Continuous-Time Fourier Transform 主講者:虞台文
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) et
Example t f(t) et
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)cos0t
Example: d/2 d/2 1 t wd(t) f(t)=wd(t)cos0t
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 (tT) f(t) f(t T) t f (t) t f (t) T
Properties of Convolution System function (tT) serves as an ideal delay or a copier. Properties of Convolution T (tT) 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