The Cramer-Rao Lower Bound

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:

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

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 $\alpha$.