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Remember the best location to observe VLF signals is out in the countryside, in an area where few cars or trucks are likely to pass You will want to get as far away from power lines as possible for optimum results The lower the power line noise is, the stronger the VLF emission signals will be Some whistlers are related phenomena and so short-lived and impossible to predict that you might want to consider using a tape recorder to record the signals in the field unattended Take a portable, battery-operated, cassette tape recorder with you and use a long audio cable to run to the tape recorder from the audio output of the VLF receiver Any tape recorder with an external microphone input can be used Best results are obtained with a recorder which has no ALC or automatic level control The VLF receiver shown has an output jack which can be coupled to a recorder input for field recording You may have to place the tape

In this chapter we ll solve a three-by-three linear system That s a triplet of linear equations in three variables There are numerous ways to tackle problems of this sort We ll look at only one example and one method The scheme presented here can be broken down into three major steps: Eliminate one variable to get a two-by-two system Solve that two-by-two system for both of its variables Solve for the original eliminated variable by substitution

For deterministic systems, Section 36 discussed the problem of state estimation This section considers state estimation for stochastic, linear, discrete-time state space systems xk+1 yk = = k xk + k uk + k Hk xk + k

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Consider three linear equations, each having three variables: x, y, and z Our mission: Find the numbers for x, y, and z that satisfy all three equations Here is the system we ll solve in this chapter: 4x + 2y 3z = 5 2x 5y = z 1 3x = 6y + 7z

48 LINEAR STATE ESTIMATION The state estimate is computed according to k+1 x yk k x+ k = k x+ + k uk k = Hk x k = x + Lk yk

15

Choose the vanishing variable Let s decide which variable we want to eliminate It doesn t make any difference whether it s x, y, or z If we do all the calculations right, we ll get the same answer in the end, no matter which variable we choose at this stage Let s get rid of z, so we are left with two equations in x and y

where yk = yk yk is called the measurement residual and Lk is a k designer speci ed gain vector The notation x indicates the estimate of x at time t = kT before correcting the estimate for the information in k the measurement yk and x+ denotes the estimate of x after correcting the estimate for the information in the measurement yk Similar interpretations apply to P , P+ , yk , and P x x y De ning the state estimation error vector at time k, prior and posterior to the measurement correction, as

Get the equations into form Now that we ve chosen the variable to eliminate, we must get all three equations into the same form The following form is as good as any:

ax + by + cz = d where a, b, c, and d are constants The first equation is already in this form The second equation is 2x 5y = z 1 Subtracting z from each side will put it into the form we want: 2x 5y z = 1 The third equation is 3x = 6y + 7z Adding 6y to each side, we get 3x + 6y = 7z We can subtract 7z from each side and it comes into the sought-after form: 3x + 6y 7z = 0 We now have the three-by-three system in this uniform condition: 4x + 2y 3z = 5 2x 5y z = 1 3x + 6y 7z = 0

k xk x k xk x+ ,

U1 Induction Loop Transmitter I TX Mono amplifier or receiver 5 10 watts output 10 ohm 10 watt 10% resistor loop coil around room (see text) HS L1

Make z vanish once Here are the first two revised equations again, for reference:

(4120) (4121)

Let s multiply the second equation through by 3, getting 6x + 15y + 3z = 3 If we add this to the first equation, we obtain the sum 4x + 2y 3z = 5 6x + 15y + 3z = 3 10x + 17y = 8

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