The Z-transform and notions of filtering
The same as Laplace, for sequences. The imaginary axis becomes the unit circle, and everything you knew about stability carries over by replacing “left of the axis” with “inside the circle”.
01Laplace, for sequences
The Z transform is to discrete signals what Laplace is to continuous ones. The definition replaces the integral with a sum and the exponential with a power:
And the relationship with the \( s \)-plane is direct: sampling with period \( T \) corresponds to the substitution
which takes the imaginary axis of the \( s \)-plane to the unit circle of the \( z \)-plane, and the left half-plane to the interior of the circle. Everything we knew about stability carries over by replacing “left of the axis” with “inside the circle”.
| In the \( s \)-plane | In the \( z \)-plane |
|---|---|
| Imaginary axis | Unit circle \( |z| = 1 \) |
| Left half-plane | Interior of the circle |
| Origin \( s = 0 \) | \( z = 1 \), DC |
| \( \omega = \pi/T \), half the sampling frequency | \( z = -1 \) |
| Stable: poles on the left | Stable: poles inside |
The fourth row has an important consequence: in the \( z \)-plane the whole frequency range fits in one turn. Going around the circle from \( z = 1 \) to \( z = -1 \) sweeps from DC up to half the sampling frequency, and continuing to turn means starting over. There is aliasing again, turned into geometry.
02Designing by placing poles and zeros
From the previous relationship comes a direct design technique: a zero on the circle cancels that frequency; a pole near the circle boosts it. With that you build filters by hand, without formulas, just by arranging points.
On the left the \( z \)-plane with the unit circle, the poles (×) and the zeros (○), and the vectors to the working point. On the right, the magnitude of the response and the impulse response.
03The region of convergence, once again
As in Laplace, \( X(z) \) alone is not enough. Except that here the regions are annuli instead of strips:
| If the region… | The sequence is… |
|---|---|
| Is the exterior of the outermost pole | Causal: it is zero for \( n \lt 0 \) |
| Is the interior of the innermost pole | Anticausal |
| Is an annulus between poles | Two-sided |
| Contains the unit circle | Has a Fourier transform and the system is stable |
And the usual conclusion, translated: a causal system is stable if and only if all of its poles are inside the unit circle. The lab shows it with the last case: raising the radius above 1, the impulse response stops dying out and starts to grow.
\( a^{n}u[n] \longleftrightarrow \dfrac{1}{1 - az^{-1}} \), with region \( |z| \gt |a| \). From this comes almost everything: the first-order filter, the geometric sum and the condition \( |a| \lt 1 \) for convergence. And its anticausal version, \( -a^{n}u[-n-1] \), has exactly the same expression with region \( |z| \lt |a| \).
04Basics of digital filtering
The geometric reading is enough to design several useful things by eye:
| What you want | What to place where |
|---|---|
| Low-pass | Pole near \( z = 1 \), zero at \( z = -1 \) |
| High-pass | Pole near \( z = -1 \), zero at \( z = 1 \) |
| Reject a frequency \( f_0 \) | Pair of zeros on the circle at \( \pm 2\pi f_0/f_s \) |
| Boost a frequency | Pair of poles near the circle at that angle |
| Remove DC | A zero exactly at \( z = 1 \) |
| Comb, for mains harmonics | Zeros spread at multiples of the 50 Hz angle |
The rule of thumb for the bandwidth of a resonator is that a pole of radius \( r \) gives a peak whose width at −3 dB is approximately \( 2(1-r) \) in radians per sample. With \( r = 0.99 \) the peak is extremely narrow; with \( r = 0.8 \), wide and smooth.
If the filter coefficients are real —and they are, because they are programmed with real numbers— the complex poles and zeros have to come in conjugate pairs. You cannot place a lone pole at \( 0.9e^{j\pi/4} \) without its twin at \( 0.9e^{-j\pi/4} \): the resulting filter would need complex arithmetic. That is why all the labs in this topic show the poles in pairs symmetric about the real axis.
05In the lab
With \( f_s = 1 \) kHz, calculate the angle that corresponds to 50 Hz and place the pair of zeros on the circle. Add a pair of poles at the same angle with \( r = 0.95 \) to narrow the rejection, and write down the coefficients of the resulting biquad.
Program \( y[n] = 2r\cos\theta\,y[n-1] - r^2 y[n-2] + x[n] \) for \( r = 0.9 \), \( 0.99 \) and \( 0.999 \), measure the −3 dB bandwidth of each and compare it with the rule \( 2(1-r) \).
With the same program, raise \( r \) to 1.01 and observe the output. Determine in how many samples a 16-bit accumulator overflows and relate it to the position of the pole.
06Common mistakes
- Placing a complex pole without its conjugate. With real coefficients they must come in pairs.
- Forgetting the region of convergence. The same \( X(z) \) describes a causal or an anticausal sequence depending on the region.
- Confusing \( |z| = 1 \) with the sampling frequency. A full turn is \( f_s \); the useful half goes up to \( z = -1 \).
- Placing a pole exactly on the circle. It is an oscillator, not a filter: it is not BIBO stable.
- Designing in the \( s \)-plane and moving to the \( z \)-plane carelessly. The bilinear transformation warps the frequency axis, and you have to prewarp.
07Self-assessment
What is the relationship between the \( s \) and \( z \) planes?
\( z = e^{sT} \). The imaginary axis goes to the unit circle and the left half-plane to the interior.
When is a causal discrete system stable?
When all of its poles are inside the unit circle.
Where does a zero go to remove DC?
At \( z = 1 \), which is the point that corresponds to zero frequency.
How do you reject a frequency \( f_0 \)?
With a pair of zeros on the circle at the angle \( \pm 2\pi f_0/f_s \).
How wide is the peak of a pole of radius \( r \)?
Approximately \( 2(1-r) \) radians per sample: the closer to the circle, the narrower.
Why do complex poles come in pairs?
Because the filter coefficients are real. A lone pole would force the use of complex arithmetic.
08Further reading
- John G. Proakis and Dimitris G. Manolakis. Digital Signal Processing: Principles, Algorithms, and Applications. 4th ed. (Spanish edition, “Tratamiento digital de señales”), Prentice Hall, 2006. The Z transform with its region of convergence, and filter design by pole and zero placement.
- Alan V. Oppenheim and Ronald W. Schafer. Discrete-Time Signal Processing. 3rd ed. (Spanish edition, “Tratamiento de señales en tiempo discreto”), Prentice Hall, 2010. The standard reference in the field. Chapter 3 is the one on the Z transform.
- Richard G. Lyons. Understanding Digital Signal Processing. 3rd ed., Prentice Hall, 2010. The geometric reading of the z-plane explained with a clarity that is hard to match.