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: $k=N+1$

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