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17-14 The half-wave,

3 2 3 2

r2 2

Antennas Many types of antennas exhibit resonant properties The simplest type of resonant antenna, and the only kind that will be mentioned here, is the center-fed, half-wavelength dipole antenna (Fig 17-14) The length Lft, in feet, for a dipole antenna at a frequency of fo, in megahertz, is given by the following formula:

p (r)dr = P rob{ c}

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L ft = 468/fo This takes into account the fact that electromagnetic fields travel along a wire at about 95 percent of the speed of light A straight, thin wire in free space has a velocity factor of approximately 095 If the length of the half-wave dipole is specified in meters as Lm, then: Lm = 143/fo A half-wave dipole has a purely resistive impedance of about 73 at its fundamental frequency fo But this type of antenna is also resonant at all harmonics of fo The dipole is a full wavelength long at 2fo; it is 3 2 wavelength long at 3fo; it is two full wavelengths long at 4fo, and so on

The random variable = v is called a Maxwell random variable The density for a Maxwell random variable is p (r) = r2 2 2 r exp 2

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The value of c such that P rob{ v c} = 05 is c = 154 A closed-form, but transcendental, solution to this equation exists [39], but this values was found by table look-up for the cumulative distribution of a Maxwell random variable We can introduce an intermediate variable x such that v = 1 x and x = U w The dimensions of the variable x are meters The components of the vector x are jointly distributed, independent, and Gaussian In the x reference frame, for a given value of c, the uncertainty ellipsoid x 2 x c has axes coincident with the x coordinate axes and P rob{w P 1 w c2 } = P rob{x 2 x c2 } In the x coordinate frame the lengths of the ellipsoid principal axes are de ned by the diagonal elements of with the ellipsoid de ned by x2 x2 x2 1 2 3 2 2 + 2 + 2 c 1 2 3 (4129)

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Radiation Resistance At fo and all of the odd harmonics, the antenna behaves like a series-resonant RLC circuit with a fairly low resistance At all even harmonics, the antenna acts like a parallel-resonant RLC circuit with a high resistance Does this confuse you There s no resistor in Fig 17-14! Where, you ask, does the resistance come from in the half-wave dipole The answer to this is rather esoteric, and it brings to light an interesting property that all antennas have It is called radiation resistance, and is a crucial factor in the design and construction of all RF antenna systems When electromagnetic energy is fed into an antenna, power is radiated into space in the form of radio waves This is a manifestation of true power, just as the dissipation of power in a pure resistance is a manifestation of true power Although there is no physical resistor in Fig 17-14, the radiation of radio waves is like power dissipation in a pure resistance In fact, if a half-wave dipole antenna were replaced with a 73- nonreactive resistor that could dissipate enough power without burning out, a radio transmitter connected to the opposite end of the line wouldn t know the difference (But a receiver would!) Problem 17-14 How many feet long is a quarter-wave section of transmission line at 705 MHz, if the velocity factor is 0800

The ellipsoid in w coordinates is simply a rotation of ellipsoid speci ed in eqn (4129)

Another signal that traders look for is the shape of the bands For example, if the upper and lower bands suddenly contract (squeeze together), this is a clue that volatility is low Don t be fooled by a quiet and low volatile market When Bollinger Bands are very narrow, it is a signal the market could suddenly explode in the opposite direction Put another way, periods of low volatility are often followed by periods of high volatility It works the other way too If the bands are expanding, it means that the market is extremely volatile There are no guaran-