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ELEN 5346/4304 DSP and Filter Design Fall 2008 1 Lecture 7: Z-transform Instructor: Dr. Gleb V. Tcheslavski Contact:

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Presentation on theme: "ELEN 5346/4304 DSP and Filter Design Fall 2008 1 Lecture 7: Z-transform Instructor: Dr. Gleb V. Tcheslavski Contact:"— Presentation transcript:

1 ELEN 5346/4304 DSP and Filter Design Fall 2008 1 Lecture 7: Z-transform Instructor: Dr. Gleb V. Tcheslavski Contact: gleb@ee.lamar.edu gleb@ee.lamar.edu Office Hours: Room 2030 Class web site: http://ee.lamar.edu/gleb/ds p/index.htm http://ee.lamar.edu/gleb/ds p/index.htm

2 ELEN 5346/4304 DSP and Filter Design Fall 2008 2 Definitions Z-transform converts a discrete-time signal into a complex frequency-domain representation. It is similar to the Laplace transform for continuous signals. If (where) it exists! (7.2.1) n is an integer time index; is a complex number;  - angular freq. When the magnitude r =1, If it exists! (7.2.2)

3 ELEN 5346/4304 DSP and Filter Design Fall 2008 3 Region of Convergence (ROC) (7.3.2) Since z is complex: The Region of convergence (ROC) is the set of points z in the complex plane, for which the summation is bounded (converges): (7.3.1) In general, z-transform exists for (7.3.3) (7.3.4) (7.3.5) r - r + Re Im

4 ELEN 5346/4304 DSP and Filter Design Fall 2008 4 Region of Convergence (ROC) Examples of ROCs from Mitra

5 ELEN 5346/4304 DSP and Filter Design Fall 2008 5 Region of Convergence (ROC) Example 7.1: Let x n = a n There are no values of z satisfying: Example 7.2: Let x n = a n u n – a causal sequence a Re Im (7.5.1) ROC We can modify (7.5.1) as (7.5.2) x roots of numerator: X(z) = 0 roots of denominator: X(z)  

6 ELEN 5346/4304 DSP and Filter Design Fall 2008 6 Region of Convergence (ROC) Example 7.3: Let x n = -a n u -n-1 – an anticausal sequence (7.6.1) a Re Im x Conclusion 1: z-transform exists only within the ROC! Conclusion 3: poles cannot exist in the ROC; only on its boundary. Note: if the ROC contains the unit circle (|z| = 1), the system is stable. Conclusion 2: z-transform and ROC uniquely specify the signal.

7 ELEN 5346/4304 DSP and Filter Design Fall 2008 7 The transfer function Consider an LCCDE: Time shift: and take z-transform utilizing time shift LTI: The system transfer function (7.7.1) (7.7.2) (7.7.3) (7.7.4) (7.7.5)

8 ELEN 5346/4304 DSP and Filter Design Fall 2008 8 Rational z-transform Frequently, a z-transform can be described as a rational function, i.e. a ratio of two polynomials in z -1 : Here M and N are the degrees of the numerator’s and denominator’s polynomials. An alternative representation is a ratio of two polynomials in z: (7.8.1) (7.8.2) Finally, a rational z-transform can be written in a factorized form: (7.8.3) zeros: numerator = 0 Poles: denominator = 0

9 ELEN 5346/4304 DSP and Filter Design Fall 2008 9 Notes on poles of a system function Positions of poles of a transfer function are used to evaluate system stability. Let assume a single real pole at z = . Therefore: The difference equation is: Therefore, the impulse response is: for Iff |  | < 1, h n decays as n  and the system is BIBO stable; otherwise, h n grows without limits. Therefore, poles of a stable system (and signals in fact) must be inside the unit circle. Zeros may be placed anywhere. Zeros at the origin produce a time delay. (7.9.1) (7.9.2) (7.9.3) (7.9.4)

10 ELEN 5346/4304 DSP and Filter Design Fall 2008 10 The transfer function and the Frequency response BIBO: where z j are zeros and p i are poles of the transfer function. BIBO: (7.10.1) (7.10.2) (7.10.3)

11 ELEN 5346/4304 DSP and Filter Design Fall 2008 11 The transfer function and the Frequency response A good way to evaluate the system’s frequency response: When the frequency approaches a pole, the frequency response has a local maximum, a zero forces the response to a local minimum. For real systems, poles and zeros are symmetrical with respect to the real axis. (7.11.1) polezero Zero-padded

12 ELEN 5346/4304 DSP and Filter Design Fall 2008 12 The transfer function and the SFG Poles of H(z) correspond to the eigenvalues of the system matrix. (7.12.1) (7.12.2) (7.12.3)

13 ELEN 5346/4304 DSP and Filter Design Fall 2008 13 More on Transfer function 1)N > M: zeros at z = 0 of multiplicity N-M 2)M > N: poles at z = 0 of multiplicity M-N zeros poles (7.13.1) (7.13.2)

14 ELEN 5346/4304 DSP and Filter Design Fall 2008 14 Types of digital filters 1. FIR (“all-zero”) filter: (7.14.1) All poles are at z = 0: a “nest of poles” ROC: the entire z-plane except of the origin (z = 0). FIR filters are stable.

