F09_midterm_review_rec_P1-2

# F09_midterm_review_rec_P1-2 - F z = z-N ˆ H z and F 1 z...

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ENEE630 ADSP Midterm Review Recitation October 21, 2009 1. All the signals and systems in this problem are real. Solution: 2. In this problem, a is a real number, and | a | < 1. Solution: (a) E 0 ( z ) = 1 2 a + z - 1 1+ az - 1 , and E 1 ( z ) = 1 2 . Take R 0 ( z ) = E 1 ( z ) and R 1 ( z ) = E 0 ( z ), we get F 0 ( z ) = 1 2 a + z - 1 + z - 2 + az - 3 1+ az - 2 and F 1 ( z ) = 1 2 - a + z - 1 - z - 2 + az - 3 1+ az - 2 . T ( z ) = 2 z - 1 E 0 ( z 2 ) E 1 ( z 2 ) = z - 1 2 a + z - 2 1+ az - 2 is all pass. (b) (c). ˆ H 0 ( z ) = H 0 * ( z - 1 ) = F 0 ( z ) = 1 2 a * + z 1 + z 2 + a * z 3 1+ a * z 2 = 1 2 a + z 1 + z 2 + az 3 1+ az 2 (since a is real), ˆ H 0 ( z ) H 0 ( z )+ ˆ H 0 ( - z ) H 0 ( - z ) = 1 4 ( a + z - 1 + z - 2 + az - 3 1+ az - 2 a + z 1 + z 2 + az 3 1+ az 2 + a - z - 1 + z - 2 - az - 3 1+ az - 2 a - z 1 + z 2 - az 3 1+ az 2 ) = 1. Let z = e , we have H 0 * ( e - ) H 0 ( e )+ H 1 * ( e - ) H 1 ( e ) = H * 0 ( e ) H 0 ( e )+ H * 1 ( e ) H 1 ( e ) = | H 0 ( e ) | 2 + | H 1 ( e ) | 2 = 1. (Convince yourself that H 0 * ( e - ) = H * 0 ( e )). This is a power complementary 1

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system (you are not necessary to use this jargon which is beyond the lecture, but you should be able to describe this situation in your own words). (d) We can ﬁnd a PR system when N is odd. Use the AC matrix " H 0 ( z ) - z - N ˆ H 0 ( - z ) H 0 ( - z ) z - N ˆ H 0 ( z ) #" F 0 ( z ) F 1 ( z ) # = " 2 T ( z ) 0 # . Choose
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Unformatted text preview: F ( z ) = z-N ˆ H ( z ) and F 1 ( z ) =-H (-z ) to get an alias-free system, and check that T ( z ) = 1 2 z-N using the result from part (c). Therefore, this is PR. Similarly, we can ﬁnd an alias-free system when N is even; however, that system is not necessarily PR. Comment: As someone seems still confused about ˆ H , I’d like to clarify it by a simple example: Assume H ( z ) = az + bz 2 , then H * ( z ) = a * z + b * z 2 (only coeﬃcients), H * ( z ) = ( az + bz 2 ) * = a * z * + b * z * 2 (all), and H * ( z-1 ) = a * z-1 + b * z-2 . 2...
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F09_midterm_review_rec_P1-2 - F z = z-N ˆ H z and F 1 z...

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