06EC44-Signals and System and System –Chapter 4.1-2009• Krupa Rasane (KLE) Page 2 REFERENCE BOOKS: 2. Alan V Oppenheim, Alan S, Willsky and A Hamid Nawab, “Signals and Systems”

06EC44-Signals and System –Chapter 4.1-2009•

Krupa Rasane (KLE) Page 1

06EC44 Signals and Systems (Chapter 4 )

Aurthored By: Prof. krupa Rasane

Asst.Prof E&C Dept.

KLE Society’s College of Engineering and Technology

Belgaum

CONTENT

Fourier Series Representation

1.1.1 Introduction to Fourier Series,

1.1.2 Brief History.

1.1.3 LTI Systems and Exponential Signal Inputs

1.1.4 Eigenfunctions and Values

1.1.5 Complex signals

1.1.6 Convergence to FS

1.1.7 Examples on FS

1.1.8 Fourier Series Properties

1.1.1 Introduction to Fourier Series

Pre-requisite knowledge

• Discrete and Continuous types of Signals.

• Complex Exponential and Sinusoidal signals.

• Time Domain Representation for Linear Time Invariant

Systems.

• Convolution: Impulse Response. Representation for LTI

Systems.

• Knowledge of Mathematical Fourier Series (not necessary

but helps)

Specified Reference Books

TEXT BOOK • Simon Haykin and Barry Van Veen “Signals and

Systems”, John Wiley & Sons, 2001.Reprint 2002



REFERENCE BOOKS:

2. Alan V Oppenheim, Alan S, Willsky and A Hamid

Nawab, “Signals and Systems” Pearson Education Asia /

PHI, 2nd edition, 1997. Indian Reprint 2002

3. H. P Hsu, R. Ranjan, “Signals and Systems”, Scham’s

outlines, TMH, 2006

4. B. P. Lathi, “Linear Systems and Signals”, Oxford

University Press, 2005

5. Ganesh Rao and Satish Tunga, “Signals and Systems”,

Sanguine Technical Publishers, 2004

Exam Question Paper pattern

• You will be able to Answer

– I Full Question from Part A and

– 2 Full Questions from Part B.

• Fourier representation for signals

Content (Unit 4)

1. DTFS - Discrete Time Periodic Signals

2. FS - Continuous Times Periodic Signals

Content (Unit 5)

1. DTFT - Discrete Times Non-Periodic Signals

2. FT - Continuous Time Non-Periodic Signals

Content (Unit 6)

1. Application of Fourier Representation



Questions you will be able to Answer at the end of session

• Fourier series

– why we use it

– how to get coefficients for each form

– Eigen functions

– what they are

– how they relate to LTI systems

– how they relate to Fourier series

• Frequency response

– what it represents

– why we use it

– how to find it

– how to use it to find the output y for any input x

4.1.2 A Historical perspective

• In 1807, Jean Baptiste Joseph Fourier Submitted a paper of using

trigonometric series to represent “any” periodic signal.

• But Lagrange rejected it!

• In 1822, Fourier published a book “The Analytical Theory of Heat”

• Fourier’s main contributions: Studied vibration, heat diffusion, etc.

and found that a series of harmonically related sinusoids is useful

in representing the temperature distribution through a body.

• He also claimed that “any” periodic signal could be represented by

Fourier series.

• These arguments were still imprecise and it remained for

P.L.Dirichlet in 1829 to provide precise conditions under which a

periodic signal could be represented by a FS.

• He however obtained a representation for aperiodic signals i.e.,

Fourier integral or transform

• Fourier did not actually contribute to the mathematical theory of

Fourier series.

• Hence out of this long history what emerged is a powerful and

cohesive framework for the analysis of continuous- time and

discrete-time signals and systems

• and an extraordinarily broad array of existing and potential

application.



• Let us see how this basic tool was developed and some important

Applications

4.1.3 The Response of LTI Systems to Complex Exponentials

We have seen in previous chapters how advantageous it is in

LTI systems to represent signals as a linear combinations of

basic signals having the following properties.

Key Properties: for Input to LTI System

1. To represent signals as linear combinations of basic

signals.

2. Set of basic signals used to construct a broad class of

signals.

3. The response of an LTI system to each signal should be

simple enough in structure.

4. It then provides us with a convenient representation for the

response of the system.

5. Response is then a linear combination of basic signal.

4.1.4 Eigenfunctions and Values

• One of the reasons the Fourier series is so important is that

it represents a signal in terms of eigenfunctions of LTI

systems.

• When I put a complex exponential function like x(t) = ejωt

through a linear time-invariant system, the output is

y(t) = H(s)x(t) = H(s) ejωt

where H(s) is a complex

constant (it does not depend on time).

