Polyphase decomposition

Polyphase decomposition - Polyphase Decompositions and QMF...

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12/1/2010 Polyphase Decompositions and QMF Channel Banks (Slides adapted from S. Mitra to accompany the text, Digital Signal Processing ) Jerry D. Gibson ECE 158 Fall 2010
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12/1/2010 Polyphase Decomposition The Decomposition • Consider an arbitrary sequence { x [ n ]} with a z - transform X ( z ) given by • We can rewrite X ( z ) as where −∞ = = n n z n x z X ] [ ) ( = = 1 0 M k M k k z X z z X ) ( ) ( −∞ = −∞ = + = = n n n n k k z k Mn x z n x z X ] [ ] [ ) ( 1 0 M k
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12/1/2010 Polyphase Decomposition • The subsequences are called the polyphase components of the parent sequence { x [ n ]} • The functions , given by the z -transforms of , are called the polyphase components of X ( z ) ]} [ { n x k ) ( z X k ]} [ { n x k
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12/1/2010 Polyphase Decomposition • The relation between the subsequences and the original sequence { x [ n ]} are given by • In matrix form we can write ]} [ { n x k 1 0 + = M k k Mn x n x k ], [ ] [ [] = ) ( ) ( ) ( . . . . ) ( ) ( M M M M M z X z X z X z z z X 1 1 0 1 1 1 . . . .
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12/1/2010 Polyphase Decomposition • A multirate structural interpretation of the polyphase decomposition is given below
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12/1/2010 Polyphase Decomposition • The polyphase decomposition of an FIR transfer function can be carried out by inspection • For example, consider a length-9 FIR transfer function: = = 8 0 n n z n h z H ] [ ) (
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12/1/2010 Polyphase Decomposition • Its 4-branch polyphase decomposition is given by where ) ( ) ( ) ( ) ( ) ( 4 3 3 4 2 2 4 1 1 4 0 z E z z E z z E z z E z H + + + = 2 1 0 8 4 0 + + = z h z h h z E ] [ ] [ ] [ ) ( 1 1 5 1 + = z h h z E ] [ ] [ ) ( 1 2 6 2 + = z h h z E ] [ ] [ ) ( 1 3 7 3 + = z h h z E ] [ ] [ ) (
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12/1/2010 Computationally Efficient Decimators • Using the cascade equivalence #1 we arrive at the computationally efficient decimator structure shown below on the right Decimator structure based on Type I polyphase decomposition y [ n ] y [ n ] x [ n ] x [ n ] ] [ n v
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12/1/2010 Quadrature-Mirror Filter Bank • In many applications, a discrete-time signal x [ n ] is split into a number of subband signals by means of an analysis filter bank • The subband signals are then processed • Finally, the processed subband signals are combined by a synthesis filter bank resulting in an output signal y [ n ] ]} [ { n v k
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12/1/2010 Quadrature-Mirror Filter Bank • If the subband signals are bandlimited to frequency ranges much smaller than that of the original input signal x [ n ], they can be down-sampled before processing • Because of the lower sampling rate, the processing of the down-sampled signals can be carried out more efficiently ]} [ { n v k
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12/1/2010 Quadrature-Mirror Filter Bank • After processing, these signals are then up- sampled before being combined by the synthesis filter bank into a higher-rate signal • The combined structure is called a quadrature-mirror filter (QMF) bank
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Polyphase decomposition - Polyphase Decompositions and QMF...

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