The continuous-time Fourier transform

The Fourier series handles periodic signals, but many of the signals Amara cares about are not periodic. A single tick of the locator tone, the burst of noise from a passing truck and a speech message announcing the street name all start and stop. The Fourier transform extends frequency analysis to such signals. Instead of a set of lines at harmonic frequencies, a non-periodic signal has a continuous spectrum that spreads its content over a range of frequencies. The transform turns convolution into multiplication, which is why it is the main working tool for filters, communication systems and audio processing, and it explains a basic trade-off of signal design: the shorter a signal is in time, the wider it is in frequency.

From series to transform

Imagine a pulse repeated every T₀ seconds and let T₀ grow. The harmonics, spaced ω₀ = 2π/T₀ apart, crowd closer together, and each coefficient c_k shrinks in proportion to 1/T₀ because the pulse occupies a smaller fraction of each period. In the limit of a single pulse, the spacing becomes infinitesimal, the lines merge into a continuous curve, and the product T₀c_k becomes a function of the continuous frequency ω. That function is the Fourier transform:

The first integral is the analysis equation, which measures how much of each frequency the signal contains; the second is the synthesis equation, which rebuilds the signal from its frequency content. X(jω) is a density: |X(jω)|² dω/2π is the energy in a small band of width dω. The notation x(t) ↔ X(jω) says the two are a transform pair. Some books use the ordinary frequency f in hertz, with ω = 2πf, which removes the 1/2π from the inverse; both conventions appear in practice, so always check which one a table uses.

Three essential pairs

The rectangular pulse of height 1 and width τ, centered at t = 0, transforms to a sinc-shaped spectrum. Integrating e^(−jωt) from −τ/2 to τ/2 gives

The spectrum has its peak value τ, the pulse's area, at ω = 0, and it first reaches zero at ω = 2π/τ, which is f = 1/τ in hertz. A 1 ms pulse has its first spectral null at 1 kHz; a 0.1 ms pulse at 10 kHz. Most of the energy lies in the main lobe between the first nulls, so the bandwidth of a pulse is roughly the inverse of its duration.

The decaying exponential e^(−at)u(t), with a > 0, transforms to 1/(a + jω), whose magnitude 1/√(a² + ω²) falls off smoothly as frequency rises. The unit impulse transforms to 1 at every frequency: an infinitely short event contains all frequencies equally, which is why a sharp click excites every resonance of a room or a structure. In the other direction, a constant contains only zero frequency, and its transform is an impulse, 2πδ(ω).

Properties

A handful of properties lets you find most transforms from a few known pairs instead of integrating. Linearity: a sum of signals transforms to the sum of their transforms. Time shift: delaying a signal by t₀ multiplies its transform by e^(−jωt₀), which changes only the phase, not the magnitude, so a delayed tick has the same spectrum magnitude as an early one. Time scaling: compressing a signal in time by a factor a stretches its spectrum by a, x(at) ↔ (1/|a|)X(jω/a), the formal version of the duration–bandwidth trade-off. Conjugate symmetry: for a real signal X(−jω) = X(jω)*, so the magnitude spectrum is even and only positive frequencies need to be plotted.

Two properties matter more than the rest. The convolution property says that convolving in time becomes multiplying in frequency:

An LTI system therefore multiplies the input spectrum by its frequency response, H(jω), the Fourier transform of its impulse response. That is the whole idea of filtering, and lesson 8 builds on it. The dual modulation property says that multiplying in time becomes convolving in frequency, scaled by 1/2π. Multiplying a signal by cos(ω₀t) shifts its spectrum up and down by ω₀:

This is how AM radio moves audio up to a carrier frequency, and it explains the spectrum of a tone burst, as the next example shows. Differentiation in time multiplies the transform by jω, which is why differentiating a signal emphasizes its high frequencies and why the derivative of noisy data is noisier still. Parseval's theorem gives the energy in either domain: ∫|x(t)|²dt = (1/2π)∫|X(jω)|²dω.

