5 Noise

When noise $N[n]$ is added in the right-hand side of Eq. (1), the solution in Eq.s (3), (7), and (8) is no longer accurate. In particular, since the noise $N[n]$ is unknown, $\cos(\phi-\theta) = (\Vert r-N\Vert^2 - A^2 - B^2)/(2AB)$ is also unknown. For small levels of noise, $D$ may be defined as in Section II as an approximation for $\cos(\theta-\phi)$. Or, the following correction can be applied to obtain a slightly improved, unbiased estimate:

$$D \triangleq (\Vert r\Vert^2 - \sigma^2 - A^2 - B^2)/(2AB) \approx \cos(\phi-\theta),$$

where $\sigma^2 = E[\Vert N\Vert^2]$. Similarly, Eq.s (7) and (8) may be bias-corrected with

$$\hat A \triangleq {1 \over 2} \left( \sqrt{X+{\pi\over 2}(Y-X) - \sigma^2} + \sqrt{X+ {\pi\over 2}(Z-X) - \sigma^2}\right)$$

$$\hat B \triangleq {1 \over 2} \left( \sqrt{X+{\pi\over 2}(Y-X) - \sigma^2} - \sqrt{X+{\pi\over 2}(Z-X) - \sigma^2}\right).$$