Fourier Series Representation of Periodic SignalsPeriodic...
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Fourier Series Representation of Periodic SignalsPeriodic Signals
Rui Wang, Assistant professorDept. of Information and Communication
T ji U i itTongji University
Email: [email protected]
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OutlineOutline
The response of LIT s stem to comple The response of LIT system to complex exponentials
Fourier series representation of continuous/discrete-time periodic signals
Properties of continuous/discrete-time Fourier series
Filtering
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3 1 Introduction3.1 Introduction
Representation & analysis of LTI using Representation & analysis of LTI using convolution sum/integral Representing signals as linear combinations of shifted Representing signals as linear combinations of shifted
impulses In chapters 3 4 and 5 we learn alternative In chapters 3, 4, and 5, we learn alternative
representation using complex exponentials.It provides us with another convenient way to analyze It provides us with another convenient way to analyze the system and gain insight into their properties.
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3 1 Introduction3.1 Introduction
In chapter 3 we focus on In chapter 3, we focus on Representation of continuous/discrete-time periodic
signalssignals In chapters 4 and 5, we extend the analysis to
Aperiodic signals ith finite energ Aperiodic signals with finite energy These representations provide us
More powerful and important tools and insights for analyzing, designing and understanding signals and LTI systemsLTI systems
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3.1 The response LTI to complex exponentials The importance of complex exponentials in the study of The importance of complex exponentials in the study of
LTI systems stems from the fact: Continuous-time
Discrete-time:
How to prove them?
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3.1 The response to LTI complex exponentials
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3.1 The response to LTI complex exponentials An example: An example:
In general: In general:
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3.3 Fourier series representation of continuous-time periodic signals 3 3 1 Linear combinations of harmonically 3.3.1 Linear combinations of harmonically
related complex exponentials
For periodic signal x(t), the minimum positive, non-zero value of T is fundamental period
is referred to as the fundamental frequency
Harmonically related complex exponentialsHarmonically related complex exponentials
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3.3 Fourier series representation of continuous-time periodic signals 3 3 1 For a signal the Fourier representation 3.3.1 For a signal, the Fourier representation
is
For k = 0, the term is a constant For K=1 or -1, the terms is called the first harmonic
component For K=2 or -2, the terms is called the second
h i tharmonic component For K=N or -N, the terms is called the N-th
harmonic componentharmonic component9
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3.3 Fourier series representation of continuous-time periodic signals For real signals we have For real signals, we have
And with , we have
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3.3 Fourier series representation of continuous-time periodic signals
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3.3 Fourier series representation of continuous-time periodic signals 3 3 2 Determine the Fourier Series 3.3.2 Determine the Fourier Series
Representation
Then we have
Referred to as Fourier series coefficients
Is the constant component of x(t)
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3.3 Fourier series representation of continuous-time periodic signals Proof: Proof:
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3.3 Fourier series representation of continuous-time periodic signals Example 1: consider a signal Example 1: consider a signal,
Determine the Fourier series coefficient.
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3.3 Fourier series representation of continuous-time periodic signals Solution: Solution:
We have
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3.3 Fourier series representation of continuous-time periodic signals Example 2: consider a signal Example 2: consider a signal,
Determine the Fourier series coefficient.
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3.3 Fourier series representation of continuous-time periodic signals Solution: Solution:
We have
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3.3 Fourier series representation of continuous-time periodic signals Example 3: consider a periodic square wave Example 3: consider a periodic square wave,
Determine the Fourier series coefficient.
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3.3 Fourier series representation of continuous-time periodic signals Solution: Solution:
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3.4 Convergence of the Fourier representation In some cases the integral in obtaining the In some cases, the integral in obtaining the
coefficient is not convergent. Fort natel there are no con ergence Fortunately, there are no convergence difficulties for large classes of periodical signalssignals.
