Spectrogram using short-time Fourier transform - MATLAB spectrogram.pdf

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2/5/2016 Spectrogram using short- time Four ier transform - MATLAB spectrogram

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Spectrogram using short-time Fourier transform

s = spectrogram(x) example

s = spectrogram(x,window)

s = spectrogram(x,window,noverlap) example

s = spectrogram(x,window,noverlap,nfft) example

[s,w,t] = spectrogram( ___ )

[s,f,t] = spectrogram( ___ ,fs) example

[s,w,t] = spectrogram(x,window,noverlap,w)

[s,f,t] = spectrogram(x,window,noverlap,f,fs)

[ ___ ,ps] = spectrogram( ___ )

[ ___ ] = spectrogram( ___ ,'reassigned') example

[ ___ ,ps,fc,tc] = spectrogram( ___ ) example

[ ___ ] = spectrogram( ___ ,freqrange)

[ ___ ] = spectrogram( ___ ,spectrumtype)

[ ___ ] = spectrogram( ___ ,'MinThreshold',thresh) example

spectrogram( ___ ) example

spectrogram( ___ ,freqloc) example

example

example

example

spectrogram

Syntax

Description

s = spectrogram(x) returns the short-time Fourier transform of the input signal, x. Each columnof s contains an estimate of the short-term, time-localized frequency content of x.

s = spectrogram(x,window) uses window to divide the signal into sections and performwindowing.

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

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

http://www.mathworks.com/help/signal/ref/spectrogram.html#bultpz_-2http://www.mathworks.com/help/signal/ref/spectrogram.html#bupm9cjhttp://www.mathworks.com/help/signal/ref/spectrogram.html#buvvogkhttp://www.mathworks.com/help/signal/ref/spectrogram.html#bumgty_http://www.mathworks.com/help/signal/ref/spectrogram.html#bupbguuhttp://www.mathworks.com/help/signal/ref/spectrogram.html#buvf01yhttp://www.mathworks.com/help/signal/ref/spectrogram.html#buu60k7http://www.mathworks.com/help/signal/ref/spectrogram.html#buu60jahttp://www.mathworks.com/help/signal/ref/spectrogram.html#bupbguqhttp://www.mathworks.com/help/signal/ref/spectrogram.html#bultpz_-2http://www.mathworks.com/help/signal/ref/spectrogram.html#bupm9cjhttp://www.mathworks.com/help/signal/ref/spectrogram.html#buvvogk


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

[s,f,t] = spectrogram( ___ ,fs) returns a vector of cyclical frequencies, f, expressed in termsof the sample rate, fs.

[s,w,t] = spectrogram(x,window,noverlap,w) returns the spectrogram at the normalizedfrequencies specified in w.

[s,f,t] = spectrogram(x,window,noverlap,f,fs) returns the spectrogram at the cyclicalfrequencies specified in f.

[ ___ ,ps] = spectrogram( ___ ) also returns a matrix, ps, containing an estimate of the power spectral density (PSD) or the power spectrum of each section.

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

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

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

[ ___ ] = spectrogram( ___ ,spectrumtype) returns PSD estimates if spectrumtype is specifiedas 'psd' and returns power spectrum estimates if spectrumtype is specified as 'power'.

[ ___ ] = spectrogram( ___ ,'MinThreshold',thresh) sets to zero those elements of ps suchthat 10 log (ps) ≤ thresh. Specify thresh in decibels.

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

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

Examples

Default Values of Spectrogram

Generate samples of a signal that consists of a sum of sinusoids. Thenormalized frequencies of the sinusoids are rad/sample and rad/sample. Thehigher 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')

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Repeat the computation.

Divide the signal into sections of length .

Window the sections using a Hamming window.

Specify 50% overlap between contiguous sections.

To compute the FFT, use points, where .

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 FFTlength as in the preceding step. Compute the short-time Fourier transform and verify that it gives the sameresult as the previous two procedures.


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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

Frequency Along x -Axis

Generate a quadratic chirp, x, sampled at 1 kHz for 2 seconds. The frequency of thechirp is 100 Hz initially and crosses 200 Hz at t = 1 s.

t = 0:0.001:2;

x = chirp(t,100,1,200,'quadratic');

Compute and display the spectrogram of x.

Divide the signal into sections of length 128, windowed with a Hamming window.

Specify 120 samples of overlap between adjoining sections.

