Z-transform & filter theory

Processing Prerequisites

Learning objectives

Define the Z-transform of a discrete sequence and interpret z = e^(iω) on the unit circle
Read poles and zeros from a transfer function H(z) and predict stability from pole locations
Predict the shape of a filter's frequency response from pole-zero geometry (nearness to unit circle)
Distinguish FIR (zeros only, always stable) from IIR (poles present, stability conditional)

The Fourier transform of §0.4 tells you what frequencies a finite sequence contains. The Z-transform is its richer cousin: it handles infinite sequences, it gives filter designers poles and zeros they can place with their hands, and it collapses every linear time-invariant filter into a single rational function.

1. Definition

For a sequence x[n],

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

where z is any complex number for which the sum converges. That set of z is the region of convergence (ROC), and it matters for causality and stability, but for the one-step-back view we need today the key fact is:

\text{On the unit circle }|z|=1,\;\text{write}\; z = e^{i\omega}.\;\text{Then } X(e^{i\omega}) = \text{DTFT of }x[n].

The Z-transform evaluated on the unit circle is the frequency response. Move z off the circle and you are probing how the system responds to signals that are growing or decaying exponentially. That extra dimension is what buys you poles, zeros, and a language for designing filters.

2. Transfer function

A linear-constant-coefficient difference equation

y[n] + \sum_{k=1}^{M} a_k\, y[n-k] \;=\; \sum_{k=0}^{N} b_k\, x[n-k]

takes the Z-domain form

H(z) \;=\; \frac{Y(z)}{X(z)} \;=\; \frac{b_0 + b_1 z^{-1} + \cdots + b_N z^{-N}}{1 + a_1 z^{-1} + \cdots + a_M z^{-M}}

The zeros are the roots of the numerator polynomial; the poles are the roots of the denominator polynomial. Every causal LTI filter — every Wiener deconvolution operator, every IIR recursion, every FIR filter — is one of these.

3. Poles and zeros on the Z-plane

Plot the complex plane. Mark every zero with a circle and every pole with a cross. The resulting picture is the pole-zero plot, and it tells you almost everything about the filter at a glance.

Drag the pole (red ×) close to the unit circle and watch the frequency response blow up in magnitude at the angle of the pole. Drag the zero (teal ○) onto the unit circle and the frequency response drops to zero at that angle — a notch. The distance from e^iω to each pole and zero sets the magnitude response:

|H(e^{i\omega})| \;=\; \frac{\prod_k |\,e^{i\omega} - z_k\,|}{\prod_k |\,e^{i\omega} - p_k\,|}

4. Stability

For a causal system, the ROC is the exterior of some circle containing all poles. For stability we need the ROC to include the unit circle — which means every pole must sit inside it:

All poles with |p| < 1 ⇔ stable causal filter.

Push a pole outside the unit circle (try the “Unstable!” preset). The impulse response grows without bound — the filter screams. This is the most basic design check: a good filter keeps all its poles safely inside.

5. FIR vs. IIR

FIR (Finite Impulse Response): all the a_k are zero. The denominator is 1, and every pole sits at the origin — always stable. The impulse response is exactly as long as the numerator polynomial. FIR filters are easy to design with linear phase, but they cost more multiplies per sample to get the same selectivity as an IIR.
IIR (Infinite Impulse Response): at least one a_k is nonzero. Poles land away from the origin, the impulse response is infinite (decaying, hopefully), and very sharp selectivity is possible with a low filter order. Stability must be checked. Phase is usually nonlinear.

Processing uses both. Minimum-phase deconvolution operators are IIR (stability matters). Band-pass filters for display are usually FIR so the seismic phase is preserved.

6. The Wiener filter in one paragraph

When you whiten a seismic trace to recover the reflectivity, the optimal least-squares filter is the Wiener filter. Its design happens in the Z-domain: invert the wavelet’s spectrum (ratio of two polynomials), stabilize by adding a tiny white-noise floor (so no pole sits exactly on the unit circle), and implement as a causal IIR. The pole-zero plot above is the same language used to design every deconvolution operator in seismic processing.

7. Three facts to carry forward

Closer to the unit circle means a bigger effect on the frequency response (a pole close to z = e^iω₀ gives a peak at ω₀; a zero close to e^iω₀ gives a notch).
Real coefficients ⇔ complex poles and zeros in conjugate pairs. That is why the widget only lets you drag the upper-half member — the mirror happens automatically, and the filter coefficients stay real.
Minimum-phase filters have all their zeros inside the unit circle. These are invertible (the inverse is stable). Non-minimum-phase wavelets are a recurring headache in seismic deconvolution.

**The one sentence to remember**

Every discrete linear filter is a rational function in z; poles make peaks, zeros make notches, and all poles must live inside the unit circle for the filter to behave.

Where this goes next

Section §0.7 switches from signals to systems: linear algebra, the language of migration, tomography, and inversion. Seismic processing beyond filtering is almost entirely linear algebra — and you need fluency with vectors, matrices, and least squares to make sense of the algorithms.

References

Oppenheim, A. V., Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall.
Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.
Robinson, E. A., Treitel, S. (2008). Digital Imaging and Deconvolution. SEG.
Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.