Qu

Laplace & Z-Transforms

The idea

A plucked guitar string rings and dies away. A microphone held too close to its own speaker rings and grows, into feedback. Both are the natural response of a system left alone after a single disturbance; one decays toward silence, the other explodes.

Lesson 4 built everything from \(e^{-j2\pi ft}\), a complex exponential that only spins in place at constant radius — a pure, undying tone. It has no way to represent decay or growth. Give it a partner: a plain real exponential \(e^{\sigma t}\), which shrinks when \(\sigma < 0\) and grows when \(\sigma > 0\), and multiply the two:

\[ e^{\sigma t}\, e^{j\omega t} = e^{(\sigma + j\omega)t} \]

The result is a spiral rather than a circle — still turning at rate \(\omega\), but its radius shrinking or growing at rate \(\sigma\) as it turns. A decaying string is a spiral winding inward; feedback is a spiral winding outward; a pure Fourier tone is the special case \(\sigma = 0\), where the spiral degenerates back into a circle.

t = (0 to 200) / 200 * 2
decaying = exp(-2 * t) .* cos(2 * pi * 5 * t)
steady = cos(2 * pi * 5 * t)
growing = exp(0.8 * t) .* cos(2 * pi * 5 * t)
figure()
subplot(1, 2, 1)
plot(t, decaying, label = "sigma < 0")
plot(t, steady, label = "sigma = 0")
plot(t, growing, label = "sigma > 0")
xlabel("time (s)")
ylabel("amplitude")

theta = (0 to 100) / 100 * 2 * pi
subplot(1, 2, 2)
plot(cos(theta), sin(theta), label = "unit circle")
scatter([0.7], [0], label = "stable pole")
scatter([1.2, 1.2], [0.3, -0.3], label = "unstable poles")
xlabel("Re")
ylabel("Im")
output
· plot: figure 1 (201 points)
· plot: figure 2 (201 points)
· plot: figure 3 (201 points)
· plot: figure 4 (101 points)
· scatter: figure 5 (1 points)
· scatter: figure 6 (2 points)
-5 -4 -3 -2 -1 0 1 2 3 4 5 0 0.6 1.2 1.8 time (s) amplitude -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 -0.6 0 0.6 1.2 x=0.7, y=0 x=1.2, y=0.3 x=1.2, y=-0.3 Re Im

Plot every value \(s = \sigma + j\omega\) on a plane — \(\sigma\) horizontal, \(\omega\) vertical — and you get the s-plane. Its vertical axis (\(\sigma = 0\)) is exactly Fourier's territory: pure oscillation, neither growing nor decaying. Left of it is decay; right of it is growth. The Laplace transform is the Fourier integral with this second axis restored:

\[ X(s) = \int_{0}^{\infty} x(t)\, e^{-st}\, dt, \qquad s = \sigma + j\omega \]

Set \(\sigma = 0\) and this collapses exactly to Lesson 4's Fourier transform: Fourier is not a separate idea, it is Laplace evaluated on one particular line of the s-plane.

Sampled systems get the same treatment, built the same way the DFT was built from the continuous integral — sum instead of integrate, over a variable z instead of s:

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

T is the sampling period. This substitution reshapes the whole picture: the s-plane's imaginary axis maps to \(|z| = 1\), the unit circle. The left half-plane (decay) maps to *inside* the circle; the right half (growth) maps to *outside*. A right-hand illustration above places two example poles this way — one inside the circle, stable, and a pair outside it, unstable — before any real filter has entered the picture at all.

A transfer function \(H(z) = B(z)/A(z)\) has zeros, where the numerator vanishes (frequencies the system blocks), and poles, where the denominator vanishes (frequencies the system rings at on its own, even with no input). Stability rule: every pole must sit strictly inside the unit circle (discrete-time) or strictly left of the imaginary axis (continuous-time). A pole exactly on the boundary rings forever, neither growing nor decaying — an idealized tuning fork, never realized exactly by anything physical.

