Computing the Fourier Transform
Compute the Fourier Transform of the signal
$ \ x(t)= t \sin(2 \pi t+ \pi/4) $
By definition the Fourier Transform of a signal is defined as:
$ X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $
First expressing the signal in as a Fourier series:
However before finding the transform we note that multiplication in the time domain is just differentiation in the frequency domain. So the game plan is to find the Fourier series of x(t)/t then differentiate it with respect to w in the frequency space.
$ \ x1(t)=\sin(2\pi t+ \pi/4) = \frac{e^{2 \pi jt + \pi/4}{2j} $
