1. Range & Doppler
Accuracy
by Ramya R

2. Doppler Effect Outline Definition Basic concept
Relation b/n Source and Observer
How a Doppler Radar works
Range and accuracy relation

3. Doppler Shift:
A frequency shift in electromagnetic waves due to the motion of scatters toward or away from the observer
Analogy: The Doppler shift for sound waves is the change in frequency one detects as race cars or airplanes approach and then recede from a stationary observer
Doppler Radar:
A radar that can determine the frequency shift through measurement of the phase change
that occurs in electromagnetic waves during a series of pulses

4. Basic Concepts
The apparent change in the frequency due to the relative motion between the source and the observer is known as Doppler effect.
Three main things are our consideration here
Source
Observer
Whether the source is moving towards or away or stationary from the observer

5. Stationary Sound Source
Sound waves are produced at a constant frequency f0, and the wavefronts propagate symmetrically away from the source at a constant speed v, which is the speed of sound in the medium. The distance between wavefronts is the wavelength. All observers will hear the same frequency, which will be equal to the actual frequency of the source.

6. Source moving with vsource < vsound
Here also the same sound source is radiating sound waves at a constant frequency in the same medium. However, now the sound source is moving to the right. The wave fronts are produced with the same frequency as before. However, since the source is moving, the center of each new wave front is now slightly displaced to the right. As a result, the wave fronts begin to bunch up on the right side (in front of) and spread further apart on the left side (behind) of the source. An observer in front of the source will hear a higher frequency f ´ > f0, and an observer behind the source will hear a lower frequency f ´ < f0.
 (Mach 0.7)

7. Source moving with vsource = vsound
 (Mach 1 – breaking the sound barrier)
Now the source is moving at the speed of sound in the medium. The speed of sound in air at sea level is about 340 m/s or about 750 mph. The wavefronts in front of the source are now all bunched up at the same point. As a result, an observer in front of the source will detect nothing until the source arrives.The pressure front will be quite intense (a shock wave), due to all the wavefronts adding together, and will not be percieved as a pitch but as a "thump" of sound as the pressure wall passes by.
Jet pilots flying at Mach 1 report that there is a noticeable "wall" or "barrier" which must be penetrated before achieving supersonic speeds. This "wall" is due to the intense pressure front, and flying within this pressure front produces a very turbulent and bouncy ride. 

8. Source moving with vsource > vsound
 (Mach Supersonic)
The sound source has now broken through the sound speed barrier, and is traveling at 1.4 times the
speed of sound .Since the source is moving faster than the sound waves it creates, it actually leads
the advancing wavefront. The sound source will pass by a stationary observer before the observer
actually hears the sound it creates.
. The figure shows a bullet travelling at velocity greater than velocity of sound. The mach cone and shock
wave fronts are very noticeable.

9. The picture at the left shows the shock wave front generated by a T-38 Talon, a twin-engine, high-altitude, supersonic jet trainer (below).
This picture shows a sonic boom created by the THRUST SSC team car as it broke the land speed record (and also broke the sound barrier on land).

10. How can we relate Doppler Effect to a RADAR ?
Doppler Radar:
A radar that can determine the frequency shift through measurement of the phase change that occurs in electromagnetic waves during a series of pulses
 If both the source and the receiver of the sound remain stationary, the receiver will hear
the same frequency sound produced by the source. This is because the receiver is
receiving the same number of waves per second that the
source is producing.

11. How can we relate Doppler Effect to a RADAR ?
Now, if either the source or the receiver or both move toward the other, the receiver will perceive a higher frequency sound. This is because the receiver will receive a greater number of sound waves per second and interpret the greater number of waves as a higher frequency sound.
Conversely, if the source and the receiver are moving apart, the receiver will receive a smaller number of sound waves per second and will perceive a lower frequency sound. In both cases, the frequency of the sound produced by the source will have remained constant

12. Calculation of Doppler frequency formulae
For example, the frequency of the whistle on a fast-moving car sounds
increasingly higher in pitch as the car is approaching than when the car is
departing.
Although the whistle is generating sound waves of a constant
frequency, and though they travel through the air at the same velocity in
all directions, the distance between the approaching car and the listener is
decreasing.

13. Calculation of Doppler frequency formulae
As a result, each wave has less distance to travel to reach the observer than the wave preceding it. Thus, the waves arrive with decreasing intervals of time between them.
fD =2·v/λ fD = Doppler Frequency [Hz] λ = wavelength [m] v = speed of the wave-source [m/s]
This equation is valid, if the speed if the source of a wave is like the radial speed. But the airplane usually flies in another direction than the direction towards to the radar. Only the radial speed is then also measured. However, this is different from the aim speed so that the following equation is valid:

15. Doppler Dilemma (Doppler ambiguity)
In pulse radar, the modulation of the carrier frequency is a periodic sequence of rectangular pulses.
The frequency spectrum of the transmitted signal is a comb-shaped line spectrum. The line spacing of the spectrum is equal to the pulse repetition frequency. These lines cannot be separated by a simple amplitude comparison.
The received frequency spectrum (subject to the Doppler Effect) can only be used for unambiguous velocity measurements when the displacement of the received spectrum is smaller than the line spacing in the spectrum i.e., the Doppler frequency must be lower than the pulse repetition frequency fPRF.

16. Doppler Dilemma (Doppler ambiguity)
Using the general equation for Doppler Frequency , we can calculate the range of the unambiguous radial speed

17. Doppler Dilemma
This equation is valid , if the direction of the Doppler shift is known ,i.e. it is known ,that the aim is moving towards or away from the radar site. If this direction is unknown the ambiguous value of the velocity is halved again
The pulse repetition frequency is a measure of the unambiguous range too. If the transmitter’s pulse width is much smaller than the pulse period, then the value of pulse repetition frequency i.e. fPRF can be substituted by the relation Co /2.Rmax

18. Doppler Dilemma
Now we can see , that the radars ability to measure an ambiguous range and unambiguous Doppler frequency (i.e. the aims radial speed) depends on the transmitter’s carrier frequency only.
In the absence of noise ,the peak of the signal term would give a time delay corresponding to the targets Range .
However this noise term will cause this peak to move around .In practical the noise term is not zero at correct time delay, the peak is displaced.
As a result the time delay of the peak will take a range of values and this noise can be related with the Gaussian noise with a mean value occurring a true delay .
Good range accuracy is not dependent on short pulse lengths, but needs large bandwidth :good Doppler accuracy needs long pulses.

19. Doppler Dilemma
Here the product of accuracies in range and velocity is inversely proportional to signal to noise ratio .This has an important interpretation that both accuracies can be improved simultameously and withoout limit by increasing the SNR.
For fixed SNR ,the product of the accuracies can be decreased by increasing the time X bandwidth product .