soln2 - max(X-Z) ). 5c. By lowpass Fltering the original...

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EECS 451 SOLUTIONS TO PROBLEM SET #2 1. # a b c d e f g h i j k l m n S Y N Y N Y Y Y Y N N Y N Y Y L N Y Y Y N N N Y Y Y N Y N Y TI Y Y N N Y Y Y N N N Y N Y Y C Y N Y N Y Y Y Y N N Y N Y Y S Y N Y Y Y Y Y Y N Y Y Y Y Y Key: S=static; L=linear; TI=time-invariant; C=causal; S=stable. 2. # a b c d e f g h True ? Y Y Y Y Y N N Y 2i. Consider T 1 { x ( n ) } = x ( n + 1); T 2 { x ( n ) } = x ( n 2) → T 1 T 2 { x ( n ) } = x ( n 1). Consider T 1 { x ( n ) } = e x ( n ) ; T 2 { x ( n ) } = log x ( n ) → T 1 T 2 { x ( n ) } = x ( n ). 3. 2.10: Since system is time-invariant, advance x 3 ( n ) by one: { 0 , 0 , 1 } → { 1 , 2 , 1 } . Suppose system linear. { 0 , 0 , 3 } → 3 { 1 , 2 , 1 } = { 3 , 6 , 3 } n = { 0 , 1 , 0 , 2 } . Nonlinear . 3. 2.11: Since system linear: x 1 ( n ) + x 2 ( n ) = δ ( n ) y 1 ( n ) + y 2 ( n ) = { 0 , 3 , 1 , 2 , 1 } . x 2 ( n ) + x 3 ( n ) = δ ( n + 1) y 2 ( n ) + y 3 ( n ) = {− 1 , 2 , 2 , 3 } . Not time invariant . 4a. y ( n ) = ∑∑ h ( i ) x ( n i ) = i h ( i )( n x ( n i )) = i h ( i ) n x ( n ). 4b. (1) { 1 , 2 , 4 } ∗ { 1 , 1 , 1 , 1 , 1 } = { 1 , 3 , 7 , 7 , 7 , 6 , 4 } . Check: (7)(5)=35. (2) { 1 , 2 , 1 } ∗ { 1 , 2 , 1 } = { 1 , 4 , 2 , 4 , 1 } . Check: (2)(2)=4. (5) { 1 , 2 , 3 } ∗ { 0 , 0 , 1 , 1 , 1 , 1 } = { 0 , 0 , 1 , 1 , 2 , 2 , 1 , 3 } . Check: (2)(4)=8. (7) { 0 , 1 , 4 , 3 } ∗ { 1 , 0 , 1 , 1 } = { 0 , 1 , 4 , 4 , 5 , 1 , 3 } . Check: (2)(-1)=-2. 5. See overleaf. Easy to conFrm Z=X to roundo± error (check
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Unformatted text preview: max(X-Z) ). 5c. By lowpass Fltering the original signal, we have halved its bandwidth. Hence the Nyquist rate is also halved, and we can downsample or subsample (take every other sample) without losing information. 1 2 3 4 5 6 x 10 4-0.5 0.5 1 2 3 4 5 6 x 10 4 200 400 600 0.5 1 1.5 2 2.5 3 x 10 4-0.5 0.5 0.5 1 1.5 2 2.5 3 x 10 4 100 200 300 1 2 3 4 5 6 x 10 4-0.5 0.5 1 2 3 4 5 6 x 10 4 200 400 600...
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This note was uploaded on 12/10/2009 for the course EECS 451 taught by Professor Andrewyagle during the Spring '08 term at University of Michigan.

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soln2 - max(X-Z) ). 5c. By lowpass Fltering the original...

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