In most of cases, the periodic signals that can be represented by Fourier series is the signal with finite energy over a single period
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3.4 Convergence of the Fourier representation Dirichlet conditions: Dirichlet conditions:
Condition1 : Over any period, x(t) must be absolutely integrable that isabsolutely integrable, that is
Which makes sure
An example:An example:
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3.4 Convergence of the Fourier representation Dirichlet conditions: Dirichlet conditions:
Condition 2 : In any finite interval of time, x(t) is of bounded variation that is there are no more than abounded variation, that is there are no more than a finite number of maxima and minima during any single period of the signal. g g
An example: p
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3.4 Convergence of the Fourier representation Dirichlet conditions: Dirichlet conditions:
Condition 3 : In any finite interval of time, there are only a finite number of discontinuities Furthermoreonly a finite number of discontinuities. Furthermore, each of these discontinuities is finite.
In practice, more of signals are convergent. F thi th ti fFor this reason, the question of convergence of Fourier series will not play a significant role i th i d f th b kin the remainder of the book.
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3.5 Properties of continuous-time Fourier Series Assumption: Fundamental period T Assumption: Fundamental period T,
fundamental frequency is
denotes a periodic signal and its Fourier series coefficient.
Condition 1: Linearity (with the same period T)
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3.5 Properties of continuous-time Fourier Series Condition 2: Time shifting Condition 2: Time shifting
Proof:Proof:
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3.5 Properties of continuous-time Fourier Series Condition 3: Time reversal Condition 3: Time reversal
Proof:Proof:
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3.5 Properties of continuous-time Fourier Series For even signal its Fourier series coefficients For even signal, its Fourier series coefficients
are also even, i.e.,
For odd signal, its Fourier series coefficients are also odd, i.e.,
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3.5 Properties of continuous-time Fourier Series Condition 4: Time scaling Condition 4: Time scaling
Proof: the fundamental period and theProof: the fundamental period and the fundamental frequency are changed.
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3.5 Properties of continuous-time Fourier Series Condition 5: multiplication Condition 5: multiplication
Proof:Proof:
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3.5 Properties of continuous-time Fourier Series Condition 6: conjugation & conjugate Condition 6: conjugation & conjugate
symmetry
With this property, we have if (realWith this property, we have if (real signal)
If x(t) is real and even we have real If x(t) is real and even, we have real
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3.5 Properties of continuous-time Fourier Series Condition 7: Parseval’s relation for Condition 7: Parseval s relation for
continuous-time periodic signal
Also
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3.5 Properties of continuous-time Fourier Series Summary: Summary:
X(t) and y(t) are periodic and with period T &
Linearity
Time shifting
Frequency shifting
Conjugation
Time reversal
Time Scaling (P i d )(Period )Periodic Convolution
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3.5 Properties of continuous-time Fourier Series
X(t) and y(t) are periodic and with period T &
Multiplication
Differentiation
Integration (finite and periodic onl ifperiodic only if Conjugate Symmetry for Real Signals real
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3.5 Properties of continuous-time Fourier Series
X(t) and y(t) are periodic and with period T &
Real and Even Signals X(t) real and even real and even
Real and Odd Signals X(t) real and odd imaginary and odd
Even-oddEven odd Decomposition of Real Signal [x(t) real]Parseval’s Relation forParseval s Relation for Periodic Signals
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3.5 Properties of continuous-time Fourier Series Example 1: consider a signal Example 1: consider a signal,
Determine the Fourier series coefficient.
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3.5 Properties of continuous-time Fourier Series Solution: with time-shift property the Fourier Solution: with time-shift property, the Fourier
coefficient x(t-1) can be expressed as
For the constant offset 1/2
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3.5 Properties of continuous-time Fourier Series Example 2: consider a signal Example 2: consider a signal,
Determine the Fourier series coefficientDetermine the Fourier series coefficient.
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3.5 Properties of continuous-time Fourier Series Solution: Solution:
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3.5 Properties of continuous-time Fourier Series Example 3: consider a signal Example 3: consider a signal,
Determine the Fourier series coefficient.
