EE 2260 Lecture 16
Fourier Series
Reading: 16.1-16.3
HW #16 AP: 16.1, 16.2
Goal: Find the steady-state response of a periodic signal
Examples of periodic signal
(a) Capacitor charging/discharging
(b) Full-wave rectifier (diode) output
(c) An ECG(EKG) signal
(a)
(b)
(c)
Any periodic signal can be represented as a sum of sinusoids.
The frequencies of the summed sinusoids are the fundamental frequency and
integer multiples of the fundamental frequency (harmonics).
Examples
http://lpsa.swarthmore.edu/Fourier/Series/WhyFS.html#Triangle_Wave
Periodic function:
§16.1 Fourier Series Analysis: An Overview
Fundamental frequency: 0  2 / T
Harmonic frequencies: 20, 30, 40, ...
Fourier coefficients: an, bn
To represent a given signal as a sum of sinusoids or the Fourier Series, we need the
coefficients av , an and bn .
§16.2 Fourier Series Analysis: An Overview
Note that av is the average value of the function.
Properties of sinusoidal integrals
To find av:
Similarly, we can obtain formulas for find ak and bk as:
Useful integrals for determining Fourier coefficients
§16.3 The Effect of symmetry on the Fourier Coefficients
Even-Function Symmetry:
An even-function is symmetrical about the y-axis.
bk  0
for all k (because bk involves sine and sine is not an even function).
Similarly, ak can be evaluated by
ak 
4
T
T /2
0
f (t ) cos(k0 t )dt .
Odd-Function Symmetry:
An odd-function is symmetrical about the x-axis.
ak  0
for all k (because ak involves cosine and cosine is not an odd-function)
Half-Wave Symmetry: f (t )   f (t  T / 2)
After delaying it by T/2 and inverting, it’s identical to the original function.
Even coefficients are zero.
Quarter-Wave Symmetry:
This is a special case of half-wave symmetry: it has symmetry about the midpoint of the positive and
negative half-cycle.
Example1
Example2
In the figure below, the signal shown in (a) has a quarter-wave symmetry. Here signal shown in (b)
doesn’t have a quarter-wave symmetry, but it has a half-wave symmetry.
Odd Function
[Using quarter-wave symmetry]
[Using quarter-wave symmetry]
Odd function, period = T, Half-wave symmetric
Fourier series: