Islamic university of Gaza Faculty of engineering Electrical engineering dept. Submitted to: Dr.Hatem Alaidy Submitted by: Ola Hajjaj2003-3005 Tahleel.

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

Contents

History

The Fourier Transform

Why STFT

Formula of STFT

Windows definition

STFT windows

Resolution concept

Comparisons

Inverse of STFT

Application for STFT

Conclusion

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.

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)

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.

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.

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.

Windows

-real and symmetric .

-Function with zero-valued outside of some chosen interval .

Definition

Windows Properties

Trade-off of time versus frequency resolution.

Detectability of sinusoidal components.

Zero phase window.

Hanning window Gaussian windows

W(t)

Windows of

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.

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.

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

Narrow window

Narrowband and Wideband Transforms.

good time resolution, poor frequency resolution.

Wide window

good frequency resolution, poor time resolution.

Spectrogram

Resolution Explanation

The Gaussian window function in the form:

w(t)=exp(-a*(t^2)/2);

Range of freq. Separated peaks in

time

Case 1:

Case 2:

Much better resolution Not se

parat

ed p

eaks

Case 3:

High frequency resolution Low

tim

e re

solu

tion

Inverse of

Time-Frequency Trade-off

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

Application for

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:

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 .

Thank you for listening.