Waves and Fourier sums

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 $32\times 32$.

Waves and Fourier sums