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 Define the impulse function: {  k } = {1, 0, 0, 0,....} Define the unit step function: {u k } = {1, 1, 1, 1,....} (Convergent for |z| < 1)

Delay/Shift Property Let y(t) = x(t-T) (delayed by T and truncated at t = T) y k = y(kT) = x(kT-T) = x((k-1)T) = x k-1 ; y 0 = 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.

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...

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 +/- j  /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  /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).

Mapping Poles and Zeros A point in the Z-plane re j  will map to a point in the S- plane according to: Conjugate roots will generate a real valued polynomial in s of the form:

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

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

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

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.