SC · Sampling and discrete time

The z-Transform

The z-transform is the Laplace transform of a discrete signal. Bundling a one-step delay into z⁻¹ turns a sequence into a function of z, where a single pole sets the response. Move the pole on the z-plane and watch the discrete response decay inside the unit circle and diverge outside it.

Move the pole inside the unit circle to settle

On the left is the z-plane, with × marking a pole pair z = r·e^(±jθ). The green disk is inside the stable unit circle, and that circle is the boundary between stable and unstable. Below is the discrete response x[n] = r^n cos(θn) that pole makes, as a stem plot. Use the two sliders to move the pole’s radius r and angle θ, pulling the pole inside the unit circle so the response decays to 0. r is the distance, θ is the oscillation speed.

z-plane (× = pole, circle = unit circle)

discrete response x[n] = r^n cos(θn)

radius r1.12

angle θ0.90 rad

Pole position and discrete response

|z| = 1.12 · θ = 0.90 rad

x[n] = 1.12^n cos(0.90 n)

The pole is outside the unit circle, so |z| = r is above 1. The discrete response r^n cos(θn) grows at every step and diverges. A pole outside the circle, an unstable system.

Unstable (outside circle)

z⁻¹ is a one-step delay

The z-transform bundles a discrete signal x[n] as X(z) = Σ x[n] z⁻ⁿ. Just as the continuous Laplace transform takes e^(st) as its basis, the discrete one takes zⁿ. The key property is delay. Push the signal back one step and x[n−1] ⟷ z⁻¹ X(z), multiplying once by z⁻¹. So a difference equation y[n] = a·y[n−1] + x[n] becomes the algebraic Y(z)(1 − a z⁻¹) = X(z), giving the transfer function H(z) = Y/X = 1/(1 − a z⁻¹) = z/(z − a). Just as a differential equation became algebra in s, a difference equation has become algebra in z.

Observedelay x[n−1] ⟷ z⁻¹ X(z)

A one-step delay is multiply by z⁻¹.

Choosepole at z = r : mode = ?

A pole at radius r gives the mode rⁿ.

Fill instable ⟺ | pole | ?

Stable means all poles inside the unit circle.

On your ownpole angle θ ⟹ oscillation cos(?)

The angle θ is the oscillation per step.

The discrete mode a pole sets

The pole of H(z) = z/(z − a) is the z = a that makes the denominator zero. The natural response this pole makes is aⁿ, a sequence built by multiplying by a at every step. If the pole is complex, a = r·e^(jθ), the response takes the form rⁿ cos(θn). The radius r is the ratio the amplitude is multiplied by each step, so r < 1 shrinks and r > 1 grows. The angle θ is the phase turned per step, the oscillation frequency. At θ = 0 it is plain growth or decay with no oscillation; the closer θ to π, the faster the oscillation, flipping sign every step. It is the continuous e^(σt)cos(ωt) with σ carried over into r and ω into θ.

The imaginary axis becomes the unit circle

A discrete system is stable when all its poles lie inside the unit circle (|z| < 1). If even one pole is outside, that mode rⁿ diverges; on the circle it oscillates without dying, marginal. What was the left half-plane for stability in continuous time has become the inside of the unit circle in discrete time. The two are linked by z = e^(sT), where T is the sampling interval. Put the imaginary axis s = jω into this and z = e^(jωT), a unit circle of radius 1. The left half-plane (σ < 0) maps to |z| = e^(σT) < 1, inside the circle; the right half maps outside. Sampling has wrapped the s-plane onto the unit circle. So the pole story of D2 repeats in discrete time, with only the boundary changed from a straight line to a circle.

Back to the first screen

With the pole radius r above 1 the pole sat outside the unit circle, so the discrete response rⁿ grew at every step and diverged; pulling r below 1 brought the pole inside and the response decayed to 0. Set r exactly to 1 and it oscillated forever at the same size on the circle. The larger the angle θ, the more it turned per step and the faster it oscillated. The pole’s distance r from the origin was the decay rate, and the angle θ was the oscillation frequency. The story from the continuous s-plane repeated in discrete time, with only the boundary of stability changed from the imaginary axis to the unit circle.

The z-transform X(z) = Σ x[n] z⁻ⁿ is the Laplace transform of a discrete signal. Because a one-step delay = multiply by z⁻¹, a difference equation becomes algebra in z, and the pole z = a of the transfer function H(z) = z/(z − a) sets the natural mode aⁿ. For a complex pole a = r·e^jθ the response is rⁿ cos(θn): r is the decay ratio (r<1 decays), θ is the oscillation per step. All poles inside the unit circle (|z|<1) means stable; on the circle marginal; outside unstable. The continuous left half-plane maps inside the unit circle by z = e^sT. The pole story of D2 repeats with only the boundary changed to the unit circle.

On to the next unit

Now you can handle discrete signals as algebra in z and read the response from poles. The next unit, discrete LTI systems, uses this tool on real digital systems. A difference equation y[n] = b·x[n] + a·y[n−1] is a digital filter mixing input samples with past outputs, and its discrete convolution with the impulse response makes the output. The convolution and impulse response of continuous time (chapter B) come back in discrete time, with the integral turned into a sum. Poles and zeros then show directly, on the unit circle, which frequencies the digital filter keeps and which it kills.