Solutions_Assignment3_w10

Solutions_Assignment3_w10 - Carleton University Department...

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1 of 7 Carleton University Department of Systems and Computer Engineering SYSC-4405: Digital Signal processing Winter Semester 2010 Solutions of Assignment # 3 Date Due: March 5, 2010, before 1:00 p.m. ______________________________________________________________________________ Problem 1 Determine the z-transform of the following signals and sketch the corresponding pole-zero pat- tern a) , a is real and N is a positive integer. x(n) can also be rewitten as: or Where , with a ROC: pole locations: and Zero locations: , for and xn () a n 2 π n N --------- ⎝⎠ ⎛⎞ sin un un N [] = a n 2 π n N sin a n 2 π n N sin = a n 2 π n N sin a N a nN 2 π N ------------------------ sin = fn a N fn N = f n a n 2 π n N sin az 1 2 π N ------ sin 12 1 2 π N cos a 2 z 2 + ---------------------------------------------------------- = Xz 1 a N z N Fz 1 a N z N 1 2 π N sin 1 2 π N cos a 2 z 2 + == za > 1 2 π N cos a 2 z 2 +0 p 1 p 2 , ae j 2 π N ----- j 2 π N , z N 1 0 N 1 poles at the origin = z N a N a N e j 2 π k z j 2 π k N -------- = k 012 N 1 ,,, , = z 0 =
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2 of 7 b) or where and From the transform tables we have: , , and 2 π N ------ 2 π N xn () 1 2 -- n 2 1 3 ⎝⎠ ⎛⎞ n 1 un 1 = 2 n 2 1 3 n 1 1 n 1 2 2 n 1 1 ++ [] 1 3 n 1 1 == 2 gn 1 = n 2 fn 2 nf n = f n 1 3 n = f n 1 1 1 3 z 1 ------------------- 1 3 z 1 1 1 3 z 1 2 --------------------------- n 2 z d dz ----- 1 3 z 1 1 1 3 z 1 2 z 1 1 3 z 1 2 1 3 z 2 2 1 3 z 1 1 1 3 z 1 1 3 z 2 1 1 3 z 1 4 ------------------------------------------------------------------------------------------------------------------- = n 2 z 1 3 z 2 1 1 3 z 1 –2 1 3 z 1 1 1 3 z 1 3 ------------------------------------------------------- 1 3 z 1 1 1 3 z 1 + 1 1 3 z 1 3 =
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Solutions_Assignment3_w10 - Carleton University Department...

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