analysis of continuous signals
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Fourier representations
A Fourier representation is unique, i.e., no two
same signals in time domain give the same
function in frequency domain
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Fourier Series versus
Fourier Transform
Fourier Series (FS): a discrete representation of a periodic signalas a linear combination of complex exponentials
The CT Fourier Series cannot represent an aperiodic signal for alltime
Fourier Transform (FT): a continuous representation of a notperiodic signal as a linear combination of complex exponentials
The CT Fourier transform represents an aperiodic signal for all time
A not-periodic signal can be viewed as a periodic signal with
an infinite period
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Fourier Series versus
Fourier Transform FS of periodic CT signals:
As the period increases T, 0
The harmonically related components kw0 become closer in
frequency
As T becomes infinite
the frequency components form a continuum and the FS
sum becomes an integral
TekXtxdtetxT
kX tjk
k
T
tjk/2;][)(;)(
1][ 0
0
00
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Fourier Series versus Fourier
Transform
k
tjk
k
T
tjk
k
eatx
dtetxT
a
0
0
)(
TPer-CTimeDFreq:SeriesFourierInverseCT
)(1
DFreqT
Per-CTime:SeriesFourierCT
0
dejXtx
dtetxjX
tj
tj
)(
2
1)(
CTimeCFreq:TransformFourierCTInverse
)()(
CFreqCTime:TransformFourierCT
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Outline
Introduction to Fourier Transform
Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform Convergence examples
Properties of CT Fourier Transform
Fourier transform of periodic signals
Summary
Appendix Transition: CT Fourier Series to CT Fourier Transform
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As , approaches
approaches zero
uncountable number of harmonics
Integral instead of
Fourier Transform of CT
aperiodic signals
0k
dtetxjX
dejXtx
tj
tj
)()(:)transform(forwardanalysisFourier
)(
2
1)(:)transform(inverseSynthesisFourier
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CT Fourier transform for
aperiodic signals
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CT Fourier Transform of
aperiodic signal
The CT FT expresses
a finite-amplitude aperiodic signal x(t)
as a linear combination (integral) of an infinite continuum of weighted, complex
sinusoids, each of which is unlimited in time
Time limited means having non-zero values
only for a finite time x(t) is, in general, time-limited but must not be
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Outline
Introduction to Fourier Transform
Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform Convergence examples
Properties of CT Fourier Transform
Fourier transform of periodic signals
Summary
Appendix Transition: CT Fourier Series to CT Fourier Transform
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CTFT: Rectangular Pulse Function
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CTFT: Exponential Decay (Right-sided)
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Outline
Introduction to Fourier Transform
Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform Convergence examples
Properties of CT Fourier Transform
Fourier transform of periodic signals
Summary
Appendix Transition: CT Fourier Series to CT Fourier Transform
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Convergence of CT Fourier
Transform
Dirichlets sufficient conditions for the
convergence of Fourier transform are similar to
the conditions for the CT-FS:
a) x(t) must be absolutely integrable
b) x(t) must have a finite number of maxima andminima within any finite interval
c) x(t) must have a finite number of discontinuities,
all of finite size, within any finite interval
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Convergence of CT Fourier
Transform
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Convergence of CT Fourier
Transform: Example 1
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Convergence of CT Fourier
Transform: Example 1
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Convergence of CT Fourier
Transform: Example 2
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Convergence of CT Fourier
Transform: Example 2
The Fourier transform for this example is real at all frequencies
The time signal and its Fourier transform are
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Convergence of CT Fourier
Transform: Example 4
discontinuities at t=T1 and t=-T1
C f CT F i
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Convergence of CT Fourier
Transform: Example 4
Reducing the width of x(t)will have an oppositeeffect on X(j)
Using the inverse Fourier
transform, we get xrec(t)which is equal to x(t) atall points exceptdiscontinuities (t=T1 & t=-
T1), where the inverseFourier transform is equalto the average of thevalues of x(t) on bothsides of the discontinuity
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Convergence of CT Fourier
Transform: Example 5
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Convergence of CT Fourier
Transform: Example 5
This example shows
the reveres effect in
the time and frequencydomains in terms of
the width of the time
signal and thecorresponding FT
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Convergence of the CT-FT:
Generalization
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Convergence of the CTFT:
Generalization
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Convergence of the CTFT:
Generalization
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Convergence of the CTFT:
Generalization
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Outline
Introduction to Fourier Transform
Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform Convergence examples
Properties of CT Fourier Transform
Fourier transform of periodic signals
Summary
Appendix Transition: CT Fourier Series to CT Fourier Transform
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Properties of the CT Fourier
Transform
The properties are useful in determining the Fourier transform
or inverse Fourier transform
They help to represent a given signal in term of operations
(e.g., convolution, differentiation, shift) on another signal for
which the Fourier transform is known
Operations on {x(t)} Operations on {X(j)}
Help find analytical solutions to Fourier transform problems of
