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spectrogram

Spectrogram using short-time Fourier transform

Description

example

s = spectrogram(x) returns the Short-Time Fourier Transform (STFT) of the input signal x. Each column of s contains an estimate of the short-term, time-localized frequency content of x. The magnitude squared of s is known as the spectrogram time-frequency representation of x [1].

example

s = spectrogram(x,window) uses window to divide the signal into segments and perform windowing.

example

s = spectrogram(x,window,noverlap) uses noverlap samples of overlap between adjoining segments.

example

s = spectrogram(x,window,noverlap,nfft) uses nfft sampling points to calculate the discrete Fourier transform.

example

[s,w,t] = spectrogram(___) returns a vector of normalized frequencies, w, and a vector of time instants, t, at which the STFT is computed. This syntax can include any combination of input arguments from previous syntaxes.

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[s,f,t] = spectrogram(___,fs) returns a vector of cyclical frequencies, f, expressed in terms of the sample rate fs. fs must be the fifth input to spectrogram. To input a sample rate and still use the default values of the preceding optional arguments, specify these arguments as empty, [].

example

[s,w,t] = spectrogram(x,window,noverlap,w) returns the STFT at the normalized frequencies specified in w. w must have at least two elements, because otherwise the function interprets it as nfft.

example

[s,f,t] = spectrogram(x,window,noverlap,f,fs) returns the STFT at the cyclical frequencies specified in f. f must have at least two elements, because otherwise the function interprets it as nfft.

example

[___,ps] = spectrogram(___,spectrumtype) also returns a matrix, ps, proportional to the spectrogram of x.

  • If you specify spectrumtype as "psd", each column of ps contains an estimate of the power spectral density (PSD) of a windowed segment.

  • If you specify spectrumtype as "power", each column of ps contains an estimate of the power spectrum of a windowed segment.

example

[___] = spectrogram(___,"reassigned") reassigns each PSD or power spectrum estimate to the location of its center of energy. If your signal contains well-localized temporal or spectral components, then this option generates a sharper spectrogram.

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[___,ps,fc,tc] = spectrogram(___) also returns two matrices, fc and tc, containing the frequency and time of the center of energy of each PSD or power spectrum estimate.

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[___] = spectrogram(___,freqrange) returns the PSD or power spectrum estimate over the frequency range specified by freqrange. Valid options for freqrange are "onesided", "twosided", and "centered".

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[___] = spectrogram(___,Name=Value) specifies additional options using name-value arguments. Options include the minimum threshold and output time dimension.

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spectrogram(___) with no output arguments plots ps in decibels in the current figure window.

example

spectrogram(___,freqloc) specifies the axis on which to plot the frequency.

Examples

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Generate Nx=1024 samples of a signal that consists of a sum of sinusoids. The normalized frequencies of the sinusoids are 2π/5 rad/sample and 4π/5 rad/sample. The higher frequency sinusoid has 10 times the amplitude of the other sinusoid.

N = 1024;
n = 0:N-1;

w0 = 2*pi/5;
x = sin(w0*n)+10*sin(2*w0*n);

Compute the short-time Fourier transform using the function defaults. Plot the spectrogram.

s = spectrogram(x);

spectrogram(x,'yaxis')

Figure contains an axes object. The axes object contains an object of type image.

Repeat the computation.

  • Divide the signal into sections of length nsc=Nx/4.5.

  • Window the sections using a Hamming window.

  • Specify 50% overlap between contiguous sections.

  • To compute the FFT, use max(256,2p) points, where p=log2nsc.

Verify that the two approaches give identical results.

Nx = length(x);
nsc = floor(Nx/4.5);
nov = floor(nsc/2);
nff = max(256,2^nextpow2(nsc));

t = spectrogram(x,hamming(nsc),nov,nff);

maxerr = max(abs(abs(t(:))-abs(s(:))))
maxerr = 0

Divide the signal into 8 sections of equal length, with 50% overlap between sections. Specify the same FFT length as in the preceding step. Compute the short-time Fourier transform and verify that it gives the same result as the previous two procedures.

ns = 8;
ov = 0.5;
lsc = floor(Nx/(ns-(ns-1)*ov));

t = spectrogram(x,lsc,floor(ov*lsc),nff);

maxerr = max(abs(abs(t(:))-abs(s(:))))
maxerr = 0

Generate a signal that consists of a complex-valued convex quadratic chirp sampled at 600 Hz for 2 seconds. The chirp has an initial frequency of 250 Hz and a final frequency of 50 Hz.

fs = 6e2;
ts = 0:1/fs:2;
x = chirp(ts,250,ts(end),50,"quadratic",0,"convex","complex");

spectrogram Function

Use the spectrogram function to compute the STFT of the signal.

