1.7 Z-transform

Definition of Z-transform

The Z-transform converts a discrete-time signal $x[n]$ into a function of the complex variable $z$, defined as:

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

This is the discrete-time counterpart to the Laplace transform. Where the Laplace transform handles continuous-time signals, the Z-transform handles sequences. It gives you a way to analyze stability, causality, and frequency response of discrete-time systems entirely through algebraic manipulation rather than working directly with sequences and convolution sums.

Region of convergence

The region of convergence (ROC) is the set of values of $z$ in the complex plane for which the Z-transform summation actually converges to a finite value. The ROC matters because:

• Two different signals can produce the same algebraic expression for $X(z)$ but differ in their ROC. The ROC is what makes the transform unique.
• The ROC takes the form of a ring (or annular region) in the z-plane: $r_1 < |z| < r_2$, where the boundaries are set by the poles of $X(z)$.
• Knowing the ROC tells you directly whether the system is stable and/or causal.

Relationship to Laplace transform

The Z-transform can be derived from the Laplace transform through the substitution:

$z = e^{sT}$

where $T$ is the sampling period and $s$ is the Laplace variable. This mapping wraps the imaginary axis of the s-plane onto the unit circle in the z-plane. Because of this relationship, many Laplace-domain techniques (partial fractions, pole-zero analysis) carry over directly to the Z-domain, just with different convergence geometry.

Unilateral vs bilateral Z-transform

• The bilateral Z-transform sums over all time indices ($-\infty < n < \infty$) and can represent both causal and non-causal signals. It requires careful specification of the ROC to ensure uniqueness.
• The unilateral Z-transform sums only for $n \geq 0$:

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

The unilateral form is far more common in practice because most systems of interest are causal, and it naturally incorporates initial conditions when solving difference equations.

Properties of Z-transform

These properties let you manipulate signals in the Z-domain without transforming back to the time domain. They're essential for simplifying system analysis and designing digital filters.

Linearity

The Z-transform is linear, satisfying both additivity and homogeneity:

• $\mathcal{Z}\{x_1[n] + x_2[n]\} = X_1(z) + X_2(z)$
• $\mathcal{Z}\{a \cdot x[n]\} = a \cdot X(z)$, where $a$ is a constant

The ROC of the sum is at least the intersection of the individual ROCs.

Time shifting

A delay or advance in the time domain corresponds to multiplication by a power of $z$:

• Right shift (delay): $\mathcal{Z}\{x[n-k]\} = z^{-k}X(z)$ for the bilateral case
• Left shift (advance), unilateral: $\mathcal{Z}\{x[n+k]\} = z^{k}X(z) - \sum_{i=0}^{k-1} x[i]z^{k-i}$

This property is why the Z-transform is so useful for difference equations: each unit delay simply becomes a factor of $z^{-1}$.

Scaling in Z-domain

Multiplying a signal by an exponential $a^n$ scales the z-variable:

$\mathcal{Z}\{a^n x[n]\} = X(z/a)$

The ROC scales accordingly: if the original ROC is $|z| > R$, the new ROC is $|z| > |a|R$.

Time reversal

Reversing a signal in time inverts $z$:

$\mathcal{Z}\{x[-n]\} = X(1/z)$

The ROC inverts as well. If the original ROC was $|z| > R$, the reversed signal's ROC is $|z| < 1/R$.

Differentiation in Z-domain

Multiplying by $n$ in the time domain corresponds to differentiation with respect to $z$:

$\mathcal{Z}\{n \cdot x[n]\} = -z \frac{dX(z)}{dz}$

Note: the original guide listed the first-difference property here. The first difference $x[n] - x[n-1]$ transforms as $(1 - z^{-1})X(z)$, which is really just an application of linearity and time shifting. The differentiation-in-z property above is the more fundamental result that carries the name "differentiation in the Z-domain."

Convolution in Z-domain

Convolution in time becomes multiplication in the Z-domain:

$\mathcal{Z}\{x[n] * h[n]\} = X(z) \cdot H(z)$

This is arguably the most powerful property for system analysis. Instead of computing a convolution sum directly, you multiply two Z-transforms and invert the result.

Correlation in Z-domain

The cross-correlation of two signals transforms as:

$\mathcal{Z}\{r_{xy}[l]\} = X(z) \cdot Y(z^{-1})$

where $r_{xy}[l] = \sum_n x[n] \cdot y[n-l]$. For autocorrelation, this becomes $X(z) \cdot X(z^{-1})$.

Parseval's relation

Parseval's relation equates total signal energy in the time domain to a contour integral in the Z-domain:

$\sum_{n=-\infty}^{\infty} |x[n]|^2 = \frac{1}{2\pi j} \oint_C X(z) X^*(1/z^*) \frac{dz}{z}$

where $C$ is a closed contour within the ROC encircling the origin. For real-valued signals, $X^*(1/z^*) = X(1/z)$, simplifying the expression.

