Signals and Systems - Electrical Engineering

4.11 What Have We Accomplished? Where Do We Go from Here? 289

for k = 2:N,

xa = xa + 2 * abs(X(k)) * cos(w0 * (k - 1). * t + angle(X(k))); % approximate signal

end n

4.11 What Have We Accomplished? Where Do We Go from Here?....................

Periodic signals are not to be found in practice, so where did Fourier get the intuition to come up with
a representation for them? As you will see, the fact that periodic signals are not found in practice does
not mean that they are not useful. The Fourier representation of periodic signals will be fundamental
in finding a representation for nonperiodic signals.

A very important concept you have learned in this chapter is that the inverse relation between time
and frequency provides complementary information for the signal. The frequency domain consti-
tutes the other side of the coin in representing signals. As mentioned before, it is the eigenfunction
property of linear time-invariant systems that holds the theory together. It will provide the funda-
mental principle for filtering. You should have started to experienced ́eja vu` in terms of the properties
of the Fourier series; some look like a version of the ones in the Laplace transform. This is due to
the connection existing between these transforms. You should have also noticed the usefulness of the
Laplace transform in finding the Fourier coefficients, avoiding integration whenever possible. Table
4.1 provides the basic properties of the Fourier series for continuous–time periodic signals.

Chapter 5 will extend some of the results obtained in this chapter, thus unifying the treatment of
periodic and nonperiodic signals and the concept of spectrum. Also the frequency representation of

Table 4.1Basic Properties of Fourier Series

Time Domain Frequency Domain

Signals and constants x(t),y(t)periodic Xk,Yk

with periodT 0 ,α,β

Linearity αx(t)+βy(t) αXk+βYk

Parseval’s power relation Px=T^10

∫

T 0 |x(t)|

(^2) dt Px=∑k|Xk| 2
Differentiation
dx(t)
dt
jk 0 Xk
Integration
∫t
−∞x(t
′)dt′only ifX 0 = 0 Xk
jk 0
k6=0,−
∑
m6= 0
Xm
jm 0
,dc
Time shifting x(t−α) e−jα^0 Xk
Frequency shifting ejM^0 tx(t) Xk−M
Symmetry x(t)real |Xk|=|X−k|even
function ofk
∠Xk=−∠X−kodd
function ofk