At first glance, a spectrum analyzer closely resembles an oscilloscope, and in fact many contemporary advanced oscilloscopes incorporate built-in spectrum analyzers. An oscilloscope is primarily a time-domain instrument where signals are accessed by means of probes or cables connected to analog inputs. These signals display along two axes. The Y-axis represents amplitude in volts and the X-axis represents the passage of time.

Alternately, the X-axis can be defined so that instead of time it corresponds to frequency. In this case the instrument is operating in the frequency domain. The X-axis still represents amplitude, but rather than voltage, it shows power, expressed in decibels, a logarithmic scale. Each division represents a ten-fold increase in power, which actually creates a well-proportioned display.

Displays of frequency spectrum make use of Fourier transforms. Joseph Fourier (1768-1830), in connection with heat transfer, developed the Fourier Transform. The mathematics of Fourier transforms is complex in the extreme, but in the 1960s it was greatly simplified by the discovery of the Fast Fourier Transform (FFT). This set of algorithms facilitates translation in either direction between time and frequency domains. In changing from one to the other, information is neither lost nor gained. But observers can grasp aspects that would otherwise be elusive by viewing the signal in the frequency domain.

A pure sine wave has all its energy concentrated at a single frequency. There are no harmonics. In the frequency domain the signal appears as a single large spike. A square wave, in contrast, has fast, near instantaneous rise and fall times. Such a signal is rich in harmonic frequencies. The frequency domain display consists of one high spike at the fundamental frequency with lower spikes at nearby frequencies. Thus, information is depicted in graphic form that aids in understanding the signal.

An oscilloscope with incorporated spectrum analyzer derives the frequency domain display from the time domain display using an FFT, as outlined above. But in the autonomous spectrum analyzer, this is not always the case. True spectrum analyzers, usually more expensive than oscilloscopes, operate as swept-tuned, FFT-based, hybrid superheterodyne-FFT or real-time FFT machines. Each of these has advantages and tradeoffs.

To understand the process it is first necessary to understand superheterodyne receivers. The superheterodyne was invented in 1918 and has been in use ever since. In a radio receiver, a sine wave generated by a local oscillator beats against the radio frequency output of the tuner so as to produce a lower intermediate frequency (IF), which retains the original modulation.

The beat frequency is the product of two frequencies that are mixed. The concept is equally applicable in acoustic sound transmission through air and RF signals in the form of electromagnetic radiation through space and electrical energy in circuitry within electronic equipment. It will be noticed that when two acoustic tones that are sufficiently close in frequency are present together, a third tone will also be heard. As one of the two original tones is adjusted to more closely approach the other, the frequency of the resultant tone will drop, acquiring a characteristic beating sound until abruptly it disappears when the two original tones exactly coincide.

This behavior is a consequence of the fact that there is alternating constructive and destructive interference that forms the resultant tone.

If the IF is to remain constant throughout the range of frequencies selected by the tuner, each of these will require a different frequency produced by the oscillator. The original solution was for each radio station to broadcast along with the RF signal the waveform required to facilitate the superheterodyne process. Soon it was found to be more efficient for the range of tones to be generated within each receiver. You may have seen the inner workings of an old tube-type radio containing a two-gang variable capacitor with two sets of different sized plates attached to a single shaft. This arrangement provided the appropriate capacitances so that the correct frequency could be synthesized to work against each broadcast carrier, the result being a single stable IF. This could be amplified without undergoing the high degree of attenuation that would accompany RF processing.

Today, rather than by means of a two-gang capacitor, the required frequencies are generated electronically. Varactors driven by a common control voltage perform this function.

In the case of a swept-tuned spectrum analyzer, a voltage-controlled oscillator gets swept through a large frequency spectrum to down-convert some of the signal of interest to facilitate depiction of it. Update speed and frequency resolution are traded off as often is the case.

Hybrid superheterodyne-FFT spectrum analyzers operate according to the FFT principle, but input frequency is reduced by mixing the signal with the output of a local oscillator so as to synthesize the signal at a workable level. The aspiration in all cases is the depict the spectrum free of gaps so that there is no loss of information.