Lecture 11_fall_2010_6tp

Lecture 11_fall_2010_6tp - Modifications to STFT...

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1 Digital Speech Processing Digital Speech Processing— Lecture 11 1 Modifications, Filter Bank Design Methods Modifications to STFT modifications to short-time spectrum can be fixed (non-time varying) or time-varying fixed modification (no variability with ) ˆ jj j n 2 () () () assume inverse DFT of ( ) exists ω ωω = kk k k nn j X eX e P e Pe 1 0 1 , then [ ] ( ) where is the number of frequencies at which ( ) is evaluated = = k N n k j pn e N N Fixed Modifications using FBS 1 0 1 using FBS methods we get ˆ [ ] [] [ ) ( ) = −∞ = ⎡⎤ = ⎢⎥ ∑∑ k k N j n n k N j mj j n yn X e e wn mxme e 3 0 1 0 ( ) ( ) [ ] ˆ [ == ∞− =−∞ = =−∞ ⎣⎦ =− =∗ km N n m mk m wn mxm e wn mxmNpn m Nxn wn ][ ] Fixed Modifications using FBS ˆ [] [] ˆ []c o n v o l v e d w i t h [] [] equivalent to linear filtering operation on [ ] - - i d e a l l y w a n t =∗⋅ •= = > ± wn pn xn xn pn 4 need duration of [] duration of [] i s ² p nw n a periodic sequence of period (sampled in frequency at frequencies) if [ ] is longer than repetitive structure in [ ] [ ] [ ] need to avoid long filters (IIR) but instead use > ⋅= = > p N N N pn wn h n 01 RW so that , =⋅ ≈ p hn pnwn pn n N Time Time-Varying Modifications Varying Modifications 1 1 represent time-varying modification as ˆ () ()() with time-varying IR, ( ), defined as ( ) =⋅ = k j n n N m Xe pm Pe e 5 0 1 0 ˆ solve for [ ] as ˆ [ ] = = = k N j k N P e e 1 0 [ ] ( ) k k k n N jn jm j n ex n m w m e P e e Time Time-Varying Modifications Varying Modifications 1 0 1 0 ˆ [ ] [ ] ( ) [) [ ]( ) [ ][] =−∞ = k k N j n N m n e xn mwme e xn mwm e N 6 ˆ [ ] [] [] [][) modified output is the window =−∞ =−∞ = = n m m xn mwm N N xn m wmp m p mwm [ ] weighted by the modification [ ] and convolved with the input [ ] window 'time limits' effects of modifications and prevents smearing in time for FBS spectral modifications lead to convol n wm p m ving the original signal with a time-limited, window weighted version of the time response due to the modification
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2 Fixed Modifications Fixed Modifications-OLA OLA 1 1 for a fixed modification we again have () () () the basic OLA method gives [ ] ( ) ωω ω ∞− = ⎡⎤ = ⎢⎥ ∑∑ ± kk k jj j nn N n r Xe Xe Pe yn Y e e N 7 0 1 0 1 1 [ ] [] [ ) =−∞ = =−∞ = ⎣⎦ = =− ± AA k rk N j n r j Pe e N xw r R e N 1 0 1 0 1 [] ( ) [ ] ∞−∞ =−∞ = =−∞ =−∞ = =−∞ ∑∑∑ A A A A k N n N n kr e xP e e w r R N Fixed Modifications Fixed Modifications-OLA OLA 1 0 00 1 1 ( ) [ ] [][ ] ( ) / ( ) [] [] =−∞ = =−∞ ⎛⎞ = −= ⎜⎟ A A ± ± N n x e wrR N y nx p nW e R W ex n p n 8 [ ] is the convolution of [ ] with the time response of the spectra =−∞ ⎝⎠ A ± R xn 1 l modification ( [ ])--with no window modifications need to use larger FFT sizes to account for (prevent) aliasing due to duration of [ ] where is the window size, is the duration of ⇒+ − pn N N N N Time Varying Modifications Time Varying Modifications-OLA OLA 1 0 1 for the case of a time-varying modification, using OLA, we obtain
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Lecture 11_fall_2010_6tp - Modifications to STFT...

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