FrequencyResolution
— Frequency resolution bandwidth
real numeric scalar
Frequency resolution bandwidth, specified as the commaseparated pair
consisting of 'FrequencyResolution'
and a real
numeric scalar, expressed in Hz if the input contains time information,
or in normalized units of rad/sample if not. This argument cannot be
specified simultaneously with 'TimeResolution'
. The
default value of this argument depends on the size of the input data.
See Spectrogram Computation
for details.
Example: pi/100
computes the spectrum of a signal
with no time information with a frequency resolution of
π
/100
rad/sample.
Leakage
— Spectral leakage
0.5
(default)  real numeric scalar between 0 and 1
Spectral leakage, specified as the commaseparated pair consisting of
'Leakage'
and a real numeric scalar between 0 and
1. 'Leakage'
controls the Kaiser window sidelobe
attenuation relative to the mainlobe width, compromising between
improving resolution and decreasing leakage:
A large leakage value resolves closely spaced tones, but masks nearby weak tones.
A small leakage value finds small tones in the vicinity of larger tones, but smears close frequencies together.
Example: 'Leakage',0
reduces leakage to a minimum at
the expense of spectral resolution.
Example: 'Leakage',0.85
approximates windowing the
data with a Hann window.
Example: 'Leakage',1
is equivalent to windowing the
data with a rectangular window, maximizing leakage but improving
spectral resolution.
MinThreshold
— Lower bound for nonzero values
Inf
(default)  real scalar
Lower bound for nonzero values, specified as the commaseparated pair
consisting of 'MinThreshold'
and a real scalar.
pspectrum
implements
'MinThreshold'
differently based on the value
of the type
argument:
'power'
or'spectrogram'
—pspectrum
sets those elements ofp
such that 10 log_{10}(p
) ≤'MinThreshold'
to zero. Specify'MinThreshold'
in decibels.'persistence'
—pspectrum
sets those elements ofp
smaller than'MinThreshold'
to zero. Specify'MinThreshold'
between 0 and 100%.
NumPowerBins
— Number of power bins for persistence spectrum
256
(default)  integer between 20 and 1024
Number of power bins for persistence spectrum, specified as the
commaseparated pair consisting of 'NumPowerBins'
and
an integer between 20 and 1024.
OverlapPercent
— Overlap between adjoining segments
real scalar in the interval [0, 100)
Overlap between adjoining segments for spectrogram or persistence
spectrum, specified as the commaseparated pair consisting of
'OverlapPercent'
and a real scalar in the
interval [0, 100). The default value of this argument depends on the
spectral window. See Spectrogram Computation
for details.
Reassign
— Reassignment option
false
(default)  true
Reassignment option, specified as the commaseparated pair consisting
of 'Reassign'
and a logical value. If this option is
set to true
, then pspectrum
sharpens the localization of spectral estimates by performing time and
frequency reassignment. The reassignment technique produces periodograms
and spectrograms that are easier to read and interpret. This technique
reassigns each spectral estimate to the center of energy of its bin
instead of the bin's geometric center. The technique provides exact
localization for chirps and impulses.
TimeResolution
— Time resolution of spectrogram or persistence spectrum
real scalar
Time resolution of spectrogram or persistence spectrum, specified as
the commaseparated pair consisting of
'TimeResolution'
and a real scalar, expressed in
seconds if the input contains time information, or as an integer number
of samples if not. This argument controls the duration of the segments
used to compute the shorttime power spectra that form spectrogram or
persistence spectrum estimates. 'TimeResolution'
cannot be specified simultaneously with
'FrequencyResolution'
. The default value of
this argument depends on the size of the input data and, if it was
specified, the frequency resolution. See Spectrogram Computation
for details.
TwoSided
— Twosided spectral estimate
false  true
Twosided spectral estimate, specified as the commaseparated pair
consisting of 'TwoSided'
and a logical value.
If this option is
true
, the function computes centered, twosided spectrum estimates over [–π, π]. If the input has time information, the estimates are computed over [–f_{s}/2, f_{s}/2], where f_{s} is the effective sample rate.If this option is
false
, the function computes onesided spectrum estimates over the Nyquist range [0, π]. If the input has time information, the estimates are computed over [0, f_{s}/2], where f_{s} is the effective sample rate. To conserve the total power, the function multiplies the power by 2 at all frequencies except 0 and the Nyquist frequency. This option is valid only for real signals.
If not specified, 'TwoSided'
defaults to false
for real input signals and to
true
for complex input signals.
