Question 10.4: Plot the power spectrum of the ± 1V square wave discussed in......
Plot the power spectrum of the ± 1V square wave discussed in Example 9.1, if the period of the wave is 1 s.
In Example 9.1, depending upon whether the square wave was treated as an even or an odd function, the Fourier coefficients were given by either a cosine series or a sine series. It was shown that this affects only the phases of the coefficients, not their magnitudes, and since the power spectrum does not depend on phase, either can be used. So taking the sine series, Eq. (H) in Example 9.1 gave
Table 10.6 lists the frequency, f_{(n)} = n/T, with T = 1; the Fourier components a_{n} \mathrm{and} b_{n} and the power, \frac{1}{2}\left(a^{2}_{n} + b^{2}_{n}\right), at each value of n.
The total ‘power’ in the waveform, which is 1 V², should be given by summing the power in all the harmonics. For the first nine terms shown in Table 10.6 this is 0.96 V², the remaining 0.04 V² therefore being accounted for by higher frequency terms.
Figure 10.7 is a plot of the discrete power spectrum of the waveform up to 10 Hz.
It can be seen from Example 10.4 that the power spectrum of a zero-mean periodic function consists of discrete amounts of power concentrated at the fundamental frequency and its harmonics, which are spaced at intervals of 1/T Hz. If T is small, then these frequency intervals may be quite large, for example as in Fig. 10.7. In cases like this it can be seen that the power spectrum should remain discrete, and should not be expressed in the form of power spectral density, discussed below, which could lead to serious errors.
| Table 10.6 | |||||||||
| 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | n |
| 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | f_{(n)} (Hz) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | a_{n} (V) |
| 4/(9\pi) | 0 | 4/(7\pi) | 0 | 4/(5\pi) | 0 | 4/(3\pi) | 0 | 4/\pi | b_{n} (V) |
| 0.0100 | 0 | 0.0165 | 0 | 0.0324 | 0 | 0.0900 | 0 | 0.8105 | \frac{1}{2}\left(a^{2}_{n} + b^{2}_{n}\right) (\mathrm{V})^{2} |