Evaluate the spectrum at frequencies andtime bins.

spectrogram(x,128,120,128,1e3)


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Replace the Hamming window with a Blackman window. Decrease the overlap to 60 samples. Plot the timeaxis so that its values increase from top to bottom.

spectrogram(x,blackman(128),60,128,1e3)

ax = gca;

ax.YDir = 'reverse';


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Open This Example

Power Spectral Densities of Chirps

Compute and display the PSD of each segment of a quadratic chirp that starts at 100 Hzand crosses 200 Hz at t = 1 s. Specify a sample rate of 1 kHz, a segment length of 128samples, and an overlap of 120 samples. Use 128 DFT points and the default Hamming

window.

t = 0:0.001:2;

x = chirp(t,100,1,200,'quadratic');

spectrogram(x,128,120,128,1e3,'yaxis')

title('Quadratic Chirp')


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Compute and display the PSD of each segment of a linear chirp that starts at DC and crosses 150 Hz at t = 1s. Specify a sample rate of 1 kHz, a segment length of 256 samples, and an overlap of 250 samples. Use thedefault Hamming window and 256 DFT points.

t = 0:0.001:2;

x = chirp(t,0,1,150);

spectrogram(x,256,250,256,1e3,'yaxis')

title('Linear Chirp')


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Compute and display the PSD of each segment of a logarithmic chirp sampled at 1 kHz that starts at 20 Hzand crosses 60 Hz at t = 1 s. Specify a segment length of 256 samples and an overlap of 250 samples. Usethe default Hamming window and 256 DFT points.

t = 0:0.001:2;

x = chirp(t,20,1,60,'logarithmic');

spectrogram(x,256,250,[],1e3,'yaxis')

title('Logarithmic Chirp')


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Use a logarithmic scale for the frequency axis. The spectrogram becomes a line.

ax = gca;

ax.YScale = 'log';


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Open This Example

Spectrogram and Instantaneous Frequency

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

Generate a quadratic chirp sampled at 1 kHz for two seconds. Specify the chirp so that its frequency is initially100 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 spectrogramfunction. Divide the signal into sections of length 100, windowed with a Hamming window. Specify 80 samplesof overlap between adjoining sect ions and evaluate the spectrum at frequencies. Suppressthe default color bar.

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

view(‐77,72)

shading interp

colorbar off


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Track the chirp frequency by finding the maximum of the power spectral density at each of the time points. View the spectrogram as a two-dimensional graphic. Restore the

color bar.

[s,f,t,p] = spectrogram(y,100,80,100,Fs);

[q,nd] = max(10*log10(p));

hold on

plot3(t,f(nd),q,'r','linewidth',4)

hold off

colorbar

view(2)


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Reassigned Spectrogram of Quadratic Chirp

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

Fs = 1000;

t = 0:1/Fs:2;


Estimate the reassigned spectrogram of the signal.

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


Evaluate the spectrum at frequencies and timebins.

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


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Spectrogram with Threshold

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

Fs = 1000;

t = 0:1/Fs:2;


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 .


Evaluate the spectrum at frequencies and timebins.

Output the frequency and time of the center of gravity of each PSD estimate. Set to zero those elements of the PSD smaller than 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),'.')


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Spectrogram Reassignment and Thresholding

Generate a signal sampled at 1024 Hz for 2 seconds.

nSamp = 2048;Fs = 1024;

t = (0:nSamp‐1)'/Fs;

During the first second, the signal consists of a 400 Hz sinusoid and a concave quadratic chirp. Specify thechirp so that it is symmetric about the interval midpoint, starting and ending at a frequency of 250 Hz andattaining a minimum of 150 Hz.

t1 = t(1:nSamp/2);

x11 = sin(2*pi*400*t1);

x12 = chirp(t1‐t1(nSamp/4),150,nSamp/Fs,1750,'quadratic');x1 = x11+x12;

The rest of the signal consists of two linear chirps of decreasing frequency. One chirp has an initial frequencyof 250 Hz that decreases to 100 Hz. The other chirp has an initial frequency of 400 Hz that decreases to 250Hz.


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t2 = t(nSamp/2+1:nSamp);

x21 = chirp(t2,400,nSamp/Fs,100);

x22 = chirp(t2,550,nSamp/Fs,250);

x2 = x21+x22;

Add white Gaussian noise to the signal. Specify a s ignal-to-noise ratio of 20 dB. Reset the random number generator for reproducible results.

