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Fourier Representations of Signals & Linear Time-Invariant Systems

Chapter 3

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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.

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

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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.

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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.

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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

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The Continuous-Time Fourier Series(CTFS)

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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.

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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.

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Periodic Excitation and Response

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Aperiodic Excitation and Response

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Basic Concept & Development of the Fourier Series

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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).

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Continuous-Time

Fourier Series

Concept

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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.

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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.

+

=

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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.

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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.

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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.

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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)

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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.

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• 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

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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

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Component of CTFS (cont…)

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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

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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

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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

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Periodicity of the CTFS

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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.

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The dash line are periodic continuations of the CTFS representation

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Linearity of the CTFS

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

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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.

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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

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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

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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.

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Numerical Computation of the CTFS (cont…)

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2

0

1X x

sF

F s

s

n TNj kf nT

snF nT

k nT e dtT

Samples from x(t)

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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

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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

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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

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Convergence of the CTFS (cont…)

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

Discontinuous Signal

Partial CTFS Sums

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Convergence of the CTFS (cont…)

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

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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

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CTFS Properties

Time Shifting 0 02

0x Xj kf tt t e k F S

0 00x Xjk tt t e k F S

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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

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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

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CTFS Properties (cont…)Time Scaling (continued)

X / , / an integerZ

0 , otherwise

k a k ak

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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

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

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CTFS Properties (cont…)

Time Differentiation

0

0

x 2 X

x X

dt j kf k

dtd

t jk kdt

F S

F S

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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…)

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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#

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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.