The z-Transform

5.1 The z-Transform

5.1.1 Definition and the z-plane

The DTFT of Chapter 3 only exists for sequences that are absolutely (or square) summable — it fails for signals as basic as the unit step $\mu[n]$ or a growing exponential $2^n\mu[n]$. The z-transform（z 轉換） generalizes the DTFT by evaluating the sum on a complex variable $z$ instead of on $e^{j\omega}$ only, which buys convergence for a much larger class of sequences and turns difference equations into algebra.

We write the pair as $g[n] \xleftrightarrow{\;\mathcal{Z}\;} G(z)$. Throughout this course the unit step is written $\mu[n]$ (identical to the $u[n]$ used in many textbooks). Since $z$ is a complex number, it is natural to picture $G(z)$ as a function defined over the z-plane（z 平面）: writing $z = re^{j\omega}$, the radius $r=|z|$ and the angle $\omega=\angle z$ play very different roles, as we now see.

5.1.2 Relation to the DTFT

Substituting $z = re^{j\omega}$ into the definition:

So the z-transform evaluated on the circle of radius $r$ is the DTFT of $g[n]$ after multiplication by the exponential weight $r^{-n}$. A decaying weight ($r>1$ for right-sided signals) can tame a sequence whose plain DTFT diverges. Two immediate consequences:

• Setting $r = 1$ (the unit circle（單位圓）, $z = e^{j\omega}$) recovers the ordinary DTFT: $$G(e^{j\omega}) = G(z)\big|_{z=e^{j\omega}}$$
• The DTFT of $g[n]$ exists exactly when the unit circle lies inside the region where the sum converges — the ROC of §5.2.

5.2 Region of Convergence (ROC)

5.2.1 Why the ROC is necessary

The critical question for any $z$: does $\sum_n g[n]z^{-n}$ converge to a finite value? The region of convergence（收斂區域，ROC） $\mathcal{R}_g$ is the set of $z$ for which it does. In general it depends only on $|z|$, so it is always a circular region — the inside of a circle, the outside of a circle, or a ring (annulus).

Figure 5-1 shows the family $a^n\mu[n]$ for three values of $a$, together with the location of the corresponding transform's pole at $z=a$. Decaying sequences ($|a|<1$) have their pole inside the unit circle; the growing sequence ($a=1.2$) has it outside — and its DTFT does not exist, although its z-transform does (ROC $|z|>1.2$).

5.2.2 ROC properties

5.2.3 Sidedness and ROC shape

Each pole $\lambda$ of a rational $G(z)$ contributes a term whose ROC is either outside $|z|=|\lambda|$ (right-sided term) or inside (left-sided term). The overall ROC tells you the sidedness（單邊性） of $g[n]$:

5.2.4 ROC intersections: four cases

Consider a transform with two poles $\lambda_1,\lambda_2$ where $|\lambda_1| < 1 < |\lambda_2|$ and no ROC specified:

Each term may independently correspond to a right-sided or a left-sided sequence, so there are 4 possible sequences/ROCs, and the overall ROC is the intersection of each term's ROC:

Only Case 3 yields an ROC containing the unit circle, so only that choice corresponds to a stable (absolutely summable) sequence with a DTFT.

5.3 Rational z-Transforms, Poles and Zeros

5.3.1 Rational and factored forms

$G(z)$ can in principle be any function of $z$, but in DSP the important class is rational（有理） z-transforms — ratios of polynomials in $z^{-1}$. By convention they are expressed in powers of $z^{-1}$, which matches the transform definition and (as §5.4.3 shows) is reminiscent of LCCDE expressions:

Factoring numerator and denominator exposes the roots:

• Zeros（零點） $z=\xi_\ell$: values where $G(z)=0$. Marked ○ on the z-plane.
• Poles（極點） $z=\lambda_\ell$: values where $G(z)\to\infty$. Marked ×. If $N>M$ there are $N-M$ additional zeros at $z=0$; if $M>N$, $M-N$ poles at $z=0$.

5.3.2 Pole–zero diagram and the z-plane surface

The pole–zero diagram（極零點圖） plots the ×'s and ○'s together with the unit circle. Up to the overall gain $p_0/d_0$, it completely determines a rational $G(z)$, and with the ROC shading added it completely determines $g[n]$. Picturing $|G(z)|$ as a surface over the z-plane: poles are tent-poles pushing the surface up to infinity, zeros pin it down to the floor. The DTFT $|G(e^{j\omega})|$ is the height profile measured while walking along the unit circle.

5.3.3 Geometric evaluation of the DTFT

Writing each factor at $z = e^{j\omega}$ as a vector difference gives a purely geometric recipe:

As $e^{j\omega}$ slides along the unit circle, the magnitude response peaks when it passes close to a pole (small denominator distance) and dips when it passes near a zero. Figure 5-3 demonstrates this for a resonator with poles at $0.9e^{\pm j\pi/4}$: the magnitude has a sharp peak near $\omega = \pi/4$.

5.4 z-Transform Properties and Common Pairs

5.4.1 Common z-transform pairs

The two sinusoidal pairs are each the sum of two complex-exponential terms with poles at $re^{\pm j\omega_0}$ — a conjugate pole pair（共軛極點對）. Real sequences always have poles and zeros in conjugate pairs (or on the real axis).

