090123Lecture08 - Multi-stage decimation ECE 700 WI 2009 3...

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Multi-stage decimation ECE 700 WI 2009 3
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Multi-stage decimation ripple Worst case passband ripple in two stage decimation: Suppose we choose δ f,s = δ g,s = δ s /3. This meets the overall stopband ECE 700 WI 2009 4 specifications. Next suppose we choose δ f,p = δ g,p = δ s /3. Then passband ripple for decimator stage is approximately δ s /2. Does this meet the overall specification?
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Two channel filterbank 1 1 1 1 2 2 2 2 2 2 2 1 1 ( ) ( ) ( ) ( ) ( ) 2 2 i i i U z H z X z H W z X W z π = + ECE 700 WI 2009 [ ] 2 2 1 1 1 1 1 2 2 2 2 2 2 0 0 1 1 1 0 0 1 0 1 Since 1 1 1 ( ) ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) ( ) ( ) 1 ( )[ ( ) ( ) ( ) ( )] 2 ( ) ( ) 1 ( ) ( ) ( ) 2 j j i i i i i i i W e e U z H z X z z H z X z Y z G z U z G z U z G z H z X z H z X z H z H z X z X z H z H π - - - = = = = - = + - - = + = + - - = - - 0 1 ( ) ( ) ( ) ( ) H z G z z G z     - dncurlybracketlefthorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketmidhorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketright
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Conditions for no aliasing The matrix H ( z ) is called the aliasing component matrix. For no aliasing, we want X (- z ) to have no contribution to Y ( z ). Why? X (- z ) = X (e j π z ) is the image due to the replica of the base spectrum at π .
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