SNR = 20;

rng('default')

sig = [x1;x2];

sig = sig + randn(size(sig))*std(sig)/db2mag(SNR);

Compute and plot the spectrogram of the signal. Specify a Kaiser window of length 63 with a shape parameter , 10 fewer samples of overlap between adjacent segments, and an FFT length of 256.

nwin = 63;

wind = kaiser(nwin,17);nlap = nwin‐10;

nfft = 256;

spectrogram(sig,wind,nlap,nfft,Fs,'yaxis')

Threshold the spectrogram so that any elements with values smaller than the SNR are set to zero.

spectrogram(sig,wind,nlap,nfft,Fs,'MinThreshold',‐SNR,'yaxis')


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Reassign each PSD estimate to the location of its center of energy.

spectrogram(sig,wind,nlap,nfft,Fs,'reassign','yaxis')


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Open This Example

Track Chirps in Audio Signal

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-samplesegments 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

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 searchto frequencies above 100 Hz and to times before the start of the wideband sound.


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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 between0.3 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

3D Spectrogram Visualization

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


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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 . 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')

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

colormap bone

view(‐45,65)


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Related Examples

Formant Estimation with LPC Coefficients

Input Arguments

x — Input signal

vector

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 — Windowinteger | vector | []

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

If window is an integer, then spectrogram divides x into sections of length window and windows eachsection with a Hamming window of that length.

If window is a vector, then spectrogram divides x into sections of the same length as the vector andwindows each section using window.

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

http://www.mathworks.com/help/signal/ug/formant-estimation-with-lpc-coefficients.html


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If you specify window as empty, then spectrogram uses a Hamming window such that x is divided into eightsections 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.

Data Types: single | double

noverlap — Number of overlapped samplespositive integer | []

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 betweensections. If the section length is unspecified, the function sets noverlap to ⌊N /4.5⌋, where N is the length of the input signal.

Data Types: double | single

nfft — Number of DFT points

positive integer scalar | []

Number of DFT points, specified as a positive integer scalar. If you specify nfft as empty, then spectrogramsets the parameter to max(256,2 ), where p = ⌈log N ⌉ for an input signal of length N .

Data Types: single | double

w — Normalized frequencies

vector

Normalized frequencies, specified as a vector. w must have at least two elements. Normalized frequencies arein rad/sample.

Example: pi./[2 4]


f — Cyclical frequencies

vector

Cyclical frequencies, specified as a vector. f must have at least two elements. The units of f are specified bythe sample rate, fs.


fs — Sample rate

1 Hz (default) | positive scalar

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

x x

p2 x x

http://www.mathworks.com/help/signal/ug/windows.html


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freqrange — Frequency range for PSD estimate

'onesided' | 'twosided' | 'centered'

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', andspecifying 'onesided' results in an error.

'onesided' — returns the one-sided spectrogram of a real input signal. If nfft is even, then ps has lengthnfft/2 + 1 and is computed over the interval [0, π ] rad/sample. If nfft is odd, then ps has length(nfft + 1)/2 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 signal. ps has length nfft and iscomputed over the interval [0, 2π ) rad/sample. If you specify fs, then the interval is [0, fs) cycles/unittime.

'centered' — returns the centered two-sided spectrogram for a real or complex signal. ps has lengthnfft. If nfft is even, then ps is computed over the interval (–π , π ] rad/sample. If nfft is odd, then ps iscomputed 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.

Data Types: char

spectrumtype — Power spectrum scaling

'psd' (default) | 'power'

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 functionintegrates the PSD over the width of each frequency bin before reassigning.

Data Types: char

thresh — Threshold

‐Inf (default) | real scalar

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

freqloc — Frequency display axis'xaxis' (default) | 'yaxis'

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.

Data Types: char

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s — Short-time Fourier transform

matrix

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

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

k = ⌊(N – noverlap)/(window – noverlap)⌋ if window is a scalar

k = ⌊(N – noverlap)/(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, then s has nfft rows.


w — Normalized frequencies

vector

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


t — Time instants

vector

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


f — Cyclical frequencies

vector

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


ps — Power spectral density or power spectrum

matrix

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

If x is real, then ps contains the one-sided modified periodogram estimate of the PSD or power spectrum of each section.

If x is complex, or if you specify a vector of frequencies, then ps contains the two-sided modifiedperiodogram estimate of the PSD or power spectrum of each section.


fc,tc — Center-of-energy frequencies and times

x

x

x

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matrices

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 normalizedfrequencies.

More About

Tips

If a short-time Fourier transform has zeros, its conversion to decibels results in negative infinities that cannotbe plotted. To avoid this potential difficulty, spectrogram adds eps to the short-time Fourier transform whenyou call it with no output arguments.

References

[1] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing . 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.

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

[3] 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.

[4] 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, January2006, pp. 360–371.

See Also

goertzel | periodogram | pwelch

Introduced before R2006a
http://www.mathworks.com/help/signal/ref/pwelch.htmlhttp://www.mathworks.com/help/signal/ref/periodogram.htmlhttp://www.mathworks.com/help/signal/ref/goertzel.html

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