RESULTS

In figure 2, we see the results generated by each estimator when noise is not present. In the plot, we use 31 positive $\alpha$ values evenly spaced on an logarithmic scale from $2\pi/F\cdot 2^{-3}$ to $2\pi/F\cdot 2^3$, where $F$ is the sampling frequency (here 16000 Hz), $K = 1024$, and $N = 201$. We see how each estimator - including the neural net estimator - functions reliably only for its specific intended range of $\alpha$. The figure reveals that the Fresnel and Taylor model estimators behave consistently when given data with $\alpha$ outside their intended regions; this will make the decision of which estimator to use for a given signal near-trivial. We presently evaluate the performance of our estimator relative to the Cramér-Rao lower bound. To obtain this bound, we first consider the Fisher information in the signal, defined as:

$$J(\alpha) = E\left[\left(\ensuremath{\frac{\partial \ln(f(\mathbf{y};\alpha))}{\partial \alpha}}\right)^2\right]$$ (20)

where $f(\mathbf{y};\alpha)$ is the PDF of our chirp signal (here, we write $y(n)$ as $\mathbf{y} = [y(-(N-1)/2) ... y(N-1)/2]$) in the presence of complex additive white Gaussian noise, namely:

$$\begin{eqnarray*} f(\mathbf{y};\alpha) = \ensuremath{\frac{1}{\sqrt{2\pi\sig... ...a^2}}\sum_n \big[(u_n - \mu_n)^2 + (v_n - \nu_n)^2\big] \Big) \end{eqnarray*}$$

where $\mu_n$ and $\nu_n$ represent the real and imaginary parts of the signal, respectively, and $u_n$ and $v_n$ represent the real and imaginary parts of the signal including noise. We assume that the noise has independent real and imaginary parts each with variance $\sigma^2$. Given the above, it may be shown that

$$J(\alpha) = \ensuremath{\frac{1}{\sigma^2}}\left(\ensuremath{\frac{N^5}{80}}- \ensuremath{\frac{N^3}{24}}+ \ensuremath{\frac{7 N}{240}}\right),$$ (21)

where we note that in this case the Fisher information is independent of $\alpha$. Since the Cramér-Rao lower bound on the variance of an estimator is given by $(1/J(\alpha))$, we plot this value along with the variance $\mathrm{var}(\hat{\alpha} - \alpha)$ of each set of estimates $\hat{\alpha}$ produced by each estimator. Figures 3 and 4 show plots for SNRs of 30dB and 15db, respectively. One thousand runs were used to determine the shown variance. Though the approximations invoked in creating the models used here lead to observable error variances, we note that the relative errors appear on the order of one to two percent. In applications where multiple sinusoidal signals and their parameters are detected [3], estimators such as those presented here are used as the first step in an iterative process, in general rendering such small errors inconsequential.

Figure 3: Each estimator is accurate in its intended range.

Figure 4: Relative estimator performance is similar at lower SNRs.