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Transcript of Continuous-Time Signal Analysisstaff.iium.edu.my/sigit/ece2221/pdf/08 Fourier Transform... ·...
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DR. SIGIT PW JAROT
ECE 2221
Continuous-Time Signal AnalysisFOURIER Transform - Applications
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Inspiring Message from Imam Shafii
Dr. Sigit PW JarotECE 2221 Signals and Systems2
Intelligence
Strong Will
Diligence
Patience
Sufficient Means
Befriend Your Teacher
You will not acquire knowledge unless you have 6 (SIX) THINGS
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Course Objectives
Dr. Sigit PW JarotECE 2221 Signals and Systems3
To provide an analysis of the continuous-time signals and systems as reflected to their roles in engineering practice.
To expose students to both the time-domain and frequency-domain methods of analyzing signals and systems.
To illustrate the potential applications of this course as a Pre-requisite course to communication engineering and principles, digital signal processing and control system.
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OBE (Outcome Based Education)Learning Outcomes
Dr. Sigit PW JarotECE 2221 Signals and Systems4
Acquire intuitive and heuristic understanding of the concepts of signals and systems, and the physical meaning of the mathematical representation.
Analyze continuous-time signals and systems in time domain.
Analyze continuous-time signals and systems in frequency domain.
Acquire introductory-level knowledge of discrete-time signals and systems, and sampling theory.
After completion of this course the students will be able to:
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Course Synopsis
Dr. Sigit PW JarotECE 2221 Signals and Systems5
Introduction to Signals
Introduction to Systems
Time-Domain Analysis of Continuous-Time Systems
Frequency-Domain System Analysis: the Laplace Transform
MID-TERM Examination
Signals Analysis using the Fourier Series
Signals Analysis using the Fourier Transform
Introduction to Discrete Time Signals and Systems Analysis
FINAL Examination
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Signals and Systems
Dr. Sigit PW JarotECE 2221 Signals and Systems6
SIGNAL A set of data or information. A function or sequence of values that represents information. A function of one or more variables (e.g. time, frequency, space,..) that conveys
information on the nature of a physical phenomenon.
SYSTEM A system is an entity that processes a set of signals (inputs) to yield another
sets of signals (outputs)
SystemInputSignal
OutputSignal
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Outline Basic [7.1 – 7.2] Motivation for using Fourier Transform Aperiodic Signal Representation by Fourier Integral Transforms of Some useful Functions
More about FT [7.3-7.4] Some Properties of the Fourier Transform Signal Transmission through LTI Systems
Applications [7.5 – 7.8] Ideal and Practical Filters Application to Communications Windowing
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Partitioning a Complex Problem into Simpler Problems
DR Sigit JarotECE2221 Signals and Systems8
A common engineering technique is the partitioning of complex problems into simpler ones.
The simpler problem are then solved, and the total solution becomes the sum of the simpler solutions.
We may have insight into the simpler problems and can thus gain insight into the complex problems.
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Three requirements
DR Sigit JarotECE2221 Signals and Systems9
1. The problem can be expressed as a number of simpler problems.
2. The problem must be linear, such that the solution for the sum of function is equal to the sum of the solutions considering only one function at a time.
3. The contributions of the simpler solutions to the total solution must become negligible after considering a few terms. adequate acuracy
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DR Sigit JarotECE2221 Signals and Systems10
As engineers we must never lose sight of the fact that the mathematics that we employ is a means to an end and is not the end itself.
Engineers apply mathematical procedures to the analysis and design of physical systems.
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Why we need frequency domain representation ?
DR Sigit JarotECE2221 Signals and Systems11
In many cases it is much easier to analyze the “frequency content” of a signal.
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Why bother with the Fourier transform?There are certain simple time functions which are more
readily represented by Fourier transforms than by Laplace transforms, e.g., x(t) = 1, x(t) = cos(2πft), periodic time functions, etc.
Certain important operations on signals are more readily analyzed with Fourier transforms, e.g., sampling, modulation, filtering.
Examination of both signals and systems in the frequency domain gives insights that complement those obtained in the“time” domain.