  • Divide the signal into segments, each M=49 samples long.

  • Specify L=11 samples of overlap between adjoining segments.

  • Discard the final, shorter segment.

  • Window each segment with a Bartlett window.

  • Evaluate the discrete Fourier transform of each segment at NDFT=1024 points. By default, spectrogram computes two-sided transforms for complex-valued signals.

M = 49;
L = 11;
g = bartlett(M);
Ndft = 1024;

[s,f,t] = spectrogram(x,g,L,Ndft,fs);

Use the waterplot function to compute and display the spectrogram, defined as the magnitude squared of the STFT.

waterplot(s,f,t)

Figure contains an axes object. The axes object contains an object of type patch.

STFT Definition

Compute the STFT of the Nx-sample signal using the definition. Divide the signal into Nx-LM-L overlapping segments. Window each segment and evaluate its discrete Fourier transform at NDFT points.

[segs,~] = buffer(1:length(x),M,L,"nodelay");

X = fft(x(segs).*g,Ndft);

Compute the time and frequency ranges for the STFT.

  • To find the time values, divide the time vector into overlapping segments. The time values are the midpoints of the segments, with each segment treated as an interval open at the lower end.

  • To find the frequency values, specify a Nyquist interval closed at zero frequency and open at the lower end.

tbuf = ts(segs);
tint = mean(tbuf(2:end,:));

fint = 0:fs/Ndft:fs-fs/Ndft;

Compare the output of spectrogram to the definition. Display the spectrogram.

maxdiff = max(max(abs(s-X)))
maxdiff = 0
waterplot(X,fint,tint)

Figure contains an axes object. The axes object contains an object of type patch.

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

Generate a signal consisting of a chirp sampled at 1.4 kHz for 2 seconds. The frequency of the chirp decreases linearly from 600 Hz to 100 Hz during the measurement time.

fs = 1400;
x = chirp(0:1/fs:2,600,2,100);

stft Defaults

Compute the STFT of the signal using the spectrogram and stft functions. Use the default values of the stft function:

  • Divide the signal into 128-sample segments and window each segment with a periodic Hann window.

  • Specify 96 samples of overlap between adjoining segments. This length is equivalent to 75% of the window length.

  • Specify 128 DFT points and center the STFT at zero frequency, with the frequency expressed in hertz.

Verify that the two results are equal.

M = 128;
g = hann(M,"periodic");
L = 0.75*M;
Ndft = 128;

[sp,fp,tp] = spectrogram(x,g,L,Ndft,fs,"centered");

[s,f,t] = stft(x,fs);

dff = max(max(abs(sp-s)))
dff = 0

Use the mesh function to plot the two outputs.

subplot(2,1,1)
mesh(tp,fp,abs(sp).^2)
title("spectrogram")
view(2), axis tight
subplot(2,1,2)
mesh(t,f,abs(s).^2)
title("stft")
view(2), axis tight

Figure contains 2 axes objects. Axes object 1 with title spectrogram contains an object of type surface. Axes object 2 with title stft contains an object of type surface.

spectrogram Defaults

Repeat the computation using the default values of the spectrogram function:

  • Divide the signal into segments of length M=Nx/4.5, where Nx is the length of the signal. Window each segment with a Hamming window.

  • Specify 50% overlap between segments.

  • To compute the FFT, use max(256,2log2M) points. Compute the spectrogram only for positive normalized frequencies.

M = floor(length(x)/4.5);
g = hamming(M);
L = floor(M/2);
Ndft = max(256,2^nextpow2(M));

[sx,fx,tx] = spectrogram(x);

[st,ft,tt] = stft(x,Window=g,OverlapLength=L,FFTLength=Ndft,FrequencyRange="onesided");

dff = max(max(sx-st))
dff = 0

Use the waterplot function to plot the two outputs. Divide the frequency axis by π in both cases. For the stft output, divide the sample numbers by the effective sample rate, 2π.

subplot(2,1,1)
waterplot(sx,fx/pi,tx)
title("spectrogram")
subplot(2,1,2)
waterplot(st,ft/pi,tt/(2*pi))
title("stft")

Figure contains 2 axes objects. Axes object 1 with title spectrogram contains an object of type patch. Axes object 2 with title stft contains an object of type patch.

function waterplot(s,f,t)
% Waterfall plot of spectrogram
    waterfall(f,t,abs(s)'.^2)
    set(gca,XDir="reverse",View=[30 50])
    xlabel("Frequency/\pi")
    ylabel("Samples")
end

Use the spectrogram function to measure and track the instantaneous frequency of a signal.

