Analyze signals in the frequency and time-frequency domains
x is a vector or a timetable with a vector
of data, then it is treated as a single channel.
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 of
pspectrum(___) with no output arguments plots
the spectral estimate in the current figure window.
Compute the power spectrum of a noisy sinusoid. Specify a sinusoid frequency of 200 Hz. Sample the sinusoid at 1 kHz for 296 milliseconds. Embed the signal in white Gaussian noise of variance 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 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.
Generate two signals, each sampled at 3 kHz for 1 second. The first 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. The second signal, also embedded in white noise, is a chirp with sinusoidally varying frequency content.
fs = 3000; t = 0:1/fs:1-1/fs; x1 = chirp(t,300,t(end),1300,'quadratic',0,'convex') + ... randn(size(t))/100; x2 = exp(2j*pi*100*cos(2*pi*2*t)) + randn(size(t))/100;
Compute and plot the two-sided power spectrum of the first signal using a rectangular window. For real signals,
pspectrum plots a one-sided spectrum by default. To plot a two-sided spectrum, set
TwoSided to true.
Compute the spectrogram of the second signal. For complex signals, the spectrogram is two-sided by default. Display the spectrogram as a waterfall plot.
[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])
Generate a two-channel 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:2-1/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 signal-to-noise 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.
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.
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.
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.
Compute the persistence spectrum of the signal. Now both signal components are clearly visible.
pspectrum(x,fs,'persistence', ... 'FrequencyLimits',[100 290],'TimeResolution',1)
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
fs = 1e3; t = 0:1/fs:2; y = chirp(t,100,1,200,'quadratic'); [sp,fp,tp] = pspectrum(y,fs,'spectrogram'); mesh(tp,fp,sp) view(-15,60) xlabel('Time (s)') ylabel('Frequency (Hz)')
Compute the reassigned spectrogram. Specify a frequency resolution of 10 Hz. Visualize the result using the
pspectrum function with no output arguments.
Recompute the spectrogram using a time resolution of 0.2 second.
Compute the reassigned spectrogram using the same time resolution.
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:0.5-1/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 half-second 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.
x— Input signal
Input signal, specified as a vector, a matrix, or a MATLAB®
x is a timetable, then it must contain
increasing finite row times.
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 (MATLAB).
x is a timetable representing a
multichannel signal, then it must have either a single variable
containing a matrix or multiple variables consisting of
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
cos(pi./[4;2]*(0:159))'+randn(160,2) is a
two-channel signal consisting of sinusoids embedded in white
timetable(seconds(0:4)',rand(5,2)) specifies a
two-channel random variable sampled at 1 Hz for 4 seconds.
specifies a two-channel random variable sampled at 1 Hz for 4
Complex Number Support: Yes
fs— Sample rate
Sample rate, specified as a positive numeric scalar.
t— Time values
seconds(0:1/100:1) is a
1 second of sampling at 100 Hz.
seconds(1) is a
a 1-second time difference between consecutive signal
type— Type of spectrum to compute
Type of spectrum to compute, specified as
'power' — Compute the power spectrum of the
input. Use this option to analyze the frequency content of a
stationary signal. See Spectrum Computation for more information.
'spectrogram' — Compute the spectrogram of
the input. Use this option to analyze how the frequency content
of a signal changes over time. See Spectrogram Computation for more information.
'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. See Persistence Spectrum Computation for more information.
'persistence' options do not support multichannel
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
'Leakage',1,'Reassigned',true,'MinThreshold',-35windows 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) | two-element numeric vector
Frequency band limits, specified as the comma-separated pair
'FrequencyLimits' and a two-element
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.
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
If the specified frequency band lies completely outside of
the Nyquist range, then
throws an error.
See Spectrum Computation for more information about the Nyquist range.
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
[0.2*pi 0.7*pi] computes the spectrum of a
signal with no time information from
'FrequencyResolution'— Frequency resolution bandwidth
Frequency resolution bandwidth, specified as the comma-separated pair
'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
default value of this argument depends on the size of the input data.
See Spectrogram Computation
pi/100 computes the spectrum of a signal
with no time information with a frequency resolution of
'Leakage'— Spectral leakage
0.5(default) | real numeric scalar between 0 and 1
Spectral leakage, specified as the comma-separated pair consisting of
'Leakage' and a real numeric scalar between 0 and
'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.
'Leakage',0 reduces leakage to a minimum at
the expense of spectral resolution.
'Leakage',0.85 approximates windowing the
data with a Hann window.
'Leakage',1 is equivalent to windowing the
data with a rectangular window, maximizing leakage but improving
'MinThreshold'— Lower bound for nonzero values
-Inf(default) | real scalar
Lower bound for nonzero values, specified as the comma-separated pair
'MinThreshold' and a real scalar.
'MinThreshold' differently based on the value
pspectrum sets those elements of
p such that 10
'MinThreshold' to zero. Specify
'MinThreshold' in decibels.
pspectrum sets those elements of
p smaller than
'MinThreshold' to zero. Specify
'MinThreshold' between 0 and
'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
comma-separated pair consisting of
an integer between 20 and 1024.
