The Z-Transform and Region of Convergence

Chapter 16 built the discrete-time Fourier transform and was careful about its convergence: the sum \(\sum_n x[n]e^{-j\Omega n}\) exists in the ordinary sense when \(\sum_n|x[n]| \lt \infty\), and in a mean-square sense when the sequence is merely square-summable. Chapter 12 and Chapter 16 between them then stretched the definition, by admitting impulses in frequency, far enough to cover constants and sinusoids.

None of that helps with \(x[n] = (1.2)^n u[n]\). The terms of the sum grow without bound; no amount of interpretation will make it converge, and no impulse will patch it. Yet such a sequence is not a mathematical curiosity — it is the impulse response of any unstable discrete-time system, and one cannot design a feedback loop by refusing to write down the responses one is trying to stabilise.

Chapter 17 met exactly this obstacle in continuous time and solved it by a device that is worth recalling in full, because the discrete case is the same idea in different notation. There the Fourier integral of \(e^{2t}u(t)\) diverged, and the repair was to multiply by a decaying real exponential \(e^{-\sigma t}\) first, choosing \(\sigma\) large enough to force convergence. The Fourier transform of the product is a function of \(\sigma\) and \(\omega\) together, which is the Laplace transform, with \(s = \sigma + j\omega\).

Do the same here. Multiply by \(r^{-n}\) for some positive real \(r\), and transform:

The two exponential factors have merged into one. Writing \(z = re^{j\Omega}\) — an arbitrary complex number in polar form, with \(r=|z|\) supplying the damping and \(\Omega = \arg z\) supplying the oscillation — gives the z-transform. It is the DTFT of the exponentially weighted sequence, and choosing \(r\) is choosing how hard to lean on the sequence to make it summable.

Three immediate observations set the ground rules.

The DTFT lives on the unit circle. Putting \(r=1\), so that \(z=e^{j\Omega}\), returns the definition of Chapter 16 exactly. The z-transform evaluated on \(|z|=1\) is the DTFT — provided the unit circle happens to lie in the region of convergence. When it does not, the sequence has no DTFT, and the z-transform is genuinely more general rather than merely dressed differently.

The transform is a power series, and its coefficients are the signal. Read \(X(z) = \cdots + x[-1]z + x[0] + x[1]z^{-1} + x[2]z^{-2} + \cdots\) and the inversion problem looks almost trivial for short sequences: expand and read off. Section 23-9 makes this into a method.

\(z^{-1}\) is the unit delay. Replacing \(x[n]\) by \(x[n-1]\) shifts every term of the series by one power, multiplying \(X(z)\) by \(z^{-1}\). This is why block diagrams of discrete-time systems label their delay elements \(z^{-1}\), and it is the property that turns difference equations into algebra. Chapter 24 proves it properly, along with what it does to the ROC.

A one-sided or unilateral version, \(\mathcal{X}(z)=\sum_{n=0}^{\infty}x[n]z^{-n}\), is also in use. It ignores everything before \(n=0\), which makes it the right tool for systems started from rest at a known instant with known initial conditions — Chapter 25 uses it for exactly that. For a causal signal the two definitions agree. Everything in this chapter is bilateral unless stated otherwise, because only the bilateral transform can represent a two-sided sequence, and only it makes the ROC an interesting object.

There remains the relationship with the Laplace transform, and it is more than an analogy. If \(x[n]\) is a sampled version of a continuous signal at period \(T\), the impulse-train construction of Chapter 20 has Laplace transform \(\sum_n x(nT)e^{-snT}\), which is the z-transform sum with

Every feature of the \(z\)-plane inherits its meaning from this map. The imaginary axis \(\sigma = 0\) becomes the unit circle \(|z|=1\); the left half-plane \(\sigma \lt 0\) becomes the interior \(|z| \lt 1\); the right half-plane becomes the exterior. The map is many-to-one — every horizontal strip of height \(2\pi/T\) in the \(s\)-plane wraps once around the circle — which is aliasing wearing yet another costume, and the reason the \(z\)-plane has no notion of "high frequency" beyond \(\Omega = \pi\).

