Large Limits Approximation Error Propagation in Magnitude Estimate

To obtain the magnitude estimate from the real and imaginary estimates, we must perform:

$$\vert Y_a(k)\vert = \sqrt{(\Re{Y_a(k)})^2+(\Im{Y_a(k)})^2},$$ (135)

which includes two squaring operations, a sum operation, and a square root operation. To make our analysis cleaner, we define:

$$A = \Re{Y_a(k)} \hspace{2cm} B = \Im{Y_a(k)}$$ (136)

$$X = A^2 \hspace{3cm} Y = B^2$$ (137)

$$Z = X+Y \hspace{2cm} \vert Y_a(k)\vert = \sqrt{Z}$$ (138)

and denote the error in any variable $V$ as $\Delta V$. The table below shows how each variable manifests error as a result of error in variables of which it is a function. Rigorous justification of the error propagation shown is given in [7].

Relating Function	Error Propagated
$X = A^2$	$\Delta X = 2A\Delta A$
$Y = B^2$	$\Delta Y = 2B\Delta B$
$Z = X + Y$	$\Delta Z = \sqrt{(\Delta X)^2 + (\Delta Y)^2}$
$\vert Y_a(k)\vert = \sqrt{Z}$	$\Delta \vert Y_a(k)\vert = \ensuremath{\frac{1}{2}}Z^{\ensuremath{\frac{-1}{2}}}\Delta Z$

Making appropriate substitutions, we thus get

$$\Delta \vert Y_a(k)\vert = \ensuremath{\frac{\sqrt{(A \Delta A)^2... ...\Delta \Re{Y_a(k)})^2+ (\Im{Y_a(k)} \Delta \Im{Y_a(k)})^2}}{\vert Y_a(k)\vert}}$$ (139)