Initial and final value theorems

These let you extract time-domain boundary values directly from $X(z)$ without performing a full inverse transform.

• Initial value theorem: $x[0] = \lim_{z \to \infty} X(z)$, valid when $x[n]$ is causal ($x[n] = 0$ for $n < 0$).
• Final value theorem: $\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z)$, valid only if the limit exists and all poles of $(z-1)X(z)$ are strictly inside the unit circle.

A common mistake: the initial value theorem does not require $\lim_{n \to \infty} x[n] = 0$. It requires causality. The final value theorem is the one with the strict convergence condition.

Z-transform of common signals

These transform pairs are the building blocks you'll use repeatedly. Memorizing them (or at least knowing where to find them quickly) saves significant time.

Unit impulse

$\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \neq 0 \end{cases}$

$\mathcal{Z}\{\delta[n]\} = 1$, with ROC: all $z$ (the entire z-plane).

Unit step

$u[n] = \begin{cases} 1, & n \geq 0 \\ 0, & n < 0 \end{cases}$

$\mathcal{Z}\{u[n]\} = \frac{1}{1 - z^{-1}} = \frac{z}{z - 1}$, with ROC: $|z| > 1$

Exponential

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

$\mathcal{Z}\{a^n u[n]\} = \frac{1}{1 - az^{-1}} = \frac{z}{z - a}$, with ROC: $|z| > |a|$

This is the most frequently used pair. The unit step is just the special case where $a = 1$.

Sinusoidal

For causal sinusoids (multiplied by $u[n]$):

$\mathcal{Z}\{\cos(\omega_0 n) u[n]\} = \frac{1 - \cos(\omega_0)z^{-1}}{1 - 2\cos(\omega_0)z^{-1} + z^{-2}}$, with ROC: $|z| > 1$

$\mathcal{Z}\{\sin(\omega_0 n) u[n]\} = \frac{\sin(\omega_0)z^{-1}}{1 - 2\cos(\omega_0)z^{-1} + z^{-2}}$, with ROC: $|z| > 1$

Both have a pair of complex conjugate poles on the unit circle at $z = e^{\pm j\omega_0}$.

Damped sinusoidal

$x[n] = a^n \cos(\omega_0 n) u[n]$ or $x[n] = a^n \sin(\omega_0 n) u[n]$, where $|a| < 1$ for a decaying signal.

$\mathcal{Z}\{a^n \cos(\omega_0 n) u[n]\} = \frac{1 - a\cos(\omega_0)z^{-1}}{1 - 2a\cos(\omega_0)z^{-1} + a^2 z^{-2}}$, with ROC: $|z| > |a|$

$\mathcal{Z}\{a^n \sin(\omega_0 n) u[n]\} = \frac{a\sin(\omega_0)z^{-1}}{1 - 2a\cos(\omega_0)z^{-1} + a^2 z^{-2}}$, with ROC: $|z| > |a|$

These generalize the sinusoidal pairs. The poles move inside the unit circle (at radius $|a|$), which is why the signal decays.

System analysis using Z-transform

Representing a discrete-time LTI system in the Z-domain turns convolution into multiplication and difference equations into algebraic equations. This section covers how to extract system properties from the Z-domain representation.

Transfer function

The transfer function is defined as:

$H(z) = \frac{Y(z)}{X(z)}$

assuming zero initial conditions. For an LTI system described by the difference equation:

$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$

the transfer function becomes:

$H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}}$

$H(z)$ completely characterizes the system's input-output behavior. It's also the Z-transform of the impulse response $h[n]$.

Poles and zeros

• Zeros are values of $z$ where $H(z) = 0$ (numerator roots).
• Poles are values of $z$ where $H(z) \to \infty$ (denominator roots).

Plotting poles (marked with $\times$) and zeros (marked with $\circ$) on the z-plane gives you a quick visual summary of system behavior. Poles near the unit circle produce strong resonances at the corresponding frequency. Zeros near the unit circle suppress those frequencies.

Stability

A discrete-time LTI system is BIBO stable (bounded input produces bounded output) if and only if:

• All poles of $H(z)$ lie strictly inside the unit circle ($|p_i| < 1$ for every pole $p_i$).
• Equivalently, the ROC of $H(z)$ includes the unit circle.

A pole exactly on the unit circle means the system is marginally stable. A pole outside the unit circle means the impulse response grows without bound.

Causality

A system is causal if $h[n] = 0$ for $n < 0$, meaning the output depends only on current and past inputs. In terms of the Z-transform:

• The ROC of a causal system is the exterior of a circle: $|z| > |p_{\max}|$, where $p_{\max}$ is the outermost pole.
• For a system to be both causal and stable, all poles must be inside the unit circle (so the ROC $|z| > |p_{\max}|$ automatically includes $|z| = 1$).

Linear phase

A system has linear phase if its frequency response satisfies:

$\angle H(e^{j\omega}) = -\alpha \omega + \beta$

where $\alpha$ is a constant delay and $\beta$ is a constant offset. This means all frequency components experience the same delay (constant group delay), so the signal shape is preserved.

Linear phase requires the impulse response to have either symmetry ($h[n] = h[N-n]$) or antisymmetry ($h[n] = -h[N-n]$). FIR filters can achieve exact linear phase; IIR filters generally cannot.

Inverse Z-transform

The inverse Z-transform recovers $x[n]$ from $X(z)$. In practice, you'll almost always use one of the first three methods below rather than the general contour integral.

Partial fraction expansion

This is the most common method for rational $X(z)$. The steps:

1. Express $X(z)/z$ (or $X(z)$) as a ratio of polynomials in $z$.
2. Factor the denominator to find the poles.
3. Decompose into partial fractions, one term per pole.
4. Identify each term with a known Z-transform pair (typically the exponential pair $a^n u[n] \leftrightarrow z/(z-a)$).
5. Use the ROC to determine whether each term corresponds to a causal (right-sided) or anti-causal (left-sided) sequence.

This method works cleanly for distinct poles. For repeated poles, the algebra gets more involved but follows the same principle.

Residue method

Based on the Cauchy residue theorem, this computes $x[n]$ as:

$x[n] = \sum_k \text{Res}\left[X(z) z^{n-1}, z = p_k\right]$

where the sum is over all poles $p_k$ inside the chosen contour. For a simple pole at $z = p_k$:

$\text{Res} = \lim_{z \to p_k} (z - p_k) X(z) z^{n-1}$

For a pole of order $m$:

$\text{Res} = \frac{1}{(m-1)!} \lim_{z \to p_k} \frac{d^{m-1}}{dz^{m-1}} \left[(z - p_k)^m X(z) z^{n-1}\right]$

This handles both distinct and repeated poles systematically.

Power series expansion

Also called long division. You expand $X(z)$ as a power series in $z^{-1}$:

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

The coefficient of $z^{-n}$ directly gives you $x[n]$. To find the coefficients:

1. Arrange the numerator and denominator as polynomials in $z^{-1}$.
2. Perform polynomial long division.
3. Read off the coefficients.

This is especially useful for finding the first several samples of $x[n]$ without needing a closed-form expression, or for non-rational transforms.

Inversion integral

The formal definition of the inverse Z-transform:

$x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz$

where $C$ is any closed contour within the ROC that encircles the origin counterclockwise. This is the most general method and works for any $X(z)$, but it requires contour integration from complex analysis. In practice, the residue method is the computational realization of this integral.

Applications of Z-transform

Discrete-time system analysis

The Z-transform provides a complete framework for characterizing LTI systems. Given a transfer function $H(z)$, you can determine stability (pole locations), frequency response (evaluate $H(z)$ on the unit circle, $z = e^{j\omega}$), and transient behavior (inverse transform of the output). This is the standard approach for analyzing any discrete-time system.

Digital filter design

FIR and IIR filters are designed by specifying a desired frequency response and then determining the filter coefficients, which correspond to the numerator and denominator polynomial coefficients of $H(z)$. The Z-transform connects the time-domain difference equation to the frequency-domain specification, making it possible to go back and forth between filter coefficients and frequency response.

Difference equations

The Z-transform converts a linear constant-coefficient difference equation into an algebraic equation in $z$. The procedure:

1. Take the Z-transform of both sides of the difference equation.
2. Use the time-shifting property to express delayed terms as powers of $z^{-1}$.
3. Solve for $Y(z)$ algebraically.
4. Apply the inverse Z-transform to get $y[n]$.

For the unilateral transform, initial conditions appear naturally in step 2, which is why it's preferred for initial-value problems.

Discrete Fourier transform

The DFT is obtained by evaluating the Z-transform at $N$ equally spaced points on the unit circle:

$X[k] = X(z)\big|_{z = e^{j2\pi k/N}}, \quad k = 0, 1, \ldots, N-1$

This connection means that properties of the Z-transform (convolution, linearity, etc.) carry over to the DFT. The FFT is an efficient $O(N \log N)$ algorithm for computing the DFT, compared to the direct $O(N^2)$ computation.

Sampling and reconstruction

When a continuous-time signal $x_c(t)$ is sampled at period $T$, the Z-transform of the resulting sequence $x[n] = x_c(nT)$ relates to the Laplace transform of $x_c(t)$ through $z = e^{sT}$. The Z-transform framework helps you analyze:

• Aliasing: frequency components above the Nyquist rate ($f_s/2$) fold back into the baseband.
• Reconstruction: the conditions under which $x_c(t)$ can be perfectly recovered from $x[n]$ (Shannon-Nyquist theorem).
• Anti-aliasing filter design: specifying the cutoff to prevent aliasing before sampling occurs.