This is an illustration of aliasing in a discrete sample.

Recall that for a sample of $N$ points, $\mathbf x = [x_0,x_1...x_{N-1}]$, the discrete Fourier transform (DFT) $\mathbf X$ will also have $N$ points. The frequencies $k$, given in the implicit units $\Delta\alpha = 2\pi/Na$, so $\alpha_k = k\Delta\alpha$, where $a$ is the spacing of samples $x$, are on the range

These $N$ DFT frequencies are equivalent to the frequencies contained in the first Brillouin zone, derived in the context of diffraction!

The only difference here being that we exclude $k = 0$ in the BZ and consider the negative Nyquist component.

The highest-frequency component (Nyquist frequency) will have a frequency $N/2$, with units of $2\pi/L = 2\pi/Na$ where $a$ is the (implicit) spacing between the sample points.

We can represent a Fourier component as

We will examine what happens if we try to sample a signal with a frequency greater than the Nyquist frequency $N/2$. Below we show a sample of 20 points, $N = 20$, over the range $x = 0...1$, at frequency $k$, taking unit amplitude $X_k = 1$. The highest frequencies that this sample can represent are $k = \pm10$, or $k = \pm N/2$ . Higher frequencies will be aliased, or undersampled (by not enough sampling points.)

The slider controls the frequency $k$ of the cosine wave we try to represent using the discrete samples. The samples represent the wave accurately if we keep $−10 \le k \le 10$. However, if we try to represent a frequency beyond this value, $N/2$, on the positive or negative side--so outside the first BZ--the waveform is represented equally well, and in many case more obviously, by its "alias," on the range $-N/2 \le k \le N/2$.

The aliasing effect can be seen most clearly for $k = \pm 19$. This frequency "aliases" to the fundamental frequencies $k = \mp 1$! Set the slider to $k = \pm 19$ to see this.

Aliasing is easy to understand mathematically. Consider that we would like to represent a unit waveform with frequency $x_{kk \pm N}$. We will then have

but since $j$ is an integer,

i.e., the waveform is exactly the same for frequency $k$ and frequency $k+N$. The same reasoning would hold if we try to represent $x_{k \pm 2N},x_{k \pm 3N}$, etc.

There are several, related conclusions:

• The frequency range $−N/2 \le k \le N/2$ (first BZ) is a complete representation of the possible frequencies contained in the signal.
• Any signal with a higher frequency has a discrete waveform which is completely identical to a that of a frequency in the first BZ. We can then always shift these frequencies $x_k \to x_{k+N}$ to those in the first BZ.

We will see that these second two statements are equivalent to the Bloch theorem. We can represent all possible electron waves in a solid with the same frequency basis, $−N/2 \le k \le N/2$, where now $N$ are the number of formula units in any direction.