Special Case: The Mth Roots of Unity

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Special Case: The Mth Roots of Unity

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Special Case: The Mth Roots of Unity

If $a=1$, we have

\

The $M$th roots of unity play an important role in the Discrete Fourier Transform. They are so important that they often have a special notation in the signal processing literature:
\

In this notation, the $M$ roots of unity can be expressed simply as$W_M^k$, for $k=0,1,2,\.

We may also call $W_M$ the generator of the mathematicalgroup consisting of the $M$th roots of unity and their products.

We will learn later that the $N$th roots of unity are used to generate all the sinusoids used by the DFT and its inverse. The $k$th sinusoid is given by

\

where $\, $t_n \, and $T$ is the sampling interval in seconds.

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