NIST Technical Note 1337: Characterization of Clocks and Oscillators

200 SAMUEL R. STEINthe window function multiplied by the desired spectrum. The use ofa windowfunction reduces the variance of the spectrum estimate at the expense ofsmearing out the spectrum to a small degree. With these changes but nowindow function, we arrive at the discrete finite transform1 N- tX(n 6.1) = - L x(kr)e-j2·ft~f't. (12-17)N .=0The spectral density of x(t) is computed from Eq. (12-17) by squaring the realand imaginary components, adding the two together, and dividing by thetotal time T:2S,,(m 6.f) = {R[X(m 6.f)J} ; {I[X(m 6.1)J}2(12-18)The digital method of estimating spectral densities has many advantagesover analog signal processing. Most important is the fact that it may becomputed from any set of equally spaced samples of a time series. As a result,the technique is compatible with other methods of characterizing the signal,that is, the sampled data can be stored and processed using a variety ofalgorithms. In addition, each record of length Tproduces a single estimate ofthe spectrum for each of the N frequencies 6.1, 2 6.1, _.. , N !1f It is thereforepossible to estimate the entire spectrum much more quickly using the digitaltechnique than it would be using analog methods. The fast Fouriertransform, a very efficient algorithm for the computation of the discrete finitetransform, has opened the way to versatile self-contained, commercialspectrum analysis. It is also very straightforward to compute the spectrumfrom data acquired by computerized digital data acquisition systems.A result of the finite sampling rate is that the upper frequency limit of thedigital spectrum analysis is 1/2T, called the Nyquist frequency (Jenkins andWatts, 1968). Power in the signal being analyzed that is at frequencies higherthan the Nyquist frequency affects the spectrum estimate for lower frequencies.This problem is called aliasing. The out-of-band signal is rejected byonly approximately 6 dB per octave above the Nyquist frequency. Thus,when significant out-of-band signals exist, they must be reduced by analogfiltering. One or more low-pass filters are usually sufficient for this purpose.As its second measure of frequency stability, the IEEE recommended thesample variance c;;(-r) of the fractional-frequency fluctuations. It is a measureof the variability of the average frequency of an oscillator between twoadjacent measurement intervals. The average fractional-frequency deviation5'. over the time interval from t. to r. + r is defined as1 J'.+t Y. = - y(r) dr, (12-19)r I.TN-70