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Because d = 0, we know that the quadratic we get by setting the trinomial equal to 0 has one real root with multiplicity 2 That means the original cubic has one more real root besides x = 5, and that root has multiplicity 2 To find it, we can use the quadratic formula with the coefficients and constant named according to the above scheme: x = [ b2 (b22 4a2c)1/2] / (2a2) Because the discriminant is equal to 0, we can simplify this to x = b2 / (2a2) Substituting in the values a2 = 9 and b2 = 24, we get x = ( 24) / [2 ( 9)] = 24/( 18) = 24/18 = 4/3 The roots of the cubic are therefore x = 5 or x = 4/3 The root x = 4/3 occurs with multiplicity 2 The solution set is X = {5, 4/3} 10 The new cubic, written in binomial-trinomial form, looks like this: (x + 3/2)(6x 2 4x + 2) = 0 Let s examine the discriminant d of the trinomial Setting a2 = 6, b2 = 4, and c = 2, we get d = b22 4a2c = ( 4)2 4 6 2 = 16 48 = 32 Because d < 0, the new cubic has only one real root, x = 3/2, exactly as the original cubic did (The two complex roots, however, differ in this equation compared with those in the final challenge in the chapter text For extra credit, you can verify this fact)

(or not-so-near) the lightning storm from which it was born! Whistlers are descending tones Their duration can range from a fraction of a second to several seconds, with the rate of frequency shift steadily decreasing as the frequency decreases A whistler s note may be pure, sounding almost as if it was produced using a laboratory audio signal generator; other whistlers are more diffuse, sounding like a breathy swoosh or composed of multiple tones On occasion, some whistlers produce echoes or long progressions of echoes, known as echo trains, that can continue for many minutes Whistlers can occur in any season Some types are more likely to be heard in summer than winter and vice-versa Statistically, the odds are especially high for good whistler activity between mid-March and mid-April Whistlers and the sounds of the dawn chorus are not heard equally well everywhere in the world Reception of these are poor in equatorial regions, with best reception at geomagnetic latitudes above 50 Fortunately the continental United States and Canada are well positioned for reception of whistlers and other signals of natural radio Considered by many listeners to be the Music of Earth , whistlers are among the accidental discoveries of science In the late 19th century, European long-distance telegraph and telephone operators were the first people to hear whistlers The long telegraph wires often picked up the snapping and crackling of lightning storms, which was mixed with the Morse code buzzes or voice audio from the sending station Sometimes, the telephone operators also heard strange whistling tones in the background They were attributed to problems in the wires and connections of the telegraph system and disregarded The first written report of this phenomenon dates back to 1886 in Austria, when whistlers were heard on a 22-km (14 mile) telephone wire without amplification The German scientist, H Barkhausen, was eavesdropping on Allied telephone conversations during World War I In order to pick up the telephone conversations, he inserted two metal probes in the ground some distance apart and connected these to the input of a sensitive audio amplifier He was surprised to hear whistling tones which lasted for one or two seconds and which glided from a high frequency in the audio range to a.

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Eqns (448) and (449) are useful for nding a linear system realization H(j ) appropriate for producing a given colored noise process y from a white noise input w Example 414 A linear system with transfer function 2 H(s) = +s

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1 In each of these situations, the trinomial can be factored into the square of a binomial Then that squared binomial is raised to the indicated power (a) Here is the original equation: (x 2 + 6x + 9)2 = 0

2 is excited by white noise w(t) with PSD de ned by Sw (j ) = w , where w , > 0 The spectral density of the output y is calculated by eqn (449) as

Sensor FGM-3

26

Sy (j ) = = =

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