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 2/13/12 Veton Këpuska 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 Këpuska 3 Short-Time Analysis (STFT) u Two approaches of STFT are explored: 1. Fourier-transform & 2. Filterbank Fourier Transform View 2/13/12 Veton Këpuska 4 2/13/12 Veton Këpuska 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 , 2/13/12 Veton Këpuska 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 Këpuska 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 , Fourier-Transform View 2/13/12 Veton Këpuska 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 Këpuska 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 n … P/2+1-P/2 P-points n 2/13/12 Veton Këpuska 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- ∞-∞ = ∞-∞ =- ∞-∞ = ∞-∞ =- ∑ ∑ ∑ ∑- =- - =- =...
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This note was uploaded on 02/11/2012 for the course ECE 5525 taught by Professor Staff during the Fall '10 term at FIT.

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Ch7-Short-Time_Fourier_Transform_Analysis_and_Synthesis -...

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