Z-transform & filter theory

Processing Prerequisites

Learning objectives

Define the Z-transform of a discrete sequence and interpret $z = e^{i\omega}$ 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: the distances from $e^{i\omega}$ to the poles and zeros
Distinguish FIR (zeros only, always stable) from IIR (poles present, stability conditional)

The Fourier transform of Section 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 this section 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 filter built from a finite difference equation, 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 an open 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. In the figure below you drag one conjugate pair of poles and one of zeros; the mirror image $p^\*$ of each follows on its own. With a sample interval of 4 ms the angle round the circle is the frequency, 0 Hz at $z = 1$ and 125 Hz at $z = -1$ , so the circle is marked in hertz. A probe point $e^{i\\omega}$ rides the circle, with a ray from it to every root.

The rays are the whole story. The gain at the probe’s frequency is the product of the zero rays divided by the product of the pole rays:

|H(e^{i\\omega})| \\;=\\; \\frac{\\prod\_k |\\,e^{i\\omega} - z\_k\\,|}{\\prod\_k |\\,e^{i\\omega} - p\_k\\,|}

In the starting setup the poles sit 0.20 inside the circle at 40 Hz, and the spectrum in (b) peaks near their angle at +17.1 dB, 21 Hz wide at −3 dB; the impulse response in (c) takes 84 ms to fall below 1 % of its peak. The readouts check the rule on every drag: the ray product at the probe and the DTFT of the drawn $h\[n\]$ agree. Now set up exercise 1, with the zeros at the origin and the probe on the pole angle at 40 Hz, and pull the poles to radius 0.97: the pole ray at the probe becomes 0.03 long, the peak rises to +26.0 dB and narrows to 2.4 Hz, and the ringing stretches to 604 ms. The width goes as $1 - r\_p$ and the ringing time as $1/(1 - r\_p)$ , so a sharp peak always rings long. Put a zero on the circle and one ray becomes exactly zero when the probe reaches it: a notch. At 60 Hz it takes the mains hum out of the trace in (d): the hum is 40 % of the input’s rms, and with the zero pair on the circle at 60 Hz (exercise 2) none of it is left. With poles just inside at the same angle the notch narrows to 4.1 Hz, because away from 60 Hz each pole ray nearly equals its zero ray.

4. Stability

For a causal system, the ROC is everything outside the circle through the outermost pole, $|z| > \\max\_k |p\_k|$ . For stability we need the ROC to include the unit circle, which means every pole must sit inside it:

A causal filter is stable exactly when every pole satisfies $|p_k| < 1$ .

Start from the Unstable filter, or set up exercise 4 and push the poles at 30 Hz from radius 0.90 to 1.10. The hatch in (a), which covers the part of the plane outside the ROC, now swallows the unit circle: there is no frequency response to draw, and $h\[n\]$ grows as $1.1^n$ , to 540 by sample 63. Poles exactly on the circle are the boundary case: $h\[n\]$ rings forever without decaying. 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.

Pull the poles to the origin in the figure and the filter is FIR: $h\[n\]$ stops after three samples, and nothing is left to ring. Processing uses both ideas. The exact inverse of a minimum-phase wavelet is IIR, and it is stable only because the wavelet’s zeros sit inside the unit circle; the deconvolution operator actually applied is a finite least-squares FIR approximation of that inverse. Band-pass filters for display are usually zero-phase 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 logic is easiest to see in the Z-domain: the ideal operator is $1/W(z)$ , whose poles are the wavelet’s zeros, so it is causal and stable only if the wavelet is minimum phase. In practice the filter is found in the time domain by solving the Toeplitz normal equations built from the trace autocorrelation, with a small prewhitening percentage added to the zero lag so that no near-zero of the spectrum becomes a near-unit-circle pole, and it is applied as a finite causal FIR operator. The pole-zero plot above is the language that explains why every deconvolution operator in seismic processing works, and when it cannot.

7. Three facts to carry forward

Closer to the unit circle means a bigger effect on the frequency response (a pole close to $e^{i\\omega\_0}$ gives a peak at $\\omega\_0$ ; a zero close to $e^{i\\omega\_0}$ gives a notch).
Real coefficients ⇒ complex poles and zeros come in conjugate pairs. That is why, in the figure, you can drag either member of a pair and its mirror image follows, so the filter coefficients stay real.
Minimum-phase filters have all their poles and zeros inside the unit circle, so their inverse $1/H(z)$ is also causal and stable. Reflect a zero pair from radius 0.8 to 1.25 in the figure and the amplitude spectrum keeps its shape, rising by a constant 3.9 dB, while $h\[n\]$ turns around in time: the spectrum alone cannot tell minimum from maximum phase, which is why non-minimum-phase wavelets are a recurring headache in seismic deconvolution.

The one sentence to remember

Every recursive or FIR 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.