Large Limits Approximation Error Propagation in Phase Estimate

We now apply a similar analysis to the phase case. We recall that in section 4.3 we base our estimate of $\alpha$ on the second order difference of the phase of $Y(k)$. Stating this in terms of the real and imaginary parts thereof, we have:

$$\ensuremath{\frac{\Delta^2 \angle Y(k)}{\Delta k^2}}= \ensuremath... ...ta^2}{\Delta k^2}}\arctan\left(\ensuremath{\frac{\Im{Y(k)}}{\Re{Y(k)}}}\right).$$ (140)

Proceeding as above, we make the notation cleaner in our error propagation analysis by defining

$$A(k) = \Re{Y_a(k)} \hspace{4cm} B(k) = \Im{Y_a(k)}$$ (141)

$$U = \ensuremath{\frac{B}{A}}\hspace{4cm} T = \arctan(U)$$ (142)

$$S = T(k) - T(k-1) \hspace{1cm} \ensuremath{\frac{\Delta^2 \angle Y(k)}{\Delta k^2}}= S(k)-S(k-1)$$ (143)

We again create an error manifestation table below. We make an important exception to the general treatment of error propagation in the case of propagating error from T to S, and from S to the final result. Instead of treating these differencing operations as the subtraction of independent errors - which results in propagating the error as a Euclidean distance (as in the addition in the previous subsection) - we acknowledge that the neighboring errors are highly correlated, and thus their combination is best modeled as a simple subtraction. This is reflected in the table below.

Relating Function	Error Propagated
$U = \ensuremath{\frac{B(k)}{A(k)}}$	$\Delta U = \sqrt{B^2\Delta A^2 + A^2\Delta B^2}$
$T = \arctan(U)$	$\Delta T = \ensuremath{\frac{\Delta U}{1+U^2}}\leq \Delta U$
$S = T(k) - T(k-1)$	$\Delta S = \Delta T(k) - \Delta T(k-1)$
$\ensuremath{\frac{\Delta^2 \angle Y(k)}{\Delta k^2}}= S(k)-S(k-1)$	$\Delta \ensuremath{\frac{\Delta^2 \angle Y(k)}{\Delta k^2}} = \Delta S(k)- \Delta S(k-1)$

Making appropriate substitutions, we obtain

$$\Delta \ensuremath{\frac{\Delta^2 \angle Y(k)}{\Delta k^2}} = \Delta T(k) - 2\Delta T(k-1)+ \Delta T(k-2)$$ (144)

$$= \sqrt{(B(k)\Delta A(k))^2+(A(k)\Delta B(k))^2}$$

$$- 2\sqrt{(B(k-1)\Delta A(k-1))^2+(A(k-1)\Delta B(k-1))^2}$$ (145)

$$+ \sqrt{ (B(k-2)\Delta A(k-2))^2+ (A(k-2)\Delta B(k-2))^2}$$

$$= \sqrt{(\Im Y(k)\Delta \Re Y(k))^2+ (\Re Y(k)\Delta \Im Y(k))^2}$$

$$- 2\sqrt{(\Im Y(k-1)\Delta \Re Y(k-1))^2+ (\Re Y(k-1)\Delta \Im Y(k-1))^2}$$ (146)

$$+ \sqrt{(\Im Y(k-2)\Delta \Re Y(k-2))^2+(\Re Y(k-2)\Delta \Im Y(k-2))^2.} \nonumber$$

It can be seen by inspection that in some cases, the error can be nearly zero, while in others, it may be double the first term. This depends on the correlation of the real part with the error in the imaginary part and vice versa. These correlations vary rapidly with $k$, as can be seen in the example plotted earlier (figure 10) where the error appeared unstable. The rapid variation does not appear to suggest an invertible error model, and as a result is in general more damaging to an estimation algorithm. The magnitude of the error in the worst cases is also problematic, as we note that the error here is not divided by $\vert Y(k)\vert$ as it was in the magnitude case. Thus, the error propagation in implementation here is more problematic than that for the magnitude case in the previous subsection. This fact is also reflected somewhat in the accuracy obtained in estimating $\alpha$ using each model.