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DR. SIGIT PW JAROT

ECE 2221

Continuous-Time Signal AnalysisFOURIER Transform - Applications

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

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.

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:

Course Synopsis

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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

Signals and Systems

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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

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

Partitioning a Complex Problem into Simpler Problems

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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.

Three requirements

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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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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.

Why we need frequency domain representation ?

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In many cases it is much easier to analyze the “frequency content” of a signal.

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.

Fourier Transform Table (1)

Fourier Transform Table (2)

Fourier Transform Table (3)

Linearity Property

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Time-Frequency Duality of Fourier Transform

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Duality Property

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Duality Property: Example

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Duality Property ExampleConsider the FT of a rectangular function:

Scaling Property

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Time-shifting Property

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Time-shifting Property: Example

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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

Frequency-shifting Property

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Frequency-Shifting Property : example

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Find and sketch the Fourier transform of the signal x(t)cos10t, where x(t)=rect(t/4)

Convolution Property

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Convolution Property: proof

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Convolution Property: example

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Find and sketch the Fourier transform of the signal x(t)cos10t, where x(t)=rect(t/4), using convolution property

Time-differentiation Property

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Signal Transmission through LTI Systems

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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

Example

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Time-domain vs. Frequency-domain

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Signal Distortion during transmission

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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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Parseval’s Theorem

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Energy Spectral Density of a Signal

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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:

Example:

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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.

Filter

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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

Filter

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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

Ideal Filters

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Has transfer function, H() that pass the frequency component of input signal within the passband and eliminate those outside it.

Ideal Filters

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PASSBAND = unity magnitude frequency response

STOPBAND = zero frequency response

Ideal Filters

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The output of the filter consist only the frequency components of input signal that are within the passband

Ideal Low-pass Filters

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0

H(j)

C-C

Passband StopbandStopband

Ideal High-pass Filters

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0

H(j)

C-C

PassbandStopbandPassband

Ideal Bandpass Filters

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0

H(j)

1-1

PassbandStopbandPassbandStopband Stopband

2-2

Ideal Bandstop Filters

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H(j)

1-1

PassbandStopbandStopbandPassband Passband

-2 2

Example

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Classify each of these transfer functions as having a lowpass, highpass, bandpass or bandstop frequency response

Lowpass

Example

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Bandpass

Bandstop

Example

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Bandpass

Bandpass

Example

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Bandstop

Ideal Lowpass Filter

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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

Impulse Response and Causality

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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

Impulse Responses and Frequency Responses of Real Causal Filters

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Impulse Responses and Frequency Responses of Real Causal Filters

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Real Filters-RC Lowpass Filter

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H j Vout j Vin j

Zc j

Zc j ZR j

1jRC 1

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Time Domain vs. Frequency Domain

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t

0

x (t) = rect (t)

1-1

Time duration/ Time interval

x (f) = sinc (f)

f

Bandwidth

F

Absolute Bandwidth

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f1 f2- f1- f2

| X(f)|

fB

3dB (or Half-Power) Bandwidth

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f1 f2- f1- f2

|H(f0 )|

f

f1- f1 0f

|H(f0 )|2

|H(f )|

|H(f0 )|

2

B

Null-to-null (or Zero Crossing) Bandwidth

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f1 f2- f1- f2

| X(f)|

f

0f

B

Bf1