The z-Transform

Definition and region of convergence, the common pairs, the unit circle and the DTFT, the shift property, and the map z = e^{sT} from the s-plane

1952 — a name at Columbia

The idea was old: Laplace had known it. In 1947 Witold Hurewicz and others re-introduced it as a way to analyse the sampled-data control systems used with radar — systems that saw the world not continuously but in samples — and in 1952 John Ragazzini and Lotfi Zadeh, in the sampled-data control group at Columbia University, gave it the name it still carries: the z-transform (Z-transform). It does for sequences what the Laplace transform does for functions of continuous time — turns shifts and convolutions into multiplication, difference equations into algebra, and stability into a question about where a few points lie.

The DTFT of Unit 6 needs sequences that die away, just as the Fourier transform does. Multiply a sequence by

r^{-n}

and many more sequences converge. Writing

z = re^{j\Omega}

folds both steps into one complex variable:

X(z) = \sum_{n=-\infty}^{\infty} x[n]\,z^{-n}, \qquad z = re^{j\Omega}

The z-transform: the DTFT of x[n]r^{−n}. On the unit circle, r = 1, it is the DTFT itself.

Derivation

A geometric series

For

x[n] = a^n u[n]

X(z) = \sum_{n=0}^{\infty} a^n z^{-n} = \sum_{n=0}^{\infty}\left(\frac az\right)^n = \frac{1}{1 - az^{-1}} = \frac{z}{z - a}, \qquad |z| > |a| .

The series converges only where

|a/z| < 1

: outside the circle of radius

|a|

. Differentiating term by term,

-z\,\dfrac{dX}{dz} = \sum n\,a^n z^{-n}

, gives the next pair,

n a^n u[n] \leftrightarrow az/(z - a)^2

💡Regions of convergence are rings

Convergence depends only on

|z|

, so the ROC is bounded by circles centred on the origin:

a right-sided sequence converges outside a circle: $a^n u[n] \leftrightarrow z/(z - a)$ for $|z| > |a|$ ;
a left-sided sequence converges inside one: $-a^n u[-n-1] \leftrightarrow z/(z - a)$ as well, but for $|z| < |a|$ ;
a two-sided sequence converges in a ring, or nowhere: $0.5^n u[n] + 2^n u[-n-1]$ only for $0.5 < |z| < 2$ .

The DTFT exists when the ROC contains the unit circle, and then

X(e^{j\Omega}) = X(z)\big|_{z = e^{j\Omega}}

💡Pairs worth knowing

| Sequence (

n \ge 0

) |

X(z)

| ROC | |---|---|---| |

\delta[n]

1

| all

z

| |

u[n]

z/(z - 1)

|z| > 1

| |

a^n

z/(z - a)

|z| > |a|

| |

n\,a^n

az/(z - a)^2

|z| > |a|

| |

\cos(\Omega_0 n)

\dfrac{z(z - \cos\Omega_0)}{z^2 - 2z\cos\Omega_0 + 1}

|z| > 1

💡A delay is a factor of z⁻¹

Shifting a sequence by one sample multiplies its transform by

z^{-1}

x[n - 1] \;\leftrightarrow\; z^{-1}X(z) .

(The unilateral version adds the sample that slides in from before

n = 0

z^{-1}X(z) + x[-1]

— initial conditions, exactly as in Lesson 7-2.) The block labelled

z^{-1}

in every signal-processing diagram is a one-sample memory. Convolution becomes multiplication,

x * h \leftrightarrow X(z)H(z)

, so a difference equation becomes a polynomial equation in

z^{-1}

💡z = e^{sT}: from the s-plane to the z-plane

Sample

e^{s_0 t}

every

T

seconds and you get

(e^{s_0 T})^n

: a continuous-time pole at

s_0

becomes a discrete-time pole at

z_0 = e^{s_0 T}

. With

s_0 = \sigma + j\omega

|z_0| = e^{\sigma T}

and

\angle z_0 = \omega T

, so

the left half-plane maps inside the unit circle, the right half-plane outside, and the imaginary axis onto the circle;
horizontal strips $2\pi/T$ high all land on the same z-plane: aliasing once more.

With

T = 0.1

s, poles at

-1 \pm 2j

land at radius

0.905

, a pole at

+0.5

at radius

1.051

, and one at

3j

exactly on the circle. Stability survives sampling intact.

God made the integers, all the rest is the work of man.

– Leopold Kronecker, lecture of 1886, as quoted by Heinrich Weber (1893), quoted on Wikiquote

🕰From radar to fuzzy logic

Lotfi Zadeh, one of the two engineers who named the z-transform, went on to found fuzzy logic. The transform itself kept developing: Eliahu Jury developed and popularised the modified, or advanced, z-transform for systems with delays that are not whole samples (Z-transform).

⚠z or z⁻¹?

Signal-processing texts and SciPy write transfer functions in powers of

z^{-1}

— b = [b0, b1, b2] means

b_0 + b_1z^{-1} + b_2z^{-2}

— because each power is a delay. Mathematics texts write the same function in powers of

z

. Multiply through by the highest power of

z

before reading off poles and zeros, and check which convention a formula uses before copying its coefficients.

💡Where you will use this

Transfer functions and stability of digital filters are Lesson 8-2; frequency responses read from pole–zero plots are Lesson 8-3. Every IIR design in Unit 9 starts in the s-plane and lands in the z-plane, and digital controllers for aircraft and spacecraft are designed directly in

z

📚Further reading

Oppenheim and Willsky, Signals and Systems, sections 10.1–10.5 — the z-transform, its ROC, inversion and properties.
John R. Ragazzini and Lotfi A. Zadeh, "The Analysis of Sampled-Data Systems", Transactions of the AIEE, Part II, 71 (1952), 225–234.
Z-transform on Wikipedia.

↓But there is a problem

Feed a filter's output back into it and three numbers can ring for ever — or grow without limit. The oldest famous example is a puzzle about rabbits, published in 1202 by a merchant's son from Pisa. Its answer is a sequence every student knows, and a pole outside the unit circle.