Question #85781
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
2019-03-04T11:55:40-0500

From the definition of the Fourier transformation


f(x)=12πF(t)eitxdtf(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}F(t)e^{-itx}dt


we obtain


f(x)=10(1+t)eitxdt+01(1t)eitxdt=f(x)=\int\limits_{-1}^{0}(1+t)e^{-itx}dt+\int\limits_{0}^{1}(1-t)e^{-itx}dt==201cos(tx)dt201tcos(tx)dt=2sin(x)x+21x22cos(x)x22sin(x)x==2\int\limits_{0}^{1}\cos(tx)dt-2\int\limits_{0}^{1}t\cos(tx)dt=2\frac{\sin(x)}{x}+2\frac{1}{x^2}-2\frac{\cos (x)}{x^2}-2\frac{\sin (x)}{x}==21cos(x)x2=2\frac{1-\cos(x)}{x^2}

Answer:

f(x)=21cos(x)x2f(x)=2\frac{1-\cos(x)}{x^2}


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