ENEE 322
Sections 0201 & 0202
Spring 2016
Homework #9
1) Consider the RLC circuit pictured in Figure P3.20 (page 254) in Oppenheim & Willsky, except that R, L,
and C are unknown (i.e. keep them as “R”, “L”, and “C”).
a) Using an input signal of V0 exp( jωt) , and knowing that because the RLC circuit is an LTI system the
output signal will be H ( jω )V0 exp( jωt) , derive an expression for the transfer function H( jω ) .
b) For the values of R, L, and C in the book ( R = 1Ω, L = 1H, C = 1F ), calculate H ( jω ) : the magnitude
of H( jω ) . Sketch it (by hand) for 0 ≤ ω < 3 (i.e. Make sure the beginning and end values are right
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and get the general trend in between). Is this a low-pass, band-pass, or high-pass filter?
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€ For each of the following signals
2)
x(t) , where a and L are unknown constants, sketch the signal and
compute its Fourier Transform X( jω ) :
a) x(t) = a ( u ( t ) − u ( t − L ))
b) x(t) = a ( u ( t + L ) − u ( t ))
(
c) x(t) = a u ( t + L 2 ) − u ( t − L 2 )
)
d) x(t) = a u ( t + L ) + a u ( t + L 2 ) − a u ( t − L 2 ) − a u ( t − L )
Note For problems 3 & 4, you may find this relation useful:
∞
∞
π exp(−b 2 4a 2 )
a
∫ exp ( jbx ) exp ( −a x ) dx = 2 ∫ cos(bx) exp ( −a x ) dx =
2
2
2
2
0
−∞
(
)
2
3) Consider the Gaussian signal x(t) = exp −t 2 2σ 2 . [ σ is called the variance and parameterizes the
width of the bell curve.]
a) Sketch x(t) for the range [–2, +2] if σ = 2 .
b) Compute its Fourier Transform X( jω ) .
c) Where is the maximum of X( jω ) ? How does X( jω ) fall off as ω → ∞ ?
(
)
4) Consider the related signal x(t) = exp jω 0t −t 2 2σ 2 .
a) Sketch Re ( x(t )) for the range [–2, +2] if σ = 2 , ω 0 = 4π .
b) Sketch Re ( x(t )) for the range [–2, +2] if σ = 2 , ω 0 = 8π .
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c) Compute its Fourier Transform X( jω ) for general σ and ω 0 .
d) Where is the maximum of X( jω ) ? How does X( jω ) fall off as ω → ∞ ?
e) Describe in a few words the difference between this X( jω ) and the X( jω ) in the previous problem.
Describe in a few words the difference between this x(t) and the x(t) in the previous problem.
5) Consider the signal x(t) = A sinc(4 f0 t) :
a) Sketch x(t) for the range [-1 f0 , +1 f0 ] .
b) Compute its Fourier Transform X( jω ) . Hint There is a hard way to do it, and an easy way to do it.
c) Sketch X( jω ) for the range [−8π f0 , +8π f0 ]
6) Compute the Fourier Transform X( jω ) for:
a) x(t) = 3sin(2π t 5)
b) x(t) = 3sin(2π t 5) + 1
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