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You can always use a tool if you know what it does without necessarily knowing how it does it Just think of this subroutine as a tool that you can use You don t need to know the details of how it works as long as you are able to utilize it when you need to The basis for this routine is the fact that the robot s perimeter is a circle and we test each boundary line for intersection with this circle A line can intersect a circle either at two points or one point (tangent to the circle) If no boundary lines intersect the extended circle around the robot then there is no violation and Violation is set to false If any of the lines intersect (at either one or two points) Violation is set to true.

The combined power received by the two crossed dipoles will be maximum when the incoming wave is circularly polarized The average power received from a sinusoidal wave is proportional to the square of the amplitude Thus, for a circularly polarized wave given by either of Eq (54) or Eq (55), the power received from each component is proportional to E 2, and the total power is twice that of one component alone The crossed dipoles would receive this total A single dipole will always receive a signal from a circularly polarized wave, but at a loss of 3 dB This is so because the single dipole will respond only to one of the linear components, and hence the received power will be half that of the crossed dipoles.

37.3 Find formulas for the inverses of the following one-one functions: (a) f (x) = 10x 4 (b) f (x) = 3ex + 1

Again, because of the symmetry of the circularly polarized wave, the dipole need only lie in the plane of polarization; its orientation with respect to the xy-axes is not a factor A grid of parallel wires will reflect a linear polarized wave when the electric field is parallel to the wires, and it will transmit the orthogonal wave This is illustrated in Fig 56 This is used in one type of dual polarized antenna, illustrated in Fig 57 Here, the grid allows the wave, the electric field of which is transverse to the wires to pass through, whereas it reflects the parallel (Ev) wave The reflector behind the grid reflects the wave that passes through Thus two orthogonal, linearly polarized waves, having high polarization isolation (see Sec 54) are transmitted from the antenna system.

The intersection is not a physical one since the lines do not actually exist The intersections are mathematically calculated as we show below In order to implement this idea we need to know about quadratic equations, slopes, intercepts, and circle formulas The equation of a line is Y mX b.

Some details of the construction of this type of antenna will be found in Maral and Bousquet (1998)..

(a) Let y = 10x 4. Then, y + 4 = 10x Hence, (b) Let y = 3ex + 1. Then, y 1 = 3ex Hence, or y 1 = ex 3 f 1 (y) = ln y 1 3 or ln y 1 =x 3

where m is the slope of the line and b is the y-axis intercept. The equation of a circle is (X Cx )2 (Y Cy )2 R2

the total number of codewords is 2n. When a given dataword is transmitted, an undetected error results when the transmission errors convert the received codeword into one that contains a permissible dataword but not the one that was transmitted. The number of such datawords is 2k 1. An upper bound on the probability of an error getting through can be made by assuming that all codewords are equiprobable. The ratio of number of possible error words to total number of codewords then gives the average probability of error. In practice, of course, all the codewords will not be equiprobable, those containing datawords being more probable than those which do not. The ratio, therefore, gives an upper bound on the probability of error: BER # 2k 2 1 2n 2sn2kd ,2 (11.28)