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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. Uniﬁed Engineering II Spring 2004 Problem S22 (Signals and Systems)
Solution:
I used Mathematica to ﬁnd some of the integrals, although you could use tables or
integrate by parts.
(a)
¯
t= � ∞ � 2 t7 e−2t/tau dt = 315 8
τ
16 t6 e−2t/tau dt = tg (t) dt = 45 7
τ
8 0 ¯
t= � � 2 ∞ g (t) dt =
0 Therefore,
¯7
t= τ
2
(b)
� ¯
(t − t)2 g 2 (t) dt = Therefore, Δt = √ 315 9
τ
32 7 τ (c)
� 9
g 2 (t) dt = τ 5
˙
8 Therefore, 2
Δω = √
5τ
(d) The durationbandwidth product is
�
Δt Δω = 2 7
≈ 2.366
5 which compares favotably with the theoretical lower bound Δt Δω ≥ 2 ...
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 Fall '05
 MarkDrela

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