The Nyquist frequency

The highest frequency in equation (11), $\omega=2\pi (N-1)/N$,is almost $2\pi$.This frequency is twice as high as the Nyquist frequency $\omega=\pi$.The Nyquist frequency is normally thought of as the ``highest possible'' frequency, because $e^{i\pi t}$, for integer t, plots as $(\cdots ,1,-1,1,-1,1,-1,\cdots)$.The double Nyquist frequency function, $e^{i2\pi t}$, for integer t, plots as $(\cdots ,1,1,1,1,1,\cdots)$.So this frequency above the highest frequency is really zero frequency! We need to recall that $B(\omega)=B(\omega -2\pi )$.Thus, all the frequencies near the upper end of the range equation (11) are really small negative frequencies. Negative frequencies on the interval $(-\pi,0)$were moved to interval $(\pi,2\pi)$by the matrix form of Fourier summation.

A picture of the Fourier transform matrix is shown in Figure 1. Notice the Nyquist frequency is the center row and center column of each matrix.

 

matrix
Figure 1 Two different graphical means of showing the real and imaginary parts of the Fourier transform matrix of size $32\times 32$.