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Unformatted text preview: UCLA Spring 20092010 Systmes and Signals Lecture 11: Frequency Response of LTI Systems April 28, 2010 EE102: Systems and Signals; Spr 0910, Lee 1 Linear TimeInvariant Systems, Revisited • A linear timeinvariant system is completely characterized by its impulse response h ( t ) . • For a linear system with an input signal x ( t ) , the output is given by the convolution y ( t ) = ( x * h )( t ) = Z ∞∞ x ( τ ) h ( t τ ) dτ y ( t ) x ( t ) * h ( t ) EE102: Systems and Signals; Spr 0910, Lee 2 • The Fourier transform of the convolution is Y ( jω ) = H ( jω ) X ( jω ) where X ( jω ) is the input spectrum, Y ( jω ) is the output spectrum, and H ( jω ) is the Fourier transform of the impulse response h ( t ) . • H ( jω ) is called the frequency response or transfer function of the system. Each frequency in the input spectrum X ( jω ) is – Scaled by the system amplitude response  H ( jω )  ,  Y ( jω )  =  H ( jω )  X ( jω )  – Phase shifted by the system phase response 6 H ( jω ) , 6 Y ( jω ) = 6 H ( jω ) + 6 X ( jω ) EE102: Systems and Signals; Spr 0910, Lee 3 to produce the output spectrum Y ( jω ) . • If the input is to a system is a complex exponential e jω t , the input spectrum is X ( jω ) = F e jω t = 2 πδ ( ω ω ) . The output spectrum is Y ( jω ) = H ( jω )(2 πδ ( ω ω )) = H ( jω )(2 πδ ( ω ω )) . The ouput signal is y ( t ) = F 1 [ Y ( jω )] = F 1 [ H ( jω )(2 πδ ( ω ω ))] EE102: Systems and Signals; Spr 0910, Lee 4 = H ( jω ) e jω t =  H ( jω )  e j ( ωt + 6 H ( jω )) A sinusoidal input e jω t to an LTI system produces a sinusoidal output at the – Same frequency, – Scaled in amplitude, and – Phase shifted. This corresponds to multiplication by a complex number H ( jω ) . EE102: Systems and Signals; Spr 0910, Lee 5 Frequency Response Example An input signal x ( t ) = 2cos( t ) + 3cos(3 t/ 2) + cos(2 t ) is applied to a system with an impulse response h ( t ) h ( t ) = 2 π sinc 2 ( t/π ) Find the output signal ( x * h )( t ) . First, the frequency response or transfer function of the system is F 2 π sinc 2 ( t/π ) = 2 π π Δ( πω/ 2 π ) = 2Δ( ω/ 2) EE102: Systems and Signals; Spr 0910, Lee 6 so H ( jω ) = 2Δ( ω/ 2) The input spectrum is X ( jω ) = 2 π [ δ ( ω 1) + δ ( ω + 1)] + 3 π [ δ ( ω 3 / 2) + δ ( ω + 3 / 2)] + π [ δ ( ω 2) + δ ( ω + 2)] The output spectrum is the product of the input spectrum, and the transfer function, as shown on the next page: EE102: Systems and Signals; Spr 0910, Lee 7 ω 2 1 1 2 ω 2 1 1 2 ω 2 1 1 2 1 2 = × 3 π 4 π π 2 π π 2 π 3 π 2 π 2 π 2 π 3 π / 2 3 π / 2 4 π 2 π X ( j ω ) H ( j ω ) = 2 Δ ( ω / 2 ) Y ( j ω ) = H ( j ω ) X ( j ω ) EE102: Systems and Signals; Spr 0910, Lee 8 The output signal spectrum is then...
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This note was uploaded on 10/21/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.
 Spring '09
 Levan
 Frequency

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