5.7 Summary#
The frequency domain describes a sound by how much of each frequency it contains, complementing the time domain waveform \(x(t)\).
For additive synthesis, the frequency-domain amplitude is read directly off the recipe: a spike of height \(a_k\) at each harmonic frequency \(k \cdot f_0\). The time domain is dense and continuous, while the frequency domain is sparse and discontinuous.
The complex plane is an analytical tool for periodic phenomena. A complex number has a rectangular form \(z = x + jy\) and a polar form \(z = r e^{j\theta}\). Under multiplication, magnitudes multiply and angles add, so multiplication models rotation. Euler’s formula \(e^{j\theta} = \cos\theta + j\sin\theta\) links the two forms.
A complex sinusoid or phasor \(a\, e^{j\omega t} : \mathbb{R} \to \mathbb{C}\) is a rotating vector whose real and imaginary parts are a cosine and a sine. It is a function of time, and it is the central building block of frequency analysis.
The Fourier transform \(X(\omega) = \int_{-\infty}^{\infty} x(t)\, e^{-j\omega t}\, dt : \mathbb{R} \to \mathbb{C}\) is a function of frequency. It probes a signal with a phasor at each frequency, multiplies to measure similarity, and integrates over time.
Converting the complex output to polar form gives the real-valued amplitude spectrum \(|X(\omega)|\) and phase spectrum \(\angle X(\omega)\). Because the ear is largely insensitive to phase, the amplitude spectrum is the more commonly used.
Intuitively, the transform winds a signal around the complex plane at a probe frequency and measures the center of mass: large when the probe matches a frequency in the signal, near zero otherwise.