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Asked Dec 11, 2019
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b

2. Given the following functions, can you have the corresponding
a) Fourier series,
b) Fourier transform and
c) Laplace transform? If yes, find them, if not, explain why you cannot.
A. x(t) = -1+cos(2t) + sin(at+1)
B. x(t) = 28(t) cos(2t) +8(t-1.57t) sin(2t)
C. x(t) = 1+cos(1.5t) + cos(4t)
%3D
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2. Given the following functions, can you have the corresponding a) Fourier series, b) Fourier transform and c) Laplace transform? If yes, find them, if not, explain why you cannot. A. x(t) = -1+cos(2t) + sin(at+1) B. x(t) = 28(t) cos(2t) +8(t-1.57t) sin(2t) C. x(t) = 1+cos(1.5t) + cos(4t) %3D

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Expert Answer

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Step 1

From the given statement, the functions are

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А. х(t) 3—1+ cos(2t) + sin (zt + +1) В. (1)-26(г)сos(2) + 8(г-15л)sin (2:) С. х(г) -1+ сos (1.5t) + сos(4t)

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Step 2

Definition used:

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.

The Laplace transforms a function of real variable t to a function of a complex variables s.

Calculation:

Note that, the range and period is not given in the functions.

So we can’t find Fourier series in this problem.

Calculation of Laplace transforms:

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(A) Consider x(t) =-1cos(2t)+ sin( tt +1). x(t) =-1cos(2t) + sin( at)cos(1) +cos ( at) sin(1) Use Laplace transform on both sides, L{ x(t)} = L{-1}+ L{cos(2t)}+ L{sin (7t)cos (1)} + L{cos(rt)sin (1)} :-L{1}+ L{cos(2t)}+ L{sin ( rt)cos(1)}+L{cos(zt)sin (1)} + sin(1)- + cos (1)- 2. s´ +2? cos(1)7 + sin(1)s s + T s? +4

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Step 3

Part (B)...

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Consider x(t) = 28(t)cos(2t)+8(t-1.57)sin(2t). 3 -T sin| 2t x(t) = 28(t)cos(2t) + 8 t- 3 Known that, L{f(t-a)u(t-a)}=e*f(s). On further it cannot possible to apply Laplace transform.

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