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113_1_113-mid03-sol

113_1_113-mid03-sol - EE113 Digital Signal Processing Prof...

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EE113: Digital Signal Processing Midterm Solutions Prof. Abeer Alwan May 15, 2003 1. (a) The difference equation that describes the system is y ( n ) + 3 y ( n 1) = x ( n ) + 2 x ( n 1) (b) To find the impulse response, we need to find the system response to the input x ( n ) = δ ( n ). In this case, the output y ( n ) = h ( n ) is the impulse response sequence. This is equivalent to solving the difference equation. h ( n ) + 3 h ( n 1) = δ ( n ) + 2 δ ( n 1) , h ( 1) = 0 This equation becomes homogeneous for n 2 hence it is described by the ho- mogeneous equation h ( n ) + 3 h ( n 1) = 0 The characteristic equation associated with this system is λ + 3 = 0 which has a single root at λ = 3. Then h ( n ) is given by h ( n ) = C ( 3) n , n 2 To find C , we need to find h (1) as follows h (0) = 3 h ( 1) + δ (0) + 2 δ ( 1) = 1 h (1) = 3 h (0) + δ (1) + 2 δ (0) = 1 Substituting by h (1) = 1, we find that C = 1 / 3 and h ( n ) is now given by h ( n ) = 1 3 ( 3) n , n 1 We know that the system is causal, then h ( n ) = 0 for n < 0. We also found that h (0) = 1 then h ( n ) is h ( n ) = δ ( n ) + 1 3 ( 3) n u ( n 1) (c) Take the z -transform of the difference equation that describes the system, we get Y ( z ) + 3 z 1 Y ( z ) = X ( z ) + 2 z 1 X ( z ) 1 1
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then Y ( z ) ( 1 + 3 z 1 ) = X ( z ) ( 1 + 2 z 1 ) and the transfer function H ( z ) is given by H ( z ) = Y ( z ) X ( z ) = 1 + 2 z 1 1 + 3 z 1 = 1 + (3 z 1 z 1 ) 1 + 3 z 1 = 1 z 1 1 + 3 z 1 Since the system is causal, the region of convergence associated with this equation is | z | > 3. Using the following z -transform pairs δ ( n ) 1 ( a ) n u ( n ) 1 1 az 1 and the following property, x ( n 1) z 1 X ( z ) We find the impulse response sequence h ( n ) to be h ( n ) = δ ( n ) ( 3) n 1 u ( n 1) (d) The system is relaxed and described be a constant coefficient difference equation, therefore, LTI. We can use find the output sequence by convolving the input sequence with the impulse response sequence.
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