PS2-2010

# PS2-2010 - c Assume H belongs to a causal system If the...

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ECE161A April 6, 2010 Discrete Time Signal Processing Professor T. Javidi Problem Set #2 Due: April 13, 2010 1. Problems 6 and 7 of the previous problem set. 2. Consider discrete-time LTI systems with input x[n] and output y[n] related by parts a-c. For each of the systems ﬁnd the impulse response if possible. In addition for each part, determine if the system is causal and stable. a) y [ n ] = n k = -∞ 3 n - k x [ k + 1] b) y [ n ] = 1 M 1 +3 2 k = - M 1 x [ n - k ] (moving average) c) y [ n ] = 1 . 1 y [ n - 1] + x [ n ] ,n 0 (i.e. the system is initially at rest) 3. Prove that the serial connection of two passive LTI systems is also passive. 4. An LTI system has impulse response h [ n ] , for which the z-transform is H ( z ) = z 2 + z ( z - 1 / 2)( z + 1 / 4) . a) Identify all possible ROC of H ( z ) . b) Identify the ROC if you are told that this system stable. Would this system be causal? Explain.
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Unformatted text preview: c) Assume H belongs to a causal system. If the input to the INVERSE system H i is x [ n ] =-1 3 (-1 4 ) n u [ n ]-4 3 2 n u [-n-1] , ﬁnd the output signal, w [ n ] . d) Draw the pole-zero plot for W ( z ) in part (b) and indicate its ROC. 5. Determine the inverse z-transform x 1 [ n ] of the following z-transform: X 1 ( z ) = 1 1-z-7 + 1 z , | z | > 1 HINT: you can use the ”useful sum” formula 6. Consider the Z-transform H ( z ) = 3 z 3 + 49 . 2 z 2 + 10 . 2 z ( z-1)( z-12)( z-. 6) a Find all possible ROC for this function. b Find the inverse transform for each region of convergence found above....
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