Output Arguments
p
— Spectrum
vector  matrix
Spectrum, returned as a vector or a matrix. The type and size of the
spectrum depends on the value of the type
argument:
'power'
—p
contains the power spectrum estimate of each channel ofx
. In this case,p
is of size N_{f} × N_{ch}, where N_{f} is the length off
and N_{ch} is the number of channels ofx
.pspectrum
scales the spectrum so that, if the frequency content of a signal falls exactly within a bin, its amplitude in that bin is the true average power of the signal. For example, the average power of a sinusoid is onehalf the square of the sinusoid amplitude. For more details, see Measure Power of Deterministic Periodic Signals.'spectrogram'
—p
contains an estimate of the shortterm, timelocalized power spectrum ofx
. In this case,p
is of size N_{f} × N_{t}, where N_{f} is the length off
and N_{t} is the length oft
.'persistence'
—p
contains, expressed as percentages, the probabilities that the signal has components of a given power level at a given time and frequency location. In this case,p
is of size N_{pwr} × N_{f}, where N_{pwr} is the length ofpwr
and N_{f} is the length off
.
f
— Spectrum frequencies
vector
Spectrum frequencies, returned as a vector. If the input signal contains
time information, then f
contains frequencies expressed
in Hz. If the input signal does not contain time information, then the
frequencies are in normalized units of rad/sample.
t
— Time values of spectrogram
vector  datetime
array  duration
array
Time values of spectrogram, returned as a vector of time values in seconds
or a duration
array. If the input does not have time
information, then t
contains sample numbers. t
contains the time values corresponding to the centers of
the data segments used to compute shorttime power spectrum estimates.
If the input to
pspectrum
is a timetable, thent
has the same format as the time values of the input timetable.If the input to
pspectrum
is a numeric vector sampled at a set of time instants specified by a numeric,duration
, ordatetime
array, thent
has the same type and format as the input time values.If the input to
pspectrum
is a numeric vector with a specified time difference between consecutive samples, thent
is aduration
array.
pwr
— Power values of persistence spectrum
vector
Power values of persistence spectrum, returned as a vector.
More About
Spectrum Computation
To compute signal spectra, pspectrum
finds a
compromise between the spectral resolution achievable with the entire length of the
signal and the performance limitations that result from computing large FFTs:
If possible, the function computes a single modified periodogram of the whole signal using a Kaiser window.
If it is not possible to compute a single modified periodogram in a reasonable amount of time, the function computes a Welch periodogram: It divides the signal into overlapping segments, windows each segment using a Kaiser window, and averages the periodograms of the segments.
Spectral Windowing
Any realworld signal is measurable only for a finite length of time. This fact introduces nonnegligible effects into Fourier analysis, which assumes that signals are either periodic or infinitely long. Spectral windowing, which assigns different weights to different signal samples, deals systematically with finitesize effects.
The simplest way to window a signal is to assume that it is identically zero outside of the measurement interval and that all samples are equally significant. This "rectangular window" has discontinuous jumps at both ends that result in spectral ringing. All other spectral windows taper at both ends to lessen this effect by assigning smaller weights to samples close to the signal edges.
The windowing process always involves a compromise between conflicting aims: improving resolution and decreasing leakage:
Resolution is the ability to know precisely how the signal energy is distributed in the frequency space. A spectrum analyzer with ideal resolution can distinguish two different tones (pure sinusoids) present in the signal, no matter how close in frequency. Quantitatively, this ability relates to the mainlobe width of the transform of the window.
Leakage is the fact that, in a finite signal, every frequency component projects energy content throughout the complete frequency span. The amount of leakage in a spectrum can be measured by the ability to detect a weak tone from noise in the presence of a neighboring strong tone. Quantitatively, this ability relates to the sidelobe level of the frequency transform of the window.
The spectrum is normalized so that a pure tone within that bandwidth, if perfectly centered, has the correct amplitude.
The better the resolution, the higher the leakage, and vice versa. At one end of the range, a rectangular window has the narrowest possible mainlobe and the highest sidelobes. This window can resolve closely spaced tones if they have similar energy content, but it fails to find the weaker one if they do not. At the other end, a window with high sidelobe suppression has a wide mainlobe in which close frequencies are smeared together.
pspectrum
uses Kaiser windows to carry out windowing. For
Kaiser windows, the fraction of the signal energy captured by the mainlobe depends
most importantly on an adjustable shape factor,
β. pspectrum
uses shape factors ranging
from β = 0, which corresponds to a rectangular window, to β = 40, where a wide mainlobe captures essentially all the spectral
energy representable in double precision. An intermediate value of β ≈ 6 approximates a Hann window quite closely. To control
β, use the 'Leakage'
namevalue pair. If
you set 'Leakage'
to ℓ, then
ℓ and β are related by β = 40(1 – ℓ). See kaiser
for more
details.


