Discrete Frequency Axis

Investigate the details of the discrete frequency axis.

With NN discrete time domain samples, there are only NN complex sinusoids that are orthogonal to each other for k=0,1,2,,N1k=0,1,2,\cdots, N-1.

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

Frequency index: k=Nk=N

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

ej2πkNnk=N=ej2πn=cos2πn+jsin2πn=1+j0 \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 sin2πn=0\sin 2\pi n = 0 and cos2πn=1\cos 2\pi n = 1. That’s why the options k=0k=0 and k=Nk=N point toward the same complex sinusoid.

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

Frequency index: k=N+1k=N+1

For k=N+1k=N+1, we have

ej2πN+1Nn=ej2πnej2π1Nn=ej2π1Nn \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=1k=1. We conclude that there are only NN distinct frequencies from 00 to N1N-1. The rest is repetition of these frequencies This leads to an interesting fact. For NN samples in the discrete time domain, there are NN samples in the discrete frequency domain!

These NN samples represent discrete frequencies of complex sinusoids that are all integer multiples of the one fundamental frequency 1/N1/N. They are all orthogonal to each other.

Construction of the frequency axis

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

0,1N,,N1N \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 00.

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

12ffs<+12 \begin{equation*} -\frac{1}{2} \le \frac{f}{f_s} < +\frac{1}{2} \end{equation*}

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

12,12+1N,,1N,0,1N,,+121N \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/Nk/N values are listed below. Consequently, the index kk of the discrete frequency axis is given by [N/2,N/21][-N/2,N/2-1] or:

kN=12,12+1N,,1N,0,1N,,+121Nk=N2,N2+1,,1,0,1,,N21 \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:

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