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Z -transforms

Z -transforms. Z -transforms. For discrete-time systems, z -transforms play the same role of Laplace transforms do in continuous-time systems As with the Laplace transform, we compute forward and inverse z -transforms by use of transforms pairs and properties. Bilateral Forward z -transform.

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Z -transforms

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  1. Z-transforms

  2. Z-transforms • For discrete-time systems, z-transforms play the same role of Laplace transforms do in continuous-time systems • As with the Laplace transform, we compute forward and inverse z-transforms by use of transforms pairs and properties Bilateral Forward z-transform Bilateral Inverse z-transform

  3. Region of the complex z-plane for which forward z-transform converges Four possibilities (z=0 is a special case and may or may not be included) Im{z} Im{z} Entire plane Disk Re{z} Re{z} Im{z} Im{z} Intersection of a disk and complement of a disk Complement of a disk Re{z} Re{z} Region of Convergence

  4. h[n] = d[n] Region of convergence: entire z-plane h[n] = d[n-1] Region of convergence: entire z-plane h[n-1]  z-1 H[z] h[n] = an u[n] Region of convergence: |z| > |a| which is the complement of a disk Z-transform Pairs

  5. Stability • Rule #1: For a causal sequence, poles are inside the unit circle (applies to z-transform functions that are ratios of two polynomials) • Rule #2: More generally, unit circle is included in region of convergence. (In continuous-time, the imaginary axis would be in the region of convergence of the Laplace transform.) • This is stable if |a| < 1 by rule #1. • It is stable if |z| > |a| and |a| < 1 by rule #2.

  6. Inverse z-transform • Yuk! Using the definition requires a contour integration in the complex z-plane. • Fortunately, we tend to be interested in only a few basic signals (pulse, step, etc.) • Virtually all of the signals we’ll see can be built up from these basic signals. • For these common signals, the z-transform pairs have been tabulated (see Lathi, Table 5.1)

  7. Example • Ratio of polynomial z-domain functions • Divide through by the highest power of z • Factor denominator into first-order factors • Use partial fraction decomposition to get first-order terms

  8. Example (con’t) • Find B0 by polynomial division • Express in terms of B0 • Solve for A1 and A2

  9. Example (con’t) • Express X[z] in terms of B0, A1, and A2 • Use table to obtain inverse z-transform • With the unilateral z-transform, or the bilateral z-transform with region of convergence, the inverse z-transform is unique

  10. Z-transform Properties • Linearity [Lathi, Section 5.1] • Right shift (delay) [Lathi, Section 5.2]

  11. Z-transform Properties • Convolution definition • Take z-transform • Z-transform definition • Interchange summation • Substitute r = n - m • Z-transform definition

  12. Example

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