DSP_ch3

# DSP_ch3 - DSP EEE 407/591 by Andreas Spanias Ph.D Chapter 3...

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Copyright ©Andreas Spanias Ch 3 -1 DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter 3 [email protected] http://www.eas.asu.edu/~spanias Copyright ©Andreas Spanias Ch 3 -2 The Z-Transform •The z-transform plays a similar role in DSP as the Laplace transform in analog circuits and systems. •It provides intuition that is sometimes not evident in time- domain analysis •Simplifies time-domain operations – time domain-convolution maps to Z-domain multiplication •Used to define transfer functions •Could be used to determine responses of systems using a table look-up process

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Copyright ©Andreas Spanias Ch 3 -3 From the Laplace Transform to the Z-Transform ) ( ) ( z H s H d a R C sRC s H a + = 1 1 ) ( () n y 1 z ( ) n x + + 1 1 1 ) ( + = z a b z H o d s-domain transfer function z-domain transfer function Copyright ©Andreas Spanias Ch 3 -4 The Z-Transform - Definition Given the sequence: ) ( n x its Z-transform is Xz x n z n n = =−∞ For causal sequences, i.e., = = 0 ) ( ) ( n n z n x z X 0 0 ) ( < = n for n x
Copyright ©Andreas Spanias Ch 3 -5 Z transform of Exponential Signals Find the Z-transform of the signal: x(n) = 2 n for n > 0 = 0 for n < 0 Xz xnz z z n n nn n n n () == = = = = ∑∑ 00 0 2 2 The above sum converges to z z 2 if z > 2 2 n Copyright ©Andreas Spanias Ch 3 -6 Double Sided Exponential Find X(z) if the signal is x(n) = (1/3) n for n > 0 = 3 n for n < 0 z z z n n n n n n n = = ⎟ + =−∞ = = 1 33 1 3 10 ) 3 1 ( 1 1 ) 3 / ( 1 3 / ) ( z z z z X + = if 13 3 << z -5 -4 -3 -2 -1 0 1 2 3 4 5 0.1 0.5 0.8 1/3

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Copyright ©Andreas Spanias Ch 3 -7 Selected Properties of the Z-Transform Linearity: if: x n X z () and yn Y z then α β xn X z Yz + + Shifting: ) ( ) ( z X z m n x m ± ± Convolution: ) ( ) ( ) ( * ) ( z Y z X n y n x Scaling: ) / ( ) ( a z X n x a n Copyright ©Andreas Spanias Ch 3 -8 Delay of 7 Samples 7 (7 ) ( ) z X z −↔
Copyright ©Andreas Spanias Ch 3 -9 Selected Z-Transform Pairs Unit-Impulse: δ () n 1 Sinusoids: {} > + 1 , ) cos( 2 1 ) sin( 0 ), sin( 2 1 1 z z z z n n > + 1 , ) cos( 2 1 ) cos( 1 0 ), cos( 2 1 1 z z z z n n Sampled Unit-Step: 10 1 1 1 1 ,, n z z ≥↔ > Exponential Signals: an z za n > 0 Copyright ©Andreas Spanias Ch 3 -10 The Transfer Function To write the transfer function put the difference equation in the z-domain xn X z yn Yz and = = = M i i L i i i n y a i n x b n y 1 0 ) ( ) ( ) ( i M i i L i i i z z Y a z z X b z Y = = = 1 0 ) ( ) ( ) ( ) ( z X 1 a M a 0 b L b ... ... . . . . . . . . . . + + + + - - - 1 b 2 a . 1 z ) ( z Y 1 z 1 z 1 z 1 z

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Copyright ©Andreas Spanias Ch 3 -11 The Transfer Function (Cont.) The transfer function H(z) is defined as: M M L L z a z a z b z b b z X z Y z H + + + + + + = = ... 1 ...
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DSP_ch3 - DSP EEE 407/591 by Andreas Spanias Ph.D Chapter 3...

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