EC-4.6 The Z-Transform

What this is and why it exists

Anything computed rather than measured arrives as a sequence: a list of values at whole-numbered instants, with nothing in between. The Laplace transform has no purchase on that, because there is no continuous function to integrate.

The Z-transform is what Laplace becomes for sequences. The integral becomes a sum, and the property that made Laplace useful reappears in a new guise. Where differentiation became multiplication, delay becomes multiplication, and a difference equation turns into algebra in exactly the same way.

It is worth meeting as mathematics before meeting it inside a filter design course. The geometry of the two planes is the part that carries over, and the part most often skipped.

The vocabulary

• Sequence — a list of values indexed by whole numbers.
• Difference equation — an equation relating a sequence's value to its earlier values.
• Delay — shifting a sequence one step later in the index.
• Region of convergence — here a ring in the complex plane rather than a half plane.
• Unit circle — the circle of radius one in the transform plane, which plays the role of the imaginary axis.
• Causal sequence — a sequence that is zero before the starting index.
• Long division — inverting a transform by dividing the polynomials and reading off the first few values.
• Mapping — the relation between the Laplace plane and the Z plane produced by sampling.

The mental model

The transform is a sum: each term is a sequence value multiplied by z raised to the power minus the index. The variable z is complex, for the same reason s was.

The property that matters is the delay. Shifting a sequence one step later multiplies its transform by z to the power minus one. Nothing else. So a difference equation, which relates a value to its predecessors, becomes a rational expression in z with no derivatives anywhere.

That is the entire method, and it is the exact counterpart of what Laplace does to a differential equation.

The region of convergence is a ring rather than a half plane, because the sum has terms in both positive and negative powers. Whether the ring includes the unit circle decides whether the sequence is stable. As with Laplace, the same expression with a different ring is a different sequence.

Now the geometry, which is the part worth carrying forward.

Sampling a continuous signal maps the Laplace plane onto the Z plane by an exponential. The left half of the s plane, where signals decay, maps to the inside of the unit circle. The imaginary axis, where signals oscillate without decaying, maps onto the unit circle itself. The right half maps outside.

So the stability question changes shape but not meaning. In continuous time, stable means poles in the left half plane. In discrete time, stable means poles inside the unit circle. Same statement, different picture.

One consequence of this mapping deserves flagging now. It is not one to one. The exponential is periodic in the imaginary direction, so many points of the s plane land on the same point of the z plane. Frequencies that differ by the sampling rate become indistinguishable after sampling. That is aliasing, and it is taken properly in the signals and systems subject rather than here. It is named here so the mapping does not look harmless.

Inverting is done two ways and it is worth having both. Partial fractions gives a closed form for the sequence, term by term, from a short table. Long division gives the first several values directly, which is often all a sanity check needs. It also works when the algebra of partial fractions is awkward.

What you should now be able to explain or do

• Say why a sequence needs a different transform from a continuous signal.
• Use the delay property to turn a difference equation into algebra.
• Say what the region of convergence looks like here and what it decides.
• Describe how the Laplace plane maps to the Z plane under sampling.
• Invert a transform both by partial fractions and by long division.

Check yourself