Unit6_LTI_FreqDomain

# Unit6_LTI_FreqDomain - EE3210 Signals and Systems Unit 6...

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EE3210 Signals and Systems Unit 6. Linear Time-Invariant Systems (Frequency Domain Analysis)

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Unit 6.1 Eigenfunction of LTI System After studying this unit, you will be able to 1. describe the eigenfunction of LTI systems 2. find the frequency response of an LTI system, given its impulse response
EE3210 Signal and Systems Unit 6 3 Eigenfunction of LTI System A signal for which the system output is just a constant (possibly complex) times the input is referred to as an eigenfunction of the system. H f(t) C f(t) C: constant the eigenvalue Objective The output to an input x(t) can be found easily if x(t) can be expressed as a weighted sum of the eigenfunctions .

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EE3210 Signal and Systems Unit 6 4 Is Cosine Function an Eigenfunction? Is cosine function an eigenfunction of an LTI system? Recall that an LTI system is completely characterized by its impulse response. H cos ω t Real: ?
EE3210 Signal and Systems Unit 6 5 Example for Illustration Consider this impulse response h(t): and input: cos ω t output

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EE3210 Signal and Systems Unit 6 6 Sum of Two Cosine Functions Conclusion: Adding 2 cosine waves of the same freq results in a wave of the same freq, with magnitude and phase changed. output [] ) cos( sin cos sin sin cos cos cos ) cos( cos 2 2 φω ωω θ ω + + = + = + = + + t D C t D t C t t B t A t B t A C D 1 tan = φ
EE3210 Signal and Systems Unit 6 7 Sum of Cosine Functions Apply the same argument recursively, the conclusion holds for any number of pulses Generalize to cases of continuous impulse response Conclusion: The output of cos( ω t) to an LTI results in magnitude and phase changed, but frequency unchanged .

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EE3210 Signal and Systems Unit 6 8 Complex Exponential A cosine wave is NOT an eigenfunction of LTI systems because there is a phase shift . Cosine wave is related to complex exponential: Euler’s formula: e j ω t cos ω t+ j sin t Is e j ω t an eigenfunction?
EE3210 Signal and Systems Unit 6 9 Complex Exponential as Input Let the input be a complex exponential. What is the corresponding output? Hint: Use convolution integral. h ( t ) t j e ω y ( t )?

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EE3210 Signal and Systems Unit 6 10 Corresponding Output The output is: ττ τ ωτ ω τω d e h e d e h d t x h t y j t j t j = = = ) ( ) ( ) ( ) ( ) ( ) ( ). ( of ansform Fourier tr : ) ( t h j H )) of (indep. constant complex : ) ( ( , ) ( ) ( t j H e j H t y t j =
EE3210 Signal and Systems Unit 6 11 Eigenfunction of LTI System Complex exponentials eigenfunctions of LTI systems. H(j ω ) Eigenvalue associated with the eigenfunction e j ω . Fourier transform of the impulse response, h(t). Frequency response ” of the system. a function of ω h ( t ) t j e ω t j e j H ) (

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EE3210 Signal and Systems Unit 6 12 Example 1: Exponential Delay Consider the following impulse response: What is the corresponding frequency response?
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Unit6_LTI_FreqDomain - EE3210 Signals and Systems Unit 6...

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