probset-four

# probset-four - (b Derive the magnitude response of H z and...

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EE533 Problem Set-4 1. Let x ( · ) be a signal with z -transform X ( z ) with RoC r 1 < | z | < r 2 . Consider the signal y ( · ) deﬁned in terms of a positive integer M according to y ( n ) = ± x ( n / M ) , n = 0, ± M , ± 2 M , . . . 0, otherwise. Determine (from ﬁrst principles) Y ( z ) and its RoC in terms of M , r 1 , and r 2 . 2. The input to an LTI system is the signal x ( n ) = 2 n u ( - n - 1 ) + 0.5 n u ( n ) . The output of the system is the signal y ( n ) = 6 ( 0.5 n - 0.75 n ) u ( n ) . Determine the system function H ( z ) , its poles, zeros, and RoC. Write the difference equation that implements the system. 3. When the input to a stable ﬁlter H ( z ) is the signal u ( n ) , its output y ( · ) (the step re- sponse ) obeys lim n y ( n ) = 0. What information does this behaviour convey about the zeros of H ( z ) ? Determine the value of lim n y ( n ) for an arbitrary (stable) ﬁlter H ( z ) with step response y ( · ) . 4. Consider the ﬁlter H ( z ) with difference equation y ( n ) - y ( n - 2 ) = x ( n ) - x ( n - 6 ) . (a) Determine the poles and zeros of H ( z ) . Is H ( z ) a causal and stable ﬁlter?
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Unformatted text preview: (b) Derive the magnitude response of H ( z ) , and sketch it for 0 ≤ ω ≤ π . 5. Consider the second-order all-pole ﬁlter: H ( z ) = 1/ ( 1-2 r cos θ z-1 + r 2 z-2 ) , 0 < r < 1, 0 < θ < π . (a) Derive an expression for the magnitude-squared response of H ( z ) . (b) Determine (in terms of r and θ ) the values of ω in the interval [ 0, π ] at which the above response is a maximum or minimum. (c) While some of the values of ω determined above are always extremal points of the response, one value is an extremal point only when r and θ satisfy a speciﬁc condition. Determine this condition. 6. Design a ﬁrst-order all-pole IIR lowpass ﬁlter whose 3-dB cutoff is ω c = 0.3 π . Ensure unity gain at ω = 0. —– END —–...
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