Discrete Fourier transform matrix in Galois field
dm = dftmtx(alph)
dm = dftmtx(alph) returns a Galois array that represents the discrete Fourier transform operation on a Galois vector, with respect to the Galois scalar alph. The element alph is a primitive nth root of unity in the Galois field GF(2m) = GF(n+1); that is, n must be the smallest positive value of k for which alph^k equals 1. The discrete Fourier transform has size n and dm is an n-by-n array. The array dm represents the transform in the sense that dm times any length-n Galois column vector yields the transform of that vector.
The example below illustrates the discrete Fourier transform and its inverse, with respect to the element gf(3,4). The example examines the first n powers of that element to make sure that only the nth power equals one. Afterward, the example transforms a random Galois vector, undoes the transform, and checks the result.
m = 4; n = 2^m-1; a = 3; alph = gf(a,m); mp = minpol(alph); if (mp(1)==1 && isprimitive(mp)) % Check that alph has order n. disp('alph is a primitive nth root of unity.') dm = dftmtx(alph); idm = dftmtx(1/alph); x = gf(randi([0 2^m-1],n,1),m); y = dm*x; % Transform x. z = idm*y; % Recover x. ck = isequal(x,z) end
The output is
alph is a primitive nth root of unity. ck = 1
The Galois field over which this function works must have 256 or fewer elements. In other words, alph must be a primitive nth root of unity in the Galois field GF(2m), where m is an integer between 1 and 8.