# Optimal pulse shape for minimal interference in adjacent frequencies

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?

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 $$$$g(\tau) = \int_{-\infty}^{\infty}h_m(t)h_n(\tau - t)dt.$$$$ The Fourier transform of $$g$$ is $$$$G(\omega) ~=~ (\textrm{Fourier transform of h_m\ast h_n}) ~=~ H_m(\omega)H_n(\omega).$$$$ Since $$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.