Discrete Frequency Axis

With $N$ discrete time domain samples, there are only $N$ complex sinusoids that are orthogonal to each other for $k=0,1,2,\cdots, N-1$.

The other option is $k=N$, which we’ll investigate next.

Frequency index: $k=N$

Let’s explore the option $k=N$ as a complex sinusoid.

$$\begin{equation*} e^{ j2\pi \frac{k}{N}n}\Big|_{k=N} =e^{j2\pi n} =\cos 2\pi n +j\sin 2\pi n = 1+j0 \end{equation*}$$

because $\sin 2\pi n = 0$ and $\cos 2\pi n = 1$. That’s why the options $k=0$ and $k=N$ point toward the same complex sinusoid.

Next, we consider $k=N+1$.

Frequency index: $k=N+1$

For $k=N+1$, we have

$$\begin{align*} e^{ j2\pi \frac{N+ 1}{N}n} &= e^{j2\pi n}\cdot e^{j2\pi \frac{1}{N}n} \\ &= e^{j 2\pi \frac{1}{N}n} \end{align*}$$

which is the same as $k=1$. We conclude that there are only $N$ distinct frequencies from $0$ to $N-1$. The rest is repetition of these frequencies This leads to an interesting fact. For $N$ samples in the discrete time domain, there are $N$ samples in the discrete frequency domain!

These $N$ samples represent discrete frequencies of complex sinusoids that are all integer multiples of the one fundamental frequency $1/N$. They are all orthogonal to each other.

Construction of the frequency axis

The discrete frequencies $k/N$ give rise to a frequency axis of the following elements:

$$\begin{equation*} 0, \frac{1}{N}, \cdots, \frac{N-1}{N} \end{equation*}$$

Due to the sampling theorem, we’ll work with an axis centered around $0$.

After the signal sampling, the unique range of the continuous frequency $f$ is $-f_s/2$ to $+f_s/2$. or:

$$\begin{equation*} -\frac{1}{2} \le \frac{f}{f_s} < +\frac{1}{2} \end{equation*}$$

Next, we can simply chop this axis at $N$ intervals because there are $N$ frequency samples in total. This is effectively sampling the frequency axis.

$$\begin{align} -\frac{1}{2},-\frac{1}{2}+\frac{1}{N},\cdots,-\frac{1}{N},0,\frac{1}{N},\cdots,+\frac{1}{2}-\frac{1}{N} \end{align}$$

The $k/N$ values are listed below. Consequently, the index $k$ of the discrete frequency axis is given by $[-N/2,N/2-1]$ or:

$$\begin{align*} \frac{k}{N} &= -\frac{1}{2},-\frac{1}{2}+\frac{1}{N},\cdots,-\frac{1}{N},0,\frac{1}{N},\cdots,+\frac{1}{2}-\frac{1}{N}\\ \\ k &= -\frac{N}{2},-\frac{N}{2}+1,\cdots,-1,0,1,\cdots,\frac{N}{2}-1 \end{align*}$$

This range is shown in the figure below: