An Introduction to Short-Time Fourier Transform...

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An Introduction toShort-Time Fourier Transform (STFT)

Advanced Structural Dynamics

M Ahmadizadeh, PhD, PE

Contents

• Scope and Goals

• Fourier Transform Review

– Advantages of Fourier Transform

– Limitations of Fourier Transform

• Short Time Fourier Transform

– Concept

– Formulation

– Examples

• Use of MATLAB

– Applications

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Scope and Goals

• To expand the capabilities of Fourier transform for time-varying signals

• In addition to showing the frequency content of the signals, it is also desirable to have an idea of when each frequency content is dominant

• Describing complicated signals with fewer number of frequencies

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Fourier Transform

• Fourier analysis expands signals or functions in terms of sinusoids (or complex exponentials).

• It reveals all frequency components present in a signal.

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Inverse DFT

DFT

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Fourier Transform

• Example: 5 Hz Signal

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0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

15 Hz

Time (s)

Sig

nal

0 20 40 60 80 1000

10

20A

mplit

ude

Output

0 20 40 60 80 100-5

0

5

Frequency (Hz)

Phase

0 20 40 60 80 1000

10

20

Am

plit

ude

Output

0 20 40 60 80 1000

5

10

Frequency (Hz)

Phase

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

115 Hz

Time (s)

Sig

nal

Fourier Transform


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0 20 40 60 80 1000

10

20A

mplit

ude

Output

0 20 40 60 80 1000

5

10

Frequency (Hz)

Phase

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

130 Hz

Time (s)

Sig

nal

Fourier Transform


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0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

33 Frequencies

Time (s)

Sig

nal

Fourier Transform

• Example: Signal Containing 5, 15 and 30 Hz Components

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0 20 40 60 80 1000

5

10

15

Am

plit

ude

Output

0 20 40 60 80 1000

5

10

Frequency (Hz)

Phase

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Fourier Transform - Limitations

• Cannot provide information about both time and frequency – i.e. cannot provide simultaneous time and frequency localization

– Not suitable for analyzing time-variant, non-stationary signals

• As an example, it cannot be used to analyze the signals obtained from nonlinear or damaged structures, since they may contain varying frequency content at different times

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• Signals: Stationary vs Non-stationary

– Stationary signals have properties (e.g frequency content) that does not change over time

– Non-stationary signals may have different frequencies at different times

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0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

33 Frequencies

Time (s)

Sig

nal

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

13 Frequencies

Time (s)

Sig

nal

Sum of 3 signals, present at all times

Combination of 3 signals occurring at different times

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0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

13 Frequencies

Time (s)

Sig

nal


• 5, 15 and 30 Hz Components at Different Times

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0 20 40 60 80 1000

2

4

Am

plit

ude

Output

0 20 40 60 80 100-100

-50

0

50

Frequency (Hz)

Phase

Excellent frequency localization (shows all existing frequencies), but no time localization


• Not appropriate for representing discontinuities or sharp corners

– Requires a large number of Fourier components to represent discontinuities).

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• El Centro Earthquake Record and its Fourier transform

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0 5 10 15 20 25 30 35-4

-3

-2

-1

0

1

2

3El Centro

Time

Accele

ration

0 5 10 15 20 25 300

1

2

3

Am

plit

ude

Output

0 5 10 15 20 25 30-500

0

500

Frequency (Hz)

Phase



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0 5 10 15 20 25 30 35-4

-3

-2

-1

0

1

2

3El Centro

Time

Acce

lera

tio

n

Included Signals: 1

0 5 10 15 20 25 30 35-1

-0.5

0

0.5

1x 10

-3 El Centro - Components: 1

Time

Acce

lera

tio

n

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0 5 10 15 20 25 30 35-4

-3

-2

-1

0

1

2

3El Centro

Time

Acce

lera

tio

n

Included Signals: 10

0 5 10 15 20 25 30 35-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15El Centro - Components: 10

