Application of the Fourier Series to Spectrum Analyzers

The Fourier series provides the spectrum of a signal. As we have seen, the spectrum consists of the amplitudes and phases of the harmonics versus frequency. By providing the spectrum of a signal $ f(t) $, the Fourier series helps us identify the pertinent features of the signal. It demonstrates which frequencies are playing an important role in the shape of the output and which ones are not. For example, audible sounds have significant components in the frequency range of $ 20 \mathrm{~Hz} $ to $ 15 \mathrm{kHz} $, while visible light signals range from $ 10^{5} \mathrm{GHz} $ to $ 10^{6} \mathrm{GHz} $. Table $ 1 $ presents some FM radio other signals and the frequency ranges of their components.

Table. 1: Frequency ranges of typical signals.

Signal	Frequency Range
Audible sounds	20 Hz to 15 kHz
AM radio	540–1600 kHz
Short-wave radio	3–36 MHz
Video signals	dc to 4.2 MHz
VHF television	54–216 MHz
UHF television	470–806 MHz
Cellular telephone	824–891.5 MHz
Microwaves	2.4–300 GHz
Visible light	$10^5 – 10^6 GHz$
X-rays	$10^8 – 10^9 GHz$

A periodic function is said to be band-limited if its amplitude spectrum contains only a finite number of coefficients $ A_{n} $ or $ c_{n} $. In this case, the Fourier series becomes $$f(t)=\sum_{n=-N}^{N} c_{n} e^{j n \omega_{0} t}=a_{0}+\sum_{n=1}^{N} A_{n} \cos \left(n \omega_{0} t+\phi_{n}\right)$$ This shows that we need only $ 2 N+1 $ terms (namely, $ a_{0}, A_{1}, A_{2}, \ldots, A_{N} $, $ \left.\phi_{1}, \phi_{2}, \ldots, \phi_{N}\right) $ to completely specify $ f(t) $ if $ \omega_{0} $ is known. This leads to the sampling theorem: a band-limited periodic function whose Fourier series contains $ N $ harmonics is uniquely specified by its values at $ 2 N+1 $ instants in one period. A spectrum analyzer is an instrument that displays the amplitude of the components of a signal versus frequency. In other words, it shows the various frequency components (spectral lines) that indicate the amount of energy at each frequency. It is unlike an oscilloscope, which displays the entire signal (all components) versus time. An oscilloscope shows the signal in the time domain, while the spectrum analyzer shows the signal in the frequency domain. There is perhaps no instrument more useful to a circuit analyst than the spectrum analyzer. An analyzer can conduct noise and spurious signal analysis, phase checks, electromagnetic interference and filter examinations, vibration measurements, radar measurements, and more. Spectrum analyzers are commercially available in various sizes and shapes.