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- 1. EC533: Digital Signal Processing 5 l l Lecture 5 The Z-Transform
- 2. 5.1 - Introduction • The Laplace Transform (s domain) is a valuable tool for representing, analyzing & designing continuos-time signals & l d l systems. • The z transform is convenient yet invaluable tool for representing, z-transform analyzing & designing discrete-time signals & systems. • The resulting transformation from s-domain to z-domain is called z-transform. z-transform • The relation between s-plane and z-plane is described below : z = esT • The z-transform maps any point s = σ + jω in the s-plane to z- plane (r θ).
- 3. 5.2 – The Z-Transform For continuous-time signal, Time Domain Ti D i S‐Domain For discrete-time signal, Ƶ Z‐Domain Time Domain Ƶ-1 Causal System where,
- 4. 5.2.1 – Z-Transform Definition • The z-transform of sequence x(n) is defined by ∞ −n X ( z ) = ∑ x ( n) z Two sided z transform Bilateral z transform n = −∞ For causal system ∞ −n X (z) = ∑ x(n)z One sided z transform Unilateral z transform n=0 • The z transform reduces to the Discrete Time Fourier transform (DTFT) if r=1; z = e−jω. DTFT ∞ X ( e jω ) = ∑ n = −∞ x ( n ) e − jω n
- 5. 5.2.2 – Geometrical interpretation of z transform z-transform • The point z = rejω is a p vector of length r from Im z origin and an angle ω with j respect to real axis. z = rejω r ω Re R z • Unit circle : The contour -1 1 |z| = 1 is a circle on the z- plane with unity radius l ith it di -j DTFT is to evaluate z-transform on a unit circle.
- 6. 5.2.3 – Pole-zero Plot • A graphical representation Im z of z-transform on z-plane j – Poles denote by “x” and – zeros denote by “o” Re z -1 1 -j
- 7. Example Find the z-transform of, Solution: It’s a geometric sequence Recall: Sum of a Geometric Sequence where, a: first term, r: common ratio, n: number of terms
- 8. 5.3 – Region Of Convergence (ROC) • ROC of X(z) is the set of all values of z for which X(z) attains a finite value. • Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|<∞, is called the region of convergence. ∞ ∞ −n | X ( z ) |= ∑ x(n) z = ∑ | x(n) || z |− n < ∞ n = −∞ n = −∞ ∞ −n Im ∑ | x(n)r |< ∞ n = −∞ ROC is an annual ring centered on r the origin. Re Rx − <| z |< Rx + ROC = {z = re jω | Rx − < r < Rx + }
- 9. Ex. 1 Find the z-transform of the following sequence x = {2 -3, 7 4 0 0 ……..} {2, 3 7, 4, 0, 0, } ∞ X ( z) = ∑ x[n]z − n = 2 − 3 z −1 + 7 z − 2 + 4 z −3 n = −∞ 2 z 3 − 3z 2 + 7 z + 4 = , |z|>0 z 3 The ROC is the entire complex z - plane except the origin. Ex. 2 Find the z-transform of δ [n] ∞ X ( z) = ∑ δ [ n] z − n = 1 n = −∞ with an ROC consisting of the entire z - plane.
- 10. Ex 3 Find the z transform of δ [n -1] Ex. z-transform 1] ∞ 1 X ( z) = n = −∞ ∑ δ [n − 1] z − n = z −1 = z with an ROC consisting of the entire z - plane except z = 0 . Ex. 4 Find the z-transform of δ [n +1] ∞ X ( z) = ∑ δ [n + 1] z − n = z n = −∞ with an ROC consisting of the entire z - plane except z = ∞, i.e., there is a pole at infinity.
