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Complex Number Vectors Complex numbers can also be represented as vectors This gives each complex number a unique magnitude and a unique direction The magnitude is the distance of the point a + jb from the origin 0 + j0 The direction is the angle of the vector, expressed counterclockwise from the positive realnumber axis This is shown in Fig 15-3 Absolute Value The absolute value of a complex number a + jb is the length, or magnitude, of its vector in the complex plane, measured from the origin (0,0) to the point (a,b) In the case of a pure real number a + j0, the absolute value is simply the real number itself, a, if a is positive If a is negative, then the absolute value of a + j0 is equal to a In the case of a pure imaginary number 0 + jb, the absolute value is equal to b, if b (a real number) is positive If b is negative, the absolute value of 0 + jb is equal to b If the number a + jb is neither pure real or pure imaginary, the absolute value must be found by using a formula First, square both a and b Then add them Finally, take the square root This is the length, c, of the vector a + jb The situation is illustrated in Fig 15-4 Problem 15-1 Find the absolute value of the complex number 22 j0 This is a pure real number Actually, it is the same as 22 + j0, because j0 = 0 Therefore, the absolute value of this complex number is ( 22) = 22 Problem 15-2 Find the absolute value of 0 j34 This is a pure imaginary number The value of b in this case is 34, because 0 j34 = 0 + j( 34) Therefore, the absolute value is ( 34) = 34.

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(8147)

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Note that receiver biases that are common to all channels, including receiver clock error, have been removed The residual ionospheric error denoted by ij Iro is typically small for baselines on the order of 10km Eqn (8147) can be solved using the approach described in Section 822 The main di erence is that the H matrix will now contain rows de ned by (hi hj ) where hi R3 is a vector pointing from satellite i to the receiver In contrast, to the GPS solution, H does not include a column for the receiver clock bias, because the receiver bias terms are canceled in the double-di erence operation At least three double di erences are required to estimate position This will still require at least four satellites While the noise on each satellite range measurement is independent, if any satellite is used in more than one double di erence, the double-di erencing operation results in the doubledi erenced ranges having correlated noise processes This measurement noise correlation must be accounted for if optimal ltering is expected The base station for a double-di erence implementation broadcasts the same information as required for a single-di erence base station The rover processing is the same except for the additional step of forming the double di erences from the single di erences and forming the double-di erence H matrix The main advantages of double-di erencing techniques is that clock modeling is not required and various small receiver error terms are removed The drawbacks are the potential processing of correlated measurements, and the possible increase in range noise The double-di erenced phase measurements formed from eqn (8141) are i,j = i ro (t) j ro (t) 1 1

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Problem 15-3 Find the absolute value of 3 j 4 In this number, a = 3 and b = 4 Squaring both of these, and adding the results, gives us 32 + ( 4)2 = 9 + 16 = 25 The square root of 25 is 5 Therefore, the absolute value of this complex number is 5

which by eqn (8145) has the error model ( i,j ) = (R(pr , pi ) R(pr , pj ) f2 ij ij ij I + ro + Nro (8148) f1 ro

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