15 ELEN 5346/4304 DSP and Filter Design Fall 2008 15 Types of digital filters 2. IIR (“all-pole”) filter: All zeros are at z = 0: a “nest of zeros” (7.15.1)

16 ELEN 5346/4304 DSP and Filter Design Fall 2008 16 Types of digital filters 3. General IIR (“zero-pole”) filter: (7.16.1)

17 ELEN 5346/4304 DSP and Filter Design Fall 2008 17 On test signals… Calculate and compare to (7.17.1) (7.17.2) (7.17.3) We don’t need any other that a delta function test signals since a unit-pulse response is a complete system’s description.

18 ELEN 5346/4304 DSP and Filter Design Fall 2008 18 Types of sequences and convergence 1. Two-sided: Converges everywhere except of z = 0 and z =  (7.18.1)

19 ELEN 5346/4304 DSP and Filter Design Fall 2008 19 Types of sequences and convergence 2. Right-sided: (7.19.1) Assume: if converges at z = z 0, converges for |z| > | z 0 | Blows up at z =  ROC: r - < |z| <  - exterior ROC For a causal sequence: |z| > r - = max|p k | - a max pole of G(z) To be causal, a sequence must be right-sided (necessary but not sufficient)

20 ELEN 5346/4304 DSP and Filter Design Fall 2008 20 Types of sequences and convergence 3. Left-sided: (7.20.1) Blows up at z = 0Converges at z 0 ROC: 0 < |z| < r + - interior ROC When encountering an interior ROC, we need to check convergence at z = 0. If the sequence “blows up” at zero – it’s an anti-causal sequence

21 ELEN 5346/4304 DSP and Filter Design Fall 2008 21 Properties from Mitra

22 ELEN 5346/4304 DSP and Filter Design Fall 2008 22 Common pairs

23 ELEN 5346/4304 DSP and Filter Design Fall 2008 23 Inverse z-transform (7.23.1) Where C is a counterclockwise closed path encircling the origin and is entirely in the ROC. Contour C must encircle all the poles of X(z). In general, there is no simple way to compute (7.23.1) A special case: C is the unit circle (can be used when the ROC includes the unit circle). The inverse z-transform reduces to the IDTFT. (7.23.2)

24 ELEN 5346/4304 DSP and Filter Design Fall 2008 24 Inverse z-transform A. Via Cauchy residue theorem For all poles of X(z)z n-1 inside C (contour of integration) Where  i are the residues of X(z)z n-1 for a pole of multiplicity k: Residue function: (7.24.1) (7.24.2) (7.24.3)

25 ELEN 5346/4304 DSP and Filter Design Fall 2008 25 Inverse z-transform: Example a Re Im x C Example:  0 is a residue of X(z)z -n-1 at z=0 – involves pole of multiplicity –n wnen n < 0. multiplicity

26 ELEN 5346/4304 DSP and Filter Design Fall 2008 26 Inverse z-transform B. Via recognition (table look-up) Example: Therefore: Sometimes, the z-transform can be modified such way that it can be found in a table…

27 ELEN 5346/4304 DSP and Filter Design Fall 2008 27 Inverse z-transform C. Via long division 1. Right-sided z-transform sequences can be expanded into a power series in z -1. The coefficient multiplying z -n is the n th sample of the inverse z-transform. Example: Lower powers first: and long division: x0x0 x1x1 x2x2 x3x3 x4x4

28 ELEN 5346/4304 DSP and Filter Design Fall 2008 28 Inverse z-transform 2. Left-sided z-transform sequences – into a power series in z 1 … Example: Multiply both numerator and denominator by z 2 … Long division… x0x0 x -1 x -2 x -3 x -4 x1x1 Non-causal

29 ELEN 5346/4304 DSP and Filter Design Fall 2008 29 Inverse z-transform Example: not suitable for long division! Example:

30 ELEN 5346/4304 DSP and Filter Design Fall 2008 30 Inverse z-transform D. Via partial fraction expansion (PFE) If the degree of the numerator is equal or greater than the degree of the denominator: M  N, G(z) is an improper polynomial. Then: A proper fraction: M 1 < N (7.30.1) (7.30.2) Then: Simple poles: multiplicity of 1. (7.30.3)

31 ELEN 5346/4304 DSP and Filter Design Fall 2008 31 Inverse z-transform Here  l is a residue poles Therefore: (7.31.1) (7.31.2) This method is suitable for complex poles. Problem: large polynomials are hard to manipulate… ??QUESTIONS??


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