TimeA

cce

lera

tio

n



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0 5 10 15 20 25 30 35-4

-3

-2

-1

0

1

2

3El Centro

Time

Acce

lera

tio

n


0 5 10 15 20 25 30 35-1.5

-1

-0.5

0

0.5

1

1.5El Centro - Components: 50

Time

Acce

lera

tio

n

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0 5 10 15 20 25 30 35-4

-3

-2

-1

0

1

2

3El Centro

Time

Acce

lera

tio

n


0 5 10 15 20 25 30 35-3

-2

-1

0

1

2

3El Centro - Components: 200

TimeA

cce

lera

tio

n



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0 5 10 15 20 25 30 35-4

-3

-2

-1

0

1

2

3El Centro

Time

Acce

lera

tio

n


0 5 10 15 20 25 30 35-4

-3

-2

-1

0

1

2


Time

Acce

lera

tio

n

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Short-Time Fourier Transform

• Basic Concept:

– Break up the signal in time domain to a number of signals of shorter duration, then transform each signal to frequency domain

• Requires fewer number of harmonics to regenerate the signal chunks

• Helps determine the time interval in which certain frequencies occur

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• Consider the first 3 seconds of the El Centro Record

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0 0.5 1 1.5 2 2.5 3 3.5-4

-3

-2

-1

0

1

2

3El Centro

Time

Accele

ration

0 5 10 15 20 25 300

2

4

Am

plit

ude

Output

0 5 10 15 20 25 30-50

0

50

100

Frequency (Hz)

Phase

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0 0.5 1 1.5 2 2.5 3 3.5-4

-3

-2

-1

0

1

2

3El Centro

Time

Accele

ration

0 0.5 1 1.5 2 2.5 3 3.5-3

-2

-1

0

1

2


Time

Accele

ration




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0 0.5 1 1.5 2 2.5 3 3.5-4

-3

-2

-1

0

1

2

3El Centro

Time

Accele

ration

0 0.5 1 1.5 2 2.5 3 3.5-3

-2

-1

0

1

2


Time

Accele

ration


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0 0.5 1 1.5 2 2.5 3 3.5-4

-3

-2

-1

0

1

2

3El Centro

Time

Accele

ration

0 0.5 1 1.5 2 2.5 3 3.5-4

-3

-2

-1

0

1

2


Time

Accele

ration



• Or consider the first chunk of non-stationary record

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0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

13 Frequencies

Time (s)

Sig

nal

Knowing the location of the considered chunk in time and its frequency content, we can get an idea of the time each frequency exists.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1Non-Stationary

Time (s)

Sig

nal


Need a local analysis scheme for a time-frequency representation (TFR).

Windowed F.T. or Short Time F.T. (STFT)

Segmenting the signal into narrow time intervals (i.e., narrow enough to be considered stationary).

Take the Fourier transform of each segment.

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• Steps:– Choose a window function of finite length

• A window function is a function that is multiplied by the signal to keep a certain portion of it

– Place the window on top of the signal at t=0

– Truncate the signal using this window

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• Steps (continued)

– Compute the FT of the truncated signal, save results.

• For each time location where the window is centered, we obtain a different FT

– Each FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency

information

– Incrementally slide the window to the right

– Repeat until window reaches the end of the signal

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• What is being Fourier-transformed:

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0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Hann Windowed Non-Stationary Signal

Time (s)

Sig

nal

0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Hann Windowed Non-Stationary Signal

Time (s)

Sig

nal

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1Hann Windowed Non-Stationary Signal

Time (s)

Sig

nal

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5


Time (s)

Sig

nal

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5


Time (s)

Sig

nal

Window Length: 0.2s

0.1s 0.5s 0.9s

0.3s 0.7s

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0 20 40 60 80 1000

0.5

1

1.5

Am

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Output

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20

Frequency (Hz)

Ph

ase

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plit

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Output

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4

Frequency (Hz)

Ph

ase


• Fourier transforms of windowed signal:

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0 20 40 60 80 1000

0.5

1

1.5

Am

plit

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Output

0 20 40 60 80 100-10

-5

0

5

Frequency (Hz)

Ph

ase

0 20 40 60 80 1000

0.5

1

Am

plit

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Output

0 20 40 60 80 100-300

-200

-100

0

Frequency (Hz)

Ph

ase

0 20 40 60 80 1000

0.5

1

Am

plit

ude

Output

0 20 40 60 80 1000

100

200

300

Frequency (Hz)

Ph

ase

0.1s 0.5s 0.9s

0.3s 0.7s


• Now we can plot the spectra next to each other to generate a surface:

