Ch7-Short-Time_Fourier_Transform_Analysis_and_Synthesis

Ch7-Short-Time_Fourier_Transform_Analysis_and_Synthesis -...

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Unformatted text preview: Speech Processing Short-Time Fourier Transform Analysis and Synthesis February 13, 2012 Veton Kpuska 2 Short-Time Fourier Transform Analysis and Synthesis Minimum-Phase Synthesis Speech & Audio Signals are varying and can be considered stochastic signals that carry information. This necessitates short-time analysis since a single Fourier transform (FT) can not characterize changes in spectral content over time (i.e., time-varying formants and harmonics) Discrete-time short-time Fourier transform (STFT) consists of separate FT of the signal in the neighborhood of that instant. FT in the STFT analysis is replaced by the discrete FT (DFT) Resulting STFT is discrete in both time and frequency. Discrete STFT vs. Discrete-time STFT which is continuous in frequency. In linear Prediction and Homomorphic Processing, underlying model of the source/filter is assumed. This leads to: Model based analysis/synthesis, also note that Analysis methods presented implicitly both used short time analysis methods (to be presented). In Short-Time Analysis systems no such restrictions apply. February 13, 2012 Veton Kpuska 3 Short-Time Analysis (STFT) Two approaches of STFT are explored: 1. Fourier-transform & 2. Filterbank February 13, 2012 Veton Kpuska 4 Fourier-Transform View Recall (from Chapter 3): w[n] is a finite-length, symmetrical sequence (i.e., window) of length N w . w[n] 0 for [0, N w-1] w[n] Analysis window or Analysis Filter ( 29 [ ] [ ] - =--= m n j e m n w m x n X , February 13, 2012 Veton Kpuska 5 Fourier-Transform View x[n] time-domain signal f n [m]=x[m]w[n-m] - Denotes short-time section of x[m] at point n. That is, signal at the frame n. X(n, ) - Fourier transform of f n [m] of short-time windowed signal data. Computing the DFT: ( 29 ( 29 k N n X k n X 2 | , , = = February 13, 2012 Veton Kpuska 6 Fourier-Transform View Thus X(n,k) is STFT for every =(2 /N)k Frequency sampling interval = (2 /N) Frequency sampling factor = N DFT: ( 29 [ ] [ ] - =--= m km N j e m n w m x k n X 2 , February 13, 2012 Veton Kpuska 7 Fourier-Transform View February 13, 2012 Veton Kpuska 8 Example 7.1 Let x[n] be a periodic impulse train sequence: Also let w[n] be a triangle of length P: - =-= l lP n n x ] [ ] [ P 2P 3P-P n P/2+1-P/2 P-points n February 13, 2012 Veton Kpuska 9 Example 7.1 lP j l m m j l m m j e lP n w e m n w lP m e m n w m x n X ) ( ] [ ) ( ] [ ] [ ) , ( -- = - =-- = - =- -=- -=-= Non-zero only for m=lP Window located at lP & Linear phase - lP February 13, 2012...
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Ch7-Short-Time_Fourier_Transform_Analysis_and_Synthesis -...

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