A complex series {y(j),0<=j<=N-1} of length N that satisfies y(0) real and y(n-j) = conj(y(j)), j = 1, ..., N-1 is said to have Hermitian symmetry, or simply to be a Hermitian series. Here conj(z) denotes the complex conjugate of complex number z. When N is even, y(N/2) is real for Hermitan y. If you define the periodic extension as the infinite complex series y<N>(j) = y(j), j = 0, ..., N-1 y<N>(j) = y<N>(j-N), j = N, N+1, ... y<N>(j) = y<N>(j+N), j = -1, -2, ... then Hermitian symmetry is equivalent to y<N>(-j) = conj(y<N>(j)), j = 0, +-1, +-2, ... If {x(t),0<=t<=N-1} is a real series, then its DFT, {x<N>(t)} is a Hermitian series. Conversely if {x(t),0<=t<=N-1} is Hermitian, then {x<N>(t)} is real. See topic 'fourier' for information on the DFT. If {x(t),0<=t<=N-1} is an unrestricted complex series, its Hermitian symmetrized form is the Hermitian series {y(t),0<=t<=N-1} where y(0) = Re(x(0)), y(t) =(1/2)(x(t)+conj(x(N-t))), t = 1, ..., N-1 Using the periodic extensions {x<N>(t)} and {y<N>(t)}, this is equivalent to y<N>(t) = (x<N>(t)+x<N>(-t))/2, j = 0, +-1, +-2, ... See topic 'complex_data' for information on how complex series are represented in MacAnova.

Gary Oehlert 2003-01-15