THEORY

To develop our model, we first consider a simple discrete time linear frequency chirp signal:

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

where $\alpha$ is the one-half the chirp rate in radians per sample. We may write the DFT of the rectangle-windowed signal as

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

where $N$ is the odd-sample zero phase window length and $K$ is the length of the optionally zero padded transform. It may be shown [8] that for sufficiently large $\alpha$ and $N$,

$$Y(k) \approx Y_W^{\mathrm{Rec}}(\omega) = \int_{-T}^T e^{j \alpha t^2} e^{-j \omega t} dt,$$ (3)

where we have applied the midpoint approximation to the definite integral of the analogous continuous time chirp,

$$y(t) = e^{j \alpha t^2}.$$

Considering the above integral, it may be shown [9] that

$$\begin{eqnarray*} \frac{d Y_W^{\mathrm{Rec}}(\omega)}{d \omega} &=& \frac{-1... ...2}2j\sin{\omega T}+ j \omega Y_W^{\mathrm{Rec}}(\omega) \big) \end{eqnarray*}$$

and we see by inspection that this expression becomes zero when $\omega = 0$, indicating a stationary point at the center frequency. When using a Hann window, it may be shown [9] that:

$$\begin{eqnarray*} \frac{d Y_W^{\mathrm{Hann}}(\omega)}{d \omega} &=& \frac{d... ...}\\ &=& \frac{-1}{2\alpha}j\omega Y_W^{\mathrm{Hann}}(\omega) \end{eqnarray*}$$

and the second derivative evaluated at $\omega = 0$ becomes

$$\frac{d^2Y_W^{\mathrm{Hann}}(\omega)}{d\omega^2} \Big\vert _{\omega=0} = \frac{-j}{2\alpha}(Y_W^{\mathrm{Hann}}(0)).$$ (4)

By choosing $K$ sufficiently large, we may approximate the second order derivative with a second order difference. Doing so and solving for $\alpha$ yields

$$\hat{\alpha} \approx \frac{-jY^{\mathrm{Hann}}(0)}{2}\left( \left... ...2\frac{\Delta^2 Y^{\mathrm{Hann}}(k)}{\Delta k^2} \Big\vert _{k=0}\right)^{-1}.$$ (5)

where we note that second order differencing operation with respect to frequency bin $k$ has been normalized by twice multiplying by $\frac{K}{2\pi}$. This expression will serve as the main part of the chirp parameter estimator.