A power series converges on some set and diverges elsewhere, and for this series the set has a shape that can be determined once and for all. The natural criterion is absolute convergence, since a power series that converges absolutely may be rearranged and manipulated freely, and every property proved in Chapter 24 relies on being allowed to do so.

Notice what has disappeared: the angle of \(z\). The condition involves \(|z|\) and nothing else, so if the series converges at one point it converges at every point of the circle through it. The region of convergence is therefore built from whole circles centred at the origin, and to determine it one need only ask which radii work.

Split the sum at the origin. The two halves make opposite demands:

The first sum is the part of the signal at non-negative times; every term is divided by \(r^n\), so increasing \(r\) can only help, and if it converges at \(r_0\) it converges for every \(r \gt r_0\). The second sum is the part at negative times; every term is multiplied by \(r^m\), so decreasing \(r\) can only help, and if it converges at \(r_1\) it converges for every \(r \lt r_1\). The two conditions together define a ring.

Two further facts follow immediately and are used constantly.

The ROC contains no poles. At a pole \(|X(z)|\) is infinite, so the series cannot converge there. Since the ROC is an open ring and poles are isolated points, the poles must lie on or outside its boundaries — which means that for a rational transform the boundaries of the ROC are pole radii. This single observation reduces the problem of finding the ROC to the problem of ordering the poles by magnitude.

The ROC can be empty. If \(R_+ \le R_-\) no radius satisfies both demands and the sequence simply has no z-transform. The sequence \(x[n] = 1\) for all \(n\) is the standard example: its right-sided half needs \(r \gt 1\) and its left-sided half needs \(r \lt 1\). The same signal has a DTFT, in the impulse sense of Chapter 16 — a useful reminder that the z-transform is more general in one direction and less in another.

One sequence carries most of the weight in this subject, for the same reason \(e^{at}u(t)\) did in Chapter 17: it is the natural response of a first-order system, and every higher-order response is a combination of such terms. Take \(x[n] = a^n u[n]\) with \(a\) any complex number.

The geometric series converges precisely when its ratio has modulus less than one, and the condition \(|az^{-1}| \lt 1\) is the statement \(|z| \gt |a|\). So the ROC is the exterior of the circle through the pole — which is what Section 23-3 predicted for a right-sided signal, whose sum needs \(r\) large.

The two forms of the answer are both worth keeping. The \(z^{-1}\) form is the one to use for partial fractions and for reading off inverse transforms, because it matches the shape of the standard pairs. The \(z\) form is the one to use for locating poles and zeros, because a ratio of polynomials in \(z\) can be factorised in the ordinary way; in \(z^{-1}\) form the zero at the origin is invisible and students routinely miscount.

Three special cases are worth naming. With \(a=1\) the sequence is the unit step and the transform is \(1/(1-z^{-1})\) with ROC \(|z|\gt 1\): the pole sits on the unit circle, so the circle is excluded and \(u[n]\) has no ordinary DTFT — precisely the difficulty Chapter 16 resolved by adding an impulse at \(\Omega=0\). With \(|a| \lt 1\) the ROC includes the unit circle and the sequence is absolutely summable, with DTFT \(1/(1-ae^{-j\Omega})\) obtained by simple substitution. With \(|a| \gt 1\) the sequence grows, the ROC excludes the unit circle, and there is no DTFT at all — yet the z-transform exists and is perfectly usable.

Now transform a sequence that lives entirely in negative time: \(x[n] = -a^n u[-n-1]\), which is \(-a^n\) for \(n \le -1\) and zero for \(n \ge 0\). The minus sign looks contrived; watch what it produces.

The geometric series in \(z/a\) converges when \(|z/a| \lt 1\), that is \(|z| \lt |a|\). The algebra has produced the same function as the right-sided exponential of Section 23-4, from a completely different signal, and the only thing distinguishing the two is the region of convergence.

The contrived-looking minus sign is now explained: it is what makes the two cases share an expression, and it appears in every table for that reason. The mnemonic is that a pole contributes a right-sided term when the ROC lies outside its circle and a left-sided term when the ROC lies inside it, with a sign change on the left-sided term.

