1 fourier representations of signals & linear time-invariant systems chapter 3

1

Fourier Representations of Signals & Linear Time-Invariant Systems

Chapter 3

2

Introduction • In the previous chapter, linearity property was

exploited to develop the convolution sum and convolution integral.

• There, the basic idea of convolution is to break up or decompose a signal into sum of elementary function.

• Then, we find the response of the system to each of those elementary function individually and add the responses to get the overall response.

• In this chapter, we will express a signal as a sum of real or complex sinusoids instead of sum of impulses.

3

• The response of LTI system to sinusoids are also sinusoids of the same frequency but with in general, different amplitude and phase.

4

Complex Sinusoids & Frequency Response of LTI System

• The response of an LTI system to a sinusoidal input leads to a characterization of system behaviour that is termed the ‘frequency response’ of the system.

5

Fourier Representation for Four Signal Classes

• There are 4 distinct Fourier representation, each applicable to a different class of signals.

• These 4 classes are defined by the periodicity properties of a signal and whether it is continuous or discrete.

6

Time property Periodic Nonperiodic

Continuous-time Fourier Series (CTFS)

Fourier Transform (CTFT)

Discrete-time Fourier Series (DTFS)

Fourier Transform (DTFT)

Relationship Between Time Properties of a Signal and the Appropriate Fourier Representations

7

The Continuous-Time Fourier Series(CTFS)

8

Objectives

• To develop methods of expressing periodic signals as linear combination of sinusoids, real or complex.

• To explore the general properties of these ways of expressing signals.

• To apply these methods to find the responses of systems to arbitrary periodic signals.

9

Representing a Signal• The Fourier series represents a signal as a linear

combination of complex sinusoids• The responses of LTI system to sinusoids are also

sinusoids of the same frequency but with, in general, different amplitude and phase.

• Expressing signals in this way leads to frequency domain concept, thinking of signals as function of frequency instead of time.

10

Periodic Excitation and Response

11

Aperiodic Excitation and Response

12

Basic Concept & Development of the Fourier Series

13

Linearity and Superposition

If an excitation can be expressed as a sum of complex sinusoidsthe response can be expressed as the sum of responses to complex sinusoids (same frequency but different multiplyingconstant).

14

Continuous-Time

Fourier Series

Concept

15

Conceptual OverviewThe Fourier series represents a signal as a sum of sinusoids.Consider original signal x(t), which we would like to present as alinear combination of sinusoids as illustrated by the dash line.

16

Conceptual Overview (cont…)The best approximation to the dashed-line signal using a constant + one sinusoid of the same fundamental frequency as the dashed-line signal is the solid line.

+

=

17

Conceptual Overview (cont…)The best approximation to the dashed-line signal using a constant + one sinusoid of the same fundamental frequency as the dashed-line signal + another sinusoid of twice the fundamentalfrequency of the dashed-line signal is the solid line.

18

Conceptual Overview (cont…)The best approximation to the dashed-line signal using a constant + three sinusoids is the solid line. In this case, the third sinusoid has zero amplitude, indicating that sinusoid at that frequency does not help the approximation.

19

Conceptual Overview (cont…)The best approximation to the dashed-line signal using a constant + four sinusoids is the solid line (the forth fundamental frequency is three times fundamental frequency of the dashed-line signal). This is a good approximation which gets better with the addition of more sinusoids at higher integer multiples of the fundamental frequency.

20

Trigonometric Form of CTFS

• In the example above, each of the sinusoids used in the approximation above is of the form cos(2ПkfFt+θ) multiplied by a constant to set its amplitude.

• So we can use trigonometry identity:cos(a+b) = cos(a)cos(b) - sin(a)sin(b)sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

• Therefore, we can reformulate this functional form into:

cos(2ПkfFt+θ)= cos(θ) cos(2ПkfFt) - sin(θ)sin(2ПkfFt)

21

Trigonometric Form of CTFS (cont…)

• The summation of all those sinusoids expressed as cosines and sines are called the continuous-time Fourier Series (CTFS).

• In the CTFS, the higher frequency sines and cosines have frequencies that are integers multiples of fundamental frequencies. The multiple is called the harmonic number, k.

22

• If we have function cos(2ПkfFt) or sin(2ПkfFt)

i) k is harmonic number

ii) kfF is highest frequency.

• If the signal to be represented is x(t), the amplitude of the kth harmonic sine will be designed Xs[k] and the amplitude of the kth harmonic cosine will be designed Xc[k].

• Xs[k] and Xc[k] are called sine and cosine harmonic function respectively.

Component of CTFS

23

Complex Sinusoids form of CTFS

• Every sine and cosine can be replaced by a linear combination of complex sinusoids

cos(2ПkfFt) = (ej2ПkfF

t+ e-j2ПkfF

t)/2

sin(2ПkfFt) = (ej2ПkfF

t - e-j2ПkfF

t)/j2

24

Component of CTFS (cont…)

25

CT Fourier Series Definition

0 0

The Fourier series representation x t of a signal x( )

over a time isF

F

t

t t t T

2x X Fj kf tF

k

t k e

where X[k] is the harmonic function, k is the harmonic

number and fF 1 / TF (pp. 240-242). The harmonic function

can be found from the signal as

0

0

21X x

F

F

t Tj kf t

F t

k t e dtT

The signal and its harmonic function form a

indicated by the notation x X .t k

Fourier series

pair F S

26

The Trigonometric CTFSThe fact that, for a real-valued function x(t)

*X Xk k

also leads to the definition of an alternate form of the CTFS, the so-called trigonometric form.

