Signals and Systems

Fourier Series Representation

of Periodic Signals

Chang-Su Kim

Introduction

Why do We Need Fourier Analysis?

The essence of Fourier analysis is to represent a

signal in terms of complex exponentials

Many reasons:

Almost any signal can be represented as a series (or sum or

integral) of complex exponentials

Signal is periodic

Fourier Series (DT, CT)

Signal is non-periodic

Fourier Transform (DT, CT)

Response of an LTI system to a complex exponential is also

a complex exponential with a scaled magnitude.

0 0 0 0 02 2

2 1 0 1 2( ) ... ...jk t j t j t j t j t

k

k

x t a e a e a e a a e a e

We will learn these

(CTFS, DTFS, CTFT, DTFT)

one by one

Response of LTI systems to Complex

Exponentials (CT)

H

( ) ( ) jwc H jw h e d

jwtejwtc e

31 2

31 2

1 2 3

1 1 2 2 3 3

( )

( ) ( ) ( ) ( )

jw tjw t jw t

jw tjw t jw t

x t a e a e a e

y t a H jw e a H jw e a H jw e

Property

This is useful because almost every signal can be represented as a sum of complex exponential functions

So we can easily compute the response of the system to almost every input signal using the above equation

Response of LTI systems to Complex

Exponentials (DT)

H

( ) [ ]jw jwk

k

c H e h k e

jwnejwnc e

31 2

3 31 1 2 2

1 2 3

1 2 3

[ ]

( ) ( ) ( ) ( )

jw njw n jw n

jw jw njw jw n jw jw n

x n a e a e a e

y t a H e e a H e e a H e e

Property

This is useful because almost every signal can be represented as a sum of complex exponential functions

So we can easily compute the response of the system to almost every input signal using the above equation

Continuous Time Fourier Series

0 0 0 0 02 2

2 1 0 1 2( ) ... ...jk t j t j t j t j t

k

k

x t a e a e a e a a e a e

How to represent a periodic function x(t) as

continuous time discrete time

periodic (series) CTFS DTFS

aperiodic (transform) CTFT DTFT

Harmonically Related Complex Exponentials

Basic periodic signal

Fundamental frequency:

Fundamental period:

The set of harmonically related complex exponentials

Each of the signals is periodic with T

Thus, a linear combination of them is also periodic with period T:

0j te

0

02 /T

0 (2 / ){ ( ) | 0, 1, 2, 3,...}jk t jk T t

k t e e k

)(tk

0jk t

k

k

a e

Continuous Time Fourier Series (CTFS)

Most (all in engineering sense) functions

with period T=2/w0 can be represented as a

CTFS

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

Example 1

10 0

0 10 0

0 1

1 1( )

sin( )sin( ) 2 .

Tjk t jk t

kT T

a x t e dt e dtT T

kk T

k k

k … -5 -4 -3 -2 -1 0 1 2 3 4 5

ak … 1/5 0 -1/3 0 1/ 1/2 1/ 0 -1/3 0 1/5

0 T1 T0

1 T1 = (1/4)T0

0 0 0 03 3

0 0

1 1 1 1 1( )

3 2 3

1 2 2cos cos3

2 3

j w t jw t jw t j w tx t e e e e

t t

Example 1 (continued)

Finite Approximation xN(t) and Gibbs phenomenon

0( )N

jk t

N k

k N

x t a e

Overshoot is always about 9% of the height of the discontinuity

Example 2

Example 3

Continuous Time Fourier Series (CTFT)

Derivation of the formula

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

Properties of CTFS

There are 15 properties in Table 3.1 in page 206 of

the textbook

Do we have to remember them all?

No

Instead, familiarize yourself with them and be able to

derive them whenever necessary

They all come from the single formula

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

Selected Properties

Given two periodic signals with same period T and

fundamental frequency 0=2/T:

1. Linearity:

2. Time-Shifting:

3. Time-Reversal (Flip):

4. Conjugate Symmetry:

k

k

bty

atx

)(

)(

kk BbAatBytAxtz )()()(

0 0

0( ) ( )jk t

kz t x t t a e

katxtz )()(

* *( ) ( ) kz t x t a

Selected Properties

5. x(t) is real and even → ak is real and even

6. x(t) is real and odd → ak is purely imaginary and odd

7. Multiplication:

8. Parseval’s Relation:

k

kT

adttxT

22

)(1

( ) ( ) ( ) l k l

l

z t x t y t a b

Other Forms of CTFS

real

Discrete Time Fourier Series

1(2 / ) (2 / ) 2(2 / ) ( 1)(2 / )

0 1 2 1

0

[ ] ...N

jk N n j N n j N n j N N n

k N

k

x n a e a a e a e a e

How to represent a periodic function x[n]

with period N as

continuous time discrete time

periodic (series) CTFS DTFS

aperiodic (transform) CTFT DTFT

Discrete Time Period Functions with

Period N

x[n]=x[n+N]

Fundamental frequency 0=2/N

{k[n] = ejk0n = ejk(2/N)n: k=0, ±1, ±2, ...} is a set of

signals, consisting of all discrete-time complex

exponentials that are periodic with period N

k[n] = k[n+N]

k[n] = k+N[n] = k+2N[n] = ...

Only N distinct signals in the set

Therefore, while an infinite number of complex exponentials are required in CTFS, only N complex exponentials are used in DTFS.

Discrete Time Fourier Series (DTFS)

All functions with period N can be

represented as a DTFS

Derive it!

2

2

( )

( )

[ ]

1[ ]

N

N

jk n

k

k N

jk n

k

n N

x n a e

a x n eN

: denotes the sum over any interval of N successive values of n.

Nn

Properties

Example

x[n]

Fourier Series and LTI Systems

CT

Frequency Response

Hjwte

jwtc e

0

0

0

( ) ( ) ,

( )

( ) ( )

jw

jk t

k

k

jk t

k

k

c H jw h e d

x t a e

y t a H jkw e

DT

Hjwne

jwnc e

2

2 2

( )

( ) ( )

( ) [ ] ,

[ ]

[ ] ( )

N

N N

jw jwk

k

jk n

k

k N

jk jk n

k

k N

c H e h k e

x n a e

y n a H e e

( ) or ( ) are called frequency responses.jwH jw H e

Example

Example 3.16 in pp. 228 in textbook

Filtering

Filtering is a process that changes the amplitude and phase of frequency components of an input signal

All LTI systems can be thought as filters

Ex) Differentiator

High frequency component is

magnified, while low frequency

component is suppressed

It is a kind of highpass filter

jejjHdt

tdxty ||)(

)()(

|H(j)|

H(j)

/2

-/2

Lowpass, Bandpass and Highpass Filters

CT

Ideal low-pass filter (LPF)

H(j) = 1 in pass band

0 in stop band

c

1

H(j)

stop band pass band stop band

c

H(j)

Ideal high-pass filter (HPF)

Ideal band-pass filter (BPF)

c1 c2

H(j)

Lowpass, Bandpass and Highpass Filters

DT

H(ej) is periodic with period 2

low frequencies: at around =0, 2 ,...

high frequencies: at around = , 3, ...

H(ej)

- c 2

LPF HPFH(ej)

- 2

H(ej)

- 2

BPF

Lowpass Filtering

Highpass Filtering

Example of CT Filter

RC lowpass filter

Vs

+ Vr -

Vc

+

-

)(:output

)(:input

)()()(

tV

tV

tVtVdt

tdVRC

c

s

scc

)(tan

2

1

)(1

1

1

1)(

)()(

)()(

RCj

tjtjtj

tjtjtj

eRCRCj

jH

eejHejHjRC

eejHejHdt

dRC

Example of CT Filter (continued)

It is a lowpass filter

1/RC

|H(j)| H(j)

/2

-/4-/2

1tan ( )

2

1 1( )

1 1 ( )

j RCH j eRCj RC

Example of DT Filter 1

y[n] - ay[n-1] = x[n]

We know if x[n]=ejn then y[n]=H(ej)ejn

j

j

njnjjnjj

aeeH

eeeaHeeH

1

1)(

)()( )1(

Example of DT Filter 1 (Continued)

|H(ej)|

|H(ej)|

(a) a=0.6 LPF (b) a=-0.6 HPF

j

j

aeeH

1

1)(

H(ej)

/2

-/2

H(ej)

Example of DT Filter 2

M

Nk

MNknxny ][][

11

,

,

0][ 1

1

otherwise

MnNfornh MN

M

Nk

kj

MN

k

kjj eekheH

11][)(

Example of DT Filter 2 (Continued)

|H(ej)|

|H(ej)|

a) N=M=16

b) N=M=32

It is a lowpass filter