## Vertical and Oblique Propagation

Before considering the behavior of a radio signal in a magne – toionic medium, we will state three theorems that relate oblique and vertical incidence propagation as depicted in Fig. 6. The first is the secant law, which relates the vertical-inci – dence frequency fv reflected at B to the oblique-incidence frequency fob reflected at the same true height. A typical derivation of this relation is given in Ref. 5, and it is usually written as

JfL 2n m |

■Bn& 2.80 x 1010Bn |

fH — |

(10) |

and the angular gyrofrequency is given by le I coH = — B0 ъ 1.76 X 10nB0 |

(11) |

fob — fv sec ф0 |

(7) |

Since electrons are much less massive than ions, the electron gyrofrequency affects the propagation of HF waves in the ionosphere more than the ion gyrofrequencies. For example, since B « 0.5 X 10-4 Wb/m2, the electron gyrofrequency is «1.40 MHz, which falls at the upper end of the medium wave band.

The Dispersion Relation. Using the recommended URSI (International Union at Radio Science) notation, the magne – toionic dispersion equation for a radio wave in a homogeneous, partially absorbing ionized gas upon which a constant magnetic field is impressed is given by

The secant