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Ch7-Short-Time_Fourier_Transform_Analysis_and_Synthesis

# Ch7-Short-Time_Fourier_Transform_Analysis_and_Synthesis -...

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Speech Processing Short-Time Fourier Transform Analysis  and Synthesis

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February 11, 2012 Veton Këpuska 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 11, 2012 Veton Këpuska 3 Short-Time Analysis (STFT) Two approaches of STFT are explored: 1. Fourier-transform & 2. Filterbank

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February 11, 2012 Veton Këpuska 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 ϖ ϖ ,
February 11, 2012 Veton Këpuska 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, ϖ ) - Fourier transform of f n [m] of short-time  windowed signal data. Computing the DFT: ( 29 ( 29 k N n X k n X π ϖ ϖ 2 | , , = =

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February 11, 2012 Veton Këpuska 6 Fourier-Transform View Thus X(n,k) is STFT for every  ϖ =(2 π /N)k Frequency sampling interval  = (2 π /N) Frequency sampling factor  = N DFT: ( 29 [ ] [ ] -∞ = - - = m km N j e m n w m x k n X π 2 ,
February 11, 2012 Veton Këpuska 7 Fourier-Transform View

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February 11, 2012 Veton Këpuska 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 ] [ ] [ δ P 2P 3P -P 0 n P/2+1 -P/2 P-points n
February 11, 2012 Veton Këpuska 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 ) ( ] [ ) ( ] [ ] [ ) , ( ϖ ϖ ϖ δ ϖ - -∞ = -∞ = - -∞ = -∞ = - - = - - = - = Non-zero only for  m=lP Window located at  lP   & Linear phase - ϖ lP

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February 11, 2012 Veton Këpuska 10 Example 7.1 Since windows w[n] do not overlap, |X(n, ϖ )| = constant and  X(n, ϖ ) is linear.
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