SC · Fourier

The Spectrum

View the same signal on the time axis and it is a waveform; on the frequency axis it is a spectrum. See that the spectrum stands each frequency component as a bar at its place, and that a pure sinusoid is a single line.

Land the line on the target frequency

The top is the time waveform of a pure sinusoid; below is its spectrum. The sinusoid appears as a single line at its frequency. Move the frequency slider to land the blue line exactly on the dashed target frequency. The wiggle rate in time and the line’s position move together.

time waveform

spectrum (magnitude vs frequency)target frequency

frequency f1.20

Frequency and magnitude

f = 1.20   |X| = 1.00

The line is far from the target. The waveform wiggles too slowly or too fast compared to the target. One frequency fixes one position.

Frequency off

The spectrum is an address book of frequencies

The spectrum is a plot with frequency across and magnitude up. It shows which frequencies a signal contains, and how much, as the height of a bar standing at that frequency. A pure sinusoid has just one frequency, so it shows as a single line whose horizontal position is the frequency and whose height is the amplitude. Change the frequency and the line slides sideways.

ObserveA cos(ω₀t + φ) ↔ line at ω₀

One sinusoid is one line at its frequency.

Chooseperiodic ↔ ?

A periodic signal has discrete lines at harmonics.

Fill inspectrum = |X| and ?

The spectrum is magnitude and phase, two plots.

On your owntime ↔ ?

Time and frequency are an invertible pair.

Magnitude and phase, two plots

To record a component fully, magnitude alone is not enough. You also need its phase — when that sinusoid reaches its peak. So the spectrum comes in two plots. The magnitude spectrum |X| shows how strong each frequency is; the phase spectrum ∠X shows how far each frequency is shifted. Both together pin the signal down completely. Here phase is set to 0, so only the magnitude plot is drawn.

Time and frequency are one pair

The time waveform and the spectrum are two faces of the same signal, and you can move between them. Add the components back from the spectrum and you get the time waveform (synthesis); measure how much of each frequency is in the waveform and you get the spectrum (analysis). A periodic signal, as in the last unit, has a discrete spectrum with lines only at the harmonics. What happens for a signal that is not periodic? That answer is the next unit, the Fourier transform.

Back to the first screen

As you moved the frequency slider, the waveform’s wiggle sped up or slowed down, and at the same time the spectral line slid to that frequency. The two changes always happened together, because both are the same signal seen on the time axis and on the frequency axis. The line’s horizontal position was the frequency, its height the magnitude. The spectrum rewrites the signal in the language of frequency, a partner holding exactly the same information as the time waveform.

The spectrum is the frequency-domain view of a signal, made of two plots: magnitude |X(ω)| (how much of each frequency) and phase ∠X(ω) (how far each is shifted). A pure sinusoid is a single line; a periodic signal is a discrete line spectrum at the harmonics. A line’s position is its frequency, its height the magnitude. The time waveform and the spectrum are an invertible pair holding the same information (analysis ↔ synthesis).

On to the next unit

A periodic signal’s spectrum was a discrete picture with lines only at the harmonics. So what spectrum does a non-repeating signal, like a single pulse, have? The next unit, the Fourier transform, follows the limit of stretching the period to infinity. As the spacing between harmonics narrows to 0, the separate lines melt into a gap-free continuous spectrum. That moment, where a discrete sum becomes an integral, is the birth of the Fourier transform.