CE 40763 Digital Signal Processing Fall 1992 Z Transform Hossein Sameti Department of Computer Engineering Sharif University of Technology

Definition of Z-transform Vs. Complex function Complex variable Q: Why do we need Z-transform? A: For some signals, Fourier transform does not converge. e.g.: Relationship between and : Z-transform evaluated at unit circle corresponds to DTFT.  The role of Z-transform in DT is similar to Laplace transform in CT.

Convergence of Z-transform For what values of z, does the z-transform converge? DTFT of DTFT of exists if is absolutely summable. Z-transform exists and converges. Convergence only depends on r. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Definition of ROC Region of Convergence (ROC): ROC is a region in “z” domain that converges. ROC is radially symmetric. It also converges for If Z-transform converges for ROC does not depend on the angle. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

ROC Possibilities There are the following possibilities: All Z-domain except one or two points (origin / infinity) Inside a circle Outside a circle Between two circles (ring) ROC does not exist Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: ROC for a finite-length Signal ROC: everywhere except at the origin: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: ROC for another finite-length Signal ROC: everywhere except at infinity: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: ROC for RHS Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: ROC for LHS Same Z-transform as the previous example. However, the ROC is different. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Importance of ROC We have shown that X(z) needs the ROC to uniquely specify a sequence. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: ROC for RHS What is the ROC? Union? Intersection? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: ROC for BHS What is the ROC? Union? Intersection? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: ROC for BHS Z-transform does not exist. What is the ROC? Union? Intersection? Z-transform does not exist. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Properties of the ROCs of Z-Transforms (1) The ROC of X(z) consists of a ring in the z-plane centered about the origin (equivalent to a vertical strip in the s-plane) (2) The ROC does not contain any poles (same as in LT). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Book Chapter10: Section 1 More ROC Properties (3) If x[n] is of finite duration, then the ROC is the entire z-plane, except possibly at z = 0 and/or z = ∞. Why? Examples: CT counterpart Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

ROC Properties Continued Book Chapter10: Section 1 ROC Properties Continued (4) If x[n] is a right-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which |z| > ro are also in the ROC. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Book Chapter10: Section 1 Side by Side If x[n] is a left-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which 0 < |z| < ro are also in the ROC. (6) If x[n] is two-sided, and if |z| = ro is in the ROC, then the ROC consists of a ring in the z-plane including the circle |z| = ro. What types of signals do the following ROC correspond to? right-sided left-sided two-sided Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example: Book Chapter10: Section 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Book Chapter10: Section 1 Example, continued Clearly, ROC does not exist if b > 1 ⇒ No z-transform for b|n|. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Summary of Observations ROC is only a function of r. FLS ROC is everywhere except at the origin or infinity. RHS ROC is outside some circle. LHS ROC is inside some circle. BHS ROC is a ring or it does not exist. FT exists if ROC includes unit circle. Let’s prove one of these properties. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Properties of the Z-Transform Linearity: ROC: at least the intersection of ROC1 and ROC2. - Linearity helps us to find the z-transform of a signal by expressing the signal as a sum of elementary signals. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Linearity Example: Solution: Using linearity property: Earlier we showed that: If we set: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Time Shifting ROC: Similar ROC with some exceptions at 0 and infinity Example: ROC? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Scaling in the z-domain Example: Earlier: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Time Reversal Example: Earlier: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Differentiation in the z-domain Example: Earlier: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Convolution of Two Sequences ROC: at least the intersection of ROC1 and ROC2 Example: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Inverse Z-Transform Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Inverse Z-Transform Methods of calculating the inverse Z-transform: Cauchy theorem for calculating the inverse Z-transform (not covered) Series expansion Partial fraction expansion and using look-up tables Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Power Series Expansion Basic concept: expand X(z) in terms of powers of z-1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Power Series Expansion Example: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Partial Fraction Expansion and Lookup Table This method is applicable if we have a ratio of two polynomials in the form of . We would like to rewrite X(z) in the following manner: This expansion is performed in a way that the inverse Z-transform of can be easily found using a lookup table and using the linearity property, we can find x(n): Example: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Polynomial Factorization Any non-constant single-variable polynomial of degree N can be factored into N terms: If the coefficients of the polynomial are real, then the roots are either real or complex conjugates. If is a complex root, then is also a root. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Partial Fraction Expansion and Lookup Table Example: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Partial Fraction Expansion and Lookup Table Example: x(n) causal In the case of complex conjugate roots, the coefficients in the partial fraction expansion will be complex conjugates. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Partial Fraction Expansion Algebra: A = 1, B = 2 Example: Partial Fraction Expansion Algebra: A = 1, B = 2 Note, particular to z-transforms: 1) When finding poles and zeros, express X(z) as a function of z. 2) When doing inverse z-transform using PFE, express X(z) as a function of z-1. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

