FOURIER SERIES
Signal Analysis
A signal is said to be continuous-time signal if it is available at all instants of time. A real time
naturally available signal is in the form of time domain. However, the analysis of a signal is far
more convenient in the frequency domain. There are three important classes of transformation
methods available for continuous-time systems. They are:
Fourier series
Fourier transform; and
Laplace transform.
Of these three methods, the Fourier series is applicable only to periodic signals; i.e. signals
which repeat periodically over -∞ < t < ∞ or -∞ < x < ∞, as the case may be.
Periodic Functions
A periodic function, with period τ, satisfies f(t + τ ) = f(t) for all values of t. Examples of
periodic functions are given in Figure 3.1.
Fig 3.1 Samples of periodic functions.
For example,
The graph of the periodic square wave is drawn in Fig 3.2
Fig 3.2 The periodic square wave.
The steps for finding the Fourier series are:
Step 1: Plot the periodic function f(t) or f(x).
Step 2: Determine its fundamental period L = 2π and its fundamental angular frequency k = 2π/L
= 1.
Step 3: Evaluate a0 , an and bn as given below.
Step 4: Write down the resulting Fourier series.
Finally, specifying a particular value of x = x1 in a Fourier series, gives a series of constants
that should equal f(x1 ).
However, if f(x) is discontinuous at this value of x, then the series converges to a value that is
half-way between the two possible function values, as shown in Fig 3.4.
Fig 3.4 Discontinuity in the function.
Example 3.1
Let f(x) be a function of period 2π such that
a) Sketch a graph of f(x) in the interval
b) Show that the Fourier series for f(x) in the interval is
c) By giving an appropriate value to x, show that
Solution
a) Sketch a graph of f(x) in the interval −2π < x < 2π
b) Fourier series representation of f(x)
c) Pick an appropriate value of x, to show that
Example 3.2
Solution
a) Sketch a graph of f(x) in the interval −3π < x < 3π
b) Fourier series representation of f(x)
c) Pick an appropriate value of x, to show that
Example 3.3
Solution
a) Sketch a graph of f(x) in the interval -2π < x < 2π
b) Fourier series representation of f(x)
C) Pick an appropriate value of x, to show that
Even functions
We find that even functions, which have the property that f(−t)= f(t), have all bn = 0 in their
Fourier series. They are represented by cosine terms only.
Another simplification in this case is
It is therefore only necessary to integrate over a half cycle.
To summarize, for an even function
Odd functions
Odd functions, where f(−t) = −f(t) have all an = 0 and only have sine terms in their Fourier
series. We only need to consider the half cycle, because
Fig 3.4 (a) An Even function (b) An Odd function.
Practice Questions
Find the Fourier series of the following functions:
