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Sec. 2–5 Fourier Series 71
2–5 FOURIER SERIES
The Fourier series is a particular type of orthogonal series that is very useful in solving engineering
problems, especially communication problems. The orthogonal functions that are
used are either sinusoids or, equivalently, complex exponential functions. †
Complex Fourier Series
The complex Fourier series uses the orthogonal exponential functions
w n (t) = e jnv 0t
(2–87)
where n ranges over all possible integer values, negative, positive, and zero; v 0 = 2p/T 0 ,
where T 0 = (b - a) is the length of the interval over which the series, Eq. (2–83), is valid;
and, from Example 2–12, K n = T 0 . The Fourier series theorem follows from Eq. (2–83).
THEOREM. A physical waveform (i.e., finite energy) may be represented over the interval
a 6 t 6 a + T 0 by the complex exponential Fourier series
w(t) =
n=q
a
n=-q
c n e jnv 0t
where the complex (phasor) Fourier coefficients are
(2–88)
c n = 1 T 0 La
and where v 0 = 2pf 0 = 2p/T 0 .
a+T 0
w(t)e -jnv 0t
dt
(2–89)
If the waveform w(t) is periodic with period T 0 , this Fourier series representation is
valid over all time (i.e., over the interval -q 6 t 6 +q), because w(t) and w n(t) are
periodic with the same fundamental period T 0 . For this case of periodic waveforms, the choice
of a value for the parameter a is arbitrary and is usually taken to be a = 0 or a = -T 0 /2 for
mathematical convenience. The frequency f 0 = 1/T 0 is said to be the fundamental frequency
and the frequency nf 0 is said to be the nth harmonic frequency, when n 7 1. The Fourier
coefficient c 0 is equivalent to the DC value of the waveform w(t), because, when n = 0,
Eq. (2–89) is identical to Eq. (2–4).
c n is, in general, a complex number. Furthermore, it is a phasor, since it is the coefficient
of a function of the type e jvt . Consequently, Eq. (2–88) is said to be a complex or phasor
Fourier series.
Some properties of the complex Fourier series are as follows:
1. If w(t) is real,
*
c n = c -n
(2–90)
† Mathematicians generally call any orthogonal series a Fourier series.