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# hermitian

Keywords: frequency domain, fourier transforms, complex numbers
```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