ROC III: ROC II: ROC I: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Linear Constant- Coefficient Difference Equation (LCCDE) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Classification of LTI Systems h(n),H (z) x(n),X(z) y(n),Y(z) LTI Systems h(n) Transfer function FIR IIR FIR has a finite number of non-zero values. It is easy to implement. IIR has infinite number of non-zero values. - why polynomials are good? They are continuous and differentiable With rational transfer function No rational transfer function Use difference equations for implementation

Example- LCCDE Cumulative Average System: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

General form of L.C.C.D.E. The general form for a linear constant-coefficient difference equation is given by: Given the difference equation of an IIR system and the initial condition of y(n), we can compute the output y(n) through the difference equation more efficiently using finite number of computations. Order of the system Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

LCCDE and LTI Systems Question: How can an LCCDE correspond to an LTI system? E.g.: It is not even a system, since a unique input does not correspond to a unique output. Q: How can we show this? A: If is a solution, is also a solution. Q: How can we resolve this problem? A: We need to add an initial condition (I.C.). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Calculating the Output using the IC Causal implementation Anti-causal implementation Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

LCCDE and Linearity By Combining the two answers, we get: Q: Does this system correspond to a linear system? A: No, zero input does not result in zero output. An LCCDE corresponds to a linear system, when IC=0. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

LCCDE and Time-invariance Q: Does this system correspond to a TI system? A: Consider . Now let’s shift the input: Output: Shift variant In order for an LCCDE to correspond to a TI system, IC=0. (for 𝑛 0 ≥0) In order for an LCCDE to correspond to an LTI system, IC=0. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

LCCDE and Causal LTI Systems In order an LCCDE to correspond to a causal LTI system : Initial rest condition (IRC)=0 E.g., if , for a first order LCCDE, the IRC should be as follows: E.g., if , for a second order LCCDE, the IRC should be as follows: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Example – Finding the impulse response Solution1:

Same Example with Z-Transform h(n),H (z) x(n),X(z) y(n),Y(z) Remember: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Poles/ Zeros Z-transform with rational transfer function: Zero: X(z)=0 Pole: X(z)=∞ Getting rid of the negative powers: Poles and zeros? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Pole-Zero plot A pole-zero plot can represent X(z) graphically. Example: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Pole-Zero plot If a polynomial has real coefficients, its roots are either real or occur in complex conjugates. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Pole-Zero Location and Time-Domain Behavior of Causal Signals Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Pole-Zero Location and Time-Domain Behavior of Causal Signals Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Pole-Zero Location and Time-Domain Behavior of Causal Signals (Double poles) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Pole-Zero Location and Time-Domain Behavior of Causal Signals (Double poles) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Pole-Zero Location and Time-Domain Behavior of Causal Signals (Two Complex-conjugate poles) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

LTI System Analysis in the z-domain: Stability For BIBO stability of an LTI system, we shall have Now, let’s see what happens in the z-domain: CONCLUSION: an LTI system is BIBO stable iff the ROC of its impulse response includes the unit circle. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

LTI System Analysis in the z-Domain: Causality Causal LTI systems have h(n) = 0 for n<0. ROC of the z-transform of a causal sequence is the exterior of a circle. CONCLUSION: an LTI system is causal iff the ROC of its system function is the exterior of a circle of radius r<∞ including the point z=∞. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

LTI System Analysis in the z-Domain: Causality and Stability Conditions for stability and causality are different and one does not imply the other. A causal system might be stable or unstable just as a non- causal system can be stable or unstable. For causal systems though, the condition of stability can be narrowed since: A causal system has z-transform with ROC outside a circle of radius r. The ROC of a stable system must contain the unit circle. Then a causal and stable system ROC must be 𝑧 >𝑟 with 𝑟<1. This means that a causal LTI system is BIBO stable iff all the poles of H(z) are inside the unit circle. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Causality and Stability of LTI Systems Example: Specify ROC and h(n) for the following cases: The system is stable: The system is causal: The system is anti-causal: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Summary Discussed z-transform, region of convergence, properties of z-transform and how the inverse z- transform can be calculated. Discussed LCCDEs and the concept of the pole-zero for their z-transform Next: we will analyze the properties of ideal filters Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology