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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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2/13/12 Veton K ë puska 2 Short-Time Fourier Transform  Analysis and Synthesis:  Minimum-Phase Synthesis u Speech & Audio Signals are varying and can be considered stochastic  signals that carry information. u 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) n Discrete-time short-time Fourier transform (STFT) consists of separate FT of  the signal in the neighborhood of that instant. n 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. u In linear Prediction and Homomorphic Processing, underlying model of  the source/filter is assumed. This leads to:  n Model based analysis/synthesis, also note that n Analysis methods presented implicitly both used short time analysis methods  (to be presented). u In Short-Time Analysis systems no such restrictions apply.
2/13/12 Veton K ë puska 3 Short-Time Analysis (STFT) u Two approaches of STFT are explored: 1. Fourier-transform & 2. Filterbank

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Fourier Transform View 2/13/12 Veton K ë puska 4
2/13/12 Veton K ë puska 5 Fourier-Transform View u Recall (from Chapter 3): u w[n] is a finite-length, symmetrical  sequence (i.e., window) of length Nw.  n w[n]   0 for [0, Nw-1] n w[n] – Analysis window or Analysis Filter ( 29 [ ] [ ] -∞ = - - = m n j e m n w m x n X ϖ ϖ ,

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2/13/12 Veton K ë puska 6 Fourier-Transform View u x[n] – time-domain signal u fn[m]=x[m]w[n-m]  - Denotes short-time section of  x[m] at point n. That is, signal at the frame n. u X(n,w ) - Fourier transform of fn[m] of short-time  windowed signal data. u Computing the DFT: ( 29 ( 29 k N n X k n X π ϖ ϖ 2 | , , = =
2/13/12 Veton K ë puska 7 Fourier-Transform View u Thus X(n,k) is STFT for every w =(2p /N)k n Frequency sampling interval  = (2p /N) n Frequency sampling factor  = N u DFT: ( 29 [ ] [ ] -∞ = - - = m km N j e m n w m x k n X π 2 ,

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Fourier-Transform View 2/13/12 Veton K ë puska 8 a) Speech waveform x[n] (blue) Window function w[n] (red) b) Windowed section of speech c) It’s Magnitude Spectrum.
2/13/12 Veton K ë puska 9 Example 7.1 u Let x[n] be a periodic impulse train sequence: u 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

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2/13/12 Veton K ë puska 10 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 -w lP
2/13/12 Veton K ë puska 11 Example 7.1 u

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