03-JDSP-Z-Transforms

03-JDSP-Z-Transforms - CHAPTER - 3 3. THE Z TRANSFORM 3.1....

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CHAPTER - 3 3. THE Z TRANSFORM 3.1. INTRODUCTION n this chapter, we review the z transform and its properties. We then use the z transform to define the transfer function of a discrete-time system. At the end of the chapter we give two sets of problems; theory problems on the z transform and a series of J-DSP computer exercises. Discrete-time signals are described in the transform domain with the z transform and the discrete- time Fourier transform (DTFT). These two transformations have similar roles as the Laplace transform and the CFT for analog signals respectively. The z transform is defined as () . n n Xz xn z =−∞ = (3.1) where z is a complex variable. For a causal (right-handed) signal, i.e., if x(n)=0 for n<0 then the lower index on the sum starts from 0. The z transform describes the signal in the complex z domain and for finite-length causal signals the transform converges everywhere in z except for z=0 . Note that a unique z transform pair requires that the range of values of z for which the z transform exists, i.e. the region of I
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convergence (ROC), is specified. For causal signals the ROC extends outwards from the outermost pole of the z domain function. Example 3 . 1 : Consider the z transform of the unit impulse, δ (n) () 1 0 0 nf o r n elsewhere = = = (3.2) () . 1 n n n z hence n ROC z δδ =−∞ = ↔∀ Example 3 . 2 : Consider a finite-length signal ( ) n x n 5 0 . 2 5 1 . 5 0 . 5 2 . 1 1 0 2 3 4 5 This signal is non-zero only between the discrete-time indexes –2 to 3. Mathematically this signal can be written in terms of unit impulses as follows . 5( 2 ) 1 . 1 ) 2() ( 1 ) . 2 ) 2 . 3 ) xn n n n nn n =− + + + + −− + The z transform of this finite-length non-causal signal is given by 21 2 3 . 5 1 . 5 2 . 5 2 . 5 X zz z z z z + + − + The z transform converges for all values of z except z=0 and z= , i.e., ROC: z 0, z .
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Example 3 . 3 : A decaying exponential signal is given by () () 0 . 5 0 00 n x nf o r n for n = > = < The z transform of the causal discrete-time signal is given by, 000 0.5 0 . 5 n nn n nnn Xz xnz z z ∞∞∞ −− === ⎛⎞ ⎜⎟ ⎝⎠ ∑∑∑ This is a geometric series that converges for |z|>0.5 to 0.5 z z = Hence this z-domain transform has a ROC: |z|>0.5. Note that this simple rational function becomes zero at z=0 and infinity at z=0.5. Hence the function has a zero at zero z=0 and a pole at z=0.5. The z transform is a linear transformation and some of the basic properties of the z transform are given below. Given the transform pairs, xn X z y nY z ↔↔ The time shifting property ( ) m xn m z X z ± ±↔ (3.3) The convolution property ()*() ()() y nX z Y z (3.4) The scaling property (/ ) n α (3.5) Note that if the z transform is evaluated on the unit circle, i.e., for
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2 j s ze f T π =Ω = then the z transform becomes the discrete-time Fourier transform (DTFT), i.e., () ( ) jj n n Xe xne −Ω =−∞ = 3.2. THE TRANSFER FUNCTION The z transform of the impulse response of a filter is called the transfer function and is given by, n n Hz hnz = (3.6) Considering the difference equation we can get the transfer function in terms of filter parameters, i.e.,
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This note was uploaded on 11/07/2009 for the course FOFL 4531 taught by Professor Haiza during the Spring '09 term at Carlos Albizu.

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03-JDSP-Z-Transforms - CHAPTER - 3 3. THE Z TRANSFORM 3.1....

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