Before making a quantitative analysis of the Fourier series of the classical waveforms, we pause to make two useful observations about symmetries in waveforms and the corresponding symmetries in the Fourier series. First, a Fourier series might consist only of even or odd-numbered harmonics; this is reflected in symmetries comparing a waveform to its displacement by half a cycle. Second, the Fourier series may contain only real-valued or pure imaginary-valued coefficients (corresponding to the cosine or sine functions). This is reflected in symmetries comparing the waveform to its reversal in time.
In this section we will assume that our waveform has an integer period , and
furthermore, for simplicity, that is even (if it isn't we can just
up-sample by a factor of two). We know from Chapter 9 that any (real or
complex valued) waveform can be written as a Fourier series (whose
coefficients we'll denote by ):
To analyze the first symmetry we delay the signal by a half-cycle. Since
Furthermore, if happens to be equal to itself shifted a half cycle, that is, if , then (looking at the definitions of and ) we get and . This implies that, in this case, has only even numbered harmonics. Indeed, this should be no surprise, since in this case would have to repeat every samples, so its fundamental frequency is twice as high as normal for period .
In the same way, if ], then can have only odd-numbered harmonics. This allows us easily to split any desired waveform into its even- and odd-numbered harmonics. (This is equivalent to using a comb filter to extract even or odd harmonics; see Chapter 7.)
To derive the second symmetry relation we compare with its time
reversal, (or, equivalently, since repeats every samples, with
). The Fourier series becomes:
So if satisfies the Fourier series consists of cosine terms only; if it consists of sine terms only; and as before we can decompose any (that repeats every samples) as a sum of the two.