'OverlapPercent'— Overlap between adjoining segments
Overlap between adjoining segments for spectrogram or persistence
spectrum, specified as the comma-separated 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
'Reassign'— Reassignment option
Reassignment option, specified as the comma-separated pair consisting
'Reassign' and a logical value. If this option is
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
Time resolution of spectrogram or persistence spectrum, specified as
the comma-separated 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 short-time power spectra that form spectrogram or
persistence spectrum estimates.
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
'TwoSided'— Two-sided spectral estimate
Two-sided spectral estimate, specified as the comma-separated pair
'TwoSided' and a logical value.
If this option is
true, the function
computes centered, two-sided spectrum estimates over [–π,
π]. If the input has time information, the
estimates are computed over [–fs/2,
fs is the
effective sample rate.
If this option is
false, the function
computes one-sided spectrum estimates over the Nyquist range [0, π]. If the input has time information, the
estimates are computed over [0,
fs is the
effective sample rate. To conserve the total power, the
function multiples the power by 2 at all frequencies except
0 and the Nyquist frequency. This option is valid only for
If not specified,
false for real input signals and to
true for complex input signals.
Spectrum, returned as a vector or a matrix. The type and size of the
spectrum depends on the value of the
contains an estimate of the short-term, time-localized power
x. In this case,
p is of size
Nf × Nt,
where Nf is the length
Nt is the length
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
Npwr × Nf,
where Npwr is the
Nf is the length
f— Spectrum frequencies
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
Time values of spectrogram, returned as a vector of time values in seconds
duration array. If the input does not have time
t contains sample numbers.
t contains the time values corresponding to the centers of
the data segments used to compute short-time power spectrum estimates.
If the input to
pspectrum is a timetable,
t has the same format as the time values of the
pwr— Power values of persistence spectrum
Power values of persistence spectrum, returned as a vector.
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.
Any real-world 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 finite-size 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 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,
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' name-value pair. If
'Leakage' to ℓ, then
ℓ and β are related by β = 40(1 – ℓ). See
kaiser for more
|51-point Hann window and 51-point Kaiser window with β = 5.7 in the time domain||51-point Hann window and 51-point 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
tmax – tmin, the record length, is the time-domain duration of the selected signal region.
ENBW is the equivalent noise
bandwidth of the spectral window. See
enbw for more
'Leakage' name-value 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
where fspan is the width of the frequency band specified using
'FrequencyLimits' is not specified, then
pspectrum uses the sample rate as fspan. RBWperformance cannot be adjusted.
To compute the spectrum of a signal, the function chooses the larger of the two values, called the target resolution bandwidth:
If the resolution bandwidth is RBWtheory, then
pspectrum computes a single
modified periodogram for the whole signal. The
function uses a Kaiser window with shape factor controlled by the
'Leakage' name-value pair and applies zero-padding
when the time limits on the axes exceed the signal duration. See
periodogram for more
If the resolution bandwidth is RBWperformance, 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
where fNyquist is the Nyquist frequency. (If there is no aliasing, the Nyquist frequency is one-half the effective sample rate, defined as the inverse of the median of the differences between adjacent time points. The Nyquist range is [0, fNyquist] for real signals and [–fNyquist, fNyquist] for complex signals.)
The stride length is found by adjusting an initial estimate,
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.
To compute the time-dependent spectrum of a nonstationary signal,
pspectrum divides the signal into overlapping segments,
windows each segment with a Kaiser window, computes the short-time Fourier
transform, and then concatenates the transforms to form a matrix.
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
pspectrum follows these steps:
Divide the signal into equal-length 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 short-time 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 side-by-side as an image with magnitude-dependent colormap.
The function can compute the spectrogram only for single-channel 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
name-value 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
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
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
|Signal Length (N)||Divisor (d)||Segment Length|
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 zero-padded.
Consider the seven-sample 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
|Number of Overlapping Samples||Resulting Segments|
s0 s1 s2 s3 s4 s5 s6 0
s0 s1 s2 s3 s3 s4 s5 s6
s0 s1 s2 s3 s2 s3 s4 s5 s4 s5 s6 0
s0 s1 s2 s3 s1 s2 s3 s4 s2 s3 s4 s5 s3 s4 s5 s6
pspectrum zero-pads the signal if the last
segment extends beyond the signal endpoint. The function returns
vector of time instants corresponding to the centers of the segments.
Window the Segments and Compute Spectra
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
'Leakage' name-value 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
Display Spectrum Power
If called with no output arguments, the function displays the power of the short-time Fourier transform in decibels, using a color bar with the default MATLAB colormap. The color bar comprises the full power range of the spectrogram.
The persistence spectrum of a signal is a time-frequency view that shows the percentage of the time that a given frequency is present in a signal. The persistence spectrum is a histogram in power-frequency 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 2-D bins. (Use the
'NumPowerBins' name-value 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 power-frequency 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 one-half of the smallest possible magnitude.
 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.
 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.