Chapter five

This chapter is on the theme of discrete time linear models, for example:

y_k+a₁y_k-1 + ...+ a_ny_k-n = b₁u_k-1 + ...+ b_nu_k-n

where y(t) is the output, u(t) the input and a_i , b_i are model parameters. The subscript 'k' denotes the sampling index.

This section focuses on the concept of sampling. why would we sample and how this is expressed through the mathematical tool of Z-transforms? We also consider common signals and the links between pole positions and behaviours.

Video overviews

• Z-transforms

• Inverse Z-transforms

1. Introduction to sampling

How do we sample a continuous time signal and how is this process captured with convenient mathematical tools?

Sampling continuous data (PDF, 553 KB)

2. Z-transforms of common signals

Using the Z-transform definition, tabulate the Z-transforms for common signals. Students should draw analogies with Laplace transforms of the same signals.

• Common signals (PDF, 533 KB)

• Use MATLAB for complex signals

3. Impact of sampling time

Unlike with Laplace transforms, changing the sampling time changes the Z-transform, and associated poles, for a fixed underlying signal.

Changing the same time and poles (PDF, 716 KB)

4. Concepts of aliasing

After sampling, very fast frequencies cannot be observed and appear like slower frequencies; this is called aliasing. The fastest observable frequency is linked to the sampling time.

Aliasing introduction (PDF, 820 KB)

5. Linking behaviours to poles

This looks at the links between Z-transforms, and specifically the poles, and the underlying signal behaviour. Students may wish to draw analogies with Laplace transforms of the same signals.

Significance of poles (PDF, 808 KB)

6. Inverse Z-transforms

Given a Z-transform, how do I find the underlying time domain signal? The best method is using tables and and/or use a computer.

• Inverse z-transforms (PDF, 535 KB)

• Inverse transform by recursion (PDF, 778 KB)

• Convolution summations (PDF, 633 KB)

7. Final value theorem

At times only the asymptotic value of a signal is required. This can be inferred very efficiently using the final value theorem (FVT).

Examples of the FVT (PDF, 643 KB)