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Unformatted text preview: EE533 Problem Set-5 1. For each of the following filtering functions, indicate which of the four linear-phase FIR filter types can be used to realize the desired function: (a) Lowpass (b) Highpass (c) Bandpass (d) Bandstop 2. Let H ( z ) be a Type-I linear-phase LPF of length M + 1 for which H ZP ( ω ) ≈ 1 for ≤ ω ≤ π /4, and H ZP ( ω ) ≈ 0 for π /3 ≤ ω ≤ π . Another filter E ( z ) is obtained from H ( z ) as E ( z ) = z- M /2- H ( z ) . (a) Express the impulse response e ( n ) in terms of h ( n ) . Show that e ( · ) is also a Type-I filter. (b) Obtain an expression for E ZP ( ω ) in terms of H ZP ( ω ) , and give a rough sketch of E ZP ( · ) for 0 ≤ ω ≤ π . (c) Compare the behaviour of E ZP ( · ) with that of F ZP ( · ) where F ( z ) = (- 1 ) M /2 H (- z ) . 3. Let H ( z ) be a Type-I or Type-II LPF of length M + 1 with H ZP ( ω ) as specified in the previous problem. Define the linear-phase filter g ( · ) as g ( n ) = 2 cos [ ω ( n- M /2 )] h ( n ) .....
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