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?

enter image description here


The root-raised-cosine pulse-shape is widely used because

  1. using the root-raised-cosine as both pulse shape and as matched filter yields the raised-cosine pulse-shape, and
  2. the root-raised-cosine and raised-cosine pulse-shapes are both bandlimited.

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.

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