complex signals
Example:
tionmultiplicaanddelaytuatyFT t })5()({
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Properties of the CT Fourier
Transform The properties of the CT Fourier transform are
very similar to those of the CT Fourier series
Consider two signals x(t) and y(t) with Fourier
transforms X(j) and Y(j), respectively (orX(f) and Y(f))
The following properties can easily been
shown using
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Properties of the CTFT:
Linearity
P ti f th CTFT
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Properties of the CTFT:
Time shift
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Properties of the CTFT:
Freq. shift
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Frequency shift property:
Example
)(2 0
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Properties of the CTFT
The time and frequency scaling properties indicate that ifa signal is expanded in one domain it is compressed in
the other domain This is also called the uncertainty
principle of Fourier analysis
P ti f th CTFT S li
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Properties of the CTFT: Scaling
Proofs
Properties of the CTFT: Scaling
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Properties of the CTFT: Scaling
Proofs
P ti f th CTFT Ti Shifti & S li E l
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Properties of the CTFT: Time Shifting & Scaling: Example
Properties of the CTFT: Time Shifting & Scaling
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Properties of the CTFT: Time Shifting & Scaling
Example
P i f h CTFT
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Properties of the CTFT: average power
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Properties of the CTFT:
Conjugation
Properties of the CTFT:
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Properties of the CTFT:
Convolution & Multiplication
x(t)y(t)
Multiplication of x(t) and y(t)
Modulation of x(t) by y(t)
Example: x(t)cos(w0t)
w0 is much higher than the max. frequency of x(t)
Properties of the CTFT: Convolution &
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ope es o e C Co o u o &
Multiplication
Properties of the CTFT:
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Properties of the CTFT:
Modulation Property Example
Properties of the CTFT:
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Properties of the CTFT:
Convolution
o h(t) is the impulse response of the LTI system
)(
)()(
jX
jYjH
Properties of the CTFT: Convolution
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Properties of the CTFT: Convolution
Property Example
Properties of the CTFT: Multiplication-
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p p
Convolution Duality Proof
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Properties of the CTFT: Integration
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Properties of the CTFT: Integration
Property Example 1
o Recall: Relation unit step and impulse:
Properties of the CTFT: Integration Property
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Properties of the CTFT: Integration Property
Example 2
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Properties of the CTFT: Differentiation Property Example 2
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Properties of the CTFT: Differentiation Property Example 2
Properties of the CTFT:
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Properties of the CTFT:
Differentiation Property Example 2
Proof of Integration Property
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Proof of Integration Property
using convolution property
Properties of the CTFT: Systems Characterized by
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p y y
linear constant-coefficient differential equations
(LCCDE)
P ti f th CTFT S t Ch t i d b li
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Properties of the CTFT: Systems Characterized by linear
constant-coefficient differential equations
M
kk
k
k
N
kk
k
kdt
txdb
dt
tyda
00
)()(
)(
)()(
jX
jYjH
N
k
k
k
M
k
kk
ja
jb
jX
jYjH
0
0
)(
)(
)(
)()(
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Properties of the CTFT: Systems Characterized by
LCCDE Example 1
)()()(
txtaydt
tyd a>0
ajja
jb
jH
h(t)
N
k
kk
M
k
k
k
1
)(
)(
)(
systemthisofresponseimpulsetheFind
0
0
)()(
2
1)( tuedejHth
attj
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Properties of the CTFT: Systems
Characterized by LCCDE Example 2a
Properties of the CTFT: Systems
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Properties of the CTFT: Systems
Characterized by LCCDE Example 2b
)(2)(
)(3)(
4)(
2
2
txdt
txdty
dt
tyd
dt
tyd
313)(4)(2)(
)(
)(
)( 21
21
2
0
0
jjjj
j
ja
jb
jHN
k
k
k
M
k
k
k
)(][)(
:responseimpulsetheFind
3
21
21 tueeth tt
Properties of the CTFT: Systems
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Properties of the CTFT: Systems
Characterized by LCCDE Example 2c
)(Find;1
1)()(Let ty
jtuetx
FTt
)3()1(
2
1
1
)3)(1(
2)()()(
2
jj
j
jjj
jjXjHjY
3)1(1)( 4
1
2
21
41
jjjjY
)(][)( 341
21
41 tueteety ttt
Properties of the CTFT:
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pDuality property
Properties of the CTFT: Duality
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p y
property
Properties of the CTFT:
Duality Property Example
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Duality Property Example
P ti f th CTFT l ti
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Properties of the CTFT: relation
between duality & other properties
P ti f th CTFT
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Properties of the CTFT:Differentiation in frequency Example
)(tute at
Properties of the CTFT:
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p
Total area-Integral
P ti f th CTFT
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Properties of the CTFT:
Area Property Illustration
P ti f th CTFT
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Properties of the CTFT:
Area Property Example 1
Properties of the CTFT: Area Property Example 2
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Properties of the CTFT:
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Properties of the CTFT:
Periodic signals
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Outline
Introduction to Fourier Transform
Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform
Convergence examples
Properties of CT Fourier Transform