51point Hann window and 51point Kaiser window with β = 5.7 in the time domain  51point Hann window and 51point Kaiser window with β = 5.7 in the frequency domain 
Parameter and Algorithm Selection
To compute signal spectra, pspectrum
initially determines the
resolution bandwidth, which measures how close two tones
can be and still be resolved. The resolution bandwidth has a theoretical value of
$${\text{RBW}}_{\text{theory}}=\frac{\text{ENBW}}{{t}_{\mathrm{max}}{t}_{\mathrm{min}}}.$$
t_{max} – t_{min}, the record length, is the timedomain duration of the selected signal region.
ENBW is the equivalent noise bandwidth of the spectral window. See
enbw
for more details.Use the
'Leakage'
namevalue pair to control the ENBW. The minimum value of the argument corresponds to a Kaiser window with β = 40. The maximum value corresponds to a Kaiser window with β = 0.
In practice, however, pspectrum
might lower the resolution.
Lowering the resolution makes it possible to compute the spectrum in a reasonable
amount of time and to display it with a finite number of pixels. For these practical
reasons, the lowest resolution bandwidth pspectrum
can use is
$${\text{RBW}}_{\text{performance}}=4\times \frac{{f}_{\text{span}}}{40961},$$
where f_{span} is the width of the frequency band specified using
'FrequencyLimits'
. If
'FrequencyLimits'
is not specified, then
pspectrum
uses the sample rate as f_{span}. RBW_{performance} cannot be adjusted.
To compute the spectrum of a signal, the function chooses the larger of the two values, called the target resolution bandwidth:
$$\text{RBW}=\mathrm{max}({\text{RBW}}_{\text{theory}},{\text{RBW}}_{\text{performance}}).$$
If the resolution bandwidth is RBW_{theory}, then
pspectrum
computes a single modified periodogram for the whole signal. The function uses a Kaiser window with shape factor controlled by the'Leakage'
namevalue pair. Seeperiodogram
for more details.If the resolution bandwidth is RBW_{performance}, then
pspectrum
computes a Welch periodogram for the signal. The function:Divides the signals into overlapping segments.
Windows each segment separately using a Kaiser window with the specified shape factor.
Averages the periodograms of all the segments.
Welch’s procedure is designed to reduce the variance of the spectrum estimate by averaging different “realizations” of the signals, given by the overlapping sections, and using the window to remove redundant data. See
pwelch
for more details.The length of each segment (or, equivalently, of the window) is computed using
$$\text{Segmentlength}=\frac{{f}_{\text{Nyquist}}\times \text{ENBW}}{\text{RBW}},$$
where f_{Nyquist} is the Nyquist frequency. (If there is no aliasing, the Nyquist frequency is onehalf the effective sample rate, defined as the inverse of the median of the differences between adjacent time points. The Nyquist range is [0, f_{Nyquist}] for real signals and [–f_{Nyquist}, f_{Nyquist}] for complex signals.)
The stride length is found by adjusting an initial estimate,
$$\text{Stridelength}\equiv \text{Segmentlength}\text{Overlap}=\frac{\text{Segmentlength}}{2\times \text{ENBW}1},$$
so that the first window starts exactly on the first sample of the first segment and the last window ends exactly on the last sample of the last segment.
Spectrogram Computation
To compute the timedependent spectrum of a nonstationary signal,
pspectrum
divides the signal into overlapping segments,
windows each segment with a Kaiser window, computes the shorttime Fourier
transform, and then concatenates the transforms to form a matrix. For more
information, see Spectrogram Computation with Signal Processing Toolbox.
A nonstationary signal is a signal whose frequency content changes with time. The
spectrogram of a nonstationary signal is an estimate of
the time evolution of its frequency content. To construct the spectrogram of a
nonstationary signal, pspectrum
follows these steps:
Divide the signal into equallength segments. The segments must be short enough that the frequency content of the signal does not change appreciably within a segment. The segments may or may not overlap.
Window each segment and compute its spectrum to get the shorttime Fourier transform.
Use the segment spectra to construct the spectrogram:
If called with output arguments, concatenate the spectra to form a matrix.
If called with no output arguments, display the power of each spectrum in decibels segment by segment. Depict the magnitudes sidebyside as an image with magnitudedependent colormap.
The function can compute the spectrogram only for singlechannel signals.
Divide Signal into Segments
To construct a spectrogram, first divide the signal into possibly overlapping
segments. With the pspectrum
function, you can control the
length of the segments and the amount of overlap between adjoining segments using
the 'TimeResolution'
and 'OverlapPercent'
namevalue pair arguments. If you do not specify the length and overlap, the
function chooses a length based on the entire length of the signal and an overlap
percentage given by
$$\left(1\frac{\text{1}}{2\times \text{ENBW}1}\right)\times 100,$$
where ENBW is the equivalent noise bandwidth of the
spectral window. See enbw
and Spectrum Computation for more
information.
