Foundations of the Z-Transform

1Introduction

This lecture explains the Z-transform as a transformation that converts a shift operator into algebraic multiplication by $z$z or $z^{-1}$z^{-1}. In continuous-time differential equations the Laplace transform converts differentiation into an expression in $s$s; in discrete-time recurrences the Z-transform performs the corresponding role for shifts and delays.

2Definitions

Assume $z\ne 0$z\ne0. For a bilateral sequence $\left(x\right[n]{)}_{n\in \mathbb{Z}}$(x[n])_{n\in\mathbb Z}, define

For a unilateral sequence $\left(x\right[n]{)}_{n\text{[PARSE ERROR: Undefined("Command("ge")")]}0}$(x[n])_{n\ge0}, define

The region of convergence (ROC) is the set of $z$z for which the corresponding series converges absolutely. A Z-transform result consists of both its expression and its ROC. The notations $X_{\mathrm{b}}$X_{\mathrm b} and $X_{+}$X_+ distinguish the bilateral and unilateral transforms throughout.

3The Same Expression Can Represent Different Sequences

Let $a\ne 0$a\ne0 and let $u\left[n\right]$u[n] be one for $n\text{[PARSE ERROR: Undefined("Command("ge")")]}0$n\ge0 and zero for $n<0$n<0. For $x\left[n\right]=a^{n}u\left[n\right]$x[n]=a^nu[n],

For $y\left[n\right]=-a^{n}u[-n-1]$y[n]=-a^nu[-n-1],

Thus the same rational expression denotes a right-sided or left-sided sequence according to the ROC. Inversion selects the Laurent expansion compatible with the ROC and extracts the coefficient of $z^{-n}$z^{-n}. Here geometric series and coefficient comparison suffice; complex contour integration is not required.

4Why the Weight Is $z^{-n}$z^{-n}

The identity $z^{-(n+1)}=z^{-1}{z}^{-n}$z^{-(n+1)}=z^{-1}z^{-n} makes a unit index shift become multiplication by $z$z or $z^{-1}$z^{-1} after transformation. This property converts equations in shift operators into algebraic equations.

5Proposition 1: The Bilateral Z-Transform Is Linear

5.1Statement

On the intersection of the ROCs of $X_{\mathrm{b}}$X_{\mathrm b} and $Y_{\mathrm{b}}$Y_{\mathrm b},

5.2Proof

Absolute convergence on the common region justifies termwise operations, and

Cancellation can enlarge the ROC of the sum, but the equality is always valid on the common region. $\mathrm{\square }$\square

6Proposition 2: Bilateral Shifts Become Multiplication by $z$z

6.1Statement

For the forward shift $\left(Ex\right)\left[n\right]=x[n+1]$(Ex)[n]=x[n+1] and delay $\left(D_{-}x\right)\left[n\right]=x[n-1]$(D_-x)[n]=x[n-1],

6.2Proof

With $m=n+1$m=n+1,

Likewise, $m=n-1$m=n-1 gives

7Initial Terms Appear in the Unilateral Transform

Because the unilateral sum begins at $n=0$n=0, shifting leaves a boundary term:

This is the discrete counterpart of $\text{[PARSE ERROR: Undefined("Command("mathcal")")]}L\left\{f^{\prime }\right\}=sF\left(s\right)-f\left(0\right)$\mathcal L\{f'\}=sF(s)-f(0). A unilateral transform includes initial data at the cost of initial terms in shift formulas.

8Application to a Recurrence

For

the unilateral transform gives

This converts an equation in the shift operator into an algebraic equation for $X_{+}$X_+. If $r\left[n\right]=0$r[n]=0,

so coefficient extraction gives $x\left[n\right]=x_0{a}^n$x[n]=x_0a^n for $n\text{[PARSE ERROR: Undefined("Command("ge")")]}0$n\ge0.

9Relation to Convolution

An LTI system preserves linear combinations and commutes with shifts. The unit impulse $\delta \left[n\right]$\delta[n] equals one at $n=0$n=0 and zero otherwise; the output generated by this input is the impulse response $h\left[n\right]$h[n]. A finitely supported sequence satisfies $x\left[n\right]=\sum _{k}x\left[k\right]\delta [n-k]$x[n]=\sum_kx[k]\delta[n-k], so linearity and time invariance give

For general sequences, this representation extends when the convolution converges and the system action commutes with limits of finite partial sums. Define

Where absolute convergence permits exchanging double sums,

The transform $H_{\mathrm{b}}$H_{\mathrm b} is the transfer function of this convolution operator. Transforming convolution into multiplication makes the Z-transform effective for difference equations and digital filters.

9.1Stability and the Unit Circle

BIBO stability is equivalent to absolute summability of the impulse response:

Sufficiency follows from $\left|\right(h*x\left)\right[n\left]\right|\text{[PARSE ERROR: Undefined("Command("le")")]}B\sum _k\left|h\right[k\left]\right|$|(h*x)[n]|\le B\sum_k|h[k]| when $\left|x\right[n\left]\right|\text{[PARSE ERROR: Undefined("Command("le")")]}B$|x[n]|\le B. Conversely, define $x_N[-k]=\stackrel{\_}{h\left[k\right]}/\left|h\right[k\left]\right|$x_N[-k]=\overline{h[k]}/|h[k]| for $\left|k\right|\text{[PARSE ERROR: Undefined("Command("le")")]}N$|k|\le N when $h\left[k\right]\ne 0$h[k]\ne0, and zero otherwise. Then $\left|x_N\right[n\left]\right|\text{[PARSE ERROR: Undefined("Command("le")")]}1$|x_N[n]|\le1 and

A stability bound independent of $N$N implies absolute summability.

Consequently the unit circle lies in the ROC and

is the frequency response. Formal substitution $z=e^{i\omega }$z=e^{i\omega} is not a frequency response when the unit circle is outside the ROC.

For a causal rational system, after pole-zero cancellation, BIBO stability is equivalent to every remaining pole lying strictly inside the unit circle. Causality makes the ROC exterior to the outermost pole, so it contains the unit circle exactly under this condition.

10Position of the Laplace, Fourier, and Z Transforms

When the bilateral ROC contains the unit circle, $X_{\mathrm{b}}\left(e^{i\omega }\right)$X_{\mathrm b}(e^{i\omega}) is the discrete-time Fourier transform. The assertion that the Fourier transform is the value of the Z-transform on the unit circle therefore requires this ROC condition.

11Scope

The ROC is part of every bilateral transform result; changing it can change the underlying sequence. The unilateral transform contains no information for $n<0$n<0 and is intended for boundary terms in initial-value problems. It must not be conflated with the bilateral ROC distinction between left- and right-sided sequences.

Before applying a Z-transform, specify whether the sequence and transform are bilateral or unilateral, whether initial data are involved, and whether a frequency response is required.

Transform contract

• Treat a bilateral result as the pair $\left(X_{\mathrm{b}}\right(z),\mathrm{R}\mathrm{O}\mathrm{C})$(X_{\mathrm b}(z),\mathrm{ROC}).
• Use $X_{+}\left(z\right)$X_+(z) and shift formulas with initial terms for unilateral transforms.
• Invert with the Laurent expansion compatible with the ROC and extract coefficients of $z^{-n}$z^{-n}.
• Use the unit circle as a frequency response only when $\left|z\right|=1$|z|=1 lies in the ROC.

12Summary

The Z-transform converts discrete-time shifts into algebraic multiplication by $z$z or $z^{-1}$z^{-1} and thereby supports the analysis of recurrences, difference equations, and discrete convolution.

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