Signals and Systems - Electrical Engineering

4.5 Trigonometric Fourier Series 253

Finally, since the exponential basis{ejk^0 t}={cos(k 0 t)+jsin(k 0 t)}, the sinusoidal bases cos(k 0 t)
and sin(k 0 t)just like the exponential basis are periodic, of periodT 0 , and orthonormal.

nExample 4.4

Find the Fourier series of a raised-cosine signal(B≥A),

x(t)=B+Acos( 0 t+θ)

which is periodic of periodT 0 and fundamental frequency 0 = 2 π/T 0. Cally(t)=B+cos( 0 t−

π/ 2 ). Find its Fourier series coefficients and compare them to those forx(t). Use symbolic MATLAB

to compute the Fourier series ofy(t)= 1 +sin( 100 t). Find and plot its magnitude and phase line

spectra.

Solution

In this case we do not need to compute the Fourier coefficients sincex(t)is already in the trigono-

metric form. From Equation (4.19) its dc value isB, andAis the coefficient of the first harmonic in

the trigonometric Fourier series, so thatX 0 =B,|X 1 |=A/2, and∠X 1 =θ. Likewise, using Euler’s

identity we obtain that

x(t)=B+

A

2

[

ej(^0 t+θ)+e−j(^0 t+θ)

]

=B+

Aejθ

2

ej^0 t+

Ae−jθ

2

e−j^0 t

which gives

X 0 =B

X 1 =

Aejθ

2

X− 1 =X∗ 1

If we letθ=−π/2 inx(t), we get

y(t)=B+Asin( 0 t)

Its Fourier series coefficients areY 0 =Band Y 1 =Ae−jπ/^2 /2 so that|Y 1 |=|Y− 1 |=A/2 and

∠Y 1 =−∠Y− 1 =−π/2. The magnitude and phase line spectra of the raised cosine(θ= 0 )and

of the raised sine(θ=−π/ 2 )are shown in Figure 4.2. For bothx(t)andy(t)there are only two

frequencies—the dc frequency and 0 —and as such the power of the signal is concentrated at

those two frequencies as shown in Figure 4.2. The difference between the line spectra ofx(t)and

y(t)is in the phase.