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 \( \Delta\alpha = 2 \pi/ (N a)\), so \( \alpha_k = k\Delta\alpha \), where \( a \) is the spacing of samples \( x \), are on the range $$ -\frac {N}{2} + 1 \le k \le \frac {N}{2}. $$

These \( N \) DFT frequencies are equivalent to the frequencies contained in the first Brillouin zone, derived in the context of diffraction: $$ -\frac {\pi}{a} \le a \le \frac {\pi}{a}. $$

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 \( 2 \pi/ L = 2 \pi / (N a) \) where \( a \) is the (implicit) spacing between the sample points.

We can represent a Fourier component as $$ x_k = X_ke^{i\alpha_kx}. $$

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 \( x = 0, 1, \ldots \), 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 \( x_{k \pm N} \). We will then have $$ x_{k \pm N} = e^{i\frac{2\pi}{N}j(k \pm N)} = e^{i \frac{2\pi}{N}jk \pm i 2\pi j}. $$

But since \( j \) is an integer, $$ x_{k \pm N} = e^{i \frac{2\pi}{N}jk} = x_k, $$ 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 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.