Unit5_FourierTransform - EE3210 Signals and Systems Unit 5....

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EE3210 Signals and Systems Unit 5. Fourier Transform (Continuous-Time and Discrete-Time)
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EE3210 Signal and Systems Unit 5 2 Objective Fourier series can be used to represent periodic signals. Q: How to represent aperiodic signals? A: Fourier Transform. Note: Fourier Transform is more general and can be used to represent periodic signals as well. Q: Why to study this? A: Fourier series and Fourier transform can be used to analyze LTI systems. See next unit.
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EE3210 Signal and Systems Unit 5 3 Outline 1. What is Fourier Transform? 2. Properties of Fourier Transform 3. Discrete-Time Fourier Transform Continuous- Time
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Unit 5.1 What is Fourier Transform? After studying this unit, you will be able to 1. describe the definition of Fourier transform 2. find the Fourier transform of some simple signals
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EE3210 Signal and Systems Unit 5 5 Definition Fourier Transform Pair: () ( ) j t X j xte d t ω −∞ = = ωω π d e j X t x t j ) ( 2 1 ) (
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EE3210 Signal and Systems Unit 5 6 Relation with Fourier Series Consider an aperiodic signal, x(t): Construct from it a periodic signal with period T: 0 x ( t ) . . . 0 T 2T ) ( ~ t x T . . .
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EE3210 Signal and Systems Unit 5 7 Limiting Case What happens to the Fourier series of when T becomes larger and larger? ). ( ) ( ~ then , let we If t x t x T T 0 x ( t ) ) ( ~ t x T
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EE3210 Signal and Systems Unit 5 8 Fourier Series Fourier series: The last equality follows from the definition of X(j ω ). −∞ = −∞ = = = = k t jk k t jk k T e jk X T T e a t x 0 0 ) ( 1 ) 2 ( , ) ( ~ 0 0 ω π
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EE3210 Signal and Systems Unit 5 9 The Effect of Doubling T We plot the F.S. coefficients, a k , against ω≡ k ω 0 , where What will happen if we double the value of T? Then ω 0 is halved. There are now twice as many components in the spectrum. The line spectrum becomes denser and denser. Also, the envelope X ( jk 0 )/ T is halved. ) ( 0 T jk X a k =
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EE3210 Signal and Systems Unit 5 10 The Effect of Doubling T -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 Continuing the process, the spectrum becomes denser while its magnitude becomes smaller. The shape of the envelope remains the same.
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EE3210 Signal and Systems Unit 5 11 X(j ω ): spectrum ” of x(t), with information concerning how x(t) is composed of sinusoids of different frequencies.
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Unit5_FourierTransform - EE3210 Signals and Systems Unit 5....

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