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Unformatted text preview: zTransform ztransform is a powerful analysis method that enables the use of polynomial and rational functions to analyze systems. By using the polynomial, the characteristics and properties of the system can be described by the roots of the polynomials. Also the algebra of the polynomials can provide systematic procedures for the operations and implementation of system. Definition The ztransform ) ( z X of a sequence ] [ n x is defined as  = = = n n z n x n x Z z X ] [ ]} [ { ) ( where the z in the summation is a complex variable whose real part and imaginary part are real while the Z in front of the bracket is considered as operator for denoting ztransform. For finite length sequence with samples from n =0 to N , the ztransform is given by = = N n n z n x z X ] [ ) ( Properties of ztransform Most of the properties of the ztransform are the same as the discrete time Fourier transform and some of the properties are summarized as follows. (1) Linearity 1 ) ( ) ( ) ( ] [ ] [ ] [ 2 1 2 1 z bX z aX z X n bx n ax n x + = + = (2) Time delay ) ( ] [ z X z n n x n  Especially, for 1 = n the ztransform of one sample delayed sequence is the original ztransform times 1 z . Hence 1 z is used to denote unit delay in the z domain and as a unitdelay operator to convert ] [ n x to ] 1 [ n x i.e ] 1 [ ]} [ { ] [ 1 = = n x n x z n y . (3) Convolution  = = = m m n h m x n h n x n y ] [ ] [ ] [ ] [ ] [ ) ( ) ( ) ( z X z H z Y = The convolution in the time domain is the multiplication in the zdomain. We can apply this property to LTI system. The ztransform of the output is the product of the ztransforms of the input and the impulse response. 2 LTI system ] [ n x ] [ n y ) ( z X ) ( ) ( ) ( zX zH zY = ]} [{ ) ( n h Z zH = 1 z ] [ n x ] 1 [ n x ) ( z X ) ( 1 z X z Example Find the ztransform of ] [ ] [ n u p n h n = 1 1 1 ] [ ) ( =  = = = = pz z p z n u p z H n n n n n n assuming 1   1 < pz or  z>p . The condition for the z variable defines a region of z that gives convergent transform. This region is called as region of convergence. Example } 2 , 1 { ] [ 1 , = = n n x and } 5 . , 1 { ] [ 1 , = = n n h The output y [ n ] is given by ] [ ] [ ] [ n h n x n y = ={1, 0.5+2, 0.5x2}={1, 2.5, 1} Using ztransform to calculate y [ n ]: 1 2 1 ) ( + = z z X and 1 5 . 1 ) ( + = z z H Hence 2 1 1 1 5 . 2 1 ) 5 . 1 )( 2 1 ( ) ( ) ( ) ( + + = + + = = z z z z z H z X z Y . This result yields y [ n ]={1, 2.5, 1} identical to the convolution. The property of convolution of two time sequences can be applied to the cascade of two systems. We have ] [ ] [ ] [ 2 1 n h n h n h = ) ( ) ( ) ( 2 1 z H z H z H = ....
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 Fall '10
 ShuHungLeung
 Digital Signal Processing, Signal Processing

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