# 2. Given the following functions, can you have the correspondinga) Fourier series,b) Fourier transform andc) 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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b help_outlineImage Transcriptionclose2. 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 fullscreen
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Step 1

From the given statement, the functions are help_outlineImage TranscriptioncloseА. 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) fullscreen
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: help_outlineImage Transcriptionclose(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 fullscreen
Step 3

Part (B)... help_outlineImage TranscriptioncloseConsider 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. fullscreen

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