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Let a be the coefficient of x 2, let b be the coefficient of x, and let c be the stand-alone constant Then a = 2, b = 0, and c = 2 The quadratic formula tells us that x = [ b (b2 4ac)1/2] / (2a) = [0 (02 4 2 2)1/2] / (2 2) = ( 16)1/2 / 4 = j4 / 4 = j The roots are x = j or x = j We can plug these into either of the original functions to get the y-values This time, let s use the quadratic function When x = j, we have y = 2j 2 3j + 3 = 2 ( 1) j3 + 3 = 2 j3 + 3 = 1 j3 When x = j, we have y = 2( j)2 3( j) + 3 = 2 ( 1) + j3 + 3 = 2 + j3 + 3 = 1 + j3 The solutions to the system are (x,y) = [j,(1 j3)] and (x,y) = [ j,(1 + j3)] 4 After we plug in the values, let s convert subtractions to negative additions to be sure we keep the signs in order First, let s check [j,(1 j3)] in the original linear equation: 3x + y 1 = 0 3j + (1 j3) 1 = 0 j3 + 1 + ( j3) + ( 1) = 0 j3 + ( j3) + 1 + ( 1) = 0 0=0 Next, let s check [ j,(1 + j3)] in that equation: 3x + y 1 = 0 3( j) + (1 + j3) 1 = 0 j3 + 1 + j3 + ( 1) = 0 j3 + j3 + 1 + ( 1) = 0 0=0

We can break this down step-by-step, paying careful attention to signs and using parentheses when we need them Here we go: 5 + ( 3) = 5 3 = 2 2 ( 6) = 2 + 6 = 8 8 10 = 2 2 + 14 = 12 12 ( 21) = 12 + 21 = 33

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where w(t) and v(t) are random variables The random variable w is called the process noise The random variable v is called the measurement noise At each time, x(t) and y(t) are also random variables The designation random variable implies that although the value of the random variable at any time is not completely predictable, the statistics of the random variable may be known Assuming that the model is constructed so that v(t) and w(t) are white, the mean and covariance of the random variables w(t) and v(t) will be denoted w (t) = E w(t) = 0 and cov (w(t), w( )) = Q(t) (t ), v (t) = E v(t) = 0 and cov (v(t), v( )) = R(t) (t ) (458) (459) (460) (461)

You cannot easily buy a radio capable of tuning 1 to 10 kHz, but it is possible to build a sensitive receiver to permit you to hear the sounds of natural radio We will explore what causes these sounds and see how you can study them yourself A whistler receiver, even though whistlers and related emissions occur at acoustical frequencies, is a receiver of radio signals To hear whistlers, you must intercept their electromagnetic energy with an antenna and transform it to the mechanical vibrations to which our ears respond The classic whistler receiving system consists of an antenna for signal collection, an amplifier to boost the signal level, and headphones or a speaker to transform the signal to sound waves (A magnetic tape recorder can be substituted for the headphones or speaker) A whistler receiver can simply be an audio amplifier connected to an antenna system However, powerful man-made interference immediately above and below the frequencies of interest tend to seriously overload receivers of this sort and make reception of whistlers and related phenomena difficult To overcome these problems, a good whistler receiver circuit includes a circuit called a low-pass filter that attenuates all signals above 7 kHz This greatly reduces interference from such sources To escape interference from AC power lines and other forms of non-natural radio noise, the receiver is designed for portable operation in the field away from such interference sources A sensitive dual FET whistler receiver is shown in the photo in Figure 14-1, and in the schematic diagram in Figure 14-2 The antenna is first fed to an RF filter

In basic arithmetic, you learned that you can add a long string of numbers backward or forward, and it doesn t matter Good accountants take advantage of this when checking their work They ll add up a column of numbers from top to bottom, then again from bottom to top, just to be sure they have done the arithmetic correctly

In the analysis that follows, it will often be accurate (and convenient) to assume that the process and measurement noise are independent of the current and previous state cov (w(t), x( )) = 0 for t cov (v(t), x( )) = 0 for t and independent of each other cov (w(t), v( )) = 0 for all t, 0 (464) (462) (463)

It works when you add The fact that you can add two integers in either order and get the same result is called the commutative law for addition It means you can commute (interchange) the two numbers you re adding, called the addends, and get the same sum either way In formal terms, a mathematician would say that for any two integers a and b,

In most applications, the random processes w(t) and v(t) will be assumed to be Gaussian The linear discrete-time model is xk+1 = k xk + wk yk = Hk xk + vk (465)

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a+b=b+a This works whether the numbers are positive, negative, or 0 It also works if there are three, four, five, or more numbers in a sum, as long as the number of addends is not infinite

It fails when you subtract In subtraction, the order does matter It s easy to find an example that shows why Consider this:

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