Math 267
Name:
Page 2 out of 10
ProbIem 1 f 16 points)
In this problem we are given n, system with input x ( f ) and outpzit y(t), characterized by the
ODE
y"@) 3y1(f3-t 2y(t) = x ( t ) .
+
(a) Find the transfer funct,ion E ( w ) of the system. [Remember that c(u)= s ( w ) ~ ( w ) ]
(b) Find the impulse response H ( t ) of t l ~ csystem.
(c) Find the output p(t) for x ( t ) = S(t - 7).
(d) Find the orltput for x ( t ) = u(t - 4 ) e ~ ~ ( ~ + " ) .
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Math 267
Page 4 out of 10
Name:
Problem 2 (14 points) CnEculnte the Fourier transform of the following functions
(a) f ( t ) = t2e-2tu(t)
(b) g ( t ) = c o s ( 5 )
*
(c) h(t)= Qrect * rect + rect jc rect * rect rcct * rect) ( t )
Gala the Inverse M e r t r - h
2
2
of the f011mvhg functiom
Math 287
Nmnc:
Problem 4 (10points)
Lct h(t) be the function whose graph is given below:
Suppose that j ( t ) = rect(S - 112) and
~vhercA nnd R nre parameters.
In) Det,crmine the values of A and B so that h ( f )= i( f * g ) ( t ) .
(b) Find %(w)
Pnge 8 out of 10
Math 267
Page 9 out of 10
Name:
Oit
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3
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