Question

Consider the signal shown in below figure, corresponds to the second derivative of a given function f(t). Then the Fourier transform of f(t) is

• A. $\frac{1-cos\omega }{{\omega }^4}$
• B. $\frac{1+cos\omega }{{\omega }^4}$
• C. $\frac{2(1+cos\omega )}{{\omega }^2}$
• D. $\frac{2(1-cos\omega )}{{\omega }^2}$

Answer 1

The correct option is D $\frac{2(1-cos\omega )}{{\omega }^2}$

$\frac{d^{2}f\left(t\right)}{d{t}^2}=\delta (t-1)-2\delta \left(t\right)+\delta (t+1)$

Taking fourier transformation

$\frac{d^{2}f\left(t\right)}{d{t}^2}\stackrel{F.T.}{⟷}{e}^{-j\omega }-2+e^{j\omega }=2cos\omega -2$

$f\left(t\right)\stackrel{F.T.}{⟷}\frac{2cos\omega -2}{(j\omega {)}^2}=\frac{2(1-cos\omega )}{{\omega }^2}$