Looks doable in a straightforward, simple, elementary way: We start with some background in 'signals'. Suppose for time t we have real number u(t). Suppose w

Looks doable in a straightforward, simple, elementary way: We start with some background in 'signals'. Suppose for time t we have real number u(t). Suppose we have a 'time invariant linear system' and send in signal u = u(t). Suppose the signal that comes out is s = s(t). Suppose the 'transfer function' of the system is H = H(w) for frequency w (H must exist because our system is time invariant and linear). Suppose U = U(w) is the Fourier transform of u and S = S(w) is the Fourier transform of s. Then, presto, for each frequency w, S(w) = H(w) U(w), and the output s = s(t) is just the inverse Fourier transform of S = S(w). If for time t we have only a finite interval and for the signal have all the power in a finite 'band' (have a 'band-limited' signal), that is, have a maximum frequency with any power, then we can discretize both time and frequency and use the fast Fourier transform (FFT) to do the Fourier transform work (actually a non-trivial signal cannot be both time limited and band-limited at the same time, but as we discretize time the signal can be band-limited and zero at all our discrete time points outside of finite interval of time -- don't worry about such things!). Suppose for some positive integer n we have n transmitters at distinct geographic locations. Suppose we also have n receivers at distinct geographic locations. Suppose for receiver j = 1, 2, ..., n we want to send to receiver j signal s_j = s_j(t) for time t. Here we are borrowing from TeX where _j indicates a subscript. Suppose by sending 'test signals' we have, for transmitter i = 1, 2, ..., n and receiver j = 1, 2, ..., n, 'transfer function' H_ij = H_ij(w) where w is frequency. So, we want signals u_i = u_i(t) so that when, all at the same time, for all i = 1, 2, ..., n, transmitter i sends signal u_i, then each receiver j receives the desired s_j. Let U_i be the Fourier transform of u_i and S_j be the Fourier transform of s_j. Then borrowing from TeX, for each frequency w S_j(w) = sum_{i = 1}^n H_ij(w) U_i(w) Now pick a particular value w of our discrete frequencies. Let's let S(w) be the n x 1 matrix with S_j(w) in component j, H(w), the n x n matrix with H_ij(w) in component i, j, and U(w), the n x 1 matrix with U_i(w) in component i. Then we have S(w) = H(w) U(w) where on the right we have matrix multiplication of n x n H(w) and n x 1 U(w) Then U(w) = H(w)^(-1) S(w) where H(w)^(-1) is the inverse of n x n matrix H(w). Do this calculation for each w and have all of U = (U_i). For each i = 1, 2, ..., n, take the inverse Fourier transform of U_i = U_i(w) and get u_i = u_i(t). Done. The math was not very advanced! Sorry 'bout that! But there is an issue with Shannon's theorem: As n grows, the total power in the signals u_i, i = 1, 2,, ..., n, stands to grow. Shannon's result was for bounded power on the communications channel or at least bounded signal to noise ratio.