Page 56 - Fundamentals of Communications Systems

2.8 Chapter Two

Properties of the Fourier Series
Property 2.1 If x(t) is real, then x n = x ∗ −n . This property is known as Hermitian
symmetry. Consequently the Fourier series of a real signal is a Hermitian symmet-
ric function of frequency. This implies that the magnitude of the Fourier series is an
even function of frequency, i.e.,

|x n |=|x −n | (2.24)
and the phase of the Fourier series is an odd function of frequency, i.e.,

arg(x n ) =− arg(x −n ) (2.25)

Property 2.2 If x(t) is real and an even function of time, i.e., x(t) = x(−t), then all the
coefficients of the Fourier series are real numbers.

EXAMPLE 2.12
Recall Example 2.4, where x(t) = cos(2π f m t). For this signal T = 1/f m and the only
nonzero Fourier coefficients are x 1 = 0.5, x −1 = 0.5.

Property 2.3 If x(t) is real and odd, i.e., x(t) =−x(−t), then all the coefficients of the
Fourier series are imaginary numbers.

EXAMPLE 2.13
For x(t) = sin(2π f m t), where T = 1/f m and the only nonzero Fourier coefficients are
x 1 =− j 0.5, x −1 = j 0.5.

Theorem 2.1 (Parseval)

∞
1 T 2 2
P x = |x(t)| dt = |x n | (2.26)
T 0
n=−∞
Parseval’s theorem states that the power of a signal can be calculated using
either the time or the frequency domain representation of the signal and the
two results are identical.

EXAMPLE 2.14
For both x 1 (t) = cos(2π f m t) and x 2 (t) = sin(2π f m t) it is apparent that

2 2
P x 1 = (0.5) + (0.5) = 0.5 = P x 2 (2.27)