Consider a single-user or multiple-access channel with a large output alphabet. A method to approximate the channel by an upgraded version having a smaller output alphabet is presented and analyzed. The gain in   symmetric channel capacity is controlled through a fidelity parameter. The larger the fidelity parameter, the better the approximation on the one hand, but the larger the new output alphabet on the other.

The approximation method is instrumental when constructing polar codes. No assumption is made on the symmetry of the original channel, and the input alphabet need not be binary.