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 z x n z =−∞ = where z is a complex variable . The inverse z-transform is found using, 1 1 [ ] [ ] 2 n x n X z z dz j π = v Symbolic representations of the z-transform and its inverse, { } [ ] [ ] X z Z x n = and { } 1 [ ] [ ] x n Z X z = Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha Or [ ] [ ] x n X z . Note that { } 1 [ ] [ ] Z Z x n x n = and { } 1 [ ] [ ] Z Z X z X z = . 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. 1 1 [ ] [ ] x n X z and 2 2 [ ] [ ] x n X z Then, 1 1 2 2 1 1 2 2 [ ] [ ] [ ] [ ] a x n a x n a X z a X z + + .
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