# Standard Deviation from one axis to another axis

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In summary: 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.If the radar to target range ##R## has variance ##\sigma_R^2## then wouldn't the usual way of projecting this variance on two
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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?

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.

.Scott said:
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.

Stephen Tashi said:
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.

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.

## 1. What is standard deviation from one axis to another axis?

Standard deviation from one axis to another axis is a statistical measure that shows how much the data points in a dataset vary from the mean value of the dataset. It is a measure of how spread out the data is from the average value.

## 2. How is standard deviation from one axis to another axis calculated?

To calculate standard deviation from one axis to another axis, you need to first calculate the mean value of the dataset. Then, for each data point, subtract the mean value and square the result. Next, find the average of all the squared values. Finally, take the square root of the average to get the standard deviation.

## 3. What does a high standard deviation from one axis to another axis indicate?

A high standard deviation from one axis to another axis indicates that the data points in the dataset are spread out over a wider range of values. This means that the data is more diverse and less consistent.

## 4. How is standard deviation from one axis to another axis useful in data analysis?

Standard deviation from one axis to another axis is useful in data analysis because it provides a measure of the variability of the data. It helps to identify outliers and understand the spread of the data, which can help in making more informed decisions and drawing more accurate conclusions from the data.

## 5. Can standard deviation from one axis to another axis be negative?

No, standard deviation from one axis to another axis cannot be negative. It is always a positive value, as it is calculated by taking the square root of the average of squared values, which are always positive.

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