ELEN 5346/4304 DSP and Filter Design Fall Lecture 7: Z-transform Instructor: Dr. Gleb V. Tcheslavski Contact: Office Hours: Room 2030 Class web site: p/index.htm p/index.htm

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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

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

ELEN 5346/4304 DSP and Filter Design Fall 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)  

ELEN 5346/4304 DSP and Filter Design Fall 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.

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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.

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

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

ELEN 5346/4304 DSP and Filter Design Fall 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.

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

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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

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

ELEN 5346/4304 DSP and Filter Design Fall Common pairs

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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

ELEN 5346/4304 DSP and Filter Design Fall 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…

ELEN 5346/4304 DSP and Filter Design Fall 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

ELEN 5346/4304 DSP and Filter Design Fall 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

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

ELEN 5346/4304 DSP and Filter Design Fall 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)

ELEN 5346/4304 DSP and Filter Design Fall 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??