Lecture 10_winter_2012_6tp

Lecture 10_winter_2012_6tp - Review of STFT 1. Digital...

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1 1 Digital Speech Processing— Lecture 10 Short-Time Fourier Analysis Methods - Filter Bank Design 2 Review of STFT 1 123 0 2 1 ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ .( ) [ ] [ ] ˆ function of (looks like a time sequence) ˆ function of (looks like a transform) ˆ ˆ ( ) defined for , , ,. ..; . Interpretations of ( ) ˆ . f i x ωω ω ωπ =−∞ =− =≤ ± ± jj m n m j n j n Xe xmwnm e n n n ˆ ˆ ˆ ed, variable; ( ) DTFT [ ] [ ] DFT View OLA implementation ˆ ˆ 2. variable, fixed; ( ) [ ] [ ] Linear Filtering view filter bank implementation FBS i == ⎡⎤ ⎣⎦ ⇒⇒ j n n n nn x n e w n mplementation 3 Review of STFT 3 2 2 ˆ ˆ . Sampling Rates in Time and Frequency ˆ recover [ ] from ( ) 1. time: ( ) has bandwidth of Hertz samples/sec rate Hamming Window: (Hz) sample =⇒ j n j S xn X e We B B F B L exactly 4 at (Hz) or every L/4 samples 2. frequency: [ ] is time limited to samples need at least frequency samples to avoid time aliasing S F L wn L L 4 Review of STFT with OLA method can recover ( ) using lower sampling rates in either time or frequency, e.g., can sample every samples (and divide by window), or can use fewer than frequency sam ± L L exactly ples (filter bank channels); but these methods are highly subject to aliasing errors with any modifications to STFT can use windows (LPF) that are longer than samples and still recover with ± L frequency channels; e.g., ideal LPF is infinite in time duration, but with zeros spaced samples apart where / is the BW of the ideal LPF <⇒ S NL N FN 5 Review of STFT H 0 (e j ω ) H 1 (e j ω ) H N-1 (e j ω ) + X(e j ω ) Y(e j ω ) 1 0 1 0 1 0 [] () [][] [ ] [ ] [ ] [ ] [ ] [ ] . .. need to design digital filters that match criteria for exact reconstr δ = = =−∞ =−∞ = = = + + % % % k jn k N k k N k k r r hn wne He H e hn h n n wnpn pn N n rN N wrN n rN w wN uction of [ ] and which stil work with modifications to STFT Tree-Decimated Filter Banks 6
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2 Tree-Decimated Filter Banks • can sample STFT in time and frequency using lowpass filter (window) which is moved in jumps of R<L samples if L N where: L is the window length, R is the window shift, and N is the number of frequency channels • for a given channel at ω = ω k , the sampling rate of the STFT need only be twice the bandwidth of the window Fourier transform – down-sample STFT estimates by a factor of R at the transmitter – up-sample back to original sampling rate at the receiver – final output formed by convolution of the up-sampled STFT with an appropriate lowpass filter, f [ n ] 7 Filter Bank Channels 8 Ful y decimated and interpolated filter bank channels; (a) analysis with bandpass filter, down- shifting frequency and down-sampling; (b) synthesis with up-sampler fol owed by lowpass interpolation filter and frequency up-shift; (c) analysis with frequency down-shift fol owed by lowpass filter fol owed by down-sampling; (d) synthesis with up-sampling fol owed by frequency up-shift fol owed by bandpass filter.
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Lecture 10_winter_2012_6tp - Review of STFT 1. Digital...

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