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(Window) Time

Frequency

Amplitude

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• Formulation:

– Continuous STFT

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STFT ( ) , , ( ) i tx t X x t w t e dt

( )x t

w t Window function, commonly a Hann window or Gaussian window bell centered around zero

Time-domain signal to be transformed

Time (slow time; lower resolution than )

Frequency

t

,X A complex function representing the phase and magnitude of the signal over time and frequency (this is essentially the Fourier Transform of ) ( )x t w t

Ofte

n p

hase u

nw

rappin

g is

em

plo

yed a

long e

ither o

r both

the tim

e

axis

, τ, a

nd fre

quency a

xis

, ω, to

suppre

ss a

ny ju

mp d

iscontin

uity

of th

e p

hase re

sult o

f the S

TFT


• Formulation:

– Discrete STFT

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STFT , , ni t

n n n m

n

x m X m x w e

nx

m

nw Sequence of discretized window function

Sequence of discretized time-domain signal to be transformed

Time index

Frequency

,X m STFT of the time-domain sequence

Time discretized, frequency continuous; but if FFT is used, they both will be discrete.

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• Inverse STFT

– The original signal can be recovered from the transform by the Inverse STFT. The most widely accepted way of inverting the STFT is by using the overlap-add (OLA) method, which also allows for modifications to the STFT complex spectrum.

• Calculating the Inverse STFT:

– First, it is required that the window function must be scaled such that the area underneath the window function is unity:

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• Calculating the Inverse STFT (Continued):

– It follows that:

– Hence:

– The continuous Fourier transform is:

– Substituting from above:

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– Swap the integration order:

– Since the inverse Fourier transform is:

– Then the time-domain signal can be recovered:

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– Or:

– Alternatively, one can obtain windowed grain or waveletof x(t) as:

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• Time and Frequency Resolution:

– The STFT has a fixed resolution

• The width of the windowing function relates to how the signal is represented

• It determines whether there is good frequency resolution (frequency components close together can be separated) or good time resolution (the time at which frequencies change).

• A wide window (wideband transform) gives better frequency resolution but poor time resolution.

• A narrower window (narrowband transform) gives good time resolution but poor frequency resolution.

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• Time and Frequency Resolution (Continued):

– That is, we cannot have good resolutions in both time and frequency.

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Good time resolution, poor frequency resolution

Good frequency resolution, poor time resolution

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– To explain this limitation, note that in Fourier transform:

• To increase the frequency resolution of the window the frequency spacing of the coefficients (sequence in frequency domain) needs to be reduced.

– Decreasing Nyquist (max) frequency (and keeping N constant) will cause the window size to increase — since there are now fewer samples per unit time.

– The other alternative is to increase N, but this again causes the window size to increase.

– So any attempt to increase the frequency resolution causes a larger window size and therefore a reduction in time resolution—and vice-versa.

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– Best simultaneous resolution of both is reached with a Gaussian window function. This STFT with some modifications for multi-resolution becomes the Morletwavelet transform.

– Window should be narrow enough to make sure that the portion of the signal falling within the window is stationary.

– Very narrow windows do not offer good localization in the frequency domain.

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– Windowing Function infinitely long:

• STFT turns into FT, providing excellent frequency localization, but no time information.

– Windowing Function infinitely short:

• gives the time signal back, with a phase factor, providing excellent time localization but no frequency information.

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, ( ) i tX x t e dt X

, ( ) (t ) ( )i t iX x t e dt x e

( ) 1w t

( ) ( )w t t


• Heisenberg (Uncertainty) Principle:

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1

4t f

Time resolution: How well two spikes in time can be separated from each other in the transform domain.

Frequency resolution:How well two spectral components can be separated from each other in the transform domain.

𝜟𝒕 𝒂𝒏𝒅 𝜟𝒇 𝒄𝒂𝒏𝒏𝒐𝒕 𝒃𝒆𝒎𝒂𝒅𝒆 𝒂𝒓𝒃𝒊𝒕𝒓𝒂𝒓𝒊𝒍𝒚 𝒔𝒎𝒂𝒍𝒍 !

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• Heisenberg (Uncertainty) Principle (Continued):

– One cannot know the exact time-frequency representation of a signal.

– We cannot precisely know at what time instance a frequency component is located.