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3.5 Properties of continuous-time Fourier Series Solution1: Solution1:
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3.5 Properties of continuous-time Fourier Series Solution 2: Solution 2:
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3.5 Properties of continuous-time Fourier Series Solution 2: Solution 2:
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3.6 Fourier series representation of discrete-time periodic signals The Fourier series representation of a The Fourier series representation of a
discrete-time periodic is a finite series.All of the follo ing comple e ponential ha e All of the following complex exponential have fundamental frequencies that are multiples of
We have
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3.6 Fourier series representation of discrete-time periodic signals The Fourier series representation of discrete- The Fourier series representation of discrete-
time signals
The problem to obtain a_k
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3.6 Fourier series representation of discrete-time periodic signals We have a closed-form expression for We have a closed-form expression for
obtaining discrete-time Fourier series pair
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3.6 Fourier series representation of discrete-time periodic signals
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3.6 Fourier series representation of discrete-time periodic signals Example 1: consider a signal Example 1: consider a signal,
Determine the Fourier series coefficient.
Solution: when is an integer,
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3.6 Fourier series representation of discrete-time periodic signals When is a ratio of integers When is a ratio of integers
Then we have
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3.6 Fourier series representation of discrete-time periodic signals Example 2: consider a discrete-time periodic Example 2: consider a discrete-time periodic
square wave
Determine the Fourier series coefficient.
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3.6 Fourier series representation of discrete-time periodic signals Solution: Solution:
Letting m = n+N_1 g
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3.6 Fourier series representation of discrete-time periodic signals And And
andand
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3.7 Properties of discrete-time Fourier series Summary Summary
X[n] and y[n] are periodic and with period N &
Linearity
Time shifting
Frequency shifting
Conjugation
Time reversal
Time Scaling( )
[ / ] if is a multiple of [ ]
x n m n mx n
(viewed as periodic kam
Periodic Convolution
( )[ ]0 if is not a multiple of mx n
n m with period )mN
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3.5 Properties of continuous-time Fourier Series
X[n] and y[n] are periodic and with period N &
Multiplication
First difference
Running sum (finite and periodic onl ifand periodic only if Conjugate Symmetry for Real Signals real
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3.5 Properties of continuous-time Fourier Series
X(t) and y(t) are periodic and with period T &
Real and Even Signals X[n] real and even real and even
Real and Odd Signals X[n] real and odd imaginary and odd
Even-oddEven odd Decomposition of Real Signal [x[n] real]Parseval’s Relation forParseval s Relation for Periodic Signals
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3.7 Properties of discrete-time Fourier series Multiplication Multiplication
Assume
We have
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3.7 Properties of discrete-time Fourier series First difference First difference
Assume
We have
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3.7 Properties of discrete-time Fourier series Parseval’s relation Parseval s relation
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3.7 Properties of discrete-time Fourier series Example 1: Example 1:
consider a signal
Determine the Fourier series coefficient.
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3.7 Properties of discrete-time Fourier series solution: solution:
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3.7 Properties of discrete-time Fourier series solution: solution:
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3.7 Properties of discrete-time Fourier series Example 2: Example 2:
Consider a signal x[n] with X[n] is periodic with period N = 6 X[n] is periodic with period N = 6
X[n] has the minimum power per period among the set of signals satisfying the preceding three conditions
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3.7 Properties of discrete-time Fourier series solution: solution:
From condition 2, we conclude
From condition 3, we have
so Since the power is , to minimize the
power
so
power
We have62
We have
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
For continuous-time case with input For continuous time case, with input
wherewhere
For discrete-time case, with input
where
We call H(s) and H(z) as system function
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
We focus on and We focus on and
st j te e
are complex exponential signals at frequency
e en j nz e
are complex exponential signals at frequency
We call and as system function
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
Based on that with input Based on that, with input
We have
[1] y(t) has the same fundamental frequency as x(t) [2] if is the set of Fourier series coefficient,
then is the set of Fourier series coefficients for the output.
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
Based on that with input Based on that, with input
We have
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
Example 1: with input Example 1: with input
and system unit impulse response being
Determine the Fourier series coefficients of outputoutput.
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
Solution: we first compute the frequency Solution: we first compute the frequency response
The output is given by
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
Example 2: with input Example 2: with input
and system unit impulse response being
Determine the Fourier series coefficients of outputoutput.
