I derived this inverse fourier transform a couple months ago but never posted. I wanted to explain what these two solved math problems were and how they are useful in great detail. But I guess I will just be short and sweet. If you want a further explanation you will have to do the work.

Basically these two transforms are useful in Ultra Wide Band (UWB) communications. UWB has a very high bandwidth because it signals extremal short Gaussian pulses at a very high data rate. The two transforms I present analyze UWB Gaussian pulses in both the frequency and time domains.

Due to the short time domain of UWB, it occupies a very large frequency spectrum.

g(t)=\int_{-W}^0e^{j\frac{\pi}{2}}e^{j2\pi{f}{t}}df \int_{0}^We^{j\frac{\pi}{2}}e^{j2\pi{f}{t}}df

g(t)=\frac{e^{j\frac{\pi}{2}}e^{j2\pi{f}{t}}}{j2\pi{t}}\biggr]_{-W}^0 \frac{e^{-j\frac{\pi}{2}}e^{j2\pi{f}{t}}}{j2\pi{t}}\biggr]_{0}^W

g(t)=\frac{e^{j\frac{\pi}{2}}-e^{j\frac{\pi}{2}}e^{-j2\pi{W}{t}}}{j2\pi{f}{t}} \frac{e^{-j\frac{\pi}{2}}e^{j2\pi{W}{t}}-e^{-j\frac{\pi}{2}}}{j2\pi{f}{t}}

\frac{e^{j\frac{\pi}{2}}}{j2\pi{t}}\biggl[1-e^{-j2\pi{Wt}}\biggr]+\frac{e^{-j\frac{\pi}{2}}}{j2\pi{t}}\biggl[e^{j2\pi{Wt}}-1\biggr]

\frac{1}{2\pi{t}}\biggl[1-e^{-j2\pi{Wt}}\biggr]+\frac{-1}{2\pi{t}}\biggl[e^{j2\pi{Wt}}-1\biggr]

\frac{1}{2\pi{t}}\biggl[1-e^{-j2\pi{Wt}}-e^{j2\pi{Wt}}+1\biggr]

\frac{1}{\pi{t}}(1-\cos(2\pi{Wt}))

\frac{1}{\pi{t}}(2\sin^2(\pi{Wt})

\frac{(\pi{t}W)^2}{\pi{t}}\frac{(2\sin^2(\pi{Wt})}{(\pi{t}W)^2}

2sinc^2(tW)\pi{t}W^2