Specified Time Resolution
If the signal does not have time information, specify the time resolution (segment length) in samples. The time resolution must be an integer greater than or equal to 1 and smaller than or equal to the signal length.
If the signal has time information, specify the time resolution in seconds. The function converts the result into a number of samples and rounds it to the nearest integer that is less than or equal to the number but not smaller than 1. The time resolution must be smaller than or equal to the signal duration.
Specify the overlap as a percentage of the segment length. The function converts the result into a number of samples and rounds it to the nearest integer that is less than or equal to the number.
Default Time Resolution
If you do not specify a time resolution, then pspectrum
uses
the length of the entire signal to choose the length of the segments. The function
sets the time resolution as ⌈N/d⌉ samples, where the ⌈⌉ symbols denote the ceiling function, N is the
length of the signal, and d is a divisor that depends on
N:
Signal Length (N)  Divisor (d)  Segment Length 

2 samples – 63
samples  2  1 sample – 32
samples 
64 samples – 255
samples  8  8 samples – 32
samples 
256 samples – 2047
samples  8  32 samples – 256
samples 
2048 samples – 4095
samples  16  128 samples – 256
samples 
4096 samples – 8191
samples  32  128 samples – 256
samples 
8192 samples – 16383
samples  64  128 samples – 256
samples 
16384 samples – N
samples  128  128 samples – ⌈N /
128 ⌉ samples 
You can still specify the overlap between adjoining segments. Specifying the overlap changes the number of segments. Segments that extend beyond the signal endpoint are zeropadded.
Consider the sevensample signal [s0 s1 s2 s3 s4 s5 s6]
.
Because ⌈7/2⌉ = ⌈3.5⌉ = 4, the function divides the signal into two segments of length four
when there is no overlap. The number of segments changes as the overlap
increases.
Number of Overlapping Samples  Resulting Segments 

0 
s0 s1 s2 s3 s4 s5 s6 0 
1 
s0 s1 s2 s3 s3 s4 s5 s6 
2 
s0 s1 s2 s3 s2 s3 s4 s5 s4 s5 s6 0 
3 
s0 s1 s2 s3 s1 s2 s3 s4 s2 s3 s4 s5 s3 s4 s5 s6 
pspectrum
zeropads the signal if the last
segment extends beyond the signal endpoint. The function returns t
, a
vector of time instants corresponding to the centers of the segments.
Window the Segments and Compute Spectra
After pspectrum
divides the signal into overlapping segments,
the function windows each segment with a Kaiser window. The shape factor
β of the window, and therefore the leakage, can be adjusted
using the 'Leakage'
namevalue pair. The function then computes
the spectrum of each segment and concatenates the spectra to form the spectrogram
matrix. To compute the segment spectra, pspectrum
follows the
procedure described in Spectrum Computation, except that
the lower limit of the resolution bandwidth is
$${\text{RBW}}_{\text{performance}}=4\times \frac{{f}_{\text{span}}}{10241}.$$
Display Spectrum Power
If called with no output arguments, the function displays the power of the shorttime Fourier transform in decibels, using a color bar with the default MATLAB colormap. The color bar comprises the full power range of the spectrogram.
Persistence Spectrum Computation
The persistence spectrum is a timefrequency view that shows the percentage of the time that a given frequency is present in a signal. The persistence spectrum is a histogram in powerfrequency space. The longer a particular frequency persists in a signal as the signal evolves, the higher its time percentage and thus the brighter or "hotter" its color in the display. Use the persistence spectrum to identify signals hidden in other signals.
To compute the persistence spectrum, pspectrum
performs these steps:
Compute the spectrogram using the specified leakage, time resolution, and overlap. See Spectrogram Computation for more details.
Partition the power and frequency values into 2D bins. (Use the
'NumPowerBins'
namevalue pair to specify the number of power bins.)For each time value, compute a bivariate histogram of the logarithm of the power spectrum. For every powerfrequency bin where there is signal energy at that instant, increase the corresponding matrix element by 1. Sum the histograms for all the time values.
Plot the accumulated histogram against the power and the frequency, with the color proportional to the logarithm of the histogram counts expressed as normalized percentages. To represent zero values, use onehalf of the smallest possible magnitude.
Power Spectra 

Histograms 

Accumulated Histogram 

References
[1] harris, fredric j. “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform.” Proceedings of the IEEE^{®}. Vol. 66, January 1978, pp. 51–83.
[2] Welch, Peter D. "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 and Electroacoustics. Vol. 15, June 1967, pp. 70–73.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
Timetables are not supported for code generation.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
Persistence spectrum is not supported.
Reassigned spectrum or spectrogram is not supported.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2017b
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