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Transcript of Chapter 4.1
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
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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
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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.
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• 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
.
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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,
,
where
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Eigenfunction and Superposition Properties
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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:
3
2)(
jH
))(4cos(|)(|)( HtHty
34
2)4()(
jHH
)1274cos(5
2)( tty
127)34(2)4()( jHH
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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
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For discrete-time signals with period N, their inner product
is defined as
Orthogonality of the Complex exponentials
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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
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Example 1 Graphical Representation
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Summaries FS
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All components have (1) the same amplitude and the same
initial phase
Example 2
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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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change) is always found. As the number of terms is increased, the
overshoot is still found; this is called the Gibbs phenomenon.
In 1899, Gibbs showed that the partial sum near discontinuity
exhibits ripples & the peak amplitude remains constant with
increasing N. Convergence of FS of a square wave to illustrate Gibbs
phenomenon Where finite series approximation for
several N.
• Still, convergence has some interesting characteristics:
As N→ ∞, xN(t) exhibiting Gibbs’ phenomenon at points of
discontinuity. Dirichlet conditions are met for the signals we will
encounter in the real world. Then The Fourier series = x(t) at points
where x(t) is continuous. The Fourier series = “midpoint” at points of
discontinuity
4.1.8 Properties of Fourier Representation
The following are the Properties for the fourier Series
1. Linearity Properties
2. Translation or Time Shift Properties
3. Frequency Shift Properties
4. Scaling Properties
5. Time Differentiation
6. Time Domain Convolution
7. Modulation or Multiplication theorem
8. Parsevals Relationships
tjkwN
Nkeatx kN
0
)(
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1) Linearity Properties
The Fourier series coefficient ck are given by the same linear combination of FS
coefficients for x(t) and y(t)
2) Frequency Shift Properties : In other words frequency shift applied to a continuous-time
signal results in a time shift of the corresponding sequence of
Fourier series coefficients
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3) Scaling Properties
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4) Time Differentiation
5) Modulation or Multiplication theorem
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6) Parsevals Relationships
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Property Summary
4.1.9 Examples using FS Properties
Example 1:
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Example 2
We know that
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Example 3 : From the following we have
,
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,
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Summary