1. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 81 Lecture 8: Z-Transforms l ROC and Causality |z|<ab<|z|b<|z|<aall |z| ( possibly excluding z=0) anticausalcausalmixedfinite-length

2. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 82 Z-Transform and Stability l The ROC for a stable sequence must contain the unit-circle! |z|<ab<|z|b<|z|<aall |z| ( possibly excluding z=0) stable, anticausalunstable, causalstable, mixedstable, finite-length

3. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 83 Poles and Zeros l A special class of Z-transforms can be expressed as a ratio of polynomials in z: {p k } are the poles; {c k } are the zeros. –All systems that are described by difference equations (all digital filters) have transfer functions that can be expressed this way.

4. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 84 Poles and Zeros l Example: x(n) = (0.8) n u(n) + (1.25) n u(n) O xx |z| > 0.8 |z| > 1.25 unstable system

5. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 85 Poles and Zeros l Example: x(n) = (0.8) n u(n) - (1.25) n u(-n-1) O xx |z| > 0.8 |z| < 1.25 stable system

6. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 86 Stability and Causality l A system (or sequence) that is both stable and causal must have all its transfer function (or z-transform) poles inside the unit circle! x x cannot be stable and causal x x can be stable and causal x x x x

7. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 87 Frequency Spectrum l Discrete-time Fourier Transform l System Frequency Response

8. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 88 Frequency Spectrum l Frequency spectrum is the z-transform evaluated on the unit circle!     low frequencies medium frequencies high frequencies    

9. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 89 Frequency Spectrum and the Z-Transform l Spectrum magnitude at a particular frequency is the ratio of the product of the distances to the zeros over the product of the distances to the poles. product of distances to zeros product of distances to poles

10. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 810 Frequency Spectrum and the Z-Transform l What does this mean? –poles “pull” the spectrum up –zeros “push” the spectrum down

11. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 08-Oct, 98EE421, Lecture 811 Moving Average Filter l Example (moving average filter): y(n) = (4/7)x(n) + (2/7)x(n-1) + (1/7)x(n-2) h(n) = (4/7)  (n) + (2/7)  (n-1) + (1/7)  (n-2) zeros poles at z = 0 (two of them)