SP_VII_P01_4p - Introduction to Signal Processing 1...

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Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha V. Discrete-Time System Analysis Using The Z-Transform The z-transform is the discrete time domain counterpart of the Laplace transform. . A. The z-Transform We define [] X z as the direct z-transform of x n . The z-transform represents signals as the sum of everlasting exponentials. Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha n n X zx n z =−∞ = where z is a complex variable . The inverse z-transform is found using, 1 1 2 n x nX z z d z j π = v Symbolic representations of the z-transform and its inverse, { } X zZ x n = a n d { } 1 x nZX z = Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha Or x z . Note that { } 1 ZZ x n x n ⎡⎤ = ⎣⎦ and { } 1 Z Z Xz Xz = . Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha Linearity of the z-Transform Like its continuous-time counterpart the z-transform too is linear, i.e. 11 x z and 22 x z Then, 2 2 1 1 ax n aX z +⇔+ .
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Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha The Unilateral z-Transform The z-transform we defined above is a very general form and is known as the bilateral z-transform . The bilateral z- transform suffers from non-uniqueness of the inverse transform. In other words, for a given bilateral z- transform [ ] X z it is possible to have more than one discrete-time domain functions, say [ ] 1 x n and [ ] 2 x n , that correspond to [ ] X z . Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha The form of the z-transform that is capable of handling causal signals only is called the unilateral z-transform and is defined as, 0 [] n n X zx n z = = . The unilateral z-transform does not suffer from the non- uniqueness problem. Introduction to Signal Processing 7 Copyright © 2005-2009 – Hayder Radha The Region of Convergence (ROC) of X z The z-transform, n n X n z =−∞ = or 0 n n X n z = = Exists only when the sum converges. The values of z in the complex plane for which the z-transform converges is called the region of convergence or ROC . Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha Example 5.1 Find the z-transform and ROC of signal n un γ Solution: The z-transform is of this signal is, 0 nn n X zu n z = = Since [] 1 = for all 0 n .
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Introduction to Signal Processing 9 Copyright © 2005-2009 – Hayder Radha 0 23 [] 1 n n Xz z zz z γ γγ = ⎛⎞ = ⎜⎟ ⎝⎠ ⎛⎞⎛⎞ ⎛⎞ =+ + + + + ⎜⎟⎜⎟ ⎜⎟ ⎝⎠⎝⎠ ⎝⎠ "" Remember the geometric progression and its sum, 1 1 1 xx x x ++ + + = " if 1 x < Introduction to Signal Processing 1 0 Copyright © 2005-2009 – Hayder Radha 1 1 1 z z z z z =< => Introduction to Signal Processing 1 1 Copyright © 2005-2009 – Hayder Radha 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 k un k Introduction to Signal Processing 1 2 Copyright © 2005-2009 – Hayder Radha Existence of the z-Transform By definition, 00 n n nn x n xnz z ∞∞ == ∑∑ The z-transform exists as long as, 0 n n xn z = ≤<
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SP_VII_P01_4p - Introduction to Signal Processing 1...

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