EE3TP4_12_FourierSeriesMotivation_Lecture 16

# EE3TP4_12_FourierSeriesMotivation_Lecture 16 - Fourier...

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Fourier Analysis (Chapter 3) These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.

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(This chapter is for C-T case only ) 3.1 Representation in terms of frequency components i.e., sinusoids It is easy to find out how sinusoids go through an LTI system! Q: Why all this attention to sinusoids ? A: Recall “sinusoidal analysis” in RLC circuits: Fundamental Result: Sinusoid In Sinusoid Out LTI System - Section 3.1 motivates the following important idea: “signals can be built from sinusoids - Then Sections 3.2 – 3.3 take this idea further to precisely answer How can we use sinusoids to build periodic signals? - Then Section 3.4 – 3.7 take this idea even further to precisely answer How can we use sinusoids to build more-general non-periodic signals?
Why Do We Study System Response to Sinusoids? Q: How does a sinusoid go through an LTI System? Consider: h ( t ) x t = A cos ω 0 t θ y t = ? LTI: Linear , Time-Invariant To make this easier to answer we use Euler’s Formula: x t = A cos ω 0 t θ = A 2 e j ω 0 t θ A 2 e j ω 0 t θ The input is now viewed as the sum of two parts… By linearity of the system we can find the response to each part and then add them together. So we now re-form our question …

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Q: How does a complex sinusoid go through an LTI System? Consider: h ( t ) y t = ? x 1 t = A 2 e j ω 0 t θ y t = x 1 t ∗ h t = −∞ x 1 t τ h τ = −∞ A 2 e j [ ω o t τ  θ ] h τ = −∞ A 2 e j [ ω o t θ ] e o τ h τ = A 2 e j [ ω o t θ ] −∞ h τ e o τ = Δ H ω o With convolution as a tool we can now easily answer this question: Plug in our input for x ( t - τ ) Use rules for exponentials Pull out part that does not depend on variable of integration… Note that it is just x 1 ( t ) Evaluates to some complex number that depends on h ( t ) and ϖ o So … the output is just this complex sinusoidal input multiplied by some complex number!
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EE3TP4_12_FourierSeriesMotivation_Lecture 16 - Fourier...

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