Fourier transform of periodic signals
Summary
Appendix Transition: CT Fourier Series to CT Fourier Transform
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Fourier transform for periodic
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signals
Note that
is the Fourier series representation of periodic signals
Fourier transform of a periodic signal is a train of
impulses with the area of the impulse at the frequencyk0equal to the k
th coefficient of the Fourier series
representation aktimes 2
Fourier transform for periodic
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Fourier transform for periodic
signals: Example 1
Fourier transform for periodic
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p
signals: Example 1
Recall: FT of square signal
Fourier transform for periodic
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p
signals: Example 2
FT of periodic time
impulse train is another
impulse train (in frequency)
Fourier Transform of Periodic
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Signals: Example 2 (details)
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Outline
Introduction to Fourier Transform
Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform
Convergence examples
Properties of CT Fourier Transform
Fourier transform of periodic signals
Summary
Appendix Transition: CT Fourier Series to CT Fourier Transform
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CTFT: Summary
Summary of CTFT Properties
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y p
Summary of CTFT Properties
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y p
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CTFT: Summary of Pairs
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O li
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Outline
Introduction to Fourier Transform Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform
Convergence examples
Fourier transform of periodic signals
Properties of CT Fourier Transform
Summary
Appendix: Transition CT Fourier Series to CT Fourier Transform
Appendix: Applications
Transition: CT Fourier Series to
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Transition: CT Fourier Series to
CT Fourier Transform
Transition: CT Fourier Series to
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Transition: CT Fourier Series to
CT Fourier Transform
Below are plots of the magnitude of X[k] for 50% and
10% duty cycles
As the period increases the sinc function widens and its
magnitude falls
As the period approaches infinity, the CT Fourier Series
harmonic function becomes an infinitely-wide sinc
function with zero amplitude (since X(k) is divided by To)
Transition: CT Fourier Series to
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Transition: CT Fourier Series to
CT Fourier Transform
This infinity-and-zero problem can be solved by normalizing the
CT Fourier Series harmonic function
Define a new modified CT Fourier Series harmonic function
Transition: CT Fourier Series to
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Transition: CT Fourier Series to
CT Fourier Transform
O tli
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Outline
Introduction to Fourier Transform
Fourier transform of CT aperiodic signals
Fourier transform examples
Convergence of the CT Fourier Transform
Convergence examples
Fourier transform of periodic signals
Properties of CT Fourier Transform
Summary
Appendix: Transition: CT Fourier Series to CT Fourier Transform Appendix: Applications
Applications of frequency- domain
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representation
Clearly shows the frequency composition a signal Can change the magnitude of any frequency
component arbitrarily by a filtering operation
Lowpass -> smoothing, noise removal
Highpass -> edge/transition detection
High emphasis -> edge enhancement
Processing of speech and music signals
Can shift the central frequency by modulation A core technique for communication, which uses
modulation to multiplex many signals into a single
composite signal, to be carried over the same physical
medium
T i l Filt
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Typical Filters
Lowpass -> smoothing, noise removal
Highpass -> edge/transition detection
Bandpass -> Retain only a certain frequency range
Low Pass Filtering
(R hi h f k i l
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(Remove high freq, make signal
smoother)
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Filtering in Temporal Domain
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g p
(Convolution)
Communication:
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Signal Bandwidth
Bandwidth of a signal is a critical feature whendealing with the transmission of this signal
A communication channel usually operates only at
certain frequency range (called channel bandwidth) The signal will be severely attenuated if it contains
frequencies outside the range of the channelbandwidth
To carry a signal in a channel, the signal needed to
be modulated from its baseband to the channelbandwidth
Multiple narrowband signals may be multiplexed touse a single wideband channel
Signal Bandwidth:
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Highest frequency estimation in a signal: Find
the shortest interval between peak and valleys
Estimation of Maximum
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Frequency
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Sample Speech Waveform 1
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Sample Speech Waveform 1
Numerical Calculation of CT
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FT The original signal is digitized, and then a Fast
Fourier Transform (FFT) algorithm is applied, which
yields samples of the FT at equally spaced intervals
For a signal that is very long, e.g. a speech signal
or a music piece, spectrogram is used.
Fourier transforms over successive overlapping short
intervals
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Sample Speech Waveform 2
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Sample Speech Waveform 2
Speech Spectrogram 2
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Speech Spectrogram 2
Sample Music Waveform
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Sample Music Waveform
Sample Music Spectrogram
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Sample Music Spectrogram