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
Solution: we first compute the Fourier series Solution: we first compute the Fourier series representation
and the frequency response
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3 8 Fourier series and LTI systems3.8 Fourier series and LTI systems
The output is given byThe output is given by
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3 9 Filtering3.9 Filtering
Filtering: change the relative amplitude of the Filtering: change the relative amplitude of the frequency component in a signal or eliminate some frequency componentssome frequency components Frequency-shaping filter: a LTI system which
changes the shape of input spectrumchanges the shape of input spectrum Frequency-selective filter: pass some frequencies
undistorted and significantly attenuate or eliminateundistorted and significantly attenuate or eliminate others
Why we can do this?Why we can do this?
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3 9 Filtering3.9 Filtering
Frequency-shaping filters Frequency-shaping filters An example: Differentiating filter
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3 9 Filtering3.9 Filtering
Differentiating filter can Differentiating filter can A complex exponential input will receive
t lifi ti f l l fa greater amplification for large values of Then, this filter will enhance the rapid
variations in a signal Often used to enhance the edges in image g g
processing
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3 9 Filtering3.9 Filtering
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3 9 Filtering3.9 Filtering
Frequency-selective filters: pass some Frequency-selective filters: pass some frequencies undistorted and significantly attenuate or eliminate othersattenuate or eliminate others An example: reduce the noise in music or
voice recording systemvoice recording system Lowpass filter: pass low frequencies and
attenuate or reject high frequencies Highpass filter: pass high frequencies and
attenuate or reject low frequencies
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3 9 Filtering3.9 Filtering
Cutoff frequency: the frequency in the Cutoff frequency: the frequency in the boundaries between frequencies that are passed and the frequencies that are rejectedpassed and the frequencies that are rejected
passband & stopband
passbandstopband stopband
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3 9 Filtering3.9 Filtering
For continuous-time case: For continuous-time case: Idea lowpass filter:
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3 9 Filtering3.9 Filtering
For continuous-time case: For continuous-time case: Idea highpass filter with cutoff frequency
Id b d filt Idea bandpass filter
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3 9 Filtering3.9 Filtering
For discrete-time case: For discrete-time case: Idea lowpass filter:
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3 9 Filtering3.9 Filtering
For discrete-time case: For discrete-time case: Idea highpass filter with cutoff frequency
Id b d filt Idea bandpass filter
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations In many applications frequency-selective In many applications, frequency-selective
filtering is accomplished by use of LTI systems described bydescribed by Linear constant-coefficient differential equations Linear constant coefficient difference equations Linear constant-coefficient difference equations
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations A simple RC lowpass filter A simple RC lowpass filter
First-order RC circuit
The output voltage is related to the input voltage through the following linear constant-coefficient differential equation
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations Determine the frequency response Determine the frequency response
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations The magnitude and phase of frequency The magnitude and phase of frequency
response are
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations A simple RC highpass filter A simple RC highpass filter
First-order RC circuit
The output voltage is related to the input voltage through the following linear constant-coefficient differential equation
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations Determine the frequency response Determine the frequency response
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3.11 Examples of discrete-time filters d ib d b diff ti l tidescribed by differential equations The discrete-time system described by The discrete-time system described by
difference equations can be Recursive and have impulse responses of infinite Recursive and have impulse responses of infinite
system (IIR systems) Nonrecursive have finite-length impulse responses Nonrecursive have finite-length impulse responses
(FIR systems)
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3.11 Examples of discrete-time filters d ib d b diff ti l tidescribed by differential equations First-order recursive discrete-time filters First-order recursive discrete-time filters
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations When When
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3.10 Examples of continuous-time filters d ib d b diff ti l tidescribed by differential equations When When
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3.11 Examples of discrete-time filters d ib d b diff ti l tidescribed by differential equations General form of nonrecursive discrete-time General form of nonrecursive discrete-time
filters (moving-average filter)
An example:
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3.11 Examples of discrete-time filters d ib d b diff ti l tidescribed by differential equations When M=N=16 When M=N=16
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3.11 Examples of discrete-time filters d ib d b diff ti l tidescribed by differential equations When M=N=32 When M=N=32
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3.11 Examples of discrete-time filters d ib d b diff ti l tidescribed by differential equations Use nonrecursive discrete-time to perform Use nonrecursive discrete-time to perform
highpass filtering operation
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