MODEL APPLICATION

We now may consider a more general and practical discrete time signal model for a given spectral peak:

$$g(n) = \exp(j(\alpha n^2 + \beta_0 n + \phi)),$$ (6)

where $\alpha$ is one-half the chirp rate in radians (increased) per sample, $\beta_0$ is the center frequency in radians per sample, and $\phi$ is the phase offset. The DFT of $g(n)$ is $G(k)$. To estimate $\alpha$ from $G(k)$, it may be shown [8] that we may use our estimator in eqn. 5, provided that we consider $k$ corresponding to $\beta_0$ rather than $k=0$. It may also be shown [9] that the estimator is valid when $\phi\neq0$. We may also use the estimator when $P$ order polynomial amplitude modulation occurs in the signal. To prove that this is valid, we first recall that taking an order 3 or higher derivative (approximated by differencing) of $Y(k)$ yields zero. Now, recalling that order $P$ polynomial amplitude modulation in the time domain corresponds to order $P$ differentiation (approximated by differencing) and multiplication by $j$ in the frequency domain, we see that such AM cannot influence our estimator. Although our estimator survives these conditions, it cannot function normally when $\alpha$ or $N$ is very small. (Because we will treat $N$ as the fixed frame length of the system, we may view this as a constraint on $\alpha$.) The error occurs because we cannot achieve the spectral resolution needed to accurately estimate the very large second order derivative in eqn. 4 with a second order difference, barring a $1/\alpha$ increase in $K$. The error manifests itself in a predictable way, however, with the estimated $\alpha$ value from eqn. 5 showing an imaginary part when in fact the estimate should be purely real. This imaginary part mimics the ratio of the real part to the correct $\alpha$ value, allowing us to solve the equation

$$x_1 + x_2 \Im\{\hat{\alpha}\} \approx \ensuremath{\frac{\Re\{\hat{\alpha}\}}{\alpha}}$$ (7)

as a least squares problem to obtain optimal coefficients $x_1$ and $x_2$. We may then substitute these and the real and imaginary parts of the eqn. 5 estimate and solve for $\alpha$ in cases where the initial estimator's guess is below our smallness threshold of 0.08 radians per $N$-sample window. (This was found to be reasonable when using zero padding to $K\leq 5N$.) Doing so creates a small-$\alpha$ estimator that is as accurate for small $\alpha$ as eqn. 5 is for larger $\alpha$. Results showing the estimator applied to a range of $\alpha$ values are shown in figure 1. In that figure, $\beta_0 = 0$, $N=201$ and $K=1024$. An arbitrary affine amplitude modulation of $0.5+0.001n$ was applied as was a phase shift of $\phi = \ensuremath{\frac{\pi}{16}}$.

Figure 1: Results achieved by the small-$\alpha$ and general estimator.

3in2intestscriptfig.eps