Signals and Systems - Electrical Engineering

320 CHAPTER 5: Frequency Analysis: The Fourier Transform

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% Example 5.8---Fourier series

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T0 = 1; N = 10; w0 = 2∗pi/T0;

m = heaviside(t)−heaviside(t−T0/2);

m1 = heaviside(t−T0/2)−heaviside(t−T0);

x = t∗m + m1∗(−t + T0); x = 2∗x; % periodic signal

[Xk, w] = fourierseries(x, T0, N); % Fourier coefficients, harmonic frequencies

% Fourier series approximation

for k = 1:N,

if k == 1;

x1 = abs(Xk(k));

else

x1 = x1 + 2∗abs(Xk(k))∗cos(w0∗(k−1)∗t + angle(Xk(k)));

end

end

% sequence of Fourier coefficients and harmonic frequencies

k = 0:N−1; Xk1 = 2∗pi∗Xk; wk = [−fliplr(k(2:N−1)) k]∗w0; Xk = [fliplr(Xk1(2:N−1)) Xk1];

In this case, the Laplace transform simplifies the computation of theXkvalues. Indeed, the Fourier

series coefficients are given by

Xk=

∫0.5

0

te−j^2 πktdt+

∫^1

0.5

( 1 −t)e−j^2 πktdt

which need to be found using integration by parts. n

5.6.3 Parseval’s Energy Conservation

We saw in Chapter 4 that for periodic signals having finite power but infinite energy, Parseval’s theo-

rem indicates how the power of the signal is distributed among the harmonic components. Likewise,

for aperiodic signals of finite energy, an energy version of Parseval’s result indicates how the signal

energy is distributed over frequencies.

For a finite-energy signalx(t)with Fourier transformX(), its energy is conserved when going from the time

to the frequency domain, or

Ex=

∫∞

−∞

|x(t)|^2 dt=

1

2 π

∫∞

−∞

|X()|^2 d (5.15)

Thus,|X()|^2 is an energy density indicating the amount of energy at each of the frequencies.

The plot|X()|^2 versusis called the energy spectrum ofx(t), and it displays how the energy of the signal is

distributed over frequency.