Generate a quadratic chirp sampled at 1 kHz for two seconds. Specify the chirp so that its frequency is initially 100 Hz and increases to 200 Hz after one second.

fs = 1000;
t = 0:1/fs:2-1/fs;
y = chirp(t,100,1,200,'quadratic');

Estimate the spectrum of the chirp using the short-time Fourier transform implemented in the spectrogram function. Divide the signal into sections of length 100, windowed with a Hamming window. Specify 80 samples of overlap between adjoining sections and evaluate the spectrum at 100/2+1=51 frequencies.

spectrogram(y,100,80,100,fs,'yaxis')

Figure contains an axes object. The axes object contains an object of type image.

Track the chirp frequency by finding the time-frequency ridge with highest energy across the (2000-80)/(100-80)=96 time points. Overlay the instantaneous frequency on the spectrogram plot.

[~,f,t,p] = spectrogram(y,100,80,100,fs);

[fridge,~,lr] = tfridge(p,f);

hold on
plot3(t,fridge,abs(p(lr)),'LineWidth',4)
hold off

Figure contains an axes object. The axes object contains 2 objects of type image, line.

Generate 512 samples of a chirp with sinusoidally varying frequency content.

N = 512;
n = 0:N-1;

x = exp(1j*pi*sin(8*n/N)*32);

Compute the centered two-sided short-time Fourier transform of the chirp. Divide the signal into 32-sample segments with 16-sample overlap. Specify 64 DFT points. Plot the spectrogram.

[scalar,fs,ts] = spectrogram(x,32,16,64,'centered');

spectrogram(x,32,16,64,'centered','yaxis')

Figure contains an axes object. The axes object contains an object of type image.

Obtain the same result by computing the spectrogram on 64 equispaced frequencies over the interval (-π,π]. The 'centered' option is not necessary.

fintv = -pi+pi/32:pi/32:pi;

[vector,fv,tv] = spectrogram(x,32,16,fintv);

spectrogram(x,32,16,fintv,'yaxis')

Figure contains an axes object. The axes object contains an object of type image.

Generate a signal that consists of a voltage-controlled oscillator and three Gaussian atoms. The signal is sampled at fs=2 kHz for 1 second.

fs = 2000;
tx = (0:1/fs:2);
gaussFun = @(A,x,mu,f) exp(-(x-mu).^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*(tx-1).^2),[0.1 0.4]*fs,fs);
x = s+x';

Short-Time Fourier Transforms

Use the pspectrum function to compute the STFT.

  • Divide the Nx-sample signal into segments of length M=80 samples, corresponding to a time resolution of 80/2000=40 milliseconds.

  • Specify L=16 samples or 20% of overlap between adjoining segments.

  • Window each segment with a Kaiser window and specify a leakage =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 g(n). The leakage and the shape factor β of the window are related by β=40×(1-).

  • pspectrum always uses NDFT=1024 points when computing the discrete Fourier transform. You can specify this number if you want to compute the transform over a two-sided or centered frequency range. However, for one-sided 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 k=Nx-LM-L segments, spectrogram truncates the signal whereas pspectrum 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 segment-by-segment power spectrum, which is already squared but is divided by a factor of ng(n) before squaring.

  • For one-sided transforms, pspectrum adds an extra factor of 2 to the spectrogram.

g = kaiser(M,40*(1-lk));

k = (length(x)-L)/(M-L);
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")

Figure contains 2 axes objects. Axes object 1 with title pspectrum contains an object of type patch. Axes object 2 with title spectrogram contains an object of type patch.

maxd = max(max(abs(abs(s).^2-S)))
maxd = 2.4419e-08

Power Spectra and Convenience Plots

The spectrogram function has a fourth argument that corresponds to the segment-by-segment power spectrum or power spectral density. Similar to the output of pspectrum, the ps argument is already squared and includes the normalization factor ng(n). For one-sided 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 = 2.4419e-08

When called with no output arguments, both pspectrum and spectrogram plot the spectrogram of the signal in decibels. Include the factor of 2 for one-sided spectrograms. Set the colormaps to be the same for both plots. Set the x-limits 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 y-axis.

