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Unformatted text preview: Uniﬁed Engineering II Spring 2004 Problem S13 (Signals and Systems)
In class, you learned about a smoother, with transfer function
G1 (s) = −a2 (s − a)(s + a) The smoother is an example of a lowpass ﬁlter, which means that it tends to attenuate
highfrequency sine waves, but “pass” lowfrequency sine waves. Unfortunately, the
smoother is noncausal, which means that it can’t be implemented in real time. A
similar causal lowpass ﬁlter is
G2 (s) = a2
(s + a)2 In this problem, you will compare these two lowpass ﬁlters, to see how they aﬀect
sinusoidal inputs. Consider an input signal
u(t) = cos ω t
1. Find the transfer function, G1 (jω ), as a function of frequency, ω .
2. Since the transfer function is complex, it can be represented as
G1 (jω ) = A1 (ω )ejφ1 (ω)
where the amplitude of the transfer function is A1 (ω ), and the phase of the trasnfer
function is φ1 (ω ). Find A1 (ω ) and φ1 (ω ).
3. Find the transfer function, G2 (jω ), as a function of frequency, ω , as well as A2 (ω )
and φ2 (ω ).
4. For the input u(t) above, show that the output of the system G1 is
y1 (t) = A1 (ω ) cos(ωt + φ1 (ω ))
and do likewise for system G2 .
5. A1 and A2 determine how much the magnitude of the input cosine wave is aﬀected
by each ﬁlter. Ideally, A1 and A2 would be 1, meaning that the ﬁlters don’t
change the magnitude of the input sine at all. Which ﬁlter (if either) changes the
magnitude the least?
6. φ1 and φ2 determine how much the phase of the input cosine wave is aﬀected
by each ﬁlter. Nonzero values of φ correspond to a shifting left or right (i.e.,
advancing or delaying) the sine wave. Ideally, φ1 and φ2 would be zero, meaning
that the ﬁlters don’t change the phase of the input sine at all. Which ﬁlter (if
either) produces the least phase shift? 7. Explain why the noncausal ﬁlter is preferred in signal processing applications
where it can be applied. ...
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.
- Fall '05