6-SYSC5602-Z_Transform_2009

6-SYSC5602-Z_Transform_2009 - SYSC5602: Digital Signal...

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SYSC5602: Digital Signal Processing Z_Transform_2009.fm Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 1 / 49 Z- Transform Definition of (the bilateral) z-transform The z-transform of the sequence x(n) is denoted by X(z), and is defined as : The inverse transform of X(z) is x(n). This is expressed as: The Region of Convergence Since the z-transform is an infinite power series, it exists only for those values of z for which this series converges. The region of convergence (ROC) of X(z) is the set of all values of z for which X(z) attains a finite value. Thus any time a z-transform is cited, the ROC should also be indicated. Example Determine the z-transforms of the following finite-duration sequences: a) b) c) d) Xz () xn z n n = = Z 1 {} = x 1 n 125701 ,,,,, = x 2 n = x 3 n 00125701 ,,,,,,, = x 4 n 245701 =
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SYSC5602: Digital Signal Processing Z_Transform_2009.fm Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 2 / 49 e) f) , g) , Solution a) , ROC: entire z-plane except b) , ROC: entire z-plane except and c) , ROC: entire z-plane except d) , ROC: entire z-plane except and e) , ROC: entire z-plane f) , ROC: entire z-plane except g) , ROC: entire z-plane except From the above example it can be concluded that the ROC of a finite duration signal is the entire z-plane except possibly for the points and/or In many cases the sum of the infinite series X(z) can be expressed in a closed form expression. Example Determine the z-transform of the real exponential sequence x 5 n () δ n = x 6 n δ nk = k 0 > x 6 n δ + = k 0 > X 1 z 12 z 1 5 z 2 7 z 3 z 5 ++++ = z 0 = X 2 z z 2 2 z 57 z 1 z 3 ++ + + = z 0 = z = X 3 z z 2 2 z 3 5 z 4 7 z 5 z 7 = z 0 = X 4 z 2 z 2 4 z 1 z 1 z 3 + + = z 0 = z = X 5 z 1 = X 6 z z k = z 0 = X 7 z z k = z = z 0 = z = xn 1 2 -- ⎝⎠ ⎛⎞ n un =
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SYSC5602: Digital Signal Processing Z_Transform_2009.fm Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 3 / 49 Solution The right hand side of the above equation is an infinite geometric series, the sum of which converges when is less than 1. , ROC: For this example we notice that 1) the sequence is causal , and 2) the region of convergence is all points in the z-plane that fall outside of a circle of radius Example Determine the z-transform of the real exponential sequence Solution ROC Xz () xn z n n = 1 2 -- ⎝⎠ ⎛⎞ n z n n 0 = 1 2 z 1 n n 0 = 1 1 2 z 1 1 2 z 1 2 1 2 z 1 n ++ + + + == = = 1 2 z 1 1 1 1 2 z 1 ------------------- = 1 2 z 1 1 z 1 2 > < r 1 2 = 2 n un –1 = z n n = 2 n z n n = 1 2 1 z l l 1 = 2 1 z 2 1 z 2 2 1 z n [] 2 1 z 12 1 z 2 1 z 2 2 1 z n + + + 2 1 z 1 z 1 z 1 = = = 2 1 z 1 z 2 < <
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SYSC5602: Digital Signal Processing Z_Transform_2009.fm Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 4 / 49
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6-SYSC5602-Z_Transform_2009 - SYSC5602: Digital Signal...

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