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In the above equation we substitute A with 1 m2 B with 2m(b Cy) 2Cx C with Cx2 (b Cy)2 R2 If B2 4AC is greater than or equal to zero then there is a solution. The line touches the circle if it is zero (Line D) or fully intersects the circle at two points (Line B or C) if it is a positive number. In the above calculations the circle s radius and center coordinates will be the robot s radius (plus a buffer) and GPS coordinates. The line s equation will be calculated from the coordinates that de ne each line in the border as follows: The two points that de ne each line-segment at the border are X1, Y1 and X2, Y2. We calculate the slope of the line m = (Y2 Y1)/(X2 X1) and the intercept b Y1 mX1. So now we have the equation for the line and can do the calculations to determine if any boundary line-segment intersects the extended circle around the robot. However there is a problem. If you observe Fig. 9.5b you will see that it is possible to have some boundary lines appear to be crossing the robot s path.

11.1. Explain in your own words how error detection and error correction differ. Why would FEC normally be used on satellite circuits 11.2. A transmission takes place where the average probability of bit error is 10 6. Given that a message containing 108 bits is transmitted, what is the average number of bit errors to be expected 11.3. A transmission utilizes a code for which k codewords and how many datawords are possible 7 and n 8. How many

CHAP. 37]

11.4. A triple redundancy coding scheme is used on a transmission where the probability of bit error is 10 5. Calculate the probability of a bit error occurring in the output when (a) error detection only is used and (b) when error correction with majority voting is implemented.

A line de ned by the previous equation has in nite length. When we derived the equation for the boundary lines we did so to enable the above calculation, however, even though the in nite line may intersect the circle (Line B in Fig. 9.5a and also as in Fig. 9.5b), the segment that we are interested in (the actual boundary line) may not do so. It is necessary to check further to see if the intercept between the line and the circle is actually within the segment being considered. This is accomplished by checking to see if the calculated intercept x-coordinates are within the x-coordinates of the two points that de ne the line-segment. If both calculated intercept x-coordinates are outside the boundary line-segment then the boundary line does not intersect the circle even though its in nite extension does, as is about to occur in Fig. 9.5b (see the Line 47 in Fig. 9.6). Figure 9.6 shows how all of the above can be implemented in code. The math can be a little complex, but once the nal equations have been developed, the implementation is fairly straightforward.

5.3 Polarization of Satellite Signals As mentioned above, the directions horizontal and vertical are easily visualized with reference to the earth. Consider, however, the situation where a geostationary satellite is transmitting a linear polarized wave. In this situation, the usual definition of horizontal polarization is where the electric field vector is parallel to the equatorial plane, and vertical polarization is where the electric field vector is parallel to the earth s polar axis. It will be seen that at the subsatellite point on the equator, both polarizations will result in electric fields that are parallel to the local horizontal plane, and care must be taken therefore not to use horizontal as defined for terrestrial systems. For other points on the earth s surface within the footprint of the satellite beam, the polarization vector (the unit vector in the direction of the electric field) will be at some angle relative to a reference plane. Following the work of Hogg and Chu (1975), the reference plane will be taken to be that which contains the direction of propagation and the local gravity direction (a plumb line ). This is shown in Fig. 5.8. With the propagation direction denoted by k and the local gravity direction at the ground station by r, the direction of the normal to the reference plane is given by the vector cross-product: f k r (5.8)

1 2 is the angle in [0, ] whose cosine is , that is, . 2 3 whose tangent is 1, that is, . (c) tan 1 ( 1) = the angle in , 2 2 4 (b) cos 1 1 2

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