- 11. Ex.5 Find the z-transform of the following right-sided sequence (causal) x [ n] = a u [ n] n ∞ ∞ −n X ( z ) = ∑ a u[n]z n = ∑ (az −1 ) n n = −∞ n =0 This f Thi form to find inverse fi d i ZT using PFE
- 12. Ex.6 Find the z-transform of the following left-sided sequence
- 13. Ex. 7 Find the z-transform of Rewriting x[n] as a sum of left-sided and right sided sequences left sided right-sided and finding the corresponding z-transforms,
- 14. where Notice from the ROC that the z-transform doesn’t exist for b > 1
- 15. 5.3.1 – Characteristic Families of Signals with Their Corresponding ROC
- 16. 5.3.2 – Properties of ROC • A ring or disk in the z-plane centered at the origin. g p g • The Fourier Transform of x(n) is converge absolutely iff the ROC includes the unit circle. • The ROC cannot include any poles • Finite Duration Sequences: The ROC is the entire z-plane except possibly z=0 or z=∞. • Right sided sequences (causal seq.): The ROC extends outward from the outermost finite pole in X(z) to z=∞. • Left sided sequences: The ROC extends inward from the innermost nonzero pole in X(z) to z=0. • Two-sided sequence: The ROC is a ring bounded by two circles passing through two pole with no poles inside the ring circles passing through two pole with no poles inside the ring
- 17. 5.4 - Properties of z-Transform (1) Linearity : a x[n] + b y[n] ←→ a X ( z ) + b Y ( z ) ⎛z⎞ (4) Z - scale Property : a x[n] ←→ X ⎜ ⎟ n ⎝a⎠ 1 (5) Time Reversal : x [−n] ←→ X ( ) l z (6) Convolution : h [n] ∗ x [n] ←→ H ( z ) X ( z ) Transfer Function
- 18. 5.5 - Rational z-Transform For most practical signals, the z-transform can be expressed as a ratio of two polynomials f l l N ( z) ( z − z1 )( z − z 2 ) L ( z − z M ) X ( z) = =G D( z ) ( z − p1 )( z − p2 ) L ( z − p N ) where G is scalar gain, z1 , z 2 , L, z M are the zeroes of X(z), i.e., the roots of the numerator polynomial and p1 , p2 , L , p N are the poles of X(z), i.e., the roots of the denominator polynomial.
- 19. 5.6 - Commonly used z-Transform pairs Sequence z‐Transform ROC δ[n] 1 All values of z All values of z u[n] 1 |z| > 1 1 − z −1 1 αnu[n] |z| > |α| 1 − αz −1 αz −1 nαnu[n] (1 − αz −1 ) 2 |z| > |α| |z| > |α| 1 (n+1) αnu[n] |z| > |α| (1 − αz −1 ) 2 1 − (r cos ω0 ) z −1 (rn cos ω on) u[n] |z| > |r| 1 − (2r cos ω0 ) z −1 + r 2 z −2 1 − (r sin ω0 ) z −1 (rn sin ωon) [n] |z| > |r| 1 − (2r cos ω0 ) z −1 + r 2 z −2
- 20. 5.7 - Z-Transform & pole-zero distribution & Stability considerations y Thus, unstable z stable R.H.S. Mapping between S-plane & Z-plane is done as follows: L.H.S. 1) Mapping of Poles on the jω‐axis of the s‐domain to the z‐domain 1) Mapping of Poles on the jω axis of the s domain to the z domain ωs/4 Maps to a unit circle & represents Marginally stable terms 1 ωs/2 ω=0 ω=ωs 3ωs/4
- 21. 5.7 - Z-Transform & pole-zero distribution & Stability considerations – cont. y 2) Mapping of Poles in the L.H.S. of the s‐plane to the z‐plane Maps to inside the unit circle & represents stable terms & the system is stable. 3) Mapping of Poles in the R.H.S. of the s‐plane to the z‐plane Outside the unit circle & represents unstable terms. Discrete Systems Stability Testing Steps 1) Find the pole positions of the z-transform. 2) If any pole is on or outside the unit circle. (Unless coincides with zero on the unit circle) The system is unstable.
- 22. 5.7.1 - Pole Location and Time-domain Behavior of Causal Signals
- 23. 5.7.2 - Stable and Causal Systems Causal Systems : ROC extends outward from the outermost pole. C lS t t d t df th t t l Im R Re Stable Systems : ROC includes the unit circle. Im A stable system requires that its Fourier transform is 1 uniformly convergent. Re

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