ECE108HW%232Sol_06

ECE108HW%232Sol_06 - ECE 108 HW#2 Solution 1 Given the...

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ECE 108 HW#2 — Solution 1. Given the discrete-time signal, x ( n ), and discrete-time system unit impulse response, h ( n ), it is desired to determine the system response, y ( n ). () ( ) ( ) () 1 25 1 1 2 n xn un un hn + =− +−  =+   (a) Sketch both signals (b) Use the graphical method of convolution to find the response. This method will require you to break the solution up into different parts (cases). How many different cases will be required for this problem? Answer: 3 (c) For each case sketch the signals that you will use in the convolution sum. The convolution sum is () ( )( ) n y nx m h n m =−∞ Case I: 12 1 nn +< ⇒ < ( ) 0 yn =
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Case II: 21 4 13 nn ≤+≤ ⇒ ≤≤ Then () ()( ) () 11 1 22 2 2 nk n n k kk k yn xkhn k −+ + ++ + == =  =− = =   ∑∑ Into this sum substitute 2 2 ik ki =− ⇒ =+ Then 1 1 1 1 2 20 0 1 1 1 2 1 4 2 1 2 2 2 1 2 2 n n n n n i n i + +− + = = = = Case III: 5 1 4 ≤+ ⇒ ≥ Then 1 1 41 1 1 25 2 5 1 1 2 2 2 n n n k k + + + = = =+ = + Now make the following substitutions,
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2 substitute into the first sum 5 substitute into the second sum ik ki mk km =− ⇒ =+ →
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This note was uploaded on 10/24/2009 for the course ECE 108 taught by Professor Li during the Spring '08 term at Lehigh University .

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ECE108HW%232Sol_06 - ECE 108 HW#2 Solution 1 Given the...

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