Generalizations of Proof to Nonzero Center Frequencies

As has been presented earlier by the author, it is possible to generalize this result to chirp signals centered about any frequency. This is of obvious importance when dealing with audio signals that are not centered about DC.

To see how this may be done, we must again consider our original expression for $Y(k)$ as first given in equation 19:

$$Y(k) = \sum_{n=-\ensuremath{\frac{N-1}{2}}}^{\ensuremath{\frac{N-1}{2}}}\exp(i( \alpha n^2 -2\pi k n/ K)).$$ (103)

When dealing with a signal of nonzero center frequency, this expression becomes

$$Y(k) = \sum_{n=-\ensuremath{\frac{N-1}{2}}}^{\ensuremath{\frac{N-1}{2}}}\exp(i( \alpha n^2 +\beta n -2\pi k n/ K)),$$ (104)

where $\beta$ is the nonzero center frequency in radians per sample. It is equivalent to say:

$$Y(k) = \sum_{n=-\ensuremath{\frac{N-1}{2}}}^{\ensuremath{\frac{N-1}{2}}}\exp(i( \alpha n^2 -2\pi \left(k-\ensuremath{\frac{K\beta}{2\pi}}\right) n/ K)).$$ (105)

Thus, if we make the substitution $p=k-\ensuremath{\frac{K\beta}{2\pi}}$, we get

$$\fbox{$ \displaystyle Y\left(p+\ensuremath{\frac{K\beta}... ...remath{\frac{N-1}{2}}}\exp(i( \alpha n^2 -2\pi p n/ K)) $}.$$ (106)

We see that we may then continue our analysis as above, treating $p$ as $k$ and noting only that our entire $Y(k)$ expression is shifted by $\ensuremath{\frac{K\beta}{2\pi}}$. Thus, we obtain all the same approximations and conclusions already seen, but recognize that they apply to a different center frequency location. Recalling that sinusoids may be expressed as the superposition of two complex exponentials, we see that we now have a way of analyzing practical signals of nonzero center frequencies.