• The LTI system scales the complex exponential ejωt

.



4.1.5 The Response of LTI Systems to Complex Exponentials

Let us analyse how an LTI system responds to complex signals

where s and z are complex Nos.

The Response of an LTI System:

For CT (Continuous Times) and DT (Discrete Times) we can say that

Where the complex amplitude factor H(s), H(z) is called the frequency

response of the system. The complex exponential est is called an eigenfunction of the system, as the output is of the same form, differing by a scaling factor.

The Response of LTI Systems to Complex Exponentials We know for LTI System Output and for CT Signals,

,

We know for LTI System Output and for CT Signals,

where



Eigenfunction and Superposition Properties



Conclusion :

• Each system has its own constant H(s) that describes how it scales eigenfunctions. It is called the frequency response.

• The frequency response H(ω)=H(s) does not depend on the input.

• If we know H(ω), it is easy to find the output when the input is an eigenfunction. y(t)=H(ω)x(t) true when x is eigenfunction.

• So, given the system response to an eigenfunction, H(s), we can compute the magnitude response |H(s)| and the phase response H(s).

• These form the scaling factor and phase shift in the output, respectively.

• The frequency of the output sinusoid will be the same as the frequency of the input sinusoid in any LTI system.

• The LTI system scales and shifts sinusoids for both continuous and discrete signals and systems.

Eigenfunction -Example: Ex :Consider the system with frequency response as given below. Find the output y for the input given by x(t) = cos(4t).

Soln:

32)(

j

H

))(4cos(|)(|)( HtHty

342)4()(

j

HH

)1274cos(52)( tty

127)34(2)4()( jHH



Need for Frequency Analysis Fast & efficient insight on signal’s building blocks. Simplifies original problem - ex.: solving Part. Diff. Eqns. Powerful & complementary to time domain analysis techniques. Several transforms in DSPing: Fourier, Laplace, z, etc.

Fourier Analysis : The following are its Applications

Telecomms - GSM/cellular phones, Electronics/IT - most DSP-based applications, Entertainment - music, audio, multimedia, Accelerator control (tune measurement for beam steering/control), Imaging, image processing, Industry/research - X-ray spectrometry, chemical analysis (FT spectrometry), PDE solution, radar design, Medical - (PET scanner, CAT scans & MRI interpretation for sleep disorder & heart malfunction diagnosis, Speech analysis (voice activated “devices”, biometry, …). Orthogonality of the Complex exponentials

Definition : Two signals are orthogonal if their inner product is zero. The inner product is defined using complex conjugation when the signals are complex valued. For continuous-time signals with period T, the inner product is defined in terms of an integral as



For discrete-time signals with period N, their inner product is defined as

Orthogonality of the Complex exponentials



Harmonically Related Complex Exponentials

Where, k=+1,-1; the first harmonic components or the fundamental Component and k=+2,-2; the second harmonic components or the fundamental Component ….. etc.

Fourier Series Representation of CT Periodic Signals Example 1



Example 1 Graphical Representation



Summaries FS



All components have (1) the same amplitude and the same initial phase Example 2



The Bar graph of the Fourier series coefficients for example

2 are real and consequently, they can be depicted

graphically with only a single graph. More generally, the

Fourier are complex so that Two graphs, corresponding to

the real and imaginary parts, or magnitude and phase, of

each coefficient, would be required.

4.1.6 Convergence for Fourier

Fourier maintained that “any” periodic signal could be

represented by a Fourier series The truth is that Fourier series

can be used to represent an extremely large class of periodic

signals. The question is that When a periodic signal x(t) does in

fact have a Fourier series representation? Convergence One

class of periodic signals: Which have finite energy over a single

period.

One class of periodic signals: Which have finite energy over a

single period. The other class of periodic signals which satisfy

Dirichlet conditions.

Dirichlets Condition

Condition 1: Krupa Over any period, x(t) must be absolutely integrable, i.e

each coefficient is to be finite.

Condition 2: In any finite interval, x(t) is of bounded variation; i.e., – There

are no more than a finite number of maxima and minima during

any single period of the signal

Condition 3: In any finite interval, x(t) has only finite number of

discontinuities. Furthermore, each of these discontinuities is

finite.

Gibbs phenomenon:

When a sudden change of amplitude occurs in a signal and the

attempt is made to represent it by a finite number of terms (N) in a

Fourier series, the overshoot at the corners (at the points of abrupt

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Documents

06EC44-Signals and System and System –Chapter 4.1-2009• Krupa Rasane (KLE) Page 2 REFERENCE BOOKS: 2. Alan V Oppenheim, Alan S, Willsky and A Hamid Nawab, “Signals and Systems”