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Fourier Transform Table (1)
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Fourier Transform Table (2)
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Fourier Transform Table (3)
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Linearity Property
Dr. Sigit PW JarotECE 2221 Signals and Systems16
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Time-Frequency Duality of Fourier Transform
Dr. Sigit PW JarotECE 2221 Signals and Systems17
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Duality Property
Dr. Sigit PW JarotECE 2221 Signals and Systems18
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Duality Property: Example
Dr. Sigit PW JarotECE 2221 Signals and Systems19
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Duality Property ExampleConsider the FT of a rectangular function:
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Scaling Property
Dr. Sigit PW JarotECE 2221 Signals and Systems21
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Time-shifting Property
Dr. Sigit PW JarotECE 2221 Signals and Systems22
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Time-shifting Property: Example
Dr. Sigit PW JarotECE 2221 Signals and Systems23
Find the Fourier transform of the gate pulse x(t) given by:
This pulse is rect(t/τ) delayed by 3τ/4 sec.
Use time-shifting theorem, we get
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Frequency-shifting Property
Dr. Sigit PW JarotECE 2221 Signals and Systems24
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Frequency-Shifting Property : example
Dr. Sigit PW JarotECE 2221 Signals and Systems25
Find and sketch the Fourier transform of the signal x(t)cos10t, where x(t)=rect(t/4)
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Convolution Property
Dr. Sigit PW JarotECE 2221 Signals and Systems26
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Convolution Property: proof
Dr. Sigit PW JarotECE 2221 Signals and Systems27
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Convolution Property: example
Dr. Sigit PW Jarot ECE 2221 Signals and Systems 28
Find and sketch the Fourier transform of the signal x(t)cos10t, where x(t)=rect(t/4), using convolution property
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Time-differentiation Property
Dr. Sigit PW JarotECE 2221 Signals and Systems29
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DR Sigit JarotECE2221 Signals and Systems30
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Signal Transmission through LTI Systems
Dr. Sigit PW JarotECE 2221 Signals and Systems31
We have seen previously that if x(t) and y(t) are input & output of a LTI system with impulse response h(t), then
We can therefore perform LTI system analysis with Fourier transform in a way similar to that of Laplace transform.
However, FT is more restrictive than Laplace transform because the system must be stable, and x(t) must itself by Fourier transformable.
Laplace transform can be used to analyse stable AND unstable system, and apply to signals that grow exponentially.
If a system is stable, it can shown that the frequency response of the system H(jω) is just the Fourier transform of h(t) (i.e. H(ω)):
Y(ω)=H(ω) X(ω)
H(ω)=H(s)|s=j ω
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Example
Dr. Sigit PW JarotECE 2221 Signals and Systems32
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Time-domain vs. Frequency-domain
Dr. Sigit PW JarotECE 2221 Signals and Systems33
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Signal Distortion during transmission
Dr. Sigit PW JarotECE 2221 Signals and Systems34
QUESTION: What is the characteristic of a system that allows signal to pass without distortion?
Transmission is distortionless if output is identical to input within a multiplicative constant, and relative delay is allowed. That is:
But Y(ω)/X(ω) = H(ω), therefore the frequency characteristic of a distortionless system is:
For distortionless transmission, amplitude response |H(ω)|must be a constant AND phase response ∠H(ω) must be linear function of ω with slope –td .
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Dr. Sigit PW JarotECE 2221 Signals and Systems35
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Parseval’s Theorem
Dr. Sigit PW JarotECE 2221 Signals and Systems36
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Energy Spectral Density of a Signal
Dr. Sigit PW JarotECE 2221 Signals and Systems37
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Dr. Sigit PW JarotECE 2221 Signals and Systems38
If x(t) is a real signal, then X(ω) and X(-ω) are conjugate :
This implies that X(ω) is an even function. Therefore
Consequently, the energy contributed by a real signal by spectral components between ω1 and ω2 is:
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Example:
Dr. Sigit PW JarotECE 2221 Signals and Systems39
Find the energy E of signal x(t) = e-at u(t). Determine the frequency W (rad/s) so that the energy contributed by the spectral component from 0 to W is 95% of the total signal energy E.
