Islamic university of Gaza Faculty of engineering Electrical engineering dept. Submitted to: Dr.Hatem Alaidy Submitted by: Ola Hajjaj Tahleel Abu seedo Short T ime F ourierT 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 of 19th 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 -2jft dt……..(1) x(t)= - X(f). e -2jft df…...(2)

Why It 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. STFT x (w) (,f)= t [x(t).w*(t- ).e -2jft 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 separated peaks

Case 3: High frequency resolution Low time resolution

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.

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