HW11_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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1 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 11 Due Wednesday, November 12, 2008 Prof. Bresler / Prof. Jones 1. (20 points) A linear-phase FIR filter has the following form: ݄ሾ݊ሿൌ݄ሾܯെ1െ݊ሿ , 0൑݊൑ܯെ1 , ܯ݅ݏ݁ݒ݁݊ Show that its amplitude response is ܪ ሺ߱ሻൌܴሺ߱ሻ݁ ି௝௄ ,whereܭൌ ெିଵ and ܴሺ߱ሻൌ෍ܾሾ݊ሿ cos ቆ߱ ൬݊െ 1 2 ൰ቇ ெ/ଶ ௡ୀଵ Give an expression for the coefficients ሼܾሾ݊ሿሽ in terms of the impulse response ݄ሾ݊ሿ . 2. (20 points) Prove the following properties of generalized linear-phase FIR filters: (a) If ܪሺݖሻ has four zeros at ݖ ൌݎ݁ ௝ఏ ݁ ௝ఏ ൌݎ݁ ି௝ఏ and ݖ ݁ ି௝ఏ , and r is a real number, then
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HW11_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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