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Unformatted text preview: Uniﬁed Engineering II Spring 2004 Problem S21 (Signals and Systems)
Solution:
1. The signal is plotted below:
1.2 1 g(t) 0.8 0.6 0.4 0.2 0
10 8 6 4 2 0
Time, t 2 4 6 8 10 The signal is very smooth, almost like a Gaussian. Therefore, I expect that the
duration bandwidth product will be close to the theoretical lower bound.
2.
� Δt
2 �2 � 22
t g (t) dt
=�2
g (t) dt The two integrals are easily evaluated for the given g (t). The result is �
7
t2 g 2 (t) dt =
2
�
5
g 2 (t) dt =
2 Therefore,
� 7 Δt = 2
5
3. The time domain formula for the bandwidth is
�
�2 � 2
g (t) dt
˙
Δω
=� 2
2
g (t) dt
The numerator integral is
� g 2 (t) dt =
˙ 1
2 Therefore,
2
Δω = √
5
4. The durationbandwidth product is
√
47
Δt Δω =
≈ 2.1166
5
which is very close to the theoretical lower limit of 2. This is not surprising, since
the shape of g (t) is close to a gaussian. ...
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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
 MarkDrela

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