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In a book about radar signal processing, they are estimating the squared distance between two objects as:

$$d^2 = \frac{(R_p-R_o)^2}{\sigma^2_R} + \frac{(\theta_p-\theta_o)^2}{\sigma^2_\theta}$$

where R is a radius, $\theta$ is an angle and $\sigma^2$ is the variance of the radius and angle respectivly. p is the predicted target and o is an observation. The book is Blackman, S. - Design and Analysis of Modern Tracking Systems

Is it just me or is the dimensional analysis for this equation a bit off?

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    What do the variables mean?2010-10-07
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    What are the $\sigma$'s supposed to be? If those other variables are indeed polar coordinates, you have the wrong formula.2010-10-07
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    More importantly... what book is this?2010-10-07
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    Polar coordinates; R is a radius, θ is an angle and σ2 is the variance. of the radius and angle respectivly. The book is Blackman, S. - Design and Analysis of Modern Tracking Systems2010-10-07
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    Doesn't $\sigma_R$ have the same units as the $R$'s and $\sigma_{\theta}$ the same units as the $\theta$'s, so $\frac{R_p - R_o}{\sigma_R}$ is a unitless quantity, ditto for the $\theta$ term.2010-10-07
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    @J.M.: You're of course right that it's the wrong formula for Euclidean distance, but the standardization terms suggest the distance is intended to be more abstract. It might (just possibly) make some sense in a model where $R_p$ and $\theta_p$ are *independent* random variables (perhaps with approximately symmetrical distributions) and the intention is to conduct a statistical hypothesis test or create a confidence interval.2010-10-07
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    "The target's reflection back towards the radar is covered in a single figure of merit, **sigma**. For some reason, sigma takes on the dimensions of **square metres** and is also called RCS (Radar Cross Section) but this is just another misnomer. Sigma somewhat depends on a target's size, but RCS calculation theories can fill whole volumes with chapter headings such as aspect angle, target size vs wavelength used, built-in corner reflectors, resonant features of a target's structure, statistical models of target behaviour, (...) " http://www.bbc.co.uk/dna/h2g2/A7438612010-10-07
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    @Américo : The "sigma" in your quotation is a scattering cross section. It is unrelated to either of the sigmas in this problem.2010-10-08
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    Voting to close as OP seems to have ignored this and I want to stop the robot from bringing it up2011-06-06
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    @Ross: I followed suit last time, this time I just voted your answer up, this should also prevent the community user from bumping it.2011-07-06

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The formula is not even close. You are right that d should have dimensions of length and the right hand side is unitless. Worse, the formula doesn't reduce properly if the errors are zero. Are you sure it isn't $d^2/\sigma(d)^2$ on the left? That would fix the units. The cosine formula gives the distance estimate you want: $d^2=R_p^2+R_o^2-2 R_p R_o \cos(\theta_p-\theta_o)$. When you mix in sigmas you are trying to estimate the error in d based on the errors in the measurements. So you should use the formulas you have for combining measurement errors to give the error in the measured quantity.

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    (btw, if you put dollars signs around your formulas, they show up as beautiful TeX.)2010-10-08
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    Done. Thanks much. I've used that elsewhere, too.2010-10-09
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I think whuber's comment is on the spot. That formula is surely not refering to a "physical distance", but it's simply a general measure of the distance (as "dissimilarity") between two multidimensional (general) $features$.

A classical example is statistical classification; for example when applying the nearest-neighbour rule, we must compute the distance from our observed feature (say ${\mathbf x} = [x_1 x_2]$) against some reference values (say ${\mathbf x^A} = [x_1^A x_2^A]$) .

Here the components $x_1 x_2$ can be any measurements, often with different dimensions (eg, weight and length f some object).

A possible way to measure the (square) distance is to compute $d^2 = (x_1 - x_1^A)^2+(x_2 - x_2^A)^2$ , but this would be dimensionally inconsistent, and hence sensitive to the scale. A more reasonable recipe is to normalize each component dividing by some "characteristic" value (eg: the standard deviation), so all components are now adimensional and approximately even distributed:

$$ d^2 = \frac{(x_1 - x_1^A)^2}{\sigma_1^2}+\frac{(x_2 - x_2^A)^2}{\sigma_2^2}$$

Of course, the distance obtained by this feature normalization is adimensional, it's only meaningful when compared with other distances.