It is informative to investigate the filtering that is implicit in the Fourier analysis of a signal. Let us consider a general periodic signal I(t) with the following (exact) Fourier series representation :

I(t)= £ Qexp(M’) (2.11.1)

к = — oo

Now let us Fourier transform that signal using n periods of data:

1 ЛлГ/2

I(co,) = F{I(t)} = — I(t) exp( —jcoit) dt

n* J — nT/2 oo £ /*пГ/2

= z exp[;(cut — 0J,)t] dt

fc= — 00 ПІ J —nT/2

(2.11.2)

The function (sinx)/x appears often in the frequency-domain analysis of signals. It is usually called the sampling function or sine function. It is shown in Fig. 2.20. (The sampling function and its square are tabulated in Appendix A.) We observe that the function is zero at all integer multiples of n except at

zero. This is a way to view the orthogonality property discussed earlier. Using orthogonality properties to evaluate Fourier coefficients is equivalent to analyzing so that all of the harmonics fall on zeros of the (sin x)/x filter except for the analysis harmonic.

This approach also illustrates how noise enters the results and what happens if the analysis is done at nonharmonic frequencies. The noise contribution will be that which is passed by the (sin x)/x filter. The effect of analysis at a nonharmonic frequency will be to mix the contributions from all the harmonics in a way that depends on the difference between the harmonic frequency and the analysis frequency (see Fig. 2.21).

The analysis also shows the effect of the number of periods of data analyzed. Figure 2.22 compares results for one and two periods of data. Clearly, using more data results in a sharper filtering effect.