A two-sided sequence combines both behaviours, and its ROC must satisfy both demands at once. Take

The first term needs \(|z| \gt \tfrac12\) and the second needs \(|z| \lt 2\), so the ROC is the annulus \(\tfrac12 \lt |z| \lt 2\) — non-empty, and containing the unit circle, so this two-sided sequence is absolutely summable despite being built from a decaying piece and a growing piece. Combining the fractions,

Poles at \(z=\tfrac12\) and \(z=2\), and the ROC is the ring between them. Had the ROC been \(|z|\gt 2\) the same expression would have described a purely right-sided, exponentially growing sequence; had it been \(|z| \lt \tfrac12\), a purely left-sided one. Worked Example 2 carries out all three inversions in full.

The pattern is now clear enough to be stated as a set of rules, each of which follows from the two-halves argument of Section 23-3 rather than needing to be memorised separately.

Finite-duration sequences. If \(x[n]\) is nonzero only for \(N_1 \le n \le N_2\), the sum has finitely many terms and converges everywhere the individual terms are finite. Terms with \(n \gt 0\) contain \(z^{-n}\), which blows up at \(z=0\); terms with \(n \lt 0\) contain positive powers of \(z\), which blow up as \(|z|\to\infty\). So the ROC is the entire plane, with the origin excluded if \(N_2 \gt 0\) and the point at infinity excluded if \(N_1 \lt 0\).

Right-sided sequences (\(x[n]=0\) for \(n \lt N_1\)). Only the "needs \(r\) large" half is present, so the ROC is an exterior \(|z| \gt R_-\), where \(R_-\) is the largest pole magnitude. It extends to infinity, and includes the point \(z=\infty\) itself exactly when \(N_1 \ge 0\) — that is, when the sequence is causal, so that the series has no positive powers of \(z\).

Left-sided sequences (\(x[n]=0\) for \(n \gt N_2\)). Only the "needs \(r\) small" half is present, so the ROC is an interior \(|z| \lt R_+\), with \(R_+\) the smallest pole magnitude. It includes \(z=0\) exactly when \(N_2 \le 0\).

Two-sided sequences. Both halves are present, so the ROC is a genuine annulus bounded by two pole radii — or empty, if the right-sided half's requirement exceeds the left-sided half's allowance.

Turning the rules around gives the practical procedure, which is worth stating as an algorithm because examiners set it repeatedly. Given a rational \(X(z)\): find the poles, sort them by magnitude, and note that consecutive distinct pole radii cut the plane into rings. If there are \(P\) distinct pole magnitudes, there are exactly \(P+1\) candidate regions — the innermost disc, the \(P-1\) rings between them, and the outermost exterior — and each is a legitimate ROC corresponding to a different signal. The innermost gives a left-sided signal, the outermost a right-sided one, and every ring in between a two-sided one. Additional information — causality, stability, or a stated ROC — is what picks one.

The pay-off for all this care about regions is that the two system properties an engineer most wants to check become statements about geometry. Let \(h[n]\) be an impulse response and \(H(z)\) its transform — the system function, and the object Chapter 25 is built around.

Start with stability. Chapter 8 established that a discrete-time LTI system is BIBO stable exactly when its impulse response is absolutely summable, \(\sum_n |h[n]| \lt \infty\). Set \(|z|=1\) in the convergence condition of Section 23-3 and that is precisely the requirement that the series converge on the unit circle.

Now causality. A causal system has \(h[n]=0\) for \(n \lt 0\), so its transform has no positive powers of \(z\) and the series converges as \(|z|\to\infty\) — every term either is constant or tends to zero. Conversely a right-sided sequence whose ROC includes the point at infinity can have no terms in positive powers of \(z\), so it must vanish for \(n \lt 0\).

For a rational \(H(z)\) written as a ratio of polynomials in \(z\), the limit is finite exactly when the numerator degree does not exceed the denominator degree — when the system has no more zeros than poles. This is a quick and reliable test, and it is the discrete counterpart of the properness condition on a transfer function in Chapter 19. A system such as \(H(z) = z^2/(z-0.5)\) has an exterior ROC and looks causal at a glance, but its impulse response begins at \(n=-2\), and the surplus zero is what gives it away.

Putting the two together produces the result that governs all of discrete-time system design.

Only a handful of pairs need to exist, and every one of them can be got from the geometric series of Section 23-4 in a line or two. Deriving them once is a better investment than memorising them, because the derivations also produce the ROCs.

The impulse. \(\mathcal{Z}\{\delta[n]\} = \sum_n \delta[n]z^{-n} = 1\), converging for every \(z\) — the only transform with no restriction at all. Shifting gives \(\delta[n-k] \leftrightarrow z^{-k}\), whose ROC excludes \(z=0\) for \(k \gt 0\) and \(z=\infty\) for \(k \lt 0\).

A finite exponential run. Truncating \(a^nu[n]\) to \(N\) terms turns the infinite geometric series into a finite one:

The apparent pole at \(z=a\) is cancelled by a zero of the numerator there, which it must be: a finite-length sequence converges everywhere except possibly at the origin, so it cannot have a pole anywhere else. Watching that cancellation happen is the best possible check on the rule.

Multiplication by \(n\). Differentiate the geometric series with respect to \(z\). Since \(\frac{d}{dz}z^{-n} = -nz^{-n-1}\), it follows that \(\sum_n n\,x[n]z^{-n} = -z\,\frac{dX}{dz}\). Applying this to \(X(z)=1/(1-az^{-1})\):

A repeated pole at \(z=a\), exactly as a repeated root of a characteristic equation produces an \(n a^n\) term in Chapter 9's solutions. The ROC is unchanged, because multiplying by \(n\) cannot alter an exponential growth rate.

Damped sinusoids. These matter most, since every second-order discrete system produces one. Write the cosine as a sum of two complex exponentials and apply the fundamental pair twice, with \(a = re^{\pm j\Omega_0}\):

Putting the two fractions over a common denominator, the cross terms give \(re^{j\Omega_0}+re^{-j\Omega_0} = 2r\cos\Omega_0\) and the product of the poles gives \(r^2\):

The formal inverse is a contour integral, obtained by multiplying the definition by \(z^{k-1}\) and integrating around a closed path inside the ROC. The only fact needed is that \(\frac{1}{2\pi j}\oint z^{k-n-1}dz\) equals 1 when \(k=n\) and 0 otherwise, which is Cauchy's theorem in its simplest form.

It is reassuring to know this exists and it is almost never evaluated by hand. Two practical routes do all the work.

Route one: partial fractions. Write \(X(z)\) as a ratio of polynomials in \(z^{-1}\), reduce it to a proper fraction if necessary by long division, and split it into first-order terms. Then, and this is the step that carries the marks, assign each term a direction by comparing the radius of its pole with the ROC. A pole inside the ROC's inner boundary contributes a right-sided term \(A a^n u[n]\); a pole outside the outer boundary contributes a left-sided term \(-A a^n u[-n-1]\).

A caution on bookkeeping: partial fractions may be done in \(z^{-1}\) or in \(z\), but not carelessly in \(z\). Expanding \(X(z)\) directly in \(z\) usually produces terms of the form \(Az/(z-a)\) rather than \(A/(z-a)\), and the standard fix is to expand \(X(z)/z\) into partial fractions and then multiply through by \(z\) at the end. Working in \(z^{-1}\) throughout avoids the issue entirely and is the habit worth acquiring.

Route two: power-series expansion. Because \(X(z)\) is the series \(\sum_n x[n]z^{-n}\), any expansion of it in powers of \(z^{-1}\) exhibits the sequence directly. For a rational function this is long division, and the ROC decides which way to divide: an exterior ROC calls for a series in ascending powers of \(z^{-1}\), which means dividing with both polynomials written in ascending \(z^{-1}\); an interior ROC calls for a series in ascending powers of \(z\), obtained by reversing both polynomials. Dividing the wrong way produces a perfectly valid series that converges in the wrong region and describes the wrong signal.

This route gives values rather than a closed form, which makes it ideal for checking the first few samples of an answer obtained by partial fractions, and for finite-length or awkward transforms where no closed form exists. For a causal signal it is equivalent to running the difference equation forward from rest, which is often the quickest check of all.

Problems 1 to 3 exercise the definition and the ROC rules; 4 and 5 concern inversion; 6 to 8 use the transform to answer questions about systems. Sketch the pole–zero plot and shade the ROC before writing any answer down — most of the errors in this material are errors of geometry, not algebra.