1

Roadmap

Signal System

InputSignal

OutputSignal

characteristics

Given input and systeminformation, solve for

the response

Solvingdifferentialequation

Calculateconvolution

LinearTime-invariant

InvertibleCausal

MemoryBound

classification

transformation

Special properties(even/odd, periodic)

Math. description

Energy & power

2

ECE310 – Lecture 13

Fourier Series - CTFS02/26/01

3

Why a New Domain? It is often much easier to analyze signals and

systems when they are represented in the frequency domain

The entire subject of signals & systems consists primarily the following concepts: Writing signals as functions of frequency Looking at how systems respond to inputs of

different frequencies Developing tools for switching between time-domain

and frequency-domain representations Learning how to determine which domain is best

suited for a particular problem

4

Scenario for Selected Legs

5two legs selected

6two legs selected

7

Leg1 Analysis (AAV from N to W)

8

Leg1 Analysis (DW from W to E)

9

Fourier Series & Fourier Transform They both represent signal in the

form of a linear combination of complex sinusoids

FS can only represent periodic signals for all time

FT can represent both periodic and aperiodic signals for all time

10

Limitations of FS Dirichlet conditions

The signal must be absolutely integrable over the time, t0 < t < t0 + TF

The signal must have a finite number of maxima and minima in the time, t0 < t < t0 + TF

The signal must have a finite number of discontinuities, all of finite size in the time, t0 < t < t0 + TF

FTt

t

dttx0

0

11

The Fourier Series of x(t) over TF Fourier series (xF) represents any function over a finite

interval TF

Outside TF, xF repeats itself periodically with period TF. xF is one period of a periodic function which matches the

function x(t) over the interval TF. If x(t) is periodic with period = T0

if TF=nT0, then the Fourier series representation (TSR) equals to x(t) everywhere;

if TF != nT0, then FSR equals to s(t) only in the time period TF, not anywhere else.

-k

2

00

kX

,

tkfjF

FF

Fetx

Tttttxtx

12

Examples

13

The Fourier Series

FF

F

FF

F

Tt

t

tkfj

F

n

tkfj

Tt

t FF

sF

Tt

tF

c

Tt

tF

c

nFsFcc

dtetxT

kX

ekXtx

dttkftxT

kXdttkftxT

kX

dttxT

X

tkfkXtkfkXXtx

0

0

0

0

0

0

0

0

2

2

1

1

, :formcomplex

2sin2 ,2cos2

,10

2sin2cos0 :form rictrigonomet

kXkXjkX

kXkXkX

kjXkXkXXX

s

c

scc

*

*

2,00

14

Some Parameters TF is the interval of signal x(t) over which

the Fourier series represents fF = 1/TF is the fundamental frequency of

the Fourier series representation n is called the “harmonic number”

2fF is the second harmonic of the fundamental frequency fF.

The Fourier series representation is always periodic and is linear combinations of sinusoids at fF and its harmonics.

15

Interpretation The FS coefficient tells us how

much of a sinusoid at the nth harmonic of fF are in the signal x(t)

In another word, how much of one signal is contained within another signal

16

Calculation of FS Sinusoidal signal (ex 1,2) Non-sinusoidal signal (ex 3) Periodic signal over a non-integer

number of periods (ex 4) Periodic signal over an integer number

of periods (ex 4) Even and odd periodic signals (ex 5) Random signal (no known mathematical

description) (ex 6)

17

Example 1 – Finite Nonzero Coef x(t) = 2cos(400t) over 0<t<10ms Band-limited signals Analytically

Trignometric form and Complex form Graphically (p6-10~6-12)

18

Example 2 – Finite Nonzero Coef x(t) = 0.5 - 0.75cos(20t) + 0.5sin(30t)

over -100ms < t < 100ms Band-limited signals (p6-14,6-15)

19

Example 3 – Infinite Nonzero Coef x(t) = rect(2t)*comb(t) over –0.5<t<0.5 When we have infinite nonzero coefficients, we

tend to use magnitude and phase of the CTFS versus harmonic to present the CTFS (p6-19)

20

Example 4 – Periodic Signal x(t) = 2cos(400pt) over 0<t<7.5ms Over a non-integer number of period p6-20

21

Example 5 – Periodic Even/Odd Signals For a periodic even function, X[k] must be real

and Xs[k] must be zero for all k For a periodic odd function, X[k] must be

imaginary and Xc[k] must be zero for all k

2

0

2

2

2

2

22

2cos22sin2cos1

11 0

0

FF

F

F

F

F

F

F

T

FF

T

TFF

F

T

T

tkfj

F

periodicTt

t

tkfj

F

dttkftxT

dttkftjxtkftxT

dtetxT

dtetxT

kX

22

Example 6 – Random Signal Is it necessary to know the mathematical

description of the signal in order to derive its CTFS?

No Graphically (p6-24, 6-25)

23

Convergence of the CTFS For continuous signal

As N increases, CTFS approaches x(t) in that interval

For signals with discontinuities As N increases, there is an overshoot or

ripple near the discontinuities which does not decrease – Gibbs phenomenon

When N goes to infinity, the height of the overshoot is constant but its width approaches zero, which does not contribute to the average power

24

Example P6-35, 6-36

25

Exercises 6.1.2

26

Response of LTI System with Periodic Excitation Represent the periodic excitation using complex

CTFS Since it’s an LTI system, the response can be

found by finding the response to each complex sinusoid

Example: RC lowpass circuit

Magnitude and phase of Vout[k]/Vin[k] (p6-53)

k

tkfjininoutout ekVtvtvtRCv 02'

27

Properties of CTFS

Linearity Time shifting Time reversal Time scaling Time differentiation Time integration Time multiplication Frequency shifting Conjugation Parseval’s theorem

kT

tfkj

q

t

k

takfj

tkfj

kXdttxT

kXtx

kkXtxe

qkXqYtytx

XkfjkXdx

kXkfjtxdtd

ekXatx

kXtxkXettx

kYkXtytx

22

0

**

02

0

0

2

20

0

00

0

00

1

00 if 2

2

28

Parseval’s Theorem Only if the signal is periodic The average power of a periodic

signal is the sum of the average powers in its harmonic components

kT

kXdttxT

22

0 0

1

29

Summary - CTFS The essence of CTFS The limitation of CTFS The calculation of CTFS The convergence of CTFS

Continuous signals Signals with discontinuities – Gibbs phenomena

Properties of CTFS Especially Parseval’s theorem

Application in LTI system

30

Test 2ECE310 Test 2 Statistics

0123456

90-100

80-89 70-79 60-69 <60

Avg. 69.9, [42, 89]

Nr. o

f Stu

dent

s

Series1