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Chapter 5 The Fourier Transform
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Basic Idea We covered the Fourier Transform which to represent periodic signals We assumed periodic continuous signals We used Fourier Series to represent periodic continuous time signals in terms of their harmonic frequency components (Ck). We want to extend this discussion to find the frequency spectra of a given signal
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Basic Idea The Fourier Transform is a method for representing signals and systems in the frequency domain We start by assuming the period of the signal is T= INF All physically realizable signals have Fourier Transform For aperiodic signals Fourier Transform pairs is described as Fourier Transforms of f(t) Inverse Fourier Transforms of F(w) Remember: notes
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Example – Rectangular Signal
Compute the Fourier Transform of an aperiodic rectangular pulse of T seconds evenly distributed about t=0. Remember this the same rectangular signal as we worked before but with T0 infinity! V -T/2 T/2 notes All physically realizable signals have Fourier Transforms
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Fourier Transform of Unit Impulse Function
Example: Plot magnitude and phase of f(t)
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Fourier Series Properties
Make sure how to use these properties!
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Fourier Series Properties - Linearity
Find F(w)
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Fourier Series Properties - Linearity
Due to linearity
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Fourier Series Properties - Time Scaling
rect(t/T) rect(t/(T/2)) Due to Time Scaling Property Remember: sinc(0)=1; sinc(2pi)=0=sinc(pi)
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Fourier Series Properties - Duality or Symmetry
Example: Find the time-domain waveform for Arect(w/2B) Refer to FTP Table Remember we had: FTP: Fourier Transfer Pair
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Fourier Series Properties - Duality or Symmetry
Example: find the frequency response Of y(t)
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Fourier Series Properties - Duality or Symmetry
Example: find the frequency response Of y(t) We know Using Fourier Transform Pairs Using duality
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Fourier Series Properties - Convolution
Proof Proof
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Fourier Series Properties - Convolution
Example: Find the Fourier Transform of x(t)=sinc2(t) In this case we have B=1, A=1 w w X1(w) X2(w) Refer to Schaum’s Prob. 2.6
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Fourier Series Properties - Convolution
Example: Find the Fourier Transform of x(t)= sinc2(t) sinc(t) We need to find the convolution of a rect and a triangle function: w Refer to Schaum’s Prob. 2.6
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Fourier Series Properties - Frequency Shifting
Example: Find the Fourier Transform of g3(t) if g1(t)=2cos(200pt), g2(t)=2cos(1000pt); g3(t)=g1(t).g2(t) ; that is [G3(w)] Remember: cosa . cosb=1/2[cos(a+b)+cos(a-b)]
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Fourier Series Properties - Time Differentiation
Example:
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More… Read your notes for applications of Fourier Transform.
Read about Power Spectral Density Read about Bode Plots
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Schaums’ Outlines Problems
Do problems Do problems 5.4, 5.5, , 5.8, 5.9, 5.10, 5.14 Do problems in the text
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