# Doppler effect to measure angle, not speed

1. Jul 17, 2007

### goodspeeler

DISCLAIMER: i may very well have exactly zero idea what i'm talking about. please feel free to berate me if i am way off base...

so i know the doppler effect is normally used to determine an unknown radial velocity, but i'm assuming that if i know the velocity of the source, i can use the frequency shift to determine the angle of the velocity vector relative to the receiver.

What i'm wondering about is the accuracy of this determination. I have heard that the standard formula for EM doppler effect is only an approximation. Does one of those assumptions require that you are parallel with the velocity vector?

Consider the following setup:

*-------->v
source moving at a known speed |v| in a known direction (angle = 0)

if $$\theta$$ is the angle between the source velocity and the source->receiver direction, can I determine this angle by simply saying the following:

$$|v|cos(\theta) = v_{rad}$$ (where $$v_{rad}$$ is the source velocity's component toward the receiver)​

and then say that
$$f_{shift} = \frac{f_{orig}*v_{rad}}{c}$$ (where $$f_{orig}$$ is the original EM frequency)​

rearranging and plugging back in for $$v_{rad}$$ i get:

$$\theta = cos^{-1}\left(\frac{f_{shift}*c}{f_{orig}*|v|}\right)$$​

so that's the setup. What i'm wondering is
1. does the doppler effect work this way, or have i violated too many of its assumptions?
2. if it works, how accurate can my angle determination be? put differently, to what precision can one measure a frequency shift? from some quick calculations with plausible $$|v|$$ and $$f_{orig}$$ values, it seemed like determining frequency to within 1-3 Hz would suffice, but i have zero idea how accurate such measurements can be. I would like to use the RF range (say, 100Mhz - 100GHz). is there some other way to get 1-3Hz accuracy in this range other than an FFT sampling at ~1THz?

hopefully the setup and my questions are relatively clear. just hoping for some input with some experience working with these sorts of things.

-gs

2. Jul 18, 2007

### bernhard.rothenstein

doppler effect and aberration

The Doppler formula in the case when the light signal propagates along a direction which makes a given angle A(A') with the line which joins source and receiver reads
f=f'g(V)(1+vcosA'/c).
I think that it is simpler to use the aberration of light formula
cosA=(cosA'+v/c)/[(1+(v/c)cosA')]
in order to solve the problem you propose.

3. Jul 18, 2007

### mgb_phys

You would have to know (and fix) the speed and direction of the source to the same accuracy which could be tricky.
Determining the frequency to 1 part in 1e9 is possible with a long enough sample time but building a source with an output this stable might be interesting.

4. Jul 18, 2007

### goodspeeler

thanks for the responses so far.

bernhard: i'm not sure i followed your explanation, though what i gathered was your agreement that it was possible?

mgb_phys: in my half-baked plan, the source speed and direction would be fixed. I was imagining several boxes sitting at defined positions around a room, each containing a microwave transmitter mounted on some kind of actuator that would move the transmitter side to side (or perhaps around in a vertically-oriented circle) at a defined speed and at a defined orientation. the receiver would then be the only thing moving around the room, and would interpret the frequency shifts from each of the transmitters (each on a slightly different frequency of course) to determine the angle with respect to each and then, in effect, triangulate the position.

after posting last night i came across http://www.freesoft.org/radar/Guardian/" [Broken] fun little link that mentioned that if you mix the shifted signals with the unshifted source signal, what you get out is actually the beat pattern that is at a frequency of exactly that shifted amount. so that seems to eliminate the needs for insanely high frequency sampling, as long as i can generate a reasonable facsimile of the known source frequency on my receiver to mix it with. that is, telling the difference between a 3Hz and a 6Hz shift now just requires telling the difference between a 3Hz and 6Hz signal. obviously, as you mentioned, I would still need the the source frequency and speeds to be stable. I don't have any clear notion of how difficult that would be. stable velocity doesn't seems like it would be prohibitively difficult, but I really have no idea what the typical frequency tolerances for microwave emitters are.

the one remaining factor would be the motion of the receiver itself. it would be small in magnitude relative to the actuator motion of the sources, but would still need to be accounted for. i haven't worked it out yet, but i imagine this can be done by coordinating between the various sources.

in any case, what i'm really hoping to do here is determine position using RF-frequency signals with millimeter or submillimeter accuracy. often people just use stationary transmitters and detect the drop in intensity and triangulate thusly, but for such accuracy this requires energy with suitably short wavelength which ends up being in the visible light range. problem with visible light range is that it will be blocked by anything sitting between the transmitters and the receiver. i was hoping that by relying on changes in frequency rather than intensity, i could eliminate the need to use such short wavelengths and thereby use signals like microwave that would penetrate obstacles and still reach the receiver.

anybody have any insight into whether using frequency in this way rather than intensity would in fact solve the accuracy problem? or is there some fundamental property of waves/energy i'm overlooking?

thanks again

-gs

Last edited by a moderator: May 3, 2017
5. Jul 18, 2007

### mgb_phys

A collegue of mine invented a system based on this 20years ago.
It uses a mobile receiver that constantly mixes the signal received from a number of fixed radio transmitters to calculate the relative distance from each. As the position of the transmitters is fixed you can work out the position of the mobile station. A big advantage was that it used commercial FM broadcast transmitters and could use phase methods to determine it's position to cm.

For various reasons to do with atmosphere to get very high accuracy you also had to a fixed receiver listening to the same signals and broadcast a small periodic correction to the mobile station - a little like Differential GPS.

The system was called "Cursor" from the company Cambridge Positioning Systems.

6. Jul 18, 2007

### BobG

Yes.

In fact, if you knew the direction and the velocity component directly towards you or away from you (the range rage), waited a little bit, took the direction of the signal and the range rate again, waited a little bit, and took the direction of the signal and the range rate again, you could compute the trajectory without even knowing the velocity beforehand.

In fact, tracking stations for satellites pass TEARR data (time, elevation, azimuth, range, and range rate) to the orbital analysts as soon as a newly launched satellite comes over the horizon, somewhere near mid pass, and right before the satellite goes over the horizon. It's not a terribly accurate trajectory, but accurate enough for the next tracking station to pick it up and it only takes a few widely separated observations to calculate the orbit for a newly launched satellite (which is never exactly what was planned since the rockets that boost the satellite into orbit never function perfectly).

Of course, even knowing the speed and the orbit, the satellite's location can't be predicted with millimeter or submilliter accuracy. For one thing, environmental perturbations are constantly changing the orbit. You'd run into the same problem, but even more so, for an object travelling on Earth. The object isn't going to maintain a constant speed, so, at best, you're only going to get an approximation for average speed, which means there's no way you're going to get the accuracy you want no matter what wavelength you use.

Last edited: Jul 18, 2007