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This is a twist we haven t seen yet! One of the coefficients is imaginary 6 Once again, for reference, let s state the general polynomial standard form of a quadratic equation: ax 2 + bx + c = 0 In the equation we derived in solution to Prob 5, we have a = 1, b = j4, and c = 21 Plugging these values into the quadratic formula and then working out the arithmetic, we get x = [ b (b2 4ac)1/2] / (2a) = { ( j4) [( j4)2 4 1 21]1/2} / (2 1) Now let s be careful with ( j 4)2 This is the sort of expression that can easily cause us to make a mistake We can break it down like this, and then solve, being careful with the signs: ( j4)2 = ( 1 j 4)2 = ( 1)2 j 2 42 = 1 ( 1) 16 = 16

As will be discussed in Section 49 and 7, navigation systems can often be designed in such a way as to allow performance analysis based on linear systems with stochastic inputs Therefore, this section considers the statistical properties of the output of a linear system when the system model is known and the input is white The results of this section are often used to model error components with signi cant time correlation via the state augmentation approach In such situations, linear stochastic systems forced by Gaussian white noise are of special interest Careful selection of the linear system and white noise statistics can often yield an output stochastic process with statistics matching those of the time correlated error process Examples of such processes are further discussed in Section 463 The output y(t) of a (deterministic) time-invariant, causal, linear system with (stochastic) input w(t) is

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Now let s substitute back in where we left off That gives us an expression that s still tricky But it can be simplified like this: x = [ ( j4) ( 16 4 1 21)1/2] / (2 1) = [j4 ( 100)1/2] / 2 = (j4 j10) / 2 This breaks down to x = (j4 + j10)/2 which simplifies to x = j14/2 and further to x = j7 or x = j3 or x = j( 6)/2 or x = (j4 j10)/2

An antenna for your new aircraft receiver can be as simple as a 21 piece of wire, an extendable whip antenna or a roof-mounted ground-plane aviation antenna Most folks near an airport will get plenty of inthe-air action from a wire or whip antenna, but if you re more than a few miles away, a decent roof-mount antenna is the way to go Radio Shack sells an ideal antenna designed for scanners which covers the aircraft band nicely and it costs around $30 A low cost TV antenna works well, even better if rotated 90 degrees (remember aircraft antennas are vertically polarized)

These are the roots we chose in Prob 4 to manufacture the quadratic 7 We ve been told to find the roots of this quadratic: (x + 2 + j3)(x 2 j3) = 0 We can convert this pair of trinomial factors to a pair of binomial factors by changing the first subtraction in the second factor to addition, and also by grouping the terms within the factors, as follows: [x + (2 + j3)][x + ( 1)(2 + j3)] = 0 which can be rewritten as [x + (2 + j3)][x (2 + j3)] = 0 Now we have an equation in binomial factor form, where the factors both consist of the variable x plus or minus a numerical constant The roots can therefore be found by solving the following two first-degree equations: x + (2 + j3) = 0 and x (2 + j3) = 0 In the top equation, we can subtract the quantity (2 + j3) from each side, getting x = (2 + j3) = 2 j3

y(t) =

Do you notice something familiar about the transformation we just performed, as if we ve done it before We have! These maneuvers are the equivalent of mirroring the whole system along the axis corresponding to the line where the values of the independent variable and the values of the dependent variable are identical (In this case, that s the line w = v) That transformation produces the graph of the inverse of a relation, as you learned in Chap 14 Look at the SI versions of the equations we graphed in Fig 17-5 For reference, here they are again: w = 7v 10 and w = ( 1/2)v 5

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