Soon after Doppler Effect discovery in 1842 (1) scientists started attempts to use wave measurements to detect objects in the fog. First measurements and first patents were based on continuous wave frequency measurements (4). In 1934 the first patent based on pulsed technique measurements was issued in the USA (3). Modern devices, such as GPS, radars, etc. use both pulsed technique and frequency change approach (5). For the stationary objects the pulsed method can provide distance to the object according to the formula D=c·Δt, where D—is distance, c—is the speed of the wave and Δt—is the time interval. In order to measure the speed of the object two or more distances should be measured over a specific time interval. For many cases the time of the measurements is too long and the resulting error prevents this method from successful implementation. If the object changes the speed or direction of its movement, this approach becomes useless. In GPS the best time measurements have an accuracy of approximately 100 nsec and produce a distance measurement accuracy of ΔD≧10−7·3·108=30 m (5). It means that the initial error is 30 meters or more. Combining the data from several satellites and following several mathematical iterations, that error can be decreased down to several millimeters, but such procedure requires significant amount of time. The approach, known as the differential GPS, or DGPS (5), as well as the incorporation of the data from weather stations and cellular transmitters can improve the result down to several meters per several seconds. For the stationary radars and relatively small speeds, such an approach can produce satisfactory results. But a plane can change its position more than 300 meters in one second. This approach cannot produce speed measurements with any reasonable accuracy especially for aviation, marine, or military applications.
The measurements of the wave frequency (or phase) change with the current technology theoretically are supposed to produce accuracy of the position measurements down to 3 mm, or 10,000 times better than the pulse method. Unfortunately, in practice it is not true. But since there is no other way to measure the instant speed and acceleration of the object, both of these methods are used (5). The time and the position resolution of the modern GPS could be considered satisfactory for city traffic, but not for marine, aviation or military application. Taking into account the absence of the cellular network in the oceans, it is clear that the error in the case of the fast boats somewhere in the ocean away from their ports will also be large.
The measurements of the frequency or phase shift of the wave and the calculations of the speed and the position of the object are currently performed according to the Doppler Effect formula
These types of measurements are supposed to produce more accurate results, but the experiments show that this is not true. Attempts to incorporate Einstein's relativistic corrections in the form of
did not improve the accuracy of that approach.
The Durandin method is based on the principle of the independence of the initial distance between the wave emitter and the moving object from the speed of the object and from the angle between the speed of the object and the direction of the wave propagation, while the Doppler Effect method is based on the projection of the speed vector of an object onto the direction of the wave propagation. The Durandin method produces a different result for the wave frequency and phase changes for the non-collinear movement of the wave and the object.
In the Durandin method the frequency change should be calculated according to the formula
for the system of coordinates associated with the wave emitter. For the GPS, when the satellites with the wave emitters are moving, the appropriate coordinates' transformations should be taken into account. Transformations also should be done for any measurements, where the wave emitter is moving, like the radar on the plane.
The definition of the angle δ in the Durandin method is different from the angle θ in the Doppler Effect method.
There are actually two important points of difference between these angles.
The GPS satellite was used as an illustration of the difference between the Doppler and the Durandin methods. The same approach can be used for a plane during evasion maneuver, or other objects, moving with acceleration, taking into account appropriate coordinates transformations.
If both the wave emitter and the object move along the same line, then cos θ=−1, when the direction of the movement is away from each other and cos θ=1, when the direction of the movement is toward each other. The Durandin formula brings us back to the Doppler formulas
for the collinear movement. In all other cases the formulas for the Durandin method are different from the Doppler Effect formulas.
In case of
For the case of the
The
should be used. This case has special importance for the time synchronization between the satellite and the ground station, because of minimum distortions from ionosphere and troposphere, when the wave propagates in the orthogonal direction. Durandin method shows that the position of the satellite, when the observed frequency is equal f=f0 or has closest value to the emitted frequency is shifted from perpendicular position (on the radius) along the direction of the satellite movement and that shift can be calculated from the Durandin method.
Advantages (improvements) of the Durandin method over existing methods are as follow:
1. The Durandin method and algorithm are more accurate and they substitute Doppler Effect method and algorithm.
2. The Durandin method does not need any relativistic corrections for speed, distance and time.
2. The Durandin method produces accurate results for curved trajectories of the object movements as well as the movements with a positive or negative acceleration. This feature is essential for GPS systems, where satellites move with rotational acceleration. The changes in two or more frequencies of the emitted waves produce the relative speed of the wave emitter and the moving object as well as the angle δ.
3. The frequency measurements can be used for more accurate time synchronization, than with the time synchronization with an atomic clock.
4. The measurements of the phase and frequency changes at the doubled frequency produce an instant value for the acceleration of the object. These include any changes in the direction of the objects movements. The straight measurements of the derivative of the signal are always more accurate than the measurements of the signal and the subsequent differentiation of the signal.
5. The combination of the Durandin method and the pulsed method can produce (approximately 1000 times) more accurate time synchronization than the currently available methods.
5. U.S. Pat. No. 6,469,663 B1; Oct. 22, 2002. Method and system for GPS and WAAS carrier phase measurements for relative positioning.
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