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EW Applications of Spherical Trigonometry.

In response to reader requests, this is the second of a two-part series on spherical trigonometry (trig). These are three typical applications of spherical trigonometry applied to EW.

Elevation-Caused Error in Azimuth-Only DF System

A direction-finding (DF) system is designed to measure only the azimuth of arrival of signals. However, signals can be located out of the plane in which the DF sensors assume the emitter is located. What is the error in the azimuth reading as a function of the elevation of the emitter above the horizontal plane?

This example assumes a simple amplitude-comparison DF system. DF systems measure the true angle from the reference direction (typically the center of the antenna baseline) to the direction from which the signal arrives. In an azimuth-only system, this measured angle is reported as the azimuth of arrival (by adding the azimuth of the reference direction to the measured angle).

As shown in Fig 1, the measured angle forms a right spherical triangle with the true azimuth and the elevation. The true azimuth is determined as follows:

cos (Az) = cos(El)/cos(M) (eqn 1)

The error in the azimuth calculation as a function of the actual elevation is then as follows:

Error = M - acos[cos(El)/cos(M)] (eqn 2)

Doppler Shift

Both the transmitter and receiver are moving. Each has a velocity vector with an arbitrary orientation. The Doppler shift is a function of the rate of change of distance between the transmitter and the receiver. To find the rate of change of range between the transmitter and the receiver as a function of the two velocity vectors, it is necessary to determine the angle between each velocity vector and the direct line between the transmitter and the receiver. The rate of change of distance is then the transmitter velocity times the cosine of this angle (at the transmitter) plus the receiver velocity times the cosine of this angle (at the receiver).

Let's place the transmitter and receiver in an orthogonal coordinate system in which the y axis is north, the x axis west and the z axis up. The transmitter is located at [X.sub.T], [Y.sub.T] [Z.sub.T]; and the receiver is located at [X.sub.R], [Y.sub.R], [Z.sub.R]. The directions of the velocity vectors are then the elevation angle (above or below the x,y plane) and azimuth (the angle clockwise from north in the x, y plane), as shown in Fig 2. We can find the azimuth and elevation of the receiver (from the transmitter) using plane trigonometry.

[az.sub.R] = Atan [([X.sub.R]-[X.sub.T])/([Y.sub.R]-[Y.sub.T])] (eqn 3)

[El.sub.R] = atan{([Z.sub.R]-[Z.sub.T])/SQRT[[([X.sub.R]-[X.sub.T]).sup.2] + [([Y.sub.R]-[Y.sub.T]).sup.2]} (eqn 4)

Now consider the angular conversions at the transmitter, as shown in Fig 3. This is a set of spherical triangles on a sphere with its origin at the transmitter. N is the direction to north; V is the direction of the velocity vector; and R is the direction toward the receiver. The angle from north to the velocity vector can be determined using the right spherical triangle formed by the velocity-vector azimuth and elevation angles. Likewise, the angle from north to the receiver can be determined from the right spherical triangle formed by its azimuth and elevation:

cos(d) = cos([Az.sub.V]) cos([El.sub.V]) (eqn 5)

cos(e) = cos ([Az.sub.R]) cos([El.sub.R]) (eqn 6)

Angles A and B can be determined from:

ctn(A) = sin([Az.sub.V])/tan([El.sub.V]) (eqn 7)

ctn(B) = sin([Az.sub.R])/tan([El.sub.R]) (eqn 8)

C = A-B (eqn 9)

Then, from the spherical triangle between N, V and R, using the law of cosines for sides, we find the angle between the transmitter's velocity vector and the receiver:

cos(VR) = cos(d) cos(e) + sin(d) sin(e) cos(C) (eqn 10)

Now, the component of the transmitter's velocity vector in the direction of the receiver is found by multiplying the velocity by cos(VR). This same

operation is performed from the receiver to determine the component of the receiver's velocity in the direction of the transmitter. The two velocity vectors are added to determine the rate of change of distance between the transmitter and receiver ([V.sub.REL]). The Doppler shift is then found from the following:

(f = f [V.sub.REL]/c (eqn 11)

Observation Angle in 3-D Engagement

Given two objects in three-dimensional (3-D) space, T is a target, and A is a maneuvering aircraft. The pilot of A is facing toward the roll axis of the aircraft, sitting perpendicular to the yaw plane. What are the observed horizontal and vertical angles of T from the pilot's point of view? This is the problem that must be solved to determine where a threat symbol would be placed on a head-up display (HUD).

Fig 4 shows the target and the aircraft in the 3-D gaming area. The target is at [X.sub.T], [Y.sub.T], [Z.sub.T]; and the aircraft is at [X.sub.A], [Y.sub.A], [Z.sub.A]. The roll axis is defined by its azimuth and elevation relative to the gaming-area coordinate system. The azimuth and elevation of the target from the aircraft location are determined as in eqns. 3 and 4 by the following:

[Az.sub.T] = atan[([X.sub.T]-[X.sub.A])/([Y.sub.T]-[Y.sub.A])] (eqn 12)

[El.sub.T] = atan{([Z.sub.T]-[Z.sub.A])/SQRT[[([X.sub.T]-[X.sub.A]).sup.2] + [([Y.sub.T]-[Y.sub.A]).sup.2]]} (eqn 13)

The two right spherical triangles and one spherical triangle of Fig 5 allow the calculation of the angular distance from the roll axis and the target (j):

cos(f) = cos ([Az.sub.T]) cos([El.sub.T]) (eqn 14)

cos(h) = cos ([Az.sub.R]) cos([El.sub.R]) (eqn 15)

ctn (C) = sin([Az.sub.T])/tan([El.sub.T]) (eqn 16)

ctn(D) = sin([Az.sub.R])/tan([El.sub.R]) (eqn 17)

J = 180[degrees] - C-D (eqn 18)

cos(j) = cos(f) cos(h) + sin(f) sin(h) cos(J) (eqn 19)

The angle E is then determined:

ctn(E) = sin ([El.sub.R])/tan ([Az.sub.R]) (eqn 20)

The angle F is determined from the law of sines:

sin(F) = sin(J) sin f/sin(j) (eqn 21)

Then the offset angle of the threat from the local vertical at the aircraft is given by the following:

G = 180[degrees] - E - F (eqn 22)

Finally, as shown in Fig 6, the location of the threat symbol on the HUD is a distance from the center of the display representing the angular distance (j), and an offset from vertical on the HUD by the sum of angle G and the roll angle of the aircraft from vertical.
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Article Details
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Author:Adamy, Dave
Publication:Journal of Electronic Defense
Article Type:Brief Article
Geographic Code:1USA
Date:Apr 1, 2000
Previous Article:A Survey of Microwave Amplifiers and Oscillators.
Next Article:Analog Module.

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