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Fourier transform of DC

Let $X[n]=1$ for all $n$ (this repeats with any desired integer period $N>1$). From the preceding discussion, we expect to find that

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

where $V$ is, as before, the unit magnitude complex number with argument $-k\omega$. This is a geometric series; as long as $V \not= 1$ 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 ${\xi^2} = V$. Then factoring appropriate powers of $\xi$ out of the numerator and denominator gives:

It's easy now to simplify the numerator:

and similarly for the denominator, giving:

Whether $V=1$ or not, we have

where ${D_N}(k),ドル known as the Dirichlet kernel, is defined as

Figure 9.1 shows the Fourier transform of $X[n]=1,ドル with $N=100$. The transform repeats every 100 samples, with a peak at $k=0,ドル another at $k=100,ドル 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 ${\cal FT} \left \{ X[n] \right \} (k)$ around the $k$ axis with a period of approximately two. The Dirichlet kernel ${D_N}(k),ドル shown in Figure 9.2, controls the magnitude of ${\cal FT} \left \{ X[n] \right \} (k)$. It has a peak, two units wide, around $k=0$. This is surrounded by one-unit-wide sidelobes, alternating in sign and gradually decreasing in magnitude as $k$ increases or decreases away from zero. The phase term rotates by almost $\pi $ radians each time the Dirichlet kernel changes sign, so that the product of the two stays roughly in the same complex half-plane for $k>1$ (and in the opposite half-plane for $k < -1$). The phase rotates by almost 2ドル\pi $ radians over the peak from $k=-1$ to $k=1$.

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next up previous contents index
Next: Shifts and phase changes Up: Properties of Fourier transforms Previous: Properties of Fourier transforms Contents Index

Miller Puckette 2006年12月30日