Question #85505
Determine the Fourier transform of the function
f(t)=1-t. 0 ≤t≤1
1+t. -1≤t≤0
0. Otherwise
1
Expert's answer
2019-03-02T11:12:16-0500
f^(k)=Rf(t)e2πiktdt\hat{f}(k)=\int\limits_{\mathbb{R}}f(t)e^{-2\pi i kt} \, dt

=10(1+t)e2πiktdt+01(1t)e2πiktdt=\int\limits_{-1}^0 (1+t) e^{-2\pi i kt} \, dt+\int\limits_0^1 (1-t) e^{-2\pi i kt} \, dt

=2πike2πik+14π2k2+2πik+e2πik14π2k2=\dfrac{2\pi i k-e^{2\pi i k}+1}{4\pi^2 k^2}+\dfrac{2\pi i k+e^{2\pi i k}-1}{4\pi^2 k^2}

=sin2(πk)π2k2=\dfrac{\sin^2(\pi k)}{\pi^2 k^2}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS