Shifted spectrum

Subroutine simpleft() sets things up in a convenient manner: The frequency range runs from minus Nyquist up to (but not including) plus Nyquist. Thus there is no problem with the many (but not all) user programs that have trouble with aliased frequencies. Subroutine ftu() , however has a frequency range from zero to double the Nyquist. Let us therefore define a friendlier ``front end'' to ftu() which looks more like simpleft().

Recall that a time shift of t₀ can be implemented in the Fourier domain by multiplication by $e^{-i\omega t_0}$.Likewise, in the Fourier domain, the frequency interval used by subroutine ftu() , namely, $0 \le \omega < 2\pi$,can be shifted to the friendlier interval $-\pi \le \omega < \pi$by a weighting function in the time domain. That weighting function is $e^{-i\omega_0 t}$where $\omega_0$ happens to be the Nyquist frequency, i.e. alternate points on the time axis are to be multiplied by -1. A subroutine for this purpose is fth().  

# FT a vector in a matrix, with first omega = - pi
#
subroutine  fth( adj,sign, m1, n12, cx)
integer i,       adj,      m1, n12
real sign
complex cx(m1,n12)
temporary complex temp(n12)
do i= 1, n12
        temp(i) = cx(1,i)
if( adj == 0)   { do i= 2, n12, 2
                        temp(i) =  -temp(i)
                  call ftu(  sign, n12, temp)
                }
else            { call ftu( -sign, n12, temp)
                  do i= 2, n12, 2
                        temp(i) =  -temp(i)
                }
do i= 1, n12
        cx(1,i) = temp(i)
return; end

To Fourier transform a 1024-point complex vector cx(1024) and then inverse transform it, we would write

call fth( 0, 1., 1, 1024, cx)
call fth( 1, 1., 1, 1024, cx)

You might wonder about the apparent redundancy of using both the argument adj and the argument sign. Having two arguments instead of one allows us to define the forward transform for a time axis with the opposite sign as the forward transform for a space axis.

The subroutine fth() is somewhat cluttered by the inclusion of a frequently needed practical feature--namely, the facility to extract vectors from a matrix, transform the vectors, and then restore them into the matrix.