seegongsik
SC · Fourier

The Spectrum

Seen on a time axis, a signal is a waveform; seen on a frequency axis, it is a spectrum. See for yourself that a spectrum stands a bar for each frequency component in its own place, and that a pure sinusoid is a single line.

Land the line on the target frequency

The upper figure is the time waveform of a pure sinusoid cos ωt, and the lower figure is its spectrum. A sinusoid appears as a single line at its own frequency. Move the angular frequency ω with the slider and the speed of the oscillation and the position of the blue line change together; when the line reaches the gold dashed target ω = 4.5, it is a match.

time waveform
spectrum (magnitude vs frequency)target frequency
angular frequency ω1.20
Frequency and magnitude
ω = 1.20 |X| = 1.00
The line is far from the target frequency, and the waveform oscillates much too slowly or too fast compared with the target. One frequency sets one line position.
Frequency off

The spectrum is an address book of frequencies

A spectrum is a plot with frequency across and magnitude up. It shows which frequencies a signal contains, and how much of each, as the height of a bar standing at that frequency, and a pure sinusoid, having only one frequency, stands as a single line. The line's horizontal position is the frequency and its height the amplitude, and changing the frequency slides the line sideways. This figure is a one-sided spectrum showing only positive frequencies, so the line's height equals the amplitude 1; in a two-sided spectrum that also shows negative frequencies, it splits into two lines of height 0.5 at ±ω.

ObserveA cos(ω₀t + φ) ↔ line at ω₀
One sinusoid is one line at its frequency.
Chooseperiodic ↔ ?
A periodic signal has discrete lines at its harmonics.
Fill inspectrum = |X| · ?
The spectrum is magnitude and phase, two plots.
On your owntime ↔ ?
Time and frequency are an invertible pair.

Magnitude and phase, two plots

To describe one component completely, magnitude alone is not enough; you also need when that sinusoid reaches its crest, its phase. So a spectrum comes as two pictures. The magnitude spectrum |X| records how strong each frequency is, the phase spectrum ∠X how far each frequency is shifted, and only together do they fix the signal completely. This screen sets the phase to 0 and draws only the magnitude picture.

Time and frequency are one pair

The time waveform and the spectrum are two faces of the same signal, and you can go back and forth between them. Adding the components of the spectrum back together gives the time waveform (synthesis), and measuring how much of each frequency is in the waveform gives the spectrum (analysis). A periodic signal, as in the previous unit, has a discrete spectrum with lines only at its harmonics; what happens for a signal with no period? The answer is the Fourier transform of the next unit.

Back to the first screen

As the angular-frequency slider moved, the waveform's oscillation sped up or slowed down, and at the same moment the spectral line slid to that frequency. The two changes always happened together because they are views of the same signal on the time axis and on the frequency axis. The line's horizontal position was the frequency and its height the magnitude. A spectrum rewrites a signal in the language of frequency, a partner that holds exactly the same information as the time waveform.

A spectrum represents a signal in the frequency domain and comes as two pictures: the magnitude |X(ω)| (how strong each frequency is) and the phase ∠X(ω) (how far each frequency is shifted). A pure sinusoid is a single line, a periodic signal a discrete line spectrum at its harmonics, and a line's position is the frequency and its height the magnitude. The time waveform and the spectrum are a pair you can go back and forth between (analysis ↔ synthesis), holding the same information.
On to the next unit

The spectrum of a periodic signal was a discrete picture with lines only at its 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 without end, and as the spacing between harmonics narrows to 0, the separate lines melt into a gap-free continuous spectrum. The moment a discrete sum turns into an integral is where the Fourier transform is born.