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You can build your own Sudden Ionospheric Disturbances (SIDs) receiver, shown in Figure 19-1, and begin your own investigation of solar flares and their effects on radio propagation The SIDs receiver is a simple VLF receiver designed to be used with a loop antenna which can be placed either inside or outside The receiver monitors the strong VLF signal from the US Navy NAA 24 kHz transmitter in Cutler Maine The Sudden Ionospheric Disturbance (SID) receiver is a modified Stokes Gyrator tuned VLF receiver based on the circuit from Communications Quarterly Spring 1994 pp 24 26 The main circuit diagram shown is Figure 19-2, it illustrates SIDs receiver which consists of two integrated circuits, a Texas Instruments RC4136 and a National LF353 operation amplifier The SIDs receiver is fed from a loop antenna at L1 The antenna is coupled to the SIDs receiver via a miniature 600 to 600 ohm matching transformer at T1 The output from the secondary of T1 is fed to two protection diodes at D1 and D2 at the front end of the receiver The output of the diode network is then coupled to a 100 pF capacitor at C1 Capacitor C2 is connected from the

Two down, one to go! We ve now solved for two of the three unknowns in our three-by-three system We have the values for x and y:

When the Kalman gain is used, a simpli ed equation can be derived for the posterior error covariance matrix P+ The derivation uses the fact that Kk Hk P Hk + Rk = P Hk k k (515)

x = 209/223 and y = 18/223 In the next section, we ll substitute these values back into one of the original three equations and solve for z Then we ll check our work Something tells me that z is going to be a fraction with a denominator of 223 What do you think

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which is easily derived from eqn (514) The derivation proceeds from eqn (513) as follows: P+ = P KHP + KRK + KHP H K P H K = [I KH] P + K R + HP H K P H K = [I KH] P + P H K P H K = [I KH] P (516)

The sunset pattern is the reverse An interesting feature of waveguide mode propagation was that the signal was split into two components which can form an interference pattern

You might again question the choice of solution processes Why, you might ask, do we use the doubleelimination method to solve the two-by-two system here Why not use morph-and-mix or rename-andreplace The answer is, of course, We can use either of those methods

Here s a challenge!

Note that the nal expression in eqn (516) is only valid for the Kalman gain K, while (512) is valid for any estimator gain vector L

Solve the preceding two-by-two system using the morph-and-mix method Consider x the dependent variable

Summary

11 12 7 V

Once again, here s our pair of equations: 10x + 17y = 8 and 11x + 41y = 7 We must get both of these into SI form, with x all by itself on the left sides of the equals signs Step-by-step, the first equation morphs like this: 10x + 17y = 8 10x = 17y + 8

This section has presented a derivation of the Kalman gain formula shown in eqn (514) and has derived a simpli ed formula for the posterior error covariance matrix that is shown in eqn (516) The Kalman gain is the time-varying gain sequence that minimizes the mean-square state estimation error at each measurement instant Computation of the Kalman gain sequence requires computation of the prior and posterior error covariance matrices The prior covariance P is used directly in eqn (514) The k posterior covariance P+ is used to compute the P for use with the next k k+1 measurement The derivation of this section is short and clearly shows the meansquared optimality The following section presents an alternative derivation that obtains the same results The alternative derivation is considerably longer, but also provides additional insight into the operation of the Kalman lter

10x = 17y 8 x = (17/10)y 8/10 The second equation morphs as follows: 11x + 41y = 7 11x = 41y + 7 11x = 41y 7 x = (41/11)y 7/11 Now we mix the right-hand sides, getting a first-degree equation in y: (17/10)y 8/10 = (41/11)y 7/11

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