 # Continuous-Time Fourier Transform

## Presentation on theme: "Continuous-Time Fourier Transform"— Presentation transcript:

Continuous-Time Fourier Transform

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

Continuous-Time Fourier Transform
Introduction

The Topic Time Discrete Fourier Periodic Series Transform Continuous
Aperiodic

Review of Fourier Series
Deal with continuous-time periodic signals. Discrete frequency spectra. A Periodic Signal T 2T 3T t f(t)

Two Forms for Fourier Series
Sinusoidal Form Complex Form:

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

Continuous-Time Fourier Transform
Fourier Integral

Fourier Integral Let

Fourier Integral F(j) Synthesis Analysis

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

Continuous-Time Fourier Transform

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

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

Continuous Spectra FR(j) FI(j) |F(j)| () Magnitude Phase

Example 1 -1 t f(t)

Example 1 -1 t f(t)

Example t f(t) et

Example t f(t) et

Continuous-Time Fourier Transform
Properties of Fourier Transform

Notation

Linearity Proved by yourselves

Time Scaling Proved by yourselves

Time Reversal Pf)

Time Shifting Pf)

Frequency Shifting (Modulation)
Pf)

Symmetry Property Pf) Interchange symbols  and t

Fourier Transform for Real Functions
If f(t) is a real function, and F(j) = FR(j) + jFI(j) F(j) = F*(j)

Fourier Transform for Real Functions
If f(t) is a real function, and F(j) = FR(j) + jFI(j) F(j) = F*(j) FR(j) = FR(j) FI(j) = FI(j) FR(j) is even, and FI(j) is odd. Magnitude spectrum |F(j)| is even, and phase spectrum () is odd.

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

Example: Sol)

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

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

d/2 d/2 1 t wd(t) Example: Sol)

Fourier Transform of f’(t)
Pf)

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

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

Fourier Transform of Integral
Let

The Derivative of Fourier Transform
Pf)

Continuous-Time Fourier Transform
Convolution

Basic Concept fi(t) fo(t)=L[fi(t)] Linear System
fi(t)=a1 fi1(t) + a2 fi2(t) fo(t)=L[a1 fi1(t) + a2 fi2(t)] A linear system satisfies fo(t) = a1L[fi1(t)] + a2L[fi2(t)] = a1fo1(t) + a2fo2(t)

Basic Concept fo(t) fi(t) Time Invariant System fi(t +t0) fo(t + t0)

Basic Concept fo(t) fi(t) Causal System A causal system satisfies
fi(t) = 0 for t < t0 fo(t) = 0 for t < t0

Basic Concept fo(t) fi(t) Causal System
Which of the following systems are causal? Basic Concept Causal System fi(t) fo(t) t fi(t) fo(t) t0 t0 t fi(t) fo(t) fi(t) t fo(t) t0

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

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

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

Convolution Definition
The convolution of two functions f1(t) and f2(t) is defined as:

Properties of Convolution

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

Properties of Convolution
Prove by yourselves

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

Properties of Convolution
f(t)

Properties of Convolution
f(t)

Properties of Convolution
T (tT) f(t) f(t T) t f (t) t f (t) T

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

Properties of Convolution

Properties of Convolution
Time Domain Frequency Domain convolution multiplication Properties of Convolution

Properties of Convolution
Time Domain Frequency Domain convolution multiplication Properties of Convolution Impulse Response LTI System h(t) f(t) f(t)*h(t) Impulse Response LTI System H(j) F(j) F(j)H(j)

Properties of Convolution
Time Domain Frequency Domain convolution multiplication Properties of Convolution F(j)H1(j) F(j)H1(j)H2(j)H3(j) H1(j) H2(j) H3(j) F(j) F(j)H1(j)H2(j)

Properties of Convolution
H(j) p p 1 Fo(j) Fi(j) An Ideal Low-Pass Filter

Properties of Convolution
H(j) p p 1 Fo(j) Fi(j) An Ideal High-Pass Filter

Properties of Convolution
Prove by yourselves

Properties of Convolution
Time Domain Frequency Domain multiplication convolution Properties of Convolution Prove by yourselves

Continuous-Time Fourier Transform
Parseval’s Theorem

Properties of Convolution
=0

Properties of Convolution
If f1(t) and f2(t) are real functions, f2(t) real

Parseval’s Theorem: Energy Preserving