Re: How does the police know how fast drive by car, using radar control

Police radar uses the "doppler effect" to measure velocity. A huge amount of information about police radar can be found at http://www.eeel.nist.gov/810.02/radar/cover.htm#MainIndex

You are probably aware that doppler effect causes the pitch of sound received from a moving object to be different than the pitch of the sound actually emitted. The pitch difference is the doppler shift, and measured as

df = -f0*v/c

where f0 = frequency actually emitted (from the emitter's frame of reference), v = the relative speed velocity between the observer and emitter, and c = speed of propagation of sound.

df describes the apparent frequency shift in the sound between the emitter's and the observer's reference frames. The sign convention chosen indicates that objects moving away from the observer have a decrease in observed pitch.

For example, a train whistle tuned to concert "A" (440 Hz) would be shifted downwards by about 37 Hz (making it just about F in the octave below) if the train were moving away from us at about 100 km/hr (61 mph), assuming the speed of sound is about 335 m/s (1100 ft/s). Conversely, if we knew the train whistle is tuned to "A", and perceived it to be at "F", we could deduce the train to be moving away from us at about 100 km/hr.

The same equation for doppler shift holds true for radar with some subtle modification: An additional factor of 2 is required because the radio waves from the radar are making a full round trip from the transmitter to an object and back. Therefore the total round trip distance that the radio waves must travel is 2*R (where R is the one-way range). The "v" in the equation above is actually intended to represent the time rate of change of the total path length. For an object moving with speed v away from the radar transmitter, the total path length is changing by 2*v.

So, what the radar does is emit a pulse at frequency f0 and measure the frequency of the reflected pulse that's eventually returned to the radar. As far as I know, it does not measure the time of flight (i.e, range to the object reflecting the radar signal)-- just velocity. We can combine these equations above to express measured velocity as a function of doppler frequency:

v = -0.5*df*c/f0

That is, for a radar frequency of 10 GHz and an observed frequency shift of +2000 Hz, we can estimate the object's speed as 108 km/hr (67 mph) towards us. Note that radio waves travel through air about c = 3*10**8 m/s.

This explanation obviously applies to stationary radars and moving objects. When the radar is moving also, (say, in a moving police car) a little more work is required, but the physics is almost all done. The radar will measure returns from all objects in its field of view. The biggest of these tends to be the stationary objects (buildings, bridges, etc). So the radar measures the speed of the strongest reflection to estimate its own speed. This speed is used as a reference against all other speed measurements are adjusted. For example, let's say the radar measures a strong return at +100 km/hr and a relatively weaker return at +20 km/hr. Control logic in the radar would conclude that the radar itself is traveling at 100 km/hr (62 mph) and that the object giving the weaker return is actually traveling at 120 km/hr (75 mph).

This leads to a source of error known as "cosine error". The relative speed that the radar measures is equal to the algebraic difference in speeds between the radar and moving object multiplied by a "direction cosine". (More succintly: the relative speed bewtween two objects is measured as dot the product of the vector velocity difference between them with a unit direction vector from one object to the other).

The practical result is that stationary radar underestimates the speed of moving objects, while moving radar underestimattes its speed relative to the ground, which *may* cause the speed of other moving objects to be overestimated. That is, the cosine error for stationary radar always works in the motorist's favor, while the cosine error for moving radar may work for or against the motorist.

So long, and happy motoring!

Steve Czarnecki