I need to find an optimal pulse shape, that convoluted with matched filter on adjacent frequency will give minimal convolution/correlation (Minimal interference) for a given pulse length.
For example Pulse Length of 10 msec, 200 Hz separation (10,000 Hz and 10,200 Hz)
Image below demonstrating the problem. Currently I have interference with adjacent signal of approx 0.05.
What would be the Optimal pulse shape for given pulse length?
The root-raised-cosine pulse-shape is widely used because
The frequency response (Fourier transform) of the root-raised-cosine is equal to the square root of the frequency response of the raised-cosine, so these Fourier transforms are nonzero in exactly the same (bounded) intervals.
Suppose that $h_1(t)$ and $h_2(t)$ are two root-raised-cosine pulses with identical rolloff parameter ($\beta$) whose Fourier transforms are nonzero on non-intersecting intervals. Then matched filtering of of $h_m(t)$ with $h_n(t)$ yields
\begin{equation}
g(\tau) = \int_{-\infty}^{\infty}h_m(t)h_n(\tau - t)dt.
\end{equation}
The Fourier transform of $g$ is
\begin{equation}
G(\omega) ~=~ (\textrm{Fourier transform of $h_m\ast h_n$}) ~=~ H_m(\omega)H_n(\omega).
\end{equation}
Since $H_m$ and $H_n$ are nonzero on non-intersecting intervals, their product is zero for all $\omega$. The inverse Fourier transform of an all-zero Fourier transforms is the zero function, so $g(\tau) = 0$ for all $\tau$. Hence, using a root-raised-cosine matched filter will filter out root-raised-cosine pulses from other frequency bands.
See the answers to this question for more information.