– We can only know what interval of frequencies are present in which time intervals.

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• Example:

– Chirp Signal (linear frequency variation 10-200 Hz)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1

-0.5

0

0.5

1

Time (sec)

x(t

)

500 points spaced at 0.002 s, variable window sizes

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• Example (Continued):

– Chirp Signal – Ordinary Fourier Transform

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0 50 100 150 200 2500

0.2

0.4

0.6

0.8

Am

plit

ude

Output

0 50 100 150 200 250-500

-400

-300

-200

-100

0

Frequency (Hz)

Phase



– Chirp Signal – Window Size: 128 points

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00.2

0.40.6

0.81

0

50

100

150

200

2500

5

10

15

20

Time (sec)

Window size = 128 Window type = Hann

Frequency (Hz)Time (sec)

Fre

qu

en

cy (

Hz)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

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00.2

0.40.6

0.81

0

50

100

150

200

2500

5

10

15

20

Time (sec)


Frequency (Hz)Time (sec)

Fre

qu

en

cy (

Hz)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250




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Time (sec)

Fre

qu

en

cy (

Hz)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

00.2

0.40.6

0.81

0

50

100

150

200

2500

5

10

15

20

Time (sec)


Frequency (Hz)




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• Use of MATLAB:

– The MATLAB command to perform STFT is spectrogram:

– Spectrogram inputs:

• x: input signal (signal to be transformed)

• window function (discrete window at the same input rate)

• window overlap (in terms of number of points)

• sampling points for FFT (number of points used in FFT)

• sampling frequency (rate of sampling for the input signal and the window function)

– Spectrogram outputs: Transformed signal, frequencies and times.

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


• Applications:

– Signal processing of any non-stationary signal (audio signals, earthquake excitations, structural responses to ambient vibrations, …)

– In structural dynamics, STFT can be used to:

• Determine the dominant modes of vibration (and their shapes and frequencies) at any time interval

• Health monitoring and damage detection through the study of dominant frequencies

– e.g. a reduction in frequency is generally indicative of damages leading to softer structures

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0 10 20 30 40-0.04

-0.02

0

0.02

0.04

Time, s

Dis

pla

cem

ent, m

-0.02 -0.01 0 0.01 0.02-2

-1

0

1

2x 10

5

Displacement, mF

orc

e, N

0 10 20 30 400

0.5

1

1.5

2

2.5x 10

4

Time, s

Energ

y, N

-m

Internal

Input

0 10 20 30 400

0.2

0.4

0.6

0.8

1

Time, s

Num

ber

of Itera

tions


• Applications:

– Damage Detection Example:

• 2-Story structure, natural frequencies 2.2 and 5.8 Hz

• Response simulated to El Centro record

• Linear Response

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• Applications:

– Damage Detection Example:

• 2-Story structure, natural frequencies 2.2 and 5.8 Hz

• Response simulated to El Centro record

• Nonlinear Response (by defining a yield displacement)

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0 10 20 30 40-0.04

-0.02

0

0.02

0.04

Time, s

Dis

pla

cem

ent, m

-0.02 -0.01 0 0.01 0.02-2

-1

0

1

2x 10

5

Displacement, m

Forc

e, N

0 10 20 30 400

0.5

1

1.5

2

2.5x 10

4

Time, s

Energ

y, N

-m

Internal

Input

0 10 20 30 400

10

20

30

Time, s

Num

ber

of Itera

tions

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• Applications:

– Damage Detection Example: Linear Response

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0 5 10 15 20 25 30 35-0.02

-0.01

0

0.01

0.02

Time (sec)

x(t

)

Time (sec)

Fre

qu

en

cy (

Hz)

Window size = 2048 Window type = Chebyshev

0 5 10 15 20 25 300

2

4

6

8

10


• Applications:

– Damage Detection Example: Nonlinear Response

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Time (sec)

Fre

qu

en

cy (

Hz)

Window size = 2048 Window type = Chebyshev

0 5 10 15 20 25 300

2

4

6

8

10

0 5 10 15 20 25 30 35-0.02

-0.01

0

0.01

0.02

Time (sec)

x(t)

An Introduction to Short-Time Fourier Transform...

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Transcript of An Introduction to Short-Time Fourier Transform...