Signals and Systems - Electrical Engineering

4.6 Fourier Coefficients from Laplace 255

0

0.2

0.4

0.6

0.8

1

Magnitude line spectrum

|Y

|k

− 200 0 200

− 1

0

1

Phase line spectrum

Ω(rad/sec)

− 200 0 200

Ω(rad/sec)

|Y

|k

0 0.1 0.2

0

0.5

1

1.5

2

y(

t)

t(sec)

FIGURE 4.3
Line spectra of Fourier series ofy(t)= 1 +sin( 100 t)(top figure). Notice the even and the odd symmetries of the
magnitude and the phase spectra. The phase is−π/ 2 at=100 rad/sec. n

RemarksJust because a signal is a sum of sinusoids, which are always periodic, is not enough for it to have a
Fourier series. The signal should be periodic. The signal x(t)=cos(t)−sin(πt)has components with periods
T 1 = 2 πand T 2 = 2 so that the ratio T 1 /T 2 =πis not a rational number. Thus, x(t)is not periodic and no
Fourier series for it is possible.

4.6 Fourier Coefficients from Laplace

The computation of theXkcoefficients (see Eq. 4.12) requires integration that for some signals
can be rather complicated. The integration can be avoided whenever we know the Laplace trans-
form of a period of the signal as we will show. In general, the Laplace transform of a period of
the signal exists over the wholes-plane, given that it is a finite-support signal. In some cases, the
dc coefficient cannot be computed with the Laplace transform, but the dc term is easy to compute
directly.