Let for all (this repeats with any desired integer period
). From the preceding discussion, we expect to find that

We will often need to know the answer for non-integer values of however, and for this there is nothing better to do than to calculate the value directly:

where is, as before, the unit magnitude complex number with argument . This is a geometric series; as long as we get:

We now symmetrize the top and bottom in the same way as we earlier did in Section 7.3. To do this let:

so that . Then factoring appropriate powers of out of the numerator and denominator gives:

It's easy now to simplify the numerator:

and similarly for the denominator, giving:

Whether or not, we have

where , known as the

Figure 9.1 shows the Fourier transform of , with . The
transform repeats every 100 samples, with a peak at , another at
, and so on. The figure endeavors to show both the magnitude and phase
behavior using a 3-dimensional graph projected onto the page. The phase
term

acts to twist the values of around the axis with a period of approximately two. The Dirichlet kernel , shown in Figure 9.2, controls the magnitude of . It has a peak, two units wide, around . This is surrounded by one-unit-wide