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)

Figure contains 2 axes objects. Axes object 1 with title pspectrum contains an object of type image. Axes object 2 with title spectrogram contains an object of type image.

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

Generate a chirp signal sampled for 2 seconds at 1 kHz. Specify the chirp so that its frequency is initially 100 Hz and increases to 200 Hz after 1 second.

Fs = 1000;
t = 0:1/Fs:2;
y = chirp(t,100,1,200,'quadratic');

Estimate the reassigned spectrogram of the signal.

  • Divide the signal into sections of length 128, windowed with a Kaiser window with shape parameter β=18.

  • Specify 120 samples of overlap between adjoining sections.

  • Evaluate the spectrum at 128/2=65 frequencies and (length(x)-120)/(128-120)=235 time bins.

spectrogram(y,kaiser(128,18),120,128,Fs,'reassigned','yaxis')

Figure contains an axes object. The axes object contains an object of type image.

Generate a chirp signal sampled for 2 seconds at 1 kHz. Specify the chirp so that its frequency is initially 100 Hz and increases to 200 Hz after 1 second.

Fs = 1000;
t = 0:1/Fs:2;
y = chirp(t,100,1,200,'quadratic');

Estimate the time-dependent power spectral density (PSD) of the signal.

  • Divide the signal into sections of length 128, windowed with a Kaiser window with shape parameter β=18.

  • Specify 120 samples of overlap between adjoining sections.

  • Evaluate the spectrum at 128/2=65 frequencies and (length(x)-120)/(128-120)=235 time bins.

Output the frequency and time of the center of gravity of each PSD estimate. Set to zero those elements of the PSD smaller than -30 dB.

[~,~,~,pxx,fc,tc] = spectrogram(y,kaiser(128,18),120,128,Fs, ...
    'MinThreshold',-30);

Plot the nonzero elements as functions of the center-of-gravity frequencies and times.

plot(tc(pxx>0),fc(pxx>0),'.')

Figure contains an axes object. The axes object contains an object of type line.

Generate a signal that consists of a real-valued chirp sampled at 2 kHz for 2 seconds.

fs = 2000;
tx = 0:1/fs:2;
x = vco(-chirp(tx,0,tx(end),2).*exp(-3*(tx-1).^2),[0.1 0.4]*fs,fs).*hann(length(tx))';

Two-Sided Spectrogram

Compute and plot the two-sided STFT of the signal.

  • Divide the signal into segments, each M=73 samples long.

  • Specify L=24 samples of overlap between adjoining segments.

  • Discard the final, shorter segment.

  • Window each segment with a flat-top window.

  • Evaluate the discrete Fourier transform of each segment at NDFT=895 points, noting that it is an odd number.

M = 73;
L = 24;
g = flattopwin(M);
Ndft = 895;
neven = ~mod(Ndft,2);

[stwo,f,t] = spectrogram(x,g,L,Ndft,fs,"twosided");

Use the spectrogram function with no output arguments to plot the two-sided spectrogram.

spectrogram(x,g,L,Ndft,fs,"twosided","power","yaxis");

Figure contains an axes object. The axes object contains an object of type image.

Compute the two-sided spectrogram using the definition. Divide the signal into M-sample segments with L samples of overlap between adjoining segments. Window each segment and compute its discrete Fourier transform at NDFT points.

[segs,~] = buffer(1:length(x),M,L,"nodelay");

Xtwo = fft(x(segs).*g,Ndft);

Compute the time and frequency ranges.

  • To find the time values, divide the time vector into overlapping segments. The time values are the midpoints of the segments, with each segment treated as an interval open at the lower end.

  • To find the frequency values, specify a Nyquist interval closed at zero frequency and open at the upper end.

tbuf = tx(segs);
ttwo = mean(tbuf(2:end,:));

ftwo = 0:fs/Ndft:fs*(1-1/Ndft);

Compare the outputs of spectrogram to the definitions. Use the waterplot function to display the spectrograms.

diffs = [max(max(abs(stwo-Xtwo))) max(abs(f-ftwo')) max(abs(t-ttwo))]
diffs = 1×3
10-12 ×

         0    0.2274    0.0002

subplot(2,1,1)
waterplot(Xtwo,ftwo,ttwo)
title("Two-Sided, Definition")
subplot(2,1,2)
waterplot(stwo,f,t)
title("Two-Sided, spectrogram Function")

Figure contains 2 axes objects. Axes object 1 with title Two-Sided, Definition contains an object of type patch. Axes object 2 with title Two-Sided, spectrogram Function contains an object of type patch.

