Fourier analysis of a periodic signal is valid at all harmonic frequencies. The harmonics in a signal with duration T are at angular frequencies 2л/T’, 4л/Т’, 6л/Т’,…. The data record may be a single period of data, or it may be an integral multiple of a period. If the data record of length T’ contains exactly n periods, then only every nth harmonic may be nonzero. This is demonstrated in Fig. 4.11 for a З-bit PRBS signal. For one period

of data, every harmonic may be nonzero. For two periods of data, every other harmonic may be nonzero. It should be emphasized that the term harmonic means the harmonic based on the total data record, not the length of a period. However, all of the harmonics obtained in a single period are also contained in a record that contains an integral number of periods.

A nonideal situation that is sometimes encountered in practical measure­ments is that the total data record does not contain an integral number of periods. This sometimes occurs in the use of the fast Fourier transform algorithm, which requires that the data contain 2" data points for integral values of n (see Section 4.6). The analyst usually picks the period T to give the desired lowest frequency and frequency spacing. The sampling time At is selected to give the desired highest frequency and to prevent aliasing. These
two quantities fix the number of points per period N:

N = T/At (4.8.1)

Then the total number of samples in M periods is MN. If the FFT algorithm is to be used and if M is to be an integer, then

MT/At = 2" (4.8.2)

or

M = (At/T)(2n) (4.8.3)

The only way the number of periods can be an integer is for the number of samples per period to be an integral power of 2. That is,

T/At = 2m (4.8.4)

Since the desired value of T is set by the test objectives and the value of At is a compromise between test objectives and the capability of the digitizing equipment, it may sometimes be inconvenient to satisfy Eq. (4.8.4). This leads us to consider the use of the FFT algorithm on a data record that contains a nonintegral number of periods.

The harmonics in the FFT analysis must be based on the length of the data record used in the analysis:

o)k = 2kn/T’ (4.8.5)

where T is the length of the data record. Since T is not necessarily equal to an integral number of periods,

T = nT + 8 (4.8.6)

where n is the number of periods, T the period, and <5 the portion of a period. The relationship between the Fourier coefficients obtained by analysis of a nonintegral number of periods, and those that would be obtained from analysis of an integral number of periods is easily derived.

where the interval — (nT 4- 8)/2 to (nT + S)/2 has been chosen for conve­nience. The signal x(t) may be represented as

Using Eq. (4.8.7) in Eq. (4.8.8) gives

sin(mn — к + md/T)n

(mn — к + md/T)n

Thus the Fourier coefficient calculated by analysis of a noninteger number of periods is related to all of the Fourier coefficients of the original signal, and the contribution of each is given by the previously encountered (sin x)/x filter (see Section 2.11). The peak of this filter occurs at m — k/(n + S/T). The displacement of the nearest harmonic from the peak is m8/T, and the first null in the filter is at m = (1 + k)/(n + 8/T). The results are shown in Fig. 4.12.

A phenomenon called the picket-fence effect appears when the Fourier analysis is based on a nonintegral number of periods. This is caused by the shift in the displacement between analysis harmonics and signal harmonics as the analysis frequency changes. For example, let us assume that the first harmonic analysis frequency is 10 % higher, than the first signal harmonic. Then the second analysis harmonic would be 20 % higher than the second signal harmonic. Figure 4.13 shows this effect. For this example, the syn­chronization between the signal harmonics and the analysis harmonics would become worse until the fifth harmonic, then would improve until they coincided at the eleventh harmonic.

Drift is also sometimes a problem. For example, the process may experience some unmeasurable input in addition to the test signal, and this additional input could cause the response to drift. One might be inclined to assume that Fourier analysis would automatically eliminate the drift effect because it is a low-frequency phenomenon and usually outside the frequency range of interest. Unfortunately, this is not true. A truly periodic signal begins and ends each period at the same value. Because of drift, the last point will not be

Fig. 4.14. Effect of drift.

the same as the initial point as shown in Fig. 4.14. However, Fourier analysis assumes that they are equal, and this would require a jump in the signal as shown in Fig. 4.14. This jump introduces high-frequency errors. High-pass filtering can be used to alleviate this problem. Also, if the drift has a simple form, such as a ramp, the drift effect can be subtracted, point by point, from the signal before Fourier analysis.