I'm trying to find the inverse Fourier transform of two impulse functions, which correspond to the Fourier transform of the function $h(t)=A\sin(2πf_0t)$.
The Fourier transform of the above sine function is as follows:
$$H(f)=j \frac{A}{2} \delta\left(f+f_0\right)-j \frac{A}{2} \delta\left(f-f_0\right)$$
When I try to find the inverse Fourier transform of $H(f)$, I keep getting a minus sign which is not in the original time-domain function. Here is my procedure:
\begin{align} h(t)&=j \frac{A}{2} \int_{-\infty}^\infty\delta(f+f_0 ) e^{j2πft} df-j \frac{A}{2}\int_{-\infty}^\infty\delta(f-f_0 ) e^{j2πft}df\\ &=j\frac{A}{2}e^{j2πf_0t}-j\frac{A}{2}e^{-j2πf_0t}\\ &=jA\left(\frac{e^{j2πf_0t}-e^{-j2πf_0t}}{2}\right)\\ &=-A\left(\frac{e^{j2πf_0t}-e^{-j2πf_0t}}{2j}\right)\\ &=-A\sin(2\pi f_0t) \end{align}
THAT is the minus sign that, so far, I've been unable to get rid of!
What is it that I am missing here? I hope you can help me!
Thank you very much!