Islamic university of Gaza

1. Islamic university of Gaza Faculty of engineering Electrical engineering dept. T ime Short F ourier T ransform Submitted to: Dr.Hatem Alaidy Submitted by: Ola Hajjaj 2003-3005 2003-4240 Tahleel Abu seedo

2. Resolution concept Contents History The Fourier Transform Why STFT Formula of STFT Windows definition STFT windows Comparisons Inverse of STFT Application for STFT Conclusion

3. 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.

4. The Fourier Transform FT: decomposes a signal to complex exponential functions of different frequencies X(f)=-∞∫ ∞ x(t).e-2j∏ft dt……..(1) x(t)= -∞∫ ∞ X(f). e-2j∏ft df…...(2) DFT: used When fs>=2fm, and the transformed signal is symmetrical. FFT: to reduce the no. of multiplications in DFT. STFT

5. 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.

6. 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.

7. Formula of STFTx(w)(,f)=t∫[x(t).w*(t- ).e-2j∏ft dt……………(3) x(t) is the signal itself, w(t) is the window function, and *is the complex conjugate Note That: The STFT of the signal is the FT of the signal multiplied by a window function. The STFT of a signal x (n) is a function of two variables: time and frequency.

8. Windows Definition -Function with zero-valued outside of some chosen interval . -real and symmetric .

9. Windows Properties Trade-off of time versus frequency resolution. Detectability of sinusoidal components. Zero phase window.

10. Windows of W(t) Hanning window Gaussian windows

11. 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.

12. 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.

13. 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

14. Narrowband and Wideband Transforms. good time resolution, poor frequency resolution. Narrow window good frequency resolution, poor time resolution. Wide window

15. Spectrogram

16. Resolution Explanation The Gaussian window function in the form: w(t)=exp(-a*(t^2)/2);

17. Case 1: Separated peaks in time Range of freq.

18. Case 2: Not separated peaks Much better resolution

19. Case 3: Low time resolution High frequency resolution

20. Inverse of

21. Time-Frequency Trade-off

23. Comparisons One window The signal multiplied by a window function. Transform is a function of both time and frequency One domain only There is resolution problem in the frequency domain no resolution problems in freq. domain Window is of finite length Its window is exp{jwt} function, from minus infinity to plus infinity

24. Application for

25. The problem of • No exact time-frequency representation of a signal • Resolution problem, time intervals in which certain band of frequencies exist. The Solution: 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.

26. 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 .

27. Thank you for listening.