Digital Signal Processing The Short-Time Fourier...

DSP: The Short-Time Fourier Transform (STFT)

Digital Signal ProcessingThe Short-Time Fourier Transform (STFT)

D. Richard Brown III

D. Richard Brown III 1 / 14


Signals with Changing Frequency Content: Motivation

For signals that have frequency content that is changing over time,e.g., music, speech, ..., taking the DFT of the whole signal usually doesn’tprovide much insight.

Example: Two second linear chirp

N = 16000;

n = 0:N-1;

x = cos(2*pi/80000*(n+1000).^2); % linear chirp

soundsc(x,8000) % listen to sound



Example Continued: FFT of Whole Signal

plot(2*n/N,20*log10(abs(fft(x)))); % FFT of whole signal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−60

−40

−20

0

20

40

60

normalized frequency (ω/π

FF

T m

agnitude



Example Continued: FFTs of Smaller Chunks

plot(2*[0:127]/128,20*log10(abs(fft(x(1:128))))); % FFT near beginning

plot(2*[0:127]/128,20*log10(abs(fft(x(8001:8128))))); % FFT near middle

plot(2*[0:127]/128,20*log10(abs(fft(x(15001:15128))))); % FFT near end

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−5

0

5

10

15

20

25

30

35


FF

T m

ag

nitu

de

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−40

−30

−20

−10

0

10

20

30

40


FF

T m

agnitude

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−40

−30

−20

−10

0

10

20

30

40


FF

T m

agnitude



Short-Time Fourier Transform

Rather than analyzing the frequency content of the whole signal, we cananalyze the frequency content of smaller snapshots. The STFT is definedas

X[n, λ) =

∞∑m=−∞

x[n+m]w[m]e−jλm

where n ∈ Z is a time index and λ ∈ R is a normalized frequency index.

Remarks:

1. Implicit in this definition is a window function w[n] of length L.

2. The STFT results in a family of DTFTs indexed by n.

3. In practice, we don’t usually compute X[n, λ) for all n = 0, 1, . . .since the frequency content of the signal does not change much fromsample to sample. Instead, we typically pick a “block shift” integerR ≤ L and compute the STFT for values of n = 0, R, 2R, . . . .



Short-Time Fourier Transform

n

x[n]

n

w[n]

L=81

n

x[n]w[n]

n

x[n+8]w[n]

n

x[n+16]w[n]

etc.

R=8

DTFT

DTFT

DTFT

X[0, λ)

X[8, λ)

X[16, λ)

λ

λ

λ



Short-Time Fourier Transform with the DFT/FFT

We can also use the DFT/FFT to compute the STFT as

X[n, k] =

L−1∑m=0

x[n+m]w[m]e−j2πkm/N .

n

x[n]

n

w[n]

L=81

n

x[n]w[n]

n

x[n+8]w[n]

n

x[n+16]w[n]

R=8

DFT

DFT

DFT

N=8 X[0, k]

X[8, k]

X[16, k]

k

k

k



Short-Time Fourier Transform Parameters

1. Window typeI Tradeoff between side lobe amplitude ASL and main lobe width ∆ML

2. Window length LI Larger L gives better frequency resolution (smaller ∆ML)I Smaller L gives less temporal averaging

3. Temporal block shift samples RI Usually R ≤ L and is related to L, e.g., R = L or R = L/2I Larger R result in less computationI Smaller R gives “smoother” results

4. FFT length NI Usually N ≥ L.I Larger N gives more frequency-domain samples of DTFT (better

location and amplitude of peaks)I Smaller N results in less computation.



Matlab Spectrogram Example

Matlab function spectrogram is useful for easily computing STFTs.

[s,f,t] = spectrogram(x,kaiser(512,2),256,1024,8000); % x = lin chirp

image(t,f,20*log10(abs(s))); set(gca,’ydir’,’normal’); colorbar;



Matlab Spectrogram Example (decrease R to 1)





Matlab Spectrogram Example (rectangular window)

[s,f,t] = spectrogram(x,ones(512,1),511,1024,8000); % x = lin chirp




Matlab Spectrogram Example (longer window)





Speech Signals Example

fre

qu

en

cy (

kH

z)

ref−mic [quiet]

0 2 40

0.2

0.4

0.6

0.8

1

time (s)

fre

qu

en

cy (

kH

z)

ref−mic [high−noise]

0 2 40

0.2

0.4

0.6

0.8

1

phys−mic [quiet]

0 2 40

0.2

0.4

0.6

0.8

1

time (s)

phys−mic [high−noise]

0 2 40

0.2

0.4

0.6

0.8

1

TERC sensor [quiet]

0 2 40

0.2

0.4

0.6

0.8

1

time (s)

TERC sensor [high−noise]

0 2 40

0.2

0.4

0.6

0.8

1



“Aphex Face” Spectrogram (log scale frequency axis)

On Aphex Twin’s Windowlicker album, track 2, ∼ 5:27–5:37.D. Richard Brown III 14 / 14

Digital Signal Processing The Short-Time Fourier...

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