Standard Deviation from one axis to another axis

[whatever]

Hi

Maybe my math/stat is very poor. I'm having trouble manipulating the standard deviation. Here's the thing.
I have radar mounted on a car. For each detection, the radar returns these variables.
- relative distance between the object/host vehicle in forward direction (in vehicle body-fixed coordinates)
- standard deviation of the relative forward distance
- relative distance between the object/host vehicle in left/right direction (in vehicle body-fixed coordinates)
- standard deviation of the relative left/right distance

I'm trying to do coordinate transform of the above data, so that I get relative distance/standard deviation in global coordinates (North, South, East, West)
Distance is easy since it only requires to rotate the axis by the amount of angle between the vehicle body-fixed axis and the global axis.

How about standard deviation? How do I transform the standard deviation from the vehicle body-fixed axis to global axis?

 

[.Scott]

I first note that the radar data is well-processed data. The normal range/azimuth (polar coordinates) has already been converted to Cartesian coordinate.

I would take the standard deviations expressed as distances and simply rotate them as you would the coordinates of the target.
The only difference is that the standard deviations are +/- values, so keep them positive.

So: SDnorth = abs(SDz*sin(bearing))+abs((SDx*cos(bearing))

Car bearing is angle clockwise of North.

 

[Stephen Tashi]

I first note that the radar data is well-processed data. The normal range/azimuth (polar coordinates) has already been converted to Cartesian coordinate.

If the radar to target range $R$ has variance $\sigma_R^2$ then wouldn't the usual way of projecting this variance on two perpendicular axes X,Y be to arrange it so $\sigma_X^2 + \sigma_Y^2 = \sigma_R^2$?

If the projection was done that way, the two given standard deviations could be used to compute $\sigma_R^2$ and then that value could be re-projected on a different pair of perpendicular axes.

 

[.Scott]

If the radar to target range $R$ has variance $\sigma_R^2$ then wouldn't the usual way of projecting this variance on two perpendicular axes X,Y be to arrange it so $\sigma_X^2 + \sigma_Y^2 = \sigma_R^2$?

If the projection was done that way, the two given standard deviations could be used to compute $\sigma_R^2$ and then that value could be re-projected on a different pair of perpendicular axes.

For range, yes. Except that the range error does not follow a Gaussian curve. With car radar (and many other radars), a linear modulated transmitted signal is mixed with the received signal and an FFT is applied. The result is all targets are laid out in range bins. Without any interpolation,the curve is roughly a rectangular curve. With interpolation, it can be triangular or some combination with rectangular.

The determination of azimuth is accomplished in a more complicated way and involves several transmit and receive antennae. I've seen (and computed) the curves - they're roughly sinusoidal, not really Gaussian. So the 2-D combination forms an arc, not an ellipse, and if the target is well left or right of the bore sight, definitely not an ellipse that aligns nicely with the X,Z axis.

So the problem is really problematic.

 

[.Scott]

I left out an important word: "linear modulated transmitted signal" should be "linear frequency modulated transmitted signal". So the frequency plot looks like a ramp.