ece306-11 - ECE 306 -11 Z Transform Rational Z-Transform...

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1 Z Transform Rational Z-Transform The inverse of the z-transform Z. Aliyazicioglu Electrical and Computer Engineering Department Cal Poly Pomona ECE 306 -11 ECE 306-11 2 Rational Z-Transform Poles and Zeros The poles of a z-transform are the values of z for which if X(z)= The zeros of a z-transform are the values of z for which if X(z)=0 M finite zeros at X(z) is in rational function form ( )( ) ( ) () ( ) 12 0 01 2 ... . . . M MN N z zzz zz b Nz Xz z D z a zpzp zp −+ −− == 1 1 M k NM k N k k z z G z Dz z p = = N finite poles at , ,..., M z zz z = , ,..., N zpp p = and |N-M| zeros if N>M or poles if M>N at the origin z=0 1 0 1 0 ... . . . M k M k Mk N N k N k k bz bb z b z a az az = = ++ + = +
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2 ECE 306-11 3 Rational Z-Transform Example: Poles and Zeros We can represent X (z) graphically by a pole-zero plot in complex plane. Shows the location of poles by ( x ) Shows the location of zeros by ( o ). Definition of ROC of a z-transform should not contain any poles. Determine the pole-zero plot for the signal Im(z) a Re(z) () n x na u n = 1 1 1 z Xz az z a == The z-transform is One zero at z 1 =0 One pole at p 1 =a . p 1 =a is not included in the ROC x o 0 ECE 306-11 4 Rational Z-Transform The z-transform is Example.2: Determine the pole-zero plot for the signal Re(z) Im(z) 07 0e l s e w h e r e = n an xn 0 a > ( ) 8 1 88 7 1 17 0 1 1 = = n n az za a z az z z a ( )( ) ( ) 12 7 7 ... −− = zz zz zz z The zeros the poles at p=0 ROC: the entire z-plane except z=0 2/ j kM k e π = ,. … 8 1 j e = 4/ 8 2 j e = 14 /8 7 j e = a o 0 o o o o o o o x
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3 Rational Z-Transform Pole Location and Time-Domain behavior for Causal Signals For the signal The z-transform is One zero at z 1 =0 . One pole at p 1 =a. () n x na u n = 0 a > 1 1 1 z Xz az z a == Causal signals with poles outside the unit circle become unbounded. The signal is decaying 0<a<1 The signal is fixed if a=1 The signal is growing if a>1 The signal alternates if a is negative n x(n) 0<a<1 n x(n) a>1 Im(z) a Re(z) x o 0 ECE 306-11 6 Rational Z-Transform Pole Location and Time-Domain behavior for Causal Signals Impulse invariance mapping is z = e s T 1 1 -1 -1 Im{ s } Re{ s } 1 Im{ z } Re{ z } s = -1 ± j z = 0.198 ± j 0.31 ( T = 1 s) s = 1 ± j z = 1.469 ± j 2.287 ( T = 1 s) f j s 2 π = 1 2 1 max > = s f f Laplace Domain Z Domain Left-hand plane Inside unit circle Imaginary axis Unit circle Right-hand plane Outside unit circle
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4 ECE 306-11 7 Rational Z-Transform The system Function of a Linear Time –Invariant System The relationship in the z-domain is called system function The input sequence x(n) The output sequence y(n) H(z) is obtained We can take the inverse of z-transform to find h(n).
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This note was uploaded on 03/11/2008 for the course ECE 306 taught by Professor Aliyazicioglu during the Winter '08 term at Cal Poly Pomona.

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ece306-11 - ECE 306 -11 Z Transform Rational Z-Transform...

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