EE3TP4_18_CTPeriodicSignalResponse_v2_Lecture 26

# EE3TP4_18_CTPeriodicSignalResponse_v2_Lecture 26 - 5.2...

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Unformatted text preview: 5.2 Response to Periodic Inputs h ( t ) H (ω) periodic x ( t ) y ( t ) = ? Since x ( t ) is periodic, write it as Fourier Series: x t = ∑ k =−∞ ∞ c k x e jk ω t H (ω) c k x e jk ω t H ω k c k x e jk ω t ( complex : magnitude & phase) Sum these to get output So, the input is a sum of terms Linear System: So… Output = Sum of Individual Responses But each individual response is to a complex sinusoid input ⇒ EASY! Sum these to get input x t = ∑ k =−∞ ∞ c k x e jk ω t y t = ∑ k =−∞ ∞ [ H ω k c k x ] e jk ω t FS coefficient of y ( t ) Indicates “for x ( t )” General Insights from this Analysis 1. periodic in, periodic out for LTI System 2. The system’s frequency response H (ω) works to modify the input FS coefficients to create the output FS coefficients: c k y = H kω c k x Example (Ex. 5.4 with Some Injected Reality) Problem: suppose you have a circuit board that has a digital clock circuit on it. It makes the rectangular pulse train shown below: . . . . . ....
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EE3TP4_18_CTPeriodicSignalResponse_v2_Lecture 26 - 5.2...

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