5.6 Spectral Representation 317
signals. When radiating a signal with an antenna, the length of the antenna is about a quarter of the
wavelength,
λ=
3 × 108
f
meters
wherefis the frequency in hertz of the signal being radiated. Thus, if we assume that frequencies
up tof=30 KHz are present in the signal (this would allow us to include music and speech in the
signal) the wavelength is 10 kilometers and the size of the antenna is 2.5 kilometers—a 1.5-mile
long antenna! Thus, for a music or a speech signal to be transmitted with a reasonable-size antenna
requires increasing the frequencies present in the signal. Modulation provides an efficient way to shift
an acoustic or speech signal to a desirable frequency.
5.6.2 Fourier Transform of Periodic Signals
By applying the frequency-shifting property to compute the Fourier transform of periodic signals, we
are able to unify the Fourier representation of aperiodic as well as periodic signals.
For a periodic signalx(t)of periodT 0 , we have the Fourier pair
x(t)=
∑
k
Xkejk^0 t ⇔ X()=
∑
k
2 πXkδ(−k 0 ) (5.14)
obtained by representingx(t)by its Fourier series.
Since a periodic signalx(t)is not absolutely integrable, its Fourier transform cannot be computed
using the integral formula. But we can use its Fourier series
x(t)=
∑
k
Xkejk^0 t
where the{Xk}are the Fourier coefficients, and 0 = 2 π/T 0 is the fundamental frequency of the peri-
odic signalx(t)of periodT 0. As such, according to the linearity and the frequency-shifting properties
of the Fourier transform, we obtain
X()=
∑
k
F[Xkejk^0 t]
=
∑
k
2 πXkδ(−k 0 )
where we used thatXkas a constant has a Fourier transform 2πXkδ(). Notice that for a periodic sig-
nal the Fourier coefficients{Xk}still characterize its frequency representation: The Fourier transform
of a periodic signal is a sequence of impulses in frequency at the harmonic frequencies,{δ(−k 0 )},
with amplitudes{ 2 πXk}.
