Time domain, frequency domain and Fourier

What it is

Any signal can be described two ways: how its voltage changes with time (the time domain, what an oscilloscope shows), or which sine waves it is made of (the frequency domain, what a spectrum analyzer shows). The bridge between them is the Fourier transform: every signal is a sum of sine waves, and each ingredient has a frequency and an amplitude.

Two views of one signal

Try it: pick Square and add components one at a time. A single sine is the fundamental. Add the 3rd harmonic and the shape starts to square off. Each additional odd harmonic sharpens the corners. The wave and its spectrum hold exactly the same information; the spectrum is simply the view that makes sense of frequency.

The AM tone setting shows why that matters in radio. Three steady lines (carrier plus two sidebands) add up to a wave whose height rises and falls: the AM envelope is the beat between them. What looks like "changing amplitude" in time is a set of constant sine waves in frequency. The same goes for every mode.

Why it matters

• Filters, mixers and sidebands are described in frequency. "SSB has one sideband" is a statement about spectrum, which you cannot see in the waveform.
• Distortion creates harmonics. A distorted sine, like an overdriven amplifier's output, contains whole-number multiples of its frequency. That is why transmitters need a low-pass filter. See Harmonics and spurious emissions.
• Speed costs bandwidth. The sharper a change in time, the wider its spectrum, which is why a key click spreads and why a pulse of short duration is wide. See Bandwidth, symbol rate and data rate.
• Instruments and software do this all the time: spectrum analyzers, SDR waterfalls and the FFT inside DSP take a window of samples and show the spectrum. See Spectrum analyzers.

Related

• What modulation isPutting information onto a carrier wave.
• Amplitude modulation (AM)Varying carrier strength: sidebands, bandwidth and efficiency.
• Sampling and the Nyquist limitTurning a continuous signal into numbers, and aliasing.
• Spectrum analyzersSeeing signal strength against frequency.
• Harmonics and spurious emissionsCleaning up your transmitter.
• Digital signal processingFiltering and demodulating with maths instead of components.