Centered Spectrogram

Compute the centered spectrogram of the signal.

  • Use the same time values that you used for the two-sided STFT.

  • Use the fftshift function to shift the zero-frequency component of the STFT to the center of the spectrum.

  • For odd-valued NDFT, the frequency interval is open at both ends. For even-valued NDFT, the frequency interval is open at the lower end and closed at the upper end.

Compare the outputs and display the spectrograms.

tcen = ttwo;

if ~neven
    Xcen = fftshift(Xtwo,1);
    fcen = -fs/2*(1-1/Ndft):fs/Ndft:fs/2;
else
    Xcen = fftshift(circshift(Xtwo,-1),1);
    fcen = (-fs/2*(1-1/Ndft):fs/Ndft:fs/2)+fs/Ndft/2;
end

[scen,f,t] = spectrogram(x,g,L,Ndft,fs,"centered");

diffs = [max(max(abs(scen-Xcen))) max(abs(f-fcen')) max(abs(t-tcen))]
diffs = 1×3
10-12 ×

         0    0.2274    0.0002

clf
subplot(2,1,1)
waterplot(Xcen,fcen,tcen)
title("Centered, Definition")
subplot(2,1,2)
waterplot(scen,f,t)
title("Centered, spectrogram Function")

Figure contains 2 axes objects. Axes object 1 with title Centered, Definition contains an object of type patch. Axes object 2 with title Centered, spectrogram Function contains an object of type patch.

One-Sided Spectrogram

Compute the one-sided spectrogram of the signal.

  • Use the same time values that you used for the two-sided STFT.

  • For odd-valued NDFT, the one-sided STFT consists of the first (NDFT+1)/2 rows of the two-sided STFT. For even-valued NDFT, the one-sided STFT consists of the first NDFT/2+1 rows of the two-sided STFT.

  • For odd-valued NDFT, the frequency interval is closed at zero frequency and open at the Nyquist frequency. For even-valued NDFT, the frequency interval is closed at both ends.

Compare the outputs and display the spectrograms. For real-valued signals, the "onesided" argument is optional.

tone = ttwo;

if ~neven
    Xone = Xtwo(1:(Ndft+1)/2,:);
else
    Xone = Xtwo(1:Ndft/2+1,:);
end

fone = 0:fs/Ndft:fs/2;

[sone,f,t] = spectrogram(x,g,L,Ndft,fs);

diffs = [max(max(abs(sone-Xone))) max(abs(f-fone')) max(abs(t-tone))]
diffs = 1×3
10-12 ×

         0    0.1137    0.0002

clf
subplot(2,1,1)
waterplot(Xone,fone,tone)
title("One-Sided, Definition")
subplot(2,1,2)
waterplot(sone,f,t)
title("One-Sided, spectrogram Function")

Figure contains 2 axes objects. Axes object 1 with title One-Sided, Definition contains an object of type patch. Axes object 2 with title One-Sided, spectrogram Function contains an object of type patch.

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

The spectrogram function has a matrix containing either the power spectral density (PSD) or the power spectrum of each segment as the fourth output argument. The power spectrum is equal to the PSD multiplied by the equivalent noise bandwidth (ENBW) of the window.

Generate a signal that consists of a logarithmic chirp sampled at 1 kHz for 1 second. The chirp has an initial frequency of 400 Hz that decreases to 10 Hz by the end of the measurement.

fs = 1000;
tt = 0:1/fs:1-1/fs;
y = chirp(tt,400,tt(end),10,"logarithmic");

Segment PSDs and Power Spectra with Sample Rate

Divide the signal into 102-sample segments and window each segment with a Hann window. Specify 12 samples of overlap between adjoining segments and 1024 DFT points.

M = 102;
g = hann(M);
L = 12;
Ndft = 1024;

Compute the spectrogram of the signal with the default PSD spectrum type. Output the STFT and the array of segment power spectral densities.

[s,f,t,p] = spectrogram(y,g,L,Ndft,fs);

Repeat the computation with the spectrum type specified as "power". Output the STFT and the array of segment power spectra.

