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Islamic university of Gaza Faculty of engineering Electrical engineering dept. Submitted to: Dr.Hatem Alaidy Submitted by: Ola Hajjaj2003-3005 Tahleel Abu seedo 2003-4240 Short T ime F ourierT 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 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.

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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 -2jft dt……..(1) x(t)= - X(f). e -2jft df…...(2)

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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.

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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. 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.

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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 separated peaks

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Case 3: High frequency resolution Low time resolution

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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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