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Filter
DR Sigit JarotECE2221 Signals and Systems40
A filter separates the wanted part of a signal from the useless part
In the signal and system analysis, it is a device which separates the signal in one frequency range from the signal in another frequency range
An ideal filter passes all signal power in its passband without distortion and completely blocks signal power outside its passband
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Filter
DR Sigit JarotECE2221 Signals and Systems41
Has transfer function, H() that allows/passed frequency component of input signal within the passband and eliminate the frequency component of input signal within the stopband
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Ideal Filters
DR Sigit JarotECE2221 Signals and Systems42
Has transfer function, H() that pass the frequency component of input signal within the passband and eliminate those outside it.
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Ideal Filters
DR Sigit JarotECE2221 Signals and Systems43
PASSBAND = unity magnitude frequency response
STOPBAND = zero frequency response
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Ideal Filters
DR Sigit JarotECE2221 Signals and Systems44
The output of the filter consist only the frequency components of input signal that are within the passband
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Ideal Low-pass Filters
DR Sigit JarotECE2221 Signals and Systems45
0
H(j)
C-C
Passband StopbandStopband
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Ideal High-pass Filters
DR Sigit JarotECE2221 Signals and Systems46
0
H(j)
C-C
PassbandStopbandPassband
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Ideal Bandpass Filters
DR Sigit JarotECE2221 Signals and Systems47
0
H(j)
1-1
PassbandStopbandPassbandStopband Stopband
2-2
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Ideal Bandstop Filters
DR Sigit JarotECE2221 Signals and Systems48
H(j)
1-1
PassbandStopbandStopbandPassband Passband
-2 2
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Example
DR Sigit JarotECE2221 Signals and Systems49
Classify each of these transfer functions as having a lowpass, highpass, bandpass or bandstop frequency response
Lowpass
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Example
DR Sigit JarotECE2221 Signals and Systems50
Bandpass
Bandstop
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Example
DR Sigit JarotECE2221 Signals and Systems51
Bandpass
Bandpass
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Example
DR Sigit JarotECE2221 Signals and Systems52
Bandstop
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Ideal Lowpass Filter
DR Sigit JarotECE2221 Signals and Systems53
The impulse response for this ideal filter implies a noncausal system, it begins long before the impulse occurs at t = 0
C C0
1
C
rectH2
)(
tth CC sinc)(F-1
t
Frequency response Impulse response
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Impulse Response and Causality
DR Sigit JarotECE2221 Signals and Systems54
All the impulse responses of ideal filters are sinc functions, or related functions, which are infinite in extent
Therefore all ideal filter impulse responses begin before time, t = 0
This makes ideal filters non-causal Ideal filters are not physically realizable
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Impulse Responses and Frequency Responses of Real Causal Filters
DR Sigit JarotECE2221 Signals and Systems55
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Impulse Responses and Frequency Responses of Real Causal Filters
DR Sigit JarotECE2221 Signals and Systems56
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Real Filters-RC Lowpass Filter
DR Sigit JarotECE2221 Signals and Systems57
H j Vout j Vin j
Zc j
Zc j ZR j
1jRC 1
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DR Sigit Jarot ECE2221 Signals and Systems 58
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Time Domain vs. Frequency Domain
DR Sigit JarotECE2221 Signals and Systems59
t
0
x (t) = rect (t)
1-1
Time duration/ Time interval
x (f) = sinc (f)
f
Bandwidth
F
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Absolute Bandwidth
DR Sigit Jarot ECE2221 Signals and Systems 60
f1 f2- f1- f2
| X(f)|
fB
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3dB (or Half-Power) Bandwidth
DR Sigit Jarot ECE2221 Signals and Systems 61
f1 f2- f1- f2
|H(f0 )|
f
f1- f1 0f
|H(f0 )|2
|H(f )|
|H(f0 )|
2
B
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Null-to-null (or Zero Crossing) Bandwidth
DR Sigit Jarot ECE2221 Signals and Systems 62
f1 f2- f1- f2
| X(f)|
f
0f
B
Bf1