[r,~,~,q] = spectrogram(y,g,L,Ndft,fs,"power");

Verify that the spectrogram is the same in both cases. Plot the spectrogram using a logarithmic scale for the frequency.

max(max(abs(s).^2-abs(r).^2))
ans = 0
waterfall(f,t,abs(s)'.^2)
set(gca,XScale="log",...
    XDir="reverse",View=[30 50])

Figure contains an axes object. The axes object contains an object of type patch.

Verify that the power spectra are equal to the power spectral densities multiplied by the ENBW of the window.

max(max(abs(q-p*enbw(g,fs))))
ans = 1.1102e-16

Verify that the matrix of segment power spectra is proportional to the spectrogram. The proportionality factor is the square of the sum of the window elements.

max(max(abs(s).^2-q*sum(g)^2))
ans = 3.4694e-18

Segment PSDs and Power Spectra with Normalized Frequencies

Repeat the computation, but now work in normalized frequencies. The results are the same when you specify the sample rate as 2π.

[~,~,~,pn] = spectrogram(y,g,L,Ndft);
[~,~,~,qn] = spectrogram(y,g,L,Ndft,"power");

max(max(abs(qn-pn*enbw(g,2*pi))))
ans = 1.1102e-16

Load an audio signal that contains two decreasing chirps and a wideband splatter sound. Compute the short-time Fourier transform. Divide the waveform into 400-sample segments with 300-sample overlap. Plot the spectrogram.

load splat

% To hear, type soundsc(y,Fs)

sg = 400;
ov = 300;

spectrogram(y,sg,ov,[],Fs,"yaxis")
colormap bone

Figure contains an axes object. The axes object contains an object of type image.

Use the spectrogram function to output the power spectral density (PSD) of the signal.

[s,f,t,p] = spectrogram(y,sg,ov,[],Fs);

Track the two chirps using the medfreq function. To find the stronger, low-frequency chirp, restrict the search to frequencies above 100 Hz and to times before the start of the wideband sound.

f1 = f > 100;
t1 = t < 0.75;

m1 = medfreq(p(f1,t1),f(f1));

To find the faint high-frequency chirp, restrict the search to frequencies above 2500 Hz and to times between 0.3 seconds and 0.65 seconds.

f2 = f > 2500;
t2 = t > 0.3 & t < 0.65;

m2 = medfreq(p(f2,t2),f(f2));

Overlay the result on the spectrogram. Divide the frequency values by 1000 to express them in kHz.

hold on
plot(t(t1),m1/1000,LineWidth=4)
plot(t(t2),m2/1000,LineWidth=4)
hold off

Figure contains an axes object. The axes object contains 3 objects of type image, line.

Generate two seconds of a signal sampled at 10 kHz. Specify the instantaneous frequency of the signal as a triangular function of time.

fs = 10e3;
t = 0:1/fs:2;
x1 = vco(sawtooth(2*pi*t,0.5),[0.1 0.4]*fs,fs);

Compute and plot the spectrogram of the signal. Use a Kaiser window of length 256 and shape parameter β=5. Specify 220 samples of section-to-section overlap and 512 DFT points. Plot the frequency on the y-axis. Use the default colormap and view.

spectrogram(x1,kaiser(256,5),220,512,fs,'yaxis')

Figure contains an axes object. The axes object contains an object of type image.

Change the view to display the spectrogram as a waterfall plot. Set the colormap to bone.

view(-45,65)
colormap bone

Figure contains an axes object. The axes object contains an object of type surface.

Input Arguments

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Input signal, specified as a row or column vector.

Example: cos(pi/4*(0:159))+randn(1,160) specifies a sinusoid embedded in white Gaussian noise.

Data Types: single | double
Complex Number Support: Yes

Window, specified as an integer or as a row or column vector. Use window to divide the signal into segments:

  • If window is an integer, then spectrogram divides x into segments of length window and windows each segment with a Hamming window of that length.

  • If window is a vector, then spectrogram divides x into segments of the same length as the vector and windows each segment using window.

If the length of x cannot be divided exactly into an integer number of segments with noverlap overlapping samples, then x is truncated accordingly.

If you specify window as empty, then spectrogram uses a Hamming window such that x is divided into eight segments with noverlap overlapping samples.

For a list of available windows, see Windows.

Example: hann(N+1) and (1-cos(2*pi*(0:N)'/N))/2 both specify a Hann window of length N + 1.

Number of overlapped samples, specified as a positive integer.

