probset-six - EE533 Problem Set-6 1 An input signal x is...

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EE533 Problem Set-6 1. An input signal x ( · ) is passed through a stable filter with transfer function H ( z ) to produce v ( · ) . The signal { v ( - n ) } is once again filtered by H ( z ) to yield w ( · ) . The final output is obtained as y ( n ) = w ( - n ) . The coefficients of H ( z ) are real. Find the overall transfer function relating y ( · ) to x ( · ) , in both the z - and ω - domains. 2. Let x ( n ) be a causal signal that is zero outside the range 0 n M . When x ( · ) is input to a causal LTI system specified by the difference equation y ( n ) - 0.36 y ( n - 2 ) = x ( n - 2 ) - 0.36 x ( n ) , the output signal y ( · ) is found to have the same duration. What information does this behaviour convey about X ( z ) ? How does the energy of y ( · ) compare with that of x ( · ) ? 3. Five transfer functions with identical magnitude responses are given below: H 1 ( z ) = 0.5 + 0.25 z - 1 + 0.4 z - 2 - 0.425 z - 3 + 0.75 z - 4 - 0.75 z - 5 + 0.6 z - 6 - 0.2 z - 7 H 2 ( z ) = 1 - 0.5 z - 1 + 0.8 z - 2 - 0.4 z - 3 + 0.25 z - 4 - 0.125 z - 5 + 0.2 z - 6 - 0.1 z - 7 H 3 ( z ) = - 0.5 + z - 1 - 0.4 z - 2 + 0.8 z - 3 - 0.125 z - 4 + 0.25 z - 5 - 0.1 z - 6 + 0.2 z - 7 H 4 ( z ) = - 0.1 + 0.2 z - 1 - 0.125 z - 2 + 0.25 z - 3 - 0.4 z - 4 + 0.8 z - 5 - 0.5 z - 6 + z - 7 H 5 ( z ) = - 0.25 + 0.25 z - 1 + 0.175 z - 2 + 0.7 z - 3 - 0.45 z - 4 + 0.9 z - 5 - 0.6 z - 6 + 0.4 z
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