5.4.2 z-Transform properties

With $g[n] \leftrightarrow G(z)$ (ROC $\mathcal{R}_g$) and $h[n] \leftrightarrow H(z)$ (ROC $\mathcal{R}_h$):

5.4.3 System function of an LCCDE; causality and stability

Apply the z-transform to a general linear constant-coefficient difference equation (LCCDE), using linearity and the time-shift property:

$H(z) = \mathcal{Z}\{h[n]\}$ is the system function / transfer function（系統函數／轉移函數）. The coefficients of the difference equation are read off directly as the coefficients of the rational polynomials — feedback coefficients $d_k$ in the denominator, feed-forward coefficients $p_k$ in the numerator. This is why rational z-transforms are written in powers of $z^{-1}$: each $z^{-k}$ is a delay of $k$ samples.

5.5 The Inverse z-Transform

Three practical methods recover $g[n]$ from $G(z)$ + ROC. Method 1 is the formal definition; methods 2 and 3 are the ones actually used by hand.

5.5.1 IZT #1 — Contour-integral definition (generalize the IDTFT)

On a circle $z = re^{j\omega}$ inside the ROC, $G(re^{j\omega})$ is the DTFT of $g[n]r^{-n}$. Applying the inverse DTFT to that weighted sequence and changing variables to $z$ gives:

where $C$ is any counterclockwise closed contour inside the ROC encircling the origin. By the residue theorem this picks out the coefficient of $z^{-n}$ in the Laurent series. The contour integral is rarely computed directly — it justifies the easier methods below.

5.5.2 Inspection（觀察法）

Recognize table entries. Given $G(z) = \dfrac{1}{1-0.8z^{-1}}$ with ROC $|z|>0.8$, match to $\alpha^n\mu[n] \leftrightarrow \frac{1}{1-\alpha z^{-1}}$ and read off $g[n] = (0.8)^n\mu[n]$. The same $G(z)$ with ROC $|z|<0.8$ instead gives $g[n] = -(0.8)^n\mu[-n-1]$. The ROC decides which pair applies.

5.5.3 IZT #2 — Power series and long division（長除法）

Since $G(z) = \sum_n g[n]z^{-n}$, expanding $G(z)$ into a power series in $z^{-1}$ (for a right-sided ROC) or in $z$ (left-sided ROC) reveals the samples one by one. For rational $G(z)$, the expansion is obtained by polynomial long division of the numerator by the denominator.

5.5.4 IZT #3 — Partial fractions（部分分式）

The workhorse method: rearrange $G(z)$ as a sum of terms recognized as simple z-transforms — especially $\dfrac{1}{1-\alpha z^{-1}} \leftrightarrow \alpha^n\mu[n]$ (or its left-sided twin). For $F(z) = \dfrac{P(z)}{\prod_{\ell=1}^{N}(1-\lambda_\ell z^{-1})}$ with order of $P(z)$ less than $N$ and no repeated poles:

The residue（留數）$\rho_\ell$ is found by covering up the pole's own factor and evaluating everything else at $z=\lambda_\ell$ — the cancelled factor is "dominated" by the remaining terms. Bookkeeping rules:

• If the numerator order $M \ge N$: first do polynomial division to pull out a finite-length part $\sum_r c_r z^{-r}$ (which inverse-transforms to impulses $c_r\delta[n-r]$), leaving a strictly proper remainder.
• Repeated pole of multiplicity $m$: include terms $\dfrac{A_k}{(1-\lambda z^{-1})^k}$, $k=1,\ldots,m$, using the $n\alpha^n\mu[n]$ pair for $k=2$.
• For each pole, the ROC decides right-sided ($\rho\,\lambda^n\mu[n]$ if $|z|>|\lambda|$) vs left-sided ($-\rho\,\lambda^n\mu[-n-1]$ if $|z|<|\lambda|$).

5.6 Summary

Exercises

All three problems of ICE503 DSP Homework #5, restated with complete worked solutions.（以下為作業五全部題目與詳解。原題的 "LTE/LCCDE" 拼寫依文意更正為 LTI 系統與脈衝響應。）

clc, clear, close all

% H(z) = (25/32)(1 - 3 z^-1) / (1 - (3/4) z^-1)
numerator   = (25/32) * [1, -3];   % feed-forward coefficients p_k
denominator = [1, -3/4];           % feedback coefficients     d_k

%% (a) pole-zero plot
figure;
zplane(numerator, denominator);
grid on;
title('Pole-Zero Plot of H(z)');
set(findobj('type','line'), 'LineWidth', 2)
set(gca, 'FontSize', 20)

%% (b) 100-point frequency response
[H, w] = freqz(numerator, denominator, 100);

figure;
subplot(2,1,1);
plot(w/pi, abs(H), 'm');
title('Magnitude Response');
xlabel('Normalized Frequency (\times\pi rad/sample)');
ylabel('Magnitude');
set(gca, 'FontSize', 20)
set(findobj('type','line'), 'LineWidth', 4)

subplot(2,1,2);
plot(w/pi, angle(H), 'm');
title('Phase Response');
xlabel('Normalized Frequency (\times\pi rad/sample)');
ylabel('Phase (radians)');
set(gca, 'FontSize', 20)
set(findobj('type','line'), 'LineWidth', 4)