SC · Fourier

Fourier Transform Properties

The most powerful property of the Fourier transform is that convolution in time becomes multiplication in frequency. See firsthand that the convolution you laboriously slid and summed in chapter B becomes, in frequency, just multiplying two spectra point by point.

Convolution and product move together

The top is the time output y = x ∗ h; below is its spectrum. In frequency the output Y is X·H, the input X times the system H point by point. Move h’s width and watch the time convolution and the frequency product change together, reaching the dashed target at the same time.

time output y = x ∗ htarget

spectrum Y = X · H

width of h2.00

Output in both domains

y = x ∗ h   ↔   Y = X · H

h’s width differs from the target. Both the time output and the frequency output are off. The two domains are two faces of the same fact, so they always move together.

Off

Convolution becomes multiplication

The key property of the Fourier transform is the convolution theorem. If y = x ∗ h in time, then Y(ω) = X(ω) H(ω) in frequency. The slide-and-sum integral becomes a simple product of matching frequencies. Here H(ω) is the Fourier transform of the impulse response h — the system’s frequency response. The LTI output computation of chapter B finishes in one multiplication in frequency.

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 turn the transform into a calculus. Linearity: sums and scalings of the input add and scale in the transform too. Time shift: the transform of x(t−t₀) is X(ω) times e^(−jωt₀), leaving the magnitude |X| unchanged and only tilting the phase. Time scaling: the transform of x(at) is (1/|a|)X(ω/a), making the reciprocity of C3 a property — compress a signal in time and it dilates in frequency.

Why this is powerful

The convolution theorem is powerful because, in frequency, an LTI system treats each frequency separately. Feed a sinusoid into an LTI system and out comes a sinusoid of the same frequency, changed in amplitude and phase by H(ω). That is, sinusoids are the eigenfunctions of LTI systems and H(ω) is the eigenvalue. So a complex convolution decomposes into one independent product per frequency. What H(ω) does at each frequency is the subject of the next unit, the frequency response.

Back to the first screen

Each time you moved h’s width, the time output y = x ∗ h and the frequency output Y = X · H always changed together and reached the target together, because they are one pair — the same signal seen in time and in frequency. In time it was a slide-and-sum convolution; in frequency it was multiplying two spectra point by point. This convolution theorem, trading a hard operation for an easy one, is the core reason Fourier is used so much in engineering.

Fourier transform properties make the transform a calculus. The key is the convolution theorem: y = x ∗ h ⟺ Y(ω) = X(ω) H(ω), where H(ω) = FT{h} is the frequency response. Also linearity (ax + by ⟺ aX + bY), 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 LTI output into a product and means sinusoids are the eigenfunctions of LTI systems.

On to the next unit

The convolution theorem compressed a system’s action into the single function H(ω). The next unit, the frequency response, looks into H(ω) itself. Its magnitude |H(ω)| says how much each frequency is amplified or attenuated; its phase ∠H(ω) says how much each frequency is delayed. Feed in one sinusoid and measure the amplitude ratio and phase shift of the sinusoid that comes out, and which frequencies the system passes and which it blocks become clear at a glance.