1. Continuous-Time Signal Analysis
Signals Analysis
and Systems
Continuous-Time Signal Analysis

2. Outline
Introduction
Fourier Series (FS) representation of Periodic Signals.
Trigonometric and Exponential Form of FS.
Parseval’s Theorem.
Simplifications Through Signal Symmetry.
LTIC System Response to Periodic Inputs.

3. The Fourier Series
Linear Circuit
I/P
O/P
Sinusoidal Inputs OK
Nonsinusoidal Inputs
Nonsinusoidal Inputs
Sinusoidal Inputs
Fourier Series

4. The Fourier Series Joseph Fourier 1768 to 1830
Fourier (French) studied the mathematical theory of heat
conduction. He established the partial differential
equation governing heat diffusion and solved it by
using infinite series of trigonometric functions.

6. Consider the periodic function
The Fourier Series
Fourier proposed in 1807
A periodic waveform f(t) could be broken down into an
infinite series of simple sinusoids which, when added
together, would construct the exact form of the original
waveform.
Consider the periodic function
T = Period, the smallest value of T that satisfies the above
Equation.

7. The Fourier Series Definition
A Fourier Series is an accurate representation of a
periodic signal and consists of the sum of sinusoids at
the fundamental and harmonic frequencies.
The waveform f(t) depends on the amplitude and phase
of every harmonic components, and we can generate any
non-sinusoidal waveform by an appropriate combination
of sinusoidal functions.

8. Sinusoidal Wave and phase
x(t) = Asin(t) = Asin(250t)
x(t) A t
T0 = 20 msec A t
td = 2.5 msec
x(t )= Asin(250[t ])
= Asin(250t-0.25)= Asin(250t-45o)
Time delay td = 25 msec correspond to phase shift =45o

9. Representation of Quantity using Basis
Any number can be represented as a linear sum of the basis number {1, 10, 100, 1000}
Ex: =10(1000) + 4(100) + 3(10) +7(1)
Any 3-D vector can be represented as a linear sum of the basis vectors {[1 0 0], [0 1 0], [0 0 1]}
Ex: [2 4 5]= 2 [1 0 0] + 4[0 1 0]+ 5[0 0 1]

10. Basis Functions for Time Signal
Any periodic signal x(t) with fundamental frequency 0 can be represented by a linear sum of the basis functions {1, cos(0t), cos(20t),…, cos(n0t), sin(0t), sin(20t),…, sin(n0t)}
Ex:
x(t) =1+ cos(2t)+ 2cos(2 2t)+ 0.5sin(23t)+ 3sin(2t)

11. x(t) =1+ cos(2t)+ 2cos(2 2t)+ 3sin(2t)+ 0.5sin(23t)

12. Purpose of the Fourier Series (FS)
FS is used to find the frequency components and their strengths for a given periodic signal x(t).

13. The Three forms of Fourier Series
Trigonometric Form
Compact Trigonometric (Polar) Form.
Complex Exponential Form.

14. Trigonometric Form
It is simply a linear combination of sines and cosines at multiples of its fundamental frequency, f0=1/T.
a0 counts for any dc offset in x(t).
a0, an, and bn are called the trigonometric Fourier Series Coefficients.
The nth harmonic frequency is nf0.

15. Trigonometric Form
How to evaluate the Fourier Series Coefficients (FSC) of x(t)?
To find a0 integrate both side of the equation over a full period

16. Trigonometric Form

17. The Fourier Series
To be described by the Fourier Series the waveform f(t)
must satisfy the following mathematical properties:
1. f(t) is a single-value function except at possibly a finite number of points.
2. The integral for any t0.
3. f(t) has a finite number of discontinuities within the period T.
4. f(t) has a finite number of maxima and minima within the period T.
In practice, f(t) = v(t) or i(t) so the above 4 conditions are
always satisfied.

18. Example Fundamental period T0 = p Fundamental frequencyp -p 1
e-t/2
f(t)
Fundamental period
T0 = p
Fundamental frequency
f0 = 1/T0 = 1/p Hz
w0 = 2p/T0 = 2 rad/s

19. Compact Trigonometric Form
Using single sinusoid,
are related to the trigonometric coefficients an and bn as:
and

20. Compact Trigonometricp -p 1
e-t/2
f(t)
Fundamental period
T0 = p
Fundamental frequency
f0 = 1/T0 = 1/p Hz
w0 = 2p/T0 = 2 rad/s

21. Line Spectra of x(t)
The amplitude spectrum of x(t) is defined as the plot of the magnitudes |Cn| versus 
The phase spectrum of x(t) is defined as the plot of the angles versus 
This results in line spectra
Bandwidth the difference between the highest and lowest frequencies of the spectral components of a signal.

22. Line Spectra p -p 1
e-t/2
f(t)
f(t)= cos(2t-75.96o) cos(4t-82.87o) +
0.084 cos(6t-85.24o) cos(8t-86.24o) + … Cn n
0.504
0.244 
0.125
0.084
0.063
-/2 

23. Line Spectra
f(t)= cos(2t-75.96o) cos(4t-82.87o) +
0.084 cos(6t-85.24o) cos(8t-86.24o) + … Cn n
0.504
0.244 
0.125
0.084
0.063
-/2 

24. Matlab Tutorials
Tutorial 1
Tutorial 2