From Difference Equations to Z-Transform | DSP-101

Why We Need an Algebraic Tool

In Module 3 we analyzed LTI systems through their impulse response h[n] and convolution. Powerful as that is, computing y[n] = x[n] * h[n] requires summing an infinite series for IIR filters, and stability analysis demands inspecting h[n] sample by sample. Designing a filter — choosing coefficients to achieve a desired frequency response — is even harder in the time domain alone.

The Z-transform solves this by lifting the problem into the complex z-plane, where convolution becomes multiplication, difference equations become algebraic equations, and stability reduces to a geometric condition about where roots of a polynomial fall. It is the discrete-time analog of the Laplace transform, and it underpins virtually every digital filter design and analysis technique you will encounter.

Z-Transform Definition

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

The Z-transform maps a sequence x[n] to a function X(z) of the complex variable z. Each sample x[n] is weighted by z raised to the power −n, compressing the entire sequence into a single algebraic object.

The variable z is complex — z = re^jω in polar form. When r = 1 the unit circle is traced, and X(z) evaluated on the unit circle equals the Discrete-Time Fourier Transform (DTFT): X(e^jω). The Z-transform therefore generalizes the DTFT to the entire complex plane, enabling analysis beyond the frequency axis.

Common Z-Transform Pairs

A handful of sequences appear so often that their Z-transforms are simply memorized. Deriving them from the definition is a useful exercise: substitute the sequence definition into the sum and apply geometric series formulas.

Unit Impulse

\mathcal{Z}\{\delta[n]\} = 1

δ[n] transforms to 1 — valid for all z. The simplest pair, confirming that the Z-transform of a unit impulse is just unity.

Unit Step

\mathcal{Z}\{u[n]\} = \frac{z}{z-1}, \quad |z|>1

u[n] sums to z/(z−1), valid for |z| > 1. The pole at z = 1 reflects the non-decaying nature of the step.

Exponential

\mathcal{Z}\{a^n u[n]\} = \frac{z}{z-a}, \quad |z|>|a|

The one-sided exponential aⁿu[n] transforms to z/(z−a), valid for |z| > |a|. Stable when |a| < 1.

Cosine

\mathcal{Z}\{\cos(\omega_0 n)\,u[n]\} = \frac{z^2 - z\cos\omega_0}{z^2 - 2z\cos\omega_0 + 1}

A cosine sequence yields two complex poles on the unit circle at ±ω₀. Used to analyze sinusoidal steady-state response.

These pairs, together with the Z-transform properties below, form a complete toolkit. Rather than evaluating the infinite sum each time, engineers build X(z) by decomposing x[n] into scaled, shifted versions of known sequences and applying linearity.

Key Properties

The Z-transform’s power lies in its properties. Three are particularly important for filter analysis.

Property 1

Linearity

Z{ax[n] + by[n]} = aX(z) + bY(z). Superposition holds — a weighted sum in time maps to a weighted sum of transforms.

Property 2

Time Shift

Z{x[n−k]} = z^−kX(z). A delay of k samples multiplies by z^−k in the z-domain — the basis of the delay operator.

Property 3

Convolution → Multiplication

Z{x[n] * h[n]} = X(z)·H(z). Convolution becomes multiplication — the central reason the Z-transform matters for LTI analysis.

Property 4

Scaling in z

Z{aⁿx[n]} = X(z/a). Multiplying by aⁿ in time scales the z-plane. Used for frequency shifting and prototype filter transformations.

Time-Shift Property

\mathcal{Z}\{x[n-k]\} = z^{-k}\,X(z)

A unit delay z⁻¹ in the z-domain corresponds to one sample delay in the time domain. This property directly converts difference equations to algebraic equations in z.

Converting Difference Equations

The time-shift property is the bridge from difference equations to the z-domain. Consider a general LCCDE with coefficients {b_k} and {a_k}. Apply the Z-transform to both sides and use linearity and the time-shift property on every term:

Z-Domain Algebraic Form

Y(z)\sum_{k=0}^{M}a_k z^{-k} = X(z)\sum_{k=0}^{N}b_k z^{-k}

Applying the Z-transform to an LCCDE converts it from a recursive time-domain equation into a rational polynomial equation — solvable by algebra.

The ratio Y(z)/X(z) is the transfer function H(z) — the Z-transform of the impulse response. It is a ratio of two polynomials in z⁻¹ (or equivalently in z), determined entirely by the filter coefficients. Every LTI system representable by an LCCDE has a rational transfer function.

First-Order IIR: Time Domain → Z Domain

Take the simple IIR filter y[n] = ay[n−1] + x[n]. Z-transform both sides:

Y(z) = az⁻¹Y(z) + X(z)

Collect Y(z): Y(z)(1 − az⁻¹) = X(z)

H(z) = Y(z)/X(z) = 1/(1 − az⁻¹) = z/(z − a)

One-Sided vs. Two-Sided Z-Transform

The definition above sums over all n from −∞ to +∞ — this is the bilateral (two-sided) Z-transform. It handles both causal and non-causal sequences but requires careful specification of the Region of Convergence (ROC) to uniquely determine the inverse.

In engineering practice, most signals are causal (zero for n < 0), so the unilateral (one-sided) Z-transform sums from n = 0 to +∞. The one-sided form is particularly useful for solving difference equations with initial conditions: non-zero initial conditions introduce additional terms that the bilateral form cannot capture cleanly.

One-Sided Z-Transform

X(z) = \sum_{n=0}^{\infty} x[n]\,z^{-n}

The unilateral Z-transform sums only over n ≥ 0. For causal sequences the result is identical to the bilateral transform. For non-causal sequences the two forms differ.

Relationship to the Laplace Transform

The Z-transform is the discrete-time counterpart of the Laplace transform. The connection becomes concrete when we recall the sampling operation: a sampled continuous-time signal x(t) can be written as x[n] = x(nT), and the Laplace transform of the sampled signal involves terms of the form e^sT. Identifying z = e^sT establishes the mapping between the s-plane and the z-plane.

s-Plane to z-Plane Mapping

z = e^{sT}

The substitution z = e^(sT) maps the s-plane to the z-plane. The imaginary axis (jω) in the s-plane maps to the unit circle in the z-plane. The left half-plane maps to the interior of the unit circle.

This mapping has profound implications. In the s-plane, stability of a continuous-time system requires poles in the left half-plane (negative real part). After mapping, left-half-plane poles land inside the unit circle in the z-plane. So discrete-time stability — all poles inside the unit circle — is the direct digital analog of continuous-time stability. This will be the central topic of Lesson 4.4.

s-Plane vs z-Plane at a Glance

Stability region: left half-plane (s) ↔ inside unit circle (z)

Frequency axis: imaginary axis jω (s) ↔ unit circle e^jω (z)

Unstable region: right half-plane (s) ↔ outside unit circle (z)

Laplace ↔ Z-Transform, just as Continuous ↔ Discrete

The next lesson explores the Region of Convergence (ROC) — the set of z values for which the Z-transform sum converges. The ROC determines whether a signal is causal, anti-causal, or two-sided, and it is essential for correctly inverting the Z-transform.

Key Takeaways

The Z-transform maps a discrete sequence x[n] to a function X(z) of the complex variable z, converting recursive difference equations into algebraic polynomial equations.
Definition: X(z) = Σ x[n]·z⁻ⁿ. When evaluated on the unit circle |z| = 1, this equals the DTFT.
Key pairs: δ[n]↔1, u[n]↔z/(z−1), aⁿu[n]↔z/(z−a).
Key properties: linearity, time shift (delay k → multiply by z⁻ᵏ), convolution → multiplication.
Any LCCDE becomes a rational transfer function H(z) = Y(z)/X(z) — a ratio of polynomials in z.
One-sided Z-transform (sum from n=0) is used for causal systems and initial-condition problems.
The mapping z = e^(sT) connects the Laplace transform to the Z-transform; left half s-plane maps to inside the unit circle in the z-plane.