SC · Sampling and discrete time

The z-Transform

The z-transform is the Laplace transform for discrete signals. Pack a one-step delay into z⁻¹ and turn a sequence into a function of z, and a single pole decides the shape of the response. In this lesson you move a pole on the z-plane and watch the discrete response die away inside the unit circle and blow up outside it.

Move the pole inside the unit circle to settle

The upper plot is the z-plane: the two gold × marks are a pole pair z = r·e^(±jθ), and the dashed line from the origin to the pole is the radius r. The green-shaded disk is the stable inside of the unit circle, and its rim, the unit circle, divides stable from unstable. The lower plot draws, as bars from n = 0 to 22, the discrete response x[n] = rⁿ cos(θn) that these poles produce. You start at r = 1.12 and θ = 0.9 rad, so the poles are outside the circle and the bars grow at every step until they leave the plot. Shrink the radius r to bring the poles inside the unit circle and the response dies away to 0; change the angle θ and the amount it turns per step, that is, how fast it oscillates, changes.

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 greater than 1. The amplitude of the discrete response rⁿ cos(θn) grows by a factor of r at every step and diverges. This is an unstable system, with its pole outside the circle.

Unstable (outside circle)

z⁻¹ is a one-step delay

The z-transform packs a discrete signal x[n] into X(z) = Σ x[n] z⁻ⁿ. Just as the Laplace transform in continuous time uses e^(st) as its basis, discrete time uses zⁿ. The most important property is the delay. Shifting a signal back by one step gives x[n−1] ⟷ z⁻¹X(z), that is, one multiplication by z⁻¹. So the difference equation y[n] = a·y[n−1] + x[n] becomes the algebraic equation in z Y(z)(1 − a z⁻¹) = X(z), and the transfer function comes out as H(z) = Y/X = 1/(1 − a z⁻¹) = z/(z − a). Just as differential equations became algebra in s, difference equations become algebra in z.

Observedelay x[n−1] ⟷ z⁻¹ X(z)

A one-step delay is multiplication by z⁻¹.

Choosepole at z = r : mode = ?

A pole of radius r produces the mode rⁿ.

Fill instable ⟺ | pole | ?

Stability requires every pole to be inside the unit circle.

On your ownpole angle θ ⟹ oscillation cos(?)

The angle θ sets the oscillation turned per step.

The discrete mode a pole sets

The pole of H(z) = z/(z − a) is z = a, which makes the denominator 0. The natural response this pole produces is aⁿ, a sequence that multiplies by a at every step. When the pole is a complex number a = r·e^(jθ) paired with its conjugate, the response takes the form rⁿ cos(θn). The radius r is the factor the amplitude is multiplied by at every step, so it shrinks when r < 1 and grows when r > 1, while the angle θ is the phase turned per step and sets how fast it oscillates. At θ = 0 the response simply shrinks or grows without oscillating, and as θ approaches π it becomes the fastest possible oscillation, flipping sign at every step. In effect, σ in the continuous-time e^(σt)cos(ωt) has moved into r = e^(σT), and ω into θ = ωT.

The imaginary axis becomes the unit circle

A discrete system is stable when every pole is inside the unit circle (|z| < 1). If even one pole is outside the circle, its mode rⁿ diverges; on the circle, it becomes a marginal state that keeps oscillating without fading. What was the stable left half-plane in continuous time has become the inside of the unit circle in discrete time. The two are linked by z = e^(sT) (T is the sampling interval): put the imaginary axis of the s-plane, s = jω, into this equation and you get z = e^(jωT), the unit circle of radius 1. The left half-plane (σ < 0) has |z| = e^(σT) < 1 and moves inside the circle, while the right half-plane moves outside. Sampling has wrapped the s-plane around and attached it to the unit circle, which is why the pole story from the D2 unit repeats in discrete time, with only the boundary changed from a line to a circle.

Back to the first screen

On the first screen, with the pole radius r above 1 the poles were outside the unit circle, so the amplitude of the discrete response rⁿ grew at every step and diverged; bringing r below 1 moved the poles inside the circle and the response died away to 0. Setting r to exactly 1 put them on the circle, and the response kept oscillating at the same size. The larger the angle θ, the more it turned per step and the faster it oscillated. The distance r from the origin to the pole was how fast the response died away, and the angle θ was how fast it oscillated. The story from the continuous-time s-plane repeated itself in discrete time, with only the stability boundary changed from the imaginary axis to the unit circle.

The z-transform X(z) = Σ x[n] z⁻ⁿ is the Laplace transform for discrete signals. A one-step delay = multiplication by z⁻¹, so difference equations become 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), where r is the factor applied at every step (it decays when r < 1) and θ is the oscillation turned per step. If every pole is inside the unit circle (|z| < 1), the system is stable; on the circle it is marginal, and outside it is unstable. The continuous-time left half-plane maps inside the unit circle through z = e^sT, so the pole story of the D2 unit repeats with only the boundary changed to the unit circle.

On to the next unit

You can now handle discrete signals as algebra in z and read the response from the poles. The next unit, Discrete LTI Systems, uses this tool on real digital systems. The difference equation y[n] = b·x[n] + a·y[n−1] is a digital filter that mixes input samples with past outputs, and the discrete convolution of its impulse response with the input produces the output. Convolution and the impulse response from continuous time (chapter B) come back to life in discrete time, with only the integral turned into a sum. Compare the locations of the poles and zeros with the unit circle, and you can read straight away which frequencies that digital filter keeps and which it kills.