Lec 11

Lec 11 - UCLA Spring 2009-2010 Systmes and Signals Lecture...

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UCLA Spring 2009-2010 Systmes and Signals Lecture 11: Frequency Response of LTI Systems April 28, 2010 EE102: Systems and Signals; Spr 09-10, Lee 1

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Linear Time-Invariant Systems, Revisited A linear time-invariant 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 ) y ( t ) x ( t ) * h ( t ) EE102: Systems and Signals; Spr 09-10, Lee 2
The Fourier transform of the convolution is Y ( ) = H ( ) X ( ) where X ( ) is the input spectrum, Y ( ) is the output spectrum, and H ( ) is the Fourier transform of the impulse response h ( t ) . H ( ) is called the frequency response or transfer function of the system. Each frequency in the input spectrum X ( ) is Scaled by the system amplitude response | H ( ) | , | Y ( ) | = | H ( ) || X ( ) | Phase shifted by the system phase response 6 H ( ) , 6 Y ( ) = 6 H ( ) + 6 X ( ) EE102: Systems and Signals; Spr 09-10, Lee 3

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to produce the output spectrum Y ( ) . If the input is to a system is a complex exponential e 0 t , the input spectrum is X ( ) = F e 0 t = 2 πδ ( ω - ω 0 ) . The output spectrum is Y ( ) = H ( )(2 πδ ( ω - ω 0 )) = H ( 0 )(2 πδ ( ω - ω 0 )) . The ouput signal is y ( t ) = F - 1 [ Y ( )] = F - 1 [ H ( 0 )(2 πδ ( ω - ω 0 ))] EE102: Systems and Signals; Spr 09-10, Lee 4
= H ( 0 ) e 0 t = | H ( 0 ) | e j ( ωt + 6 H ( 0 )) A sinusoidal input e 0 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 ( 0 ) . EE102: Systems and Signals; Spr 09-10, Lee 5

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Frequency Response Example An input signal x ( t ) = 2 cos( t ) + 3 cos (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 09-10, Lee 6
so H ( ) = 2Δ( ω/ 2) The input spectrum is X ( ) = 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 09-10, Lee 7

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ω - 2 0 - 1 1 2 ω - 2 0 - 1 1 2 ω - 2 0 - 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 09-10, Lee 8
The output signal spectrum is then Y ( ) = 2 π [ δ ( ω - 1) + δ ( ω + 1)] + 3 π 2 [ δ ( ω - 3 / 2) + δ ( ω + 3 / 2)] and the output signal is y ( t ) = 2 cos( t ) + 3 2 cos(3 t/ 2) .

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