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s13_ps11_spring04

s13_ps11_spring04 - Unied Engineering II Spring 2004...

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Unified Engineering II Spring 2004 Problem S13 (Signals and Systems) In class, you learned about a smoother , with transfer function 2 G 1 ( s ) = a ( s a )( s + a ) The smoother is an example of a low-pass filter , which means that it tends to attenuate high-frequency sine waves, but “pass” low-frequency sine waves. Unfortunately, the smoother is non-causal, which means that it can’t be implemented in real time. A similar causal low-pass filter is 2 a G 2 ( s ) = ( s + a ) 2 In this problem, you will compare these two low-pass filters, to see how they affect sinusoidal inputs. Consider an input signal u ( t ) = cos ωt 1. Find the transfer function, G 1 ( ), as a function of frequency, ω . 2. Since the transfer function is complex, it can be represented as G 1 ( ) = A 1 ( ω ) e 1 ( ω ) where the amplitude of the transfer function is A 1 ( ω ), and the phase of the trasnfer function is φ 1 ( ω ). Find A 1 ( ω ) and φ 1 ( ω ). 3. Find the transfer function, G 2 ( ), as a function of frequency, ω , as well as A 2 ( ω ) and φ 2 ( ω ). 4. For the input u ( t ) above, show that the output of the system G 1 is y 1 ( t ) = A 1 ( ω ) cos(
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