  • If window is scalar, then noverlap must be smaller than window.

  • If window is a vector, then noverlap must be smaller than the length of window.

If you specify noverlap as empty, then spectrogram uses a number that produces 50% overlap between segments. If the segment length is unspecified, the function sets noverlap to Nx/4.5⌋, where Nx is the length of the input signal and the ⌊⌋ symbols denote the floor function.

Number of DFT points, specified as a positive integer scalar. If you specify nfft as empty, then spectrogram sets the parameter to max(256,2p), where p = ⌈log2 Nw, the ⌈⌉ symbols denote the ceiling function, and

  • Nw = window if window is a scalar.

  • Nw = length(window) if window is a vector.

Normalized frequencies, specified as a vector. w must have at least two elements, because otherwise the function interprets it as nfft. Normalized frequencies are in rad/sample.

Example: pi./[2 4]

Cyclical frequencies, specified as a vector. f must have at least two elements, because otherwise the function interprets it as nfft. The units of f are specified by the sample rate, fs.

Sample rate, specified as a positive scalar. The sample rate is the number of samples per unit time. If the unit of time is seconds, then the sample rate is in Hz.

Frequency range for the PSD estimate, specified as "onesided", "twosided", or "centered". For real-valued signals, the default is "onesided". For complex-valued signals, the default is "twosided", and specifying "onesided" results in an error.

  • "onesided" — returns the one-sided spectrogram of a real input signal. If nfft is even, then ps has nfft/2 + 1 rows and is computed over the interval [0, π] rad/sample. If nfft is odd, then ps has (nfft + 1)/2 rows and the interval is [0, π) rad/sample. If you specify fs, then the intervals are respectively [0, fs/2] cycles/unit time and [0, fs/2) cycles/unit time.

  • "twosided" — returns the two-sided spectrogram of a real or complex-valued signal. ps has nfft rows and is computed over the interval [0, 2π) rad/sample. If you specify fs, then the interval is [0, fs) cycles/unit time.

  • "centered" — returns the centered two-sided spectrogram of a real or complex-valued signal. ps has nfft rows. If nfft is even, then ps is computed over the interval (–π, π] rad/sample. If nfft is odd, then ps is computed over (–π, π) rad/sample. If you specify fs, then the intervals are respectively (–fs/2, fs/2] cycles/unit time and (–fs/2, fs/2) cycles/unit time.

Power spectrum scaling, specified as "psd" or "power".

  • Omitting spectrumtype, or specifying "psd", returns the power spectral density.

  • Specifying "power" scales each estimate of the PSD by the equivalent noise bandwidth of the window. The result is an estimate of the power at each frequency. If the "reassigned" option is on, the function integrates the PSD over the width of each frequency bin before reassigning.

Frequency display axis, specified as "xaxis" or "yaxis".

  • "xaxis" — displays frequency on the x-axis and time on the y-axis.

  • "yaxis" — displays frequency on the y-axis and time on the x-axis.

This argument is ignored if you call spectrogram with output arguments.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: spectrogram(x,100,OutputTimeDimension="downrows") divides x into segments of length 100 and windows each segment with a Hamming window of that length The output of the spectrogram has time dimension down the rows.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: spectrogram(x,100,'OutputTimeDimension','downrows') divides x into segments of length 100 and windows each segment with a Hamming window of that length The output of the spectrogram has time dimension down the rows.

Threshold, specified as a real scalar expressed in decibels. spectrogram sets to zero those elements of s such that 10 log10(s) ≤ thresh.

Output time dimension, specified as "acrosscolumns" or "downrows". Set this value to "downrows", if you want the time dimension of s, ps, fc, and tc down the rows and the frequency dimension along the columns. Set this value to "acrosscolumns", if you want the time dimension of s, ps, fc, and tc across the columns and frequency dimension along the rows. This input is ignored if the function is called without output arguments.

Output Arguments

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Short-time Fourier transform, returned as a matrix. Time increases across the columns of s and frequency increases down the rows, starting from zero.

  • If x is a signal of length Nx, then s has k columns, where

    • k = ⌊(Nxnoverlap)/(windownoverlap)⌋ if window is a scalar.

    • k = ⌊(Nxnoverlap)/(length(window)noverlap)⌋ if window is a vector.

  • If x is real and nfft is even, then s has (nfft/2 + 1) rows.

  • If x is real and nfft is odd, then s has (nfft + 1)/2 rows.