In Qu

Qu does not yet expose a general continuous-time transfer-function tool — no s-domain tf() sitting next to the filter designers — but its discrete-time toolkit turns pole location from algebra into numbers you can read off a real design:

fs = 1000
lp_wide = butter(2, "low", 100, fs)
lp_narrow = butter(2, "low", 10, fs)

pw = poles(lp_wide)
pn = poles(lp_narrow)
print("wide-band poles: {pw:.6f}")
print("narrow-band poles: {pn:.6f}")
print("wide |pole| = {abs(pw[0]):.4f}")
print("narrow |pole| = {abs(pn[0]):.4f}")
print("wide stable: {is_stable(lp_wide)}, narrow stable: {is_stable(lp_narrow)}")

theta = (0 to 100) / 100 * 2 * pi
figure()
plot(cos(theta), sin(theta), label = "unit circle")
scatter(real(pw), imag(pw), label = "wide poles")
scatter(real(pn), imag(pn), label = "narrow poles")
xlabel("Re")
ylabel("Im")
output
wide-band poles: [0.57149 + 0.293599j, 0.57149 - 0.293599j]
narrow-band poles: [0.955599 + 0.042512j, 0.955599 - 0.042512j]
wide |pole| = 0.6425
narrow |pole| = 0.9565
wide stable: true, narrow stable: true
· plot: figure 7 (101 points)
· scatter: figure 8 (2 points)
· scatter: figure 9 (2 points)
-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 x=0.571, y=0.294 x=0.571, y=-0.294 x=0.956, y=0.043 x=0.956, y=-0.043 Re Im 
wide-band poles: [0.57149 + 0.293599i, 0.57149 - 0.293599i]
narrow-band poles: [0.955599 + 0.042512i, 0.955599 - 0.042512i]
wide |pole| = 0.6425
narrow |pole| = 0.9565
wide stable: true, narrow stable: true

Both designs are stable — butter never hands you an unstable filter — but not equally close to trouble. Narrowing the passband from 100 Hz to 10 Hz, out of the same 1000 Hz sample rate, pulls the pole magnitude from 0.64 to 0.96, dragging it toward the unit circle's edge. A pole near that edge means a longer, more sharply resonant impulse response: the filter rings more and settles more slowly. This is the general shape of the trade-off between selectivity and stability margin, visible directly in a printed number rather than asserted.

poles/is_stable describe an existing design. You can also watch decay versus growth happen, by building the two one-pole systems this lesson opened with and feeding each an impulse:

impulse_x = impulse(12)
decay = filter_ba([1], [1, -0.8], impulse_x)
grow  = filter_ba([1], [1, -1.2], impulse_x)
print("decaying (pole at 0.8): {decay:.4f}")
print("growing (pole at 1.2): {grow:.4f}")

n = 0 to 11
figure()
plot(n, decay, label = "pole at 0.8")
plot(n, grow, label = "pole at 1.2")
xlabel("sample index n")
ylabel("amplitude")
output
decaying (pole at 0.8): [1, 0.8, 0.64, 0.512, 0.4096, 0.32768, 0.262144, 0.209715, ... (12 elements)]
growing (pole at 1.2): [1, 1.2, 1.44, 1.728, 2.0736, 2.48832, 2.985984, 3.583181, ... (12 elements)]
· plot: figure 10 (12 points)
· plot: figure 11 (12 points)
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 10 11 sample index n amplitude 
decaying (pole at 0.8): [1, 0.8, 0.64, 0.512, 0.4096, 0.32768, 0.262144, 0.209715, ... (12 elements)]
growing (pole at 1.2): [1, 1.2, 1.44, 1.728, 2.0736, 2.48832, 2.985984, 3.583181, ... (12 elements)]

filter_ba(b, a, x) runs y[n] = x[n] + 0.8*y[n-1] in the first case and y[n] = x[n] + 1.2*y[n-1] in the second — the difference equation behind a single real pole at z = 0.8 and z = 1.2. One sits inside the unit circle and the response is a guitar string, each sample 0.8 times the last, approaching zero. The other sits outside, each sample 1.2 times the last, forever: mic feedback, laid out as plain numbers. Nothing here needed a spectrum — the pole location alone predicted the shape of the answer before a single sample was computed.

Every claim in this lesson assumed a linear, time-invariant system with poles fixed in place for all time. That describes a designed filter well. It does not describe a chirp sweeping from 20 Hz to 20 kHz, a spoken word, or a bird call, where the frequency content is not one fixed set of poles but something that changes while you listen. The s-plane and z-plane answer "will this system's own behavior blow up," and cannot answer "what is happening in this signal right now, compared to a moment ago." Lesson 6 takes on that question directly.