e the output of a real LTI system is a cosine at the same frequency as the

E the output of a real lti system is a cosine at the

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i.e. the output of a real LTI system is a cosine at the same frequency as the input ( ω 0 ), with scaling and phase shift determined by the magnitude and phase of the frequency response. EE 224, #14 7
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Input Consisting of Multiple Complex Sinusoids Suppose that we can express a CT signal x ( t ) as x ( t ) = X k a k e j ω k t ω k in rad/s . Then, the output y ( t ) can be easily obtained as y ( t ) = T { x ( t ) } = X k a k H F ( ω k ) e j ω k t . EE 224, #14 8
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Example: RC Circuit Calculate the frequency response of an RC circuit. We obtained the impulse response of this RC circuit in handout #12 [see (8) in handout #12]: h ( t ) = 1 R C e - t/ ( R C ) u ( t ) = α e - α t u ( t ) where α = 1 / ( R C ) . The circuit’s frequency response is then H F ( ω ) = Z + -∞ e - j ω τ h ( τ ) = Z + -∞ e - j ω τ α e - α τ u ( τ ) = α Z + 0 e - ( α + j ω ) τ = α e - ( α + j ω ) τ α + j ω 0 τ =+ = α α + j ω . EE 224, #14 9
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An alternative approach for computing the frequency response H F ( ω ) . By Kirchoff’s voltage law, x ( t ) = R i ( t ) + y ( t ) . Using i ( t ) = C y 0 ( t ) , we have y 0 ( t ) + α y ( t ) = α x ( t ) (4) which is an LCCD equation. Now, consider the input x ( t ) = e j ω t . Since the system is linear, y ( t ) = H F ( ω ) e j ω t y 0 ( t ) = H F ( ω ) j ω e j ω t . Substitute the above into (4): H F ( ω ) j ω e j ω t + α H F ( ω ) e j ω t = α e j ω t cancel e j ω t and solve for H F ( ω ) : H F ( ω ) = α α + j ω .
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