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Introduction to Digital Signal and System Analysis Z Domain Analysis 5.2 Relationship between z-transform and Fourier transform Let z = exp( jW) , i.e. the complex variable z is only allowed on the unit circle, the z-transform becomes a unilateral Fourier transform X W = ∑ ∞ ( ) = n 0 x[ n]exp ( − jWn) Obviously, apart from on the unit circle, the complex operator z can be specified into other curves or region, if necessary. Later, it will be shown the unit circle is important boundary on the z-domain. 5.3 Z as time shift operator Multiplying by z implies a time advance and dividing by z, or multiplying by impulse, −1 z , is to cause a time delay. For the unit X ( z) = ∞ ∑ n= 0 d [ n] z −n = z −n n= 0 = 1 Please click the advert Download free ebooks at bookboon.com 62
Introduction to Digital Signal and System Analysis Z Domain Analysis For the delayed unit impulse, ∞ −n −n X ( z) = ∑d [ n −1] z = z = z n= 0 n= 1 −1 n For more general cases of shifting by 0 samples, ∞ X ( z) = ∑d [ n − n n= 0 0 ] z −n = z −n n= n0 = z −n0 For shifted signal x[n], i.e. delayed by n samples, the z-transform 0 ∞ ∑ n= 0 x [ n − n0 ] u[ n − n0 ] z −n = X ( z) z −n0 5.4 Transfer function The transfer function describes the input-output relationship, or the transmissibility between input and output, in the z-domain. Applying the z-transform to the output of a system, the relationship between the z-transforms of input and output can be found: Y ( z) = = = ∞ ∑ r=−∞ ∞ ∑ r=−∞ ∞ ∑ n= 0 x( r) z x( r) z y[ n] z ∞ −r −r ∑ n= 0 ∞ ∑ −n m=−r = ∞ ∑ n= 0 h[ n − r] z h[ m] z r=−∞ −m ∞ ∑ x[ r] h[ n − r] z −( n−r) = X ( z) H ( z) −n Therefore, H ( z) = Y ( z) X ( z) (5.3) i.e., the transfer function can be obtained from the z-transforms of input and output. Alternatively, the transfer function H (z) can be obtained by applying z-transform directly to the impulse response h[n] . The relationships of input ( x [n] and X (z) ), output ( y [n] and Y (z) ) and system function ( h [n] and H (z) the time and z domains are depicted in Figure 5.2. ) in 63 Download free ebooks at bookboon.com
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