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Continuous-Time Fourier Transform

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Content Introduction Fourier Integral Fourier Transform Properties of Fourier Transform Convolution Parseval s Theorem

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Continuous-Time Fourier Transform Introduction

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The Topic Fourier Series Fourier Series Discrete Fourier Transform Discrete Fourier Transform Continuous Fourier Transform Continuous Fourier Transform Fourier Transform Fourier Transform Continuous Time Discrete Time Periodic Aperiodic

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Review of Fourier Series Deal with continuous-time periodic signals. Discrete frequency spectra. A Periodic Signal T2T3T t f(t)f(t)

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Two Forms for Fourier Series Sinusoidal Form Complex Form:

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How to Deal with Aperiodic Signal? A Periodic Signal T t f(t)f(t) If T, what happens?

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Continuous-Time Fourier Transform Fourier Integral

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Let

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Fourier Integral F ( j ) Synthesis Analysis

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Fourier Series vs. Fourier Integral Fourier Series: Fourier Integral: Period Function Discrete Spectra Non-Period Function Continuous Spectra

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Continuous-Time Fourier Transform Fourier Transform

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Fourier Transform Pair Synthesis Analysis Fourier Transform: Inverse Fourier Transform:

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Existence of the Fourier Transform Sufficient Condition: f(t) is absolutely integrable, i.e.,

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Continuous Spectra F R (j ) F I (j ) |F(j )| ( ) Magnitude Phase

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Example 1 1 t f(t)f(t)

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Example 1 1 t f(t)f(t)

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Example t f(t)f(t) e t

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Example t f(t)f(t) e t

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Continuous-Time Fourier Transform Properties of Fourier Transform

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Notation

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Linearity Proved by yourselves

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Time Scaling Proved by yourselves

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Time Reversal Pf)

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Time Shifting Pf)

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Frequency Shifting (Modulation) Pf)

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Symmetry Property Pf) Interchange symbols and t

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Fourier Transform for Real Functions If f(t) is a real function, and F(j ) = F R (j ) + jF I (j ) F( j ) = F*(j )

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Fourier Transform for Real Functions If f(t) is a real function, and F(j ) = F R (j ) + jF I (j ) F( j ) = F*(j ) F R (j ) is even, and F I (j ) is odd. F R ( j ) = F R ( j ) F I ( j ) = F I ( j ) Magnitude spectrum |F(j )| is even, and phase spectrum ( ) is odd.

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Fourier Transform for Real Functions If f(t) is real and even F(j ) is real If f(t) is real and odd F(j ) is pure imaginary Pf) Even Real Odd Real

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Example: Sol)

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Example: d/2 1 t wd(t)wd(t) t f(t)=w d (t)cos 0 t

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Example: d/2 1 t wd(t)wd(t) t f(t)=w d (t)cos 0 t

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Example: Sol) d/2 1 t wd(t)wd(t)

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Fourier Transform of f(t) Pf)

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Fourier Transform of f (n) (t) Proved by yourselves

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Fourier Transform of f (n) (t) Proved by yourselves

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Fourier Transform of Integral Let

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The Derivative of Fourier Transform Pf)

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Continuous-Time Fourier Transform Convolution

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Basic Concept Linear System fi(t)fi(t) f o (t)=L[f i (t)] f i (t)=a 1 f i1 (t) + a 2 f i2 (t)f o (t)=L[a 1 f i1 (t) + a 2 f i2 (t)] A linear system satisfies f o (t) = a 1 L[f i1 (t)] + a 2 L[f i2 (t)] = a 1 f o1 (t) + a 2 f o2 (t)

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Basic Concept Time Invariant System Time Invariant System fi(t)fi(t) fo(t)fo(t) f i (t +t 0 ) f o (t + t 0 ) f i (t t 0 ) f o (t t 0 ) t fi(t)fi(t) t fo(t)fo(t) t fi(t+t0)fi(t+t0) t fo(t+t0)fo(t+t0) t f i (t t 0 ) t f o (t t 0 )

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Basic Concept Causal System Causal System fi(t)fi(t) fo(t)fo(t) A causal system satisfies f i (t) = 0 for t < t 0 f o (t) = 0 for t < t 0

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Basic Concept Causal System Causal System fi(t)fi(t) fo(t)fo(t) t fi(t)fi(t) t fo(t)fo(t) t0t0 t0t0 t0t0 t fi(t)fi(t) t fo(t)fo(t) t0t0 fi(t)fi(t) tt fo(t)fo(t) t0t0 t0t0 Which of the following systems are causal?

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Unit Impulse Response LTI System LTI System (t) h(t)=L[ (t)] f(t)f(t) L[f(t)]=? Facts: Convolution

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Unit Impulse Response LTI System LTI System (t) h(t)=L[ (t)] f(t)f(t) L[f(t)]=? Facts: Convolution

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Unit Impulse Response Impulse Response LTI System h(t) Impulse Response LTI System h(t) f(t)f(t) f(t)*h(t)

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Convolution Definition The convolution of two functions f 1 (t) and f 2 (t) is defined as:

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Properties of Convolution

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Impulse Response LTI System h(t) Impulse Response LTI System h(t) f(t)f(t)f(t)*h(t) Properties of Convolution Impulse Response LTI System f(t) Impulse Response LTI System f(t) h(t)h(t) h(t)*f(t)

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Properties of Convolution Prove by yourselves

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Properties of Convolution h1(t)h1(t) h1(t)h1(t) h2(t)h2(t) h2(t)h2(t) h3(t)h3(t) h3(t)h3(t) h2(t)h2(t) h2(t)h2(t) h3(t)h3(t) h3(t)h3(t) h1(t)h1(t) h1(t)h1(t) The following two systems are identical

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Properties of Convolution (t) f(t)f(t)f(t)f(t)

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Properties of Convolution (t) f(t)f(t)f(t)f(t)

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Properties of Convolution f(t)f(t) f(t T) 0T (t T) t f (t)f (t) 0 t f (t)f (t) 0T

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Properties of Convolution f(t)f(t) f(t T) 0T (t T) t f (t)f (t) 0 t f (t)f (t) 0T System function (t T) serves as an ideal delay or a copier.

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Properties of Convolution

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Time DomainFrequency Domain convolutionmultiplication

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Properties of Convolution Time DomainFrequency Domain convolutionmultiplication Impulse Response LTI System h(t) Impulse Response LTI System h(t) f(t)f(t) f(t)*h(t) Impulse Response LTI System H(j ) Impulse Response LTI System H(j ) F(j ) F(j )H(j )

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Properties of Convolution Time DomainFrequency Domain convolutionmultiplication F(j ) F(j )H 1 (j ) H 2 (j ) H 1 (j ) H 3 (j ) F(j )H 1 (j )H 2 (j ) F(j )H 1 (j )H 2 (j )H 3 (j )

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Properties of Convolution An Ideal Low-Pass Filter 0 F i (j ) 0 F o (j ) 0 H(j ) p p 1

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Properties of Convolution An Ideal High-Pass Filter 0 F i (j ) 0 H(j ) p p 1 0 F o (j )

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Properties of Convolution Prove by yourselves

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Properties of Convolution Prove by yourselves Time DomainFrequency Domain multiplicationconvolution

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Continuous-Time Fourier Transform Parseval s Theorem

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Properties of Convolution =0

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Properties of Convolution If f 1 (t) and f 2 (t) are real functions, f 2 (t) real

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Parseval s Theorem: Energy Preserving

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