DFT Sinusoids

The sampled sinusoids generated by integer powers of the $N$ roots of unity are plotted in Fig.6.2. These are the sampled sinusoids $(W_N^k)^n = e^{j 2 \pi k n / N} = e^{j\omega_k nT}$ used by the DFT. Note that taking successively higher integer powers of the point $W_N^k$ on the unit circle generates samples of the $k$th DFT sinusoid, giving $[W_N^k]^n$, $n=0,1,2,\ldots,N-1$. The $k$th sinusoid generator $W_N^k$ is in turn the $k$th $N$th root of unity ($k$th power of the primitive $N$th root of unity $W_N$).

Figure 6.2: Complex sinusoids used by the DFT for $N=8$.

Note that in Fig.6.2 the range of $k$ is taken to be $[-N/2,N/2-1] = [-4,3]$ instead of $[0,N-1]=[0,7]$. This is the most ``physical'' choice since it corresponds with our notion of ``negative frequencies.'' However, we may add any integer multiple of $N$ to $k$ without changing the sinusoid indexed by $k$. In other words, $k\pm mN$ refers to the same sinusoid $\exp(j\omega_k nT)$ for all integers $m$.