The Frequency Domain: Why Everything Is Sinusoids

A recording is a list of numbers, one per instant. That is the time domain, and it is the natural way to store a signal and almost the worst way to reason about one. Nothing in a waveform tells you there is a 440 Hz note under a 60 Hz hum.

The frequency domain rewrites the identical signal as a recipe: how much of each frequency it contains, and with what phase. No information is added or lost. It is a change of coordinates, exactly like describing a point by latitude and longitude instead of by distance and bearing.

Key idea: Every question in this course, filtering, sampling, compression, noise removal, is easy in one of the two domains and awkward in the other. The skill being taught is knowing which one you are in and how to move.

Joseph Fourier made the claim in 1822, in his treatise on the propagation of heat: any reasonable periodic function can be written as a sum of sines and cosines at integer multiples of its fundamental frequency.

$f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(n\omega t) + b_n \sin(n\omega t) \right)$

Contemporaries including Lagrange objected, and they were not being obtuse: the claim is that a sum of perfectly smooth curves can reproduce a square wave with a vertical edge in it. It can, in the limit, though the way it fails on the way there matters and shows up again in the filters lesson.

Definition: The coefficients $a_n$ and $b_n$ are the spectrum. Finding them is analysis; adding the terms back up is synthesis. A synthesiser and a spectrum analyser are the same machine run in opposite directions.

You could decompose a signal onto square waves, wavelets, or any other basis, and people do. Sinusoids get special treatment for a structural reason, not an aesthetic one.

Feed a sinusoid into any linear, time-invariant system, an amplifier, a room, a cable, a filter, and what comes out is a sinusoid at the same frequency. Only the amplitude and the phase change. No other family of signals survives an arbitrary LTI system so cleanly; feed in a square wave and you generally get something that is no longer square.

Key idea: Sinusoids are the eigenfunctions of linear time-invariant systems. That is why the frequency domain turns the hard operation, convolution, into the easy one, multiplication. Every result in the rest of this course is downstream of that single fact.

Real signals are finite lists of samples, so the practical tool is the discrete Fourier transform. For $N$ samples $x_0 \dots x_{N-1}$:

$X_k = \sum_{n=0}^{N-1} x_n \, e^{-2\pi i k n / N}$

Each $X_k$ is a complex number obtained by multiplying the signal against a complex sinusoid at bin frequency $k$ and summing. Large magnitude means the signal has a lot of that frequency in it.

The complex exponential is doing the work of the sine and cosine at once, via Euler's identity $e^{i\theta} = \cos\theta + i\sin\theta$. The real part correlates against the cosine, the imaginary part against the sine, and the pair carries both how much and when.

• $|X_k|$ is the magnitude: how much of that frequency is present
• $\arg(X_k)$ is the phase: where in its cycle that component starts.

import numpy as np

fs = 8000                      # sample rate, Hz
N = 8000                       # one second of audio
t = np.arange(N) / fs
x = np.sin(2*np.pi*440*t) + 0.5*np.sin(2*np.pi*1000*t)

X = np.fft.rfft(x)             # real input -> half spectrum
freqs = np.fft.rfftfreq(N, 1/fs)
mag = np.abs(X) / (N/2)        # scale to amplitude

peaks = np.argsort(mag)[-2:]
print(freqs[peaks], mag[peaks])  # [1000. 440.] [0.5 1.0]

Two tones go in, two peaks come out at the right frequencies with the right amplitudes. rfft is used rather than fft because a real-valued signal has a symmetric spectrum, so the upper half is redundant and computing it wastes half the work.

The bins are evenly spaced, and the spacing follows from the two numbers you chose:

$\Delta f = \frac{f_s}{N}$

Since $N / f_s$ is just the duration $T$ of the recording, this is the same as $\Delta f = 1/T$. Resolution is set by how long you listened, not by how fast you sampled.

Gotcha: To separate two tones 1 Hz apart you need at least one second of signal, whatever your sample rate. Recording at 192 kHz instead of 48 kHz raises the highest frequency you can see, four times higher, and does nothing at all for your ability to tell 440 Hz from 441 Hz. This is the single most common misconception about spectrum analysis, and it costs people money in hardware that cannot fix their problem.

Look again at the DFT sum. Each of the $N$ outputs is a sum over all $N$ inputs, so the direct computation costs on the order of $N^2$ operations. For a one-second CD-quality clip, $N$ is about 44,100 and $N^2$ is around two billion multiply-accumulates for one spectrum.

The algorithm exploits redundancy: the same complex exponentials recur throughout the sum, so a transform of length $N$ splits into two of length $N/2$, recursively.

The advantage is not a constant, it grows with the transform size, which is why the FFT did not merely speed things up but changed what was worth attempting at all.

Cooley and Tukey published the algorithm in 1965. Gauss had worked out essentially the same decomposition in 1805 while interpolating the orbits of the asteroids Pallas and Juno, in a manuscript he never published and where he never analysed its cost. The 1965 paper landed when there were digital computers to run it on, and that timing is most of why the name attached there.

Almost every spectrum you see plots magnitude only. Phase is discarded because the eye reads magnitude easily and phase plots look like noise. That discard is usually harmless and occasionally destroys everything.

The classic demonstration: take two images, swap their Fourier phases while keeping their magnitudes, and each result looks like the image that donated the phase. Where things are, in other words, lives in the part that gets thrown away.

In practice: If you reconstruct a signal from a modified spectrum, you must handle phase deliberately. Discarding it and guessing on the way back is what makes badly done audio processing sound smeared and metallic.

The DFT of a whole recording answers "which frequencies are in this signal" and refuses to answer "when". A five-minute track containing one brief 3 kHz beep produces a spectrum with a small 3 kHz component, indistinguishable from a faint 3 kHz tone running throughout.

That is not a defect to be fixed but a consequence of the transform: each basis sinusoid extends over the entire window, so it has no notion of location. Every real analysis tool works around it the same way, by chopping the signal into short overlapping windows and transforming each one, which is what a spectrogram is.

That workaround introduces its own arithmetic, a hard trade between time resolution and frequency resolution, and a set of artefacts that will make you misread a spectrum if you do not know they are there. Those are the subject of the last lesson in this course.