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SC · Fourier

Fourier Transform Properties

The most powerful property of the Fourier transform is that convolution in the time domain becomes multiplication in the frequency domain. See for yourself how the convolution you laboriously slid and summed in chapter B turns, in frequency, into multiplying two spectra point by point.

Convolution and product move together

The upper figure is the time output y = x ∗ h, the convolution of a rectangular pulse x of width 1 with a rectangular pulse h whose width you can change, and the gold dashed curve is its target. In the lower figure, the green curve is the output spectrum Y = X·H, the faint gray X and the faint blue H multiplied point by point. Move the width of h and the convolution above and the product below always change together; when the output in the upper figure reaches its target, the product below becomes that target's spectrum too.

time output y = x ∗ htarget
spectrum Y = X · H
width of h2.00
Output in both domains
y = x ∗ h ↔ Y = X · H
The width of h differs from the target, so the time output is off and the product Y below differs to the same extent. The two domains are two faces of the same fact and always move together.
Off

Convolution becomes multiplication

The key property of the Fourier transform is the convolution theorem. If y = x ∗ h in the time domain, then Y(ω) = X(ω) H(ω) in the frequency domain, so the slide-and-sum integral turns into a simple product of matching frequencies. Here H(ω) is the Fourier transform of the impulse response h, that is, the system's frequency response. The LTI output calculation seen in chapter B ends, in frequency, with a single multiplication.

Observex ∗ h ↔ X · H
Convolution in time is multiplication in frequency.
Choosea x + b y ↔ ?
Sums and scalings carry straight over.
Fill inx(t − t₀) ↔ X(ω) · ?
A time shift is the phase e−jωt₀.
On your ownx(at) ↔ ?
Compress in time, dilate in frequency.

Shift is phase, scale is reciprocal

Other properties also turn the transform into a calculating tool. By linearity, sums and scalings of inputs carry over unchanged into the transform. For a time shift, the transform of x(t − t₀) is X(ω) multiplied by e^(−jωt₀), so the magnitude |X| stays the same and only the phase tilts in proportion to frequency. For time scaling, the transform of x(at) is (1/|a|)X(ω/a), so the reciprocal relation seen in C3 becomes a property in its own right: compressing a signal in time stretches it in frequency.

Why this is powerful

The convolution theorem is powerful because, in the frequency domain, an LTI system treats each frequency separately. Feed a sinusoid into an LTI system and a sinusoid of the same frequency comes out with its size and phase changed by H(ω), so sinusoids are the eigenfunctions of LTI systems and H(ω) is their eigenvalue. That is why a complicated convolution breaks into one independent product per frequency. What H(ω) does at each frequency is the topic of the next unit, the frequency response.

Back to the first screen

Every time the width of h moved, the time output y = x ∗ h and the frequency output Y = X·H changed together. That is because they are one pair, the same signal seen in time and in frequency. In time it was a slide-and-sum convolution, but in frequency it was multiplying two spectra point by point, and the moment the widths matched, the pair of a triangle and a sinc² appeared. This convolution theorem, which turns a hard operation into an easy one, is the key reason Fourier is used so much in engineering.

The Fourier transform properties turn the transform into a calculating tool. The key one is the convolution theorem: y = x ∗ h ⟺ Y(ω) = X(ω) H(ω), where H(ω) = FT{h} is the frequency response. There are also linearity (ax + by ⟺ aX + bY), the time shift (x(t − t₀) ⟺ e−jωt₀ X(ω), magnitude unchanged) and time scaling (x(at) ⟺ (1/|a|) X(ω/a)). The convolution theorem turns an LTI output into a multiplication, and it also means that sinusoids are the eigenfunctions of LTI systems.
On to the next unit

The convolution theorem packed what a system does into a single function, H(ω). The next unit, the frequency response, looks into that H(ω) itself: the magnitude |H(ω)| shows how much each frequency is boosted or cut, and the phase ∠H(ω) how much each frequency is delayed. Feed in one sinusoid and measure the amplitude ratio and phase difference of the sinusoid that comes out, and which frequencies the system passes and which it blocks becomes clear at a glance.