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Islamic university of Gaza
Faculty of engineering
Electrical engineering dept.
Submitted to:
Dr.Hatem Alaidy
Submitted by:
Ola Hajjaj 2003-3005
Tahleel Abu seedo 2003-4240
ShortT ime F ourier T ransform
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Contents
History
The Fourier Transform
Why STFT
Formula of STFT
Windows definition
STFT windows
Resolution concept
Comparisons
Inverse of STFT
Application for STFT
Conclusion
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History of19th century, J. Fourier, reach to the formula of periodic
function as an infinite sum of periodic complex exponential functions.
Many years after, non-periodic functions were generalized.
Then periodic & non-periodic discrete time signals were known.
In 1965, (FFT) was known.
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The Fourier Transform
DFT: used When fs>=2fm, and the transformed signal is symmetrical.
FT: decomposes a signal to complex exponential functions of different frequencies
FFT: to reduce the no. of multiplications in DFT.
STFT
X(f)=-∞∫ ∞ x(t).e-2j∏ft dt……..(1)
x(t)= -∞∫ ∞ X(f). e-2j∏ft df…...(2)
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WhyIt gives a suitable description for the local change in frequency content because the frequency component which defined by FT have infinite time support.
STFT provides a means of joint time-frequency analysis.
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Continue.
In STFT, the signal is divided into small enough segments.
For this purpose, a window function "w" is chosen. The width of this window must be equal to the segment of the signal.
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Formula of
x(t) is the signal itself,
w(t) is the window function, and
* is the complex conjugate
The STFT of the signal is the FT of the signal multiplied by a window function.
STFTx(w)(,f)=t∫[x(t).w*(t- ).e-2j∏ft dt……………(3)
Note That:
The STFT of a signal x (n) is a function of two variables: time and frequency.
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Windows
-real and symmetric .
-Function with zero-valued outside of some chosen interval .
Definition
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Windows Properties
Trade-off of time versus frequency resolution.
Detectability of sinusoidal components.
Zero phase window.
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Hanning window Gaussian windows
W(t)
Windows of
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Transforming steps in
This window function is located at the beginning of the signal At (t=0).
The window function will overlap with the first T/2 seconds of the original signal
The window function and the signal are then multiplied.
Taking the FT of the product.
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The window would be shifted by t1 to a new location multiplying with the signal.
Repeat from step 3 Until the end of the signal.
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Window & Resolution
STFT has a fixed resolution.
The width of the windowing function relates to the how the signal is represented.
It determines whether there is good frequency resolution or good time resolution
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Narrow window
Narrowband and Wideband Transforms.
good time resolution, poor frequency resolution.
Wide window
good frequency resolution, poor time resolution.
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Spectrogram
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Resolution Explanation
The Gaussian window function in the form:
w(t)=exp(-a*(t^2)/2);
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Range of freq. Separated peaks in
time
Case 1:
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Case 2:
Much better resolution Not se
parat
ed p
eaks
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Case 3:
High frequency resolution Low
tim
e re
solu
tion
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Inverse of
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Time-Frequency Trade-off
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Comparisons
The signal multiplied by a window function.
Transform is a function of both time and frequency
There is resolution problem in the frequency domain
Window is of finite length
Its window is exp{jwt} function, from minus infinity to plus infinity
no resolution problems in freq. domain
One domain only
One window
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Application for
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The problem of
No exact time-frequency representation of a signal
Resolution problem, time intervals in which certain band of frequencies exist.
Wavelet transform (or multi resolution analysis) high-frequency gives good time resolution for events, and good frequency resolution for low-frequency events, which is the type of analysis best suited for many real signals.
The Solution:
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Conclusion
STFT is a Fourier related transform & it is a Function of two variable (time & frequency).
Used to determined the freq. and phase content of local section of a signal over time.
It deals with two windows (hanning & Gaussian).
There is a relation between window and resolution .
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Thank you for listening.