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Handout20

# Handout20 - Course 18.327 and 1.130 Wavelets and Filter...

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Course 18.327 and 1.130 Wavelets and Filter Banks M-band wavelets: DFT filter banks and cosine modulated filter banks. Multiwavelets. 2 M-channel Filter B anks Used in communication e.g. DSL 1 Scaling function, M-1 wavelets H 0 ( ω ) M X( ω ) H 1 ( ω ) M H M-1 ( ω ) M ± F 0 ( ω ) M F 1 ( ω ) M F M-1 ( ω ) M X( ω ) ^ ( 2)( 2)X( ω ) = ½ [X( ω )+X( ω + π )] ( M)( M)X( ω ) = [ X( ω + )] M M-1 n=0 2 π n M 1 1

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3 Perfect Reconstruction F k ( ω ) X( ω + )H k ( ω + ) = e -i ω ± X( ω ) i.e. X( ω + ) F k ( ω )H k ( ω + ) = e -i ω ± X( ω ) Matching terms on either side n = 0 F k ( ω )H k ( ω ) = Me -i ω ± no distortion n 0 F k ( ω )H k ( ω + ) = 0 no aliasing 1 M M-1 n=0 2 π n M 2 π n M M-1 k=0 2 π n M M-1 n=0 M-1 k=0 2 π n M 2 π n M M-1 k=0 M-1 k=0 1 M 4 e.g. M = 3 F 0 ( ω )H 0 ( ω ) + F 1 ( ω )H 1 ( ω )+F 2 ( ω )H 2 ( ω ) = 3e -i ω ± F 0 ( ω )H 0 ( ω + ) + F 1 ( ω )H 1 ( ω + ) + F 2 ( ω )H 2 ( ω + ) = 0 F 0 ( ω )H 0 ( ω + ) + F 1 ( ω )H 1 ( ω + ) + F 2 ( ω )H 2 ( ω + ) = 0 Cast in matrix form [F 0 ( ω ) F 1 ( ω ) F 2 ( ω )] H m ( ω ) = [3e -i ω ± 0 0] H 0 ( ω ) H 0 ( ω + ) H 0 ( ω + ) H 1 ( ω ) H 1 ( ω + ) H 1 ( ω + ) H 2 ( ω ) H 2 ( ω + ) H 2 ( ω + ) 2 π 3 2 π 3 2 π 3 4 π 3 4 π 3 4 π 3 2 π 3 H m ( ω ) = 2 π 3 2 π 3 4
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