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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 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 13, 2012 Veton Këpuska 3 Short-Time Analysis (STFT) Two approaches of STFT are explored: 1. Fourier-transform & 2. Filterbank February 13, 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 , February 13, 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 2 | , , = = February 13, 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 13, 2012 Veton Këpuska 7 Fourier-Transform View February 13, 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 n … P/2+1-P/2 P-points n February 13, 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 δ-∞-∞ = ∞-∞ =-∞-∞ = ∞-∞ =-∑ ∑ ∑ ∑-=- -=-= Non-zero only for m=lP Window located at lP & Linear phase -ϖ lP February 13, 2012...
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Ch7-Short-Time_Fourier_Transform_Analysis_and_Synthesis -...

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