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Several types of analyzers exist today that allow a time-domain signal to be converted to a frequency-domain spectrum. The resulting spectrum of allspectrum analyzers is equivalent to the amplitudejfrequencyplot， which is obtained by passing the given signal across a set of constant bandwidth filters and noting the output of each filter at its center frequency.
Unfortunately， such a simple procedure cannot beused because， foradequate resolution， each filter can cover only a very narrow frequencyband， and because of the cost involved. In the so-called ""wave analyzer"" or ""tracking filter"" one filter is utilized by manually incrementing the filter across the time input to determine which frequencies exhibit a large ampli-tude. In time-compression real-time analyzers (RTA) the filter is swept electronically across the input. The term ""real time"" as applied here means the instrument takes the time-domain signal and converts it to a frequencydomain while the event is actually taking place. In technical terms， real time is viewed when the rate of sampling is equal to or greater than the bandwidth of the filters taking the measurements. RTAs use an analog-to-digital converter and digital circuits to speed up the data signal effectively andimprove the sweeping filter scanrate， thus creating an apparent time com-pression. Both of the previous analyzers are basically analog instrumentsand， because of the characteristics of analog filtering， may be quite slow at lower frequencies.
The Fourier analyzer is a digital device based on the conversion of time-domain data to a frequency domain by the use of the fast Fourier transform. The fast Fourier transform (FFT) analyzers employ a minicomputer to solve a set of simultaneous equations by matrix methods.
Time domains and frequency domains are related through Fourier series and Fourier transforms.By Fourier analysis， a variable expressed as a function of time may be decomposed into a series of oscillatory functions (each with a characteristicfrequency)， which when superpositioned orsummed at eachtime， will equal the original expression of the variable. This process is shown graphically in Figure 16-1. Since each of the oscillatory signals has acharacteristic frequency， the frequency domain reflects the amplitude of the oscillatory function at that corresponding frequency.

The breakdown of a given signal into a sum of oscillatory functions is accomplished by application of Fourier series techniques or by Fourier transforms. For a periodic function F(t) with a period t， a Fourier series may be expressed as
 
F(t)二號+ (號n cos nωt +bn sin nωt)(16-卻)
卻 n
二1
Here號 and b are amplitudes of the oscillatory functions cos (nωt) and sin (nωt)， respectively. The value of ω is related to the characteristic frequency f by
ω二卻πf (16-3) The previous function may also be written in a complex form as
 
F(t)二G(ω)忡Jωt ω (16-4)
 
where:
 
t)ωt t
G(ω)二卻1πF(忡 J(16-5)
 
The function G(ω) is the exponential Fourier transform of F(t) and is a function of the circular frequency ω. In practice the function F(t) is not given over the entire time domain but is known from time zero to some finite time T， as shown in Figure16-卻. The time span T may be divided into K equal increments of罵teach. For computational reasons， let K二卻p where p is an integer.Also， let the circular frequency span ωn be divided into N parts where N二卻Q. (In practice， N is often set equal to K.) By setting f二K川NT， the frequency interval罵ω becomes
罵ω二卻π罵f二卻πK (16-6)
NT
Now， discrete equations analogous to Equations (16-3) and (16-4) may be defined
N 1 Jω t
n
F(t川)二罵ω G(ωn)忡川 (16-7)n二
and
　
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