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where S is the Doppler shift, in hertz R is the radius of rotation, in meters is the angular velocity of the antenna, in radians per second Fc is the carrier frequency of the incoming signal, in hertz c is the velocity of light (3 108 m/s) In theory this antenna works nicely, but in practice there are problems One of the big problems is getting a large enough Doppler shift to easily measure Unfortunately, the rotational speed required of the antenna is very high too high for practical use However, the effect can be simulated by using a number of antennas, arranged in a circle, that are sequentially scanned The result is a piecewise approximation of the effect seen when the antenna is rotated at high speed
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One of the problems associated with small RDF antennas is that they have such a small aperture that relatively large distortions of their pattern result from even small anomalies Follow that What it means is that the pattern is all messed up by small defects If you build a wide-aperture direction finder (WADF), however, you can average the signals from a large number of antenna elements distributed over a large-circumference circle The Wullenweber array (Fig 23-14) is such an antenna It consists of a circle of vertical elements In the HF band the circle can be 500 to 2000 ft in diameter A goniometer rotor spins inside the ring to produce an output that will indicate the direction of arrival of the signal as a function of the position of the goniometer The theoretical resolution of the Wullenweber array is on the order of 01 , although practical resolutions of about 28 are commonly seen
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If you erect two antennas with distance d apart, then arriving signals can be detected by examining the time-of-arrival difference Figure 23-15A shows an example signal If the advancing wavefront is parallel to the line between the antennas, then it will arrive at both antennas at the same time The TDOA is zero in that case But if the signal arrives at an angle (as in Fig 23-15A), it will arrive at one antenna first From the difference between the time of arrival at the two antennas we can discern the direction of arrival There is an ambiguity in the basic TDOA array in that the combined output will be the same for conjugate angles, ie at the same angle from opposite directions This problem can be resolved by the system shown in Fig 23-15B The signals from ANT1 and ANT2 are designated V1 and V2, respectively These signals are detected by receivers (RCVR1 and RCVR2), and are then threshold detected in order to prevent signal-to-noise problems from interfering with the operation The outputs of the threshold detectors are used to trigger a sawtooth generator that controls the horizontal sweep on an oscilloscope The two signals are then delayed, and one is inverted The reason for inverting one signal is to allow the operator to distinguish them on the CRT screen The reason why this is necessary is the ambiguity If the antennas in Fig 23-15A are arrayed east to west, then the line perpendicular to the line between them is north to south If we designate north as 0 , then the signal shown arrives at an angle of 330 That means it will arrive at ANT1 before it arrives at ANT2 A signal arriving from a bearing of 30 will produce the same output signal, even though it arrives at ANT2 before ANT1 All signals of bearing 0 x < 180 will arrive at ANT2 first, while all
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