  • If x is complex-valued, then s has nfft rows.

Note

When freqrange is set to "onesided", spectrogram outputs the s values in the positive Nyquist range and does not conserve the total power.

s is not affected by the "reassigned" option.

Normalized frequencies, returned as a vector. w has a length equal to the number of rows of s.

Time instants, returned as a vector. The time values in t correspond to the midpoint of each segment.

Cyclical frequencies, returned as a vector. f has a length equal to the number of rows of s.

Power spectral density (PSD) or power spectrum, returned as a matrix.

  • If x is real and freqrange is left unspecified or set to "onesided", then ps contains the one-sided modified periodogram estimate of the PSD or power spectrum of each segment. The function multiplies the power by 2 at all frequencies except 0 and the Nyquist frequency to conserve the total power.

  • If x is complex-valued or if freqrange is set to "twosided" or "centered", then ps contains the two-sided modified periodogram estimate of the PSD or power spectrum of each segment.

  • If you specify a vector of normalized frequencies in w or a vector of cyclical frequencies in f, then ps contains the modified periodogram estimate of the PSD or power spectrum of each segment evaluated at the input frequencies.

Center-of-energy frequencies and times, returned as matrices of the same size as the short-time Fourier transform. If you do not specify a sample rate, then the elements of fc are returned as normalized frequencies.

More About

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Short-Time Fourier Transform

The short-time Fourier transform (STFT) is used to analyze how the frequency content of a nonstationary signal changes over time. The magnitude squared of the STFT is known as the spectrogram time-frequency representation of the signal. For more information about the spectrogram and how to compute it using Signal Processing Toolbox™ functions, see Spectrogram Computation with Signal Processing Toolbox.

The STFT of a signal is computed by sliding an analysis window g(n) of length M over the signal and calculating the discrete Fourier transform (DFT) of each segment of windowed data. The window hops over the original signal at intervals of R samples, equivalent to L = MR samples of overlap between adjoining segments. Most window functions taper off at the edges to avoid spectral ringing. The DFT of each windowed segment is added to a complex-valued matrix that contains the magnitude and phase for each point in time and frequency. The STFT matrix has

k=NxLML

columns, where Nx is the length of the signal x(n) and the ⌊⌋ symbols denote the floor function. The number of rows in the matrix equals NDFT, the number of DFT points, for centered and two-sided transforms and an odd number close to NDFT/2 for one-sided transforms of real-valued signals.

The mth column of the STFT matrix X(f)=[X1(f)X2(f)X3(f)Xk(f)] contains the DFT of the windowed data centered about time mR:

Xm(f)=n=x(n)g(nmR)ej2πfn.

  • The short-time Fourier transform is invertible. The inversion process overlap-adds the windowed segments to compensate for the signal attenuation at the window edges. For more information, see Inverse Short-Time Fourier Transform.

  • The istft function inverts the STFT of a signal.

  • Under a specific set of circumstances it is possible to achieve "perfect reconstruction" of a signal. For more information, see Perfect Reconstruction.

  • The stftmag2sig returns an estimate of a signal reconstructed from the magnitude of its STFT.

Tips

If a short-time Fourier transform has zeros, its conversion to decibels results in negative infinities that cannot be plotted. To avoid this potential difficulty, spectrogram adds eps to the short-time Fourier transform when you call it with no output arguments.

References

[1] Boashash, Boualem, ed. Time Frequency Signal Analysis and Processing: A Comprehensive Reference. Second edition. EURASIP and Academic Press Series in Signal and Image Processing. Amsterdam and Boston: Academic Press, 2016.

[2] Chassande-Motin, Éric, François Auger, and Patrick Flandrin. "Reassignment." In Time-Frequency Analysis: Concepts and Methods. Edited by Franz Hlawatsch and François Auger. London: ISTE/John Wiley and Sons, 2008.

[3] Fulop, Sean A., and Kelly Fitz. "Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram, with applications." Journal of the Acoustical Society of America. Vol. 119, January 2006, pp. 360–371.

[4] Oppenheim, Alan V., and Ronald W. Schafer, with John R. Buck. Discrete-Time Signal Processing. Second edition. Upper Saddle River, NJ: Prentice Hall, 1999.

[5] Rabiner, Lawrence R., and Ronald W. Schafer. Digital Processing of Speech Signals. Englewood Cliffs, NJ: Prentice-Hall, 1978.

Extended Capabilities

Version History

Introduced before R2006a