THEORY

To develop our models, we first consider a simple discrete time linear frequency chirp signal with center frequency zero:

$$y(n) = \exp(j\alpha n^2).$$ (1)

where $\alpha$ is the one-half the chirp rate in radians per sample squared. (It may be shown that the results obtained for such a signal may be generalized to arbitrary center frequency and amplitude scaling.) We may write the DFT of the rectangle-windowed signal as

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

where we implicitly apply a zero-phase window of odd sample length $N$, and where $K$ is the length of the optionally zero padded transform.