Periodic signals and the transform

The Fourier transform also covers periodic signals, which ties it back to lesson 6. A periodic signal with Fourier coefficients c_k has a transform made of impulses at the harmonic frequencies: X(jω) = Σ 2πc_k δ(ω − kω₀). Each line of the line spectrum becomes an impulse whose area is 2π times the coefficient. The cosine entry in the table is the simplest case, with c_{±1} = ½. Writing periodic and non-periodic signals in one framework means a single set of tools handles a signal that is partly both, such as the steady 60 Hz hum, which is a pair of impulses at ±60 Hz, plus the occasional truck, which is a continuous spread. It also shows something useful about the square wave from lesson 6: its coefficients c_k = d·sin(πkd)/(πkd) are samples of the transform of a single pulse, taken at the harmonic frequencies and divided by the period. Repeating a signal in time samples its spectrum in frequency, a fact that returns in lesson 9 with the roles of time and frequency swapped.

The frequency response is a transform

Because the output spectrum is X(jω)H(jω), the frequency response H(jω) is simply the Fourier transform of the impulse response. For the RC low-pass filter, h(t) = (1/RC)e^(−t/RC)u(t), and the table gives H(jω) = 1/(1 + jωRC). The magnitude starts at 1 at DC and falls as frequency rises, which is exactly what a low-pass filter should do. There are therefore two equivalent ways to find a filter's output: convolve the input with h(t) in time, or multiply the spectra and transform back. For simple signals the second is often far easier, and for long signals on a computer the fast Fourier transform of lesson 10 makes it far faster.

Duration, bandwidth and uncertainty

The pairs above share a pattern: whatever shape a signal has, squeezing it in time spreads it in frequency. No signal can be both very short and very narrow in frequency. Defined carefully with root-mean-square widths, the product of duration and bandwidth has a minimum value that only a Gaussian pulse achieves. Engineers meet this limit everywhere. A radar that wants to resolve close targets needs short pulses and therefore wide bandwidth; a radio channel with narrow bandwidth cannot carry fast symbols; and a noise meter that wants to tell 880 Hz from 900 Hz must listen for tens of milliseconds. For the pedestrian signal it means the locator tick cannot be both very short and very pure in pitch.

import math, cmath

def ft(x, t, w):
    """Approximate X(jw) = integral x(t) exp(-j w t) dt with a Riemann sum."""
    dt = t[1] - t[0]
    return sum(xv * cmath.exp(-1j * w * tv) for xv, tv in zip(x, t)) * dt

tau, f0 = 20e-3, 880.0
dt = 1e-5
t = [-tau / 2 + n * dt for n in range(int(tau / dt))]
x = [math.cos(2 * math.pi * f0 * tv) for tv in t]

def exact(f):
    def half(u):
        return tau / 2 if abs(u) < 1e-12 else tau / 2 * math.sin(u * tau / 2) / (u * tau / 2)
    w = 2 * math.pi * f
    return half(w - 2 * math.pi * f0) + half(w + 2 * math.pi * f0)

for f in (0, 800, 830, 860, 880, 900, 930, 1000):
    num = abs(ft(x, t, 2 * math.pi * f))
    print(f"{f:5d} Hz: |X| numeric {num*1e3:7.3f} ms   exact {abs(exact(f))*1e3:7.3f} ms")

    0 Hz: |X| numeric   0.346 ms   exact   0.344 ms
  800 Hz: |X| numeric   1.985 ms   exact   1.982 ms
  830 Hz: |X| numeric   0.056 ms   exact   0.055 ms
  860 Hz: |X| numeric   7.624 ms   exact   7.622 ms
  880 Hz: |X| numeric   9.946 ms   exact   9.947 ms
  900 Hz: |X| numeric   7.481 ms   exact   7.483 ms
  930 Hz: |X| numeric   0.053 ms   exact   0.052 ms
 1000 Hz: |X| numeric   1.178 ms   exact   1.181 ms

The numerical and exact magnitudes agree to within a fraction of a percent. The peak at 880 Hz is very nearly τ/2 = 10 ms; the small shortfall comes from the tail of the mirror-image lobe centered at −880 Hz, which overlaps slightly. The near-zeros at 830 and 930 Hz mark the ends of the main lobe, and at 860 and 900 Hz, 20 Hz from the center, the magnitude has already dropped to about 75 % of the peak. Content at 800 and 1,000 Hz comes from the side lobes. When Amara's noise detector listens for traffic noise, she excludes a band around 880 Hz wide enough to hold the tick's main lobe, so the detector does not mistake the signal's own tick for street noise.