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Z Transform Primer

Z Transform Primer. Basic Concepts. Consider a sequence of values: {x k : k = 0,1,2,... } These may be samples of a function x(t), sampled at instants t = kT; thus x k = x(kT). The Z transform is simply a polynomial in z having the x k as coefficients:. Fundamental Functions.

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Z Transform Primer

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  1. Z Transform Primer

  2. Basic Concepts • Consider a sequence of values: {xk : k = 0,1,2,... } • These may be samples of a function x(t), sampled at instants t = kT; thus xk = x(kT). • The Z transform is simply a polynomial in z having the xk as coefficients:

  3. Fundamental Functions • Define the impulse function: {dk} = {1, 0, 0, 0,....} • Define the unit step function: {uk} = {1, 1, 1, 1,....} (Convergent for |z| < 1)

  4. Delay/Shift Property • Let y(t) = x(t-T) (delayed by T and truncated at t = T) yk = y(kT) = x(kT-T) = x((k-1)T) = xk-1; y0 = 0 • Let j = k-1 ; k = j + 1 • The values in the sequence, the coefficients of the polynomial, slide one position to the right, shifting in a zero.

  5. The Laplace Connection • Consider the Laplace Transforms of x(t) and y(t): • Equate the transform domain delay operators: • Examine s-plane to z-plane mapping . . .

  6. S-Plane to Z-Plane Mapping Anything in the Alias/Overlay region in the S-Plane will be overlaid on the Z-Plane along with the contents of the strip between +/- jp/T. In order to avoid aliasing, there must be nothing in this region, i.e. there must be no signals present with radian frequencies higher than w = p/T, or cyclic frequencies higher than f = 1/2T. Stated another way, the sampling frequency must be at least twice the highest frequency present (Nyquist rate).

  7. Mapping Poles and Zeros A point in the Z-plane rejq will map to a point in the S-plane according to: Conjugate roots will generate a real valued polynomial in s of the form:

  8. Example 1: Running Average Algorithm (Non-Recursive) Z Transform Block Diagram Transfer Function Note: Each [Z-1] block can be thought of as a memory cell, storing the previously applied value.

  9. Example 2: Trapezoidal Integrator (Recursive) Z Transform Block Diagram Transfer Function

  10. Ex. 2 (cont) Block Diagram Manipulation Intuitive Structure Explicit representation of xk-1 and yk-1 has been lost, but memory element usage has been reduced from two to one. Equivalent Structure

  11. Ex. 2 (cont)More Block Diagram Manipulation Note that the final form is equivalent to a rectangular integrator with an additive forward path. In a PI compensator, this path can be absorbed by the proportional term, so there is no advantage to be gained by implementing a trapezoidal integrator.

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