1

x X 0 X cos 2 X sin 2F c c F s Fk

t k kf t k kf t

0

0

2X x cos 2

Ft T

c FF t

k t kf t dtT

0

0

2X x sin 2

Ft T

s FF t

k t kf t dtT

where

27

The Trigonometric CTFSSince both the complex and trigonometric forms of theCTFS represent a signal, there must be relationships between the harmonic functions. Those relationships are

*

*

X 0 X 0

X 0 0, 1,2,3,

X X X

X X X

c

s

c

s

kk k k

k j k k

*

X 0 X 0

X XX , 1,2,3,

2X X

X X2

c

c s

c s

k j kk k

k j kk k

28

Periodicity of the CTFS

29

The dash line are periodic continuations of the CTFS representation

The illustrations show how various kinds of signals are represented by CTFS over a finite time.

30

The dash line are periodic continuations of the CTFS representation

31

Linearity of the CTFS

These relations hold only if the harmonic functions X of allthe component functions x are based on the samerepresentation time.

32

Magnitude and Phase of X[k]A graph of the magnitude and phase of the harmonic functionas a function of harmonic number is a good way of illustrating it.

33

CTFS of Even and Odd Functions

For an , the complex CTFS harmonic function

X is and the sine harmonic function X is

zero.

For an , the complex CTFS harmonic function

X is and

sk k

k

even function

purely real

odd function

purely imaginary

the cosine harmonic function

X is zero.c k

34

Numerical Computation of the CTFSHow could we find the CTFS of this signal which has noknown functional description?

Numerically.

21X x F

F

j kf t

TF

k t e dtT

Unknown

35

Numerical Computation of the CTFS

We don’t know the function x(t), but if we set of NF samples over one period starting at t=0, the time between the samples is Ts TF/NF, and we can approximate the integral by the sum of several integrals, each covering a time of lenght Ts.

36

Numerical Computation of the CTFS (cont…)

11

2

0

1X x

sF

F s

s

n TNj kf nT

snF nT

k nT e dtT

Samples from x(t)

37

Numerical Computation of the CTFS (cont…)

X 1/ x , F s Fk N nT k N DFT

where

1

2 /

0

x xF

F

Nj nk N

s sn

nT nT e

D F T

38

Convergence of the CTFS

• To examine how the CTFS summation approaches the signal it represents as the number of terms used in the sum approaches infinity.

• We do this by examining the partial sum.

02x XN

j kf tN

k N

t k e

39

Convergence of the CTFS (cont…)

For continuous signals, convergence is exact at every point.

A Continuous Signal

Partial CTFS Sums

02x XN

j kf tN

k N

t k e

40

Convergence of the CTFS (cont…)

For discontinuous signals, convergence is exact at every point of continuity.

Discontinuous Signal

Partial CTFS Sums

41

Convergence of the CTFS (cont…)

At points of discontinuitythe Fourier seriesrepresentation convergesto the mid-point of thediscontinuity.

42

CTFS Properties

Linearity

x y X Yt t k k F S

0

0

Let a signal x( ) have a fundamental period and let a

signal y( ) have a fundamental period . Let the CTFS

harmonic functions, each using a common period as the

representation time, be X[ ] a

x

y

F

t T

t T

T

k nd Y[ ]. Then the following

properties apply.

k

43

CTFS Properties

Time Shifting 0 02

0x Xj kf tt t e k F S

0 00x Xjk tt t e k F S

44

CTFS Properties (cont…)

Frequency Shifting (Harmonic Number

Shifting)

0 020x Xj k f te t k k F S

0 00x Xjk te t k k F S

A shift in frequency (harmonic number) corresponds to multiplication of the time function by a complex exponential.

Time Reversal x Xt k F S

45

CTFS Properties (cont…)Time Scaling

Let z x , 0t at a

0 0Case 1. / for zF x zT T a T t

Z Xk k

0Case 2. for zF xT T t

If a is an integer,

X / , / an integerZ

0 , otherwise

k a k ak

46

CTFS Properties (cont…)Time Scaling (continued)

X / , / an integerZ

0 , otherwise

k a k ak

47

CTFS Properties (cont…)

Change of Representation Time

0With , x XF xT T t k F S

X / , / an integerX

0 , otherwisem

k m k mk

(m is any positive integer)

0With , x XF x mT mT t k F S

48

CTFS Properties (cont…)Change of Representation Time (cont..)

49

CTFS Properties (cont…)

Time Differentiation

0

0

x 2 X

x X

dt j kf k

dtd

t jk kdt

F S

F S

50

Time Integration

Case 1. X 0 0

0

Xx

2

t kd

j kf

F S

0

Xx

t kd

j k

F S

Case 2. X 0 0

xt

d is not periodic

CTFS Properties (cont…)

51

CTFS Properties (cont…)Multiplication-Convolution Duality

x y X Yt t k k F S

(The harmonic functions, X[ ] and Y[ ], must be based

on the same representation period .)F

k k

T

0x y X Yt t T k kF S#

0

x y x yT

t t t d #

x t y t xap t y t where xap t is any single period of x t

The symbol indicates .

Periodic convolution is defined mathematically by

periodic convolution#

52

CTFS Properties (cont…)

Conjugation

* *x Xt k F S

Parseval’s Theorem

0

2 2

0

1x X

Tk

t dt kT

The average power of a periodic signal is the sum of theaverage powers in its harmonic components.