By M. Schetzen

The booklet starts off with a simple dialogue of the Doppler impression and its a variety of functions, and the way Doppler radar can be utilized for the stabilization and navigation of plane. A quasi-static approximation of the Doppler spectrum is gifted in addition to illustrations and discussions to aid the reader achieve an intuitive figuring out of the approximation and its barriers. A precis of the mathematical techniques required for improvement of an actual idea is then offered utilizing the case of a slim beam antenna. this is often by way of the improvement of the precise thought for the overall case, that is graphically illustrated and in comparison with the quasi-static approximation. normal stipulations for which the quasi-static approximation blunders will be over the top – particularly as utilized to laser Doppler radars and low-flying plane – are presented.

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**Sample text**

9). To reduce the algebraic manipulations, the port and starboard beams were chosen to be at right angles to the aircraft. However, note that if the drift angle fd is small, then there could be unacceptable error because then dp and dS will be small. In practice, it thus may be desirable to choose jca j , p=2 radians. Note that the plane can be kept on level ﬂight by using the control described above to hold the pitch angle at zero degrees. Observe that a rise of the terrain is equivalent to a nose down of the aircraft and a fall of the terrain is equivalent to a nose up of the aircraft.

2 depicts the case for which the drift angle is positive. The velocity v is the aircraft ground speed, which is the aircraft velocity relative to the ground. Because there are four quantities to determine, we require at least four Doppler measurements. For this, the airborne Doppler radar is designed with four antennae. To begin, assume the aircraft roll, pitch, and drift are zero. Then, referring to Fig. 1, let one antenna beam be directed toward the terrain fore of the aircraft with the angles cr ¼ ue and ca ¼ 0; let a second antenna beam be directed toward the terrain aft of the aircraft with the angles cr ¼ Àue and ca ¼ 0; let a third antenna beam be directed toward the terrain on the starboard side of the aircraft with the angles cr ¼ ua and ca ¼ p=2; and let a fourth antenna beam be directed toward the terrain on the port side of the aircraft with the angles cr ¼ ua and ca ¼ Àp=2.

The Doppler shifts in hertz thus are: 1) For the fore antenna beam, df ¼ 2v sin (ue þ f p ) cos (fd ) l hertz (3:9a) 2) For the aft antenna beam, 2v sin (Àue þ f p ) cos (fd ) l 2v ¼ À sin (ue À f p ) cos (fd ) l da ¼ hertz (3:9b) 3) For the starboard antenna beam, p 2v sin (ua þ fr ) cos þ fd l 2 2v ¼ À sin (ua þ fr ) sin (fd ) hertz l dS ¼ (3:9c) 4) For the port antenna beam, p 2v sin (ua À fr ) cos À þ fd l 2 2v ¼ sin (ua À fr ) sin (fd ) hertz l dp ¼ (3:9d) Using standard trigonometric identities we then obtain 4v cos fd sin f p cos ue l 4v d f À da ¼ cos fd cos f p sin ue l 4v d p þ dS ¼ À sin fd sin fr cos ua l 4v d p À dS ¼ sin fd cos fr sin ua l d f þ da ¼ (3:10a) (3:10b) (3:10c) (3:10d) We then obtain by dividing one equation by another tan f p ¼ d f þ da tan ue d f À da (3:11a) 36 AIRBORNE DOPPLER RADAR tan fr ¼ À d p þ ds tan ua d p À ds (3:11b) tan fd ¼ À d p þ ds cos ue sin f p d f þ da cos ua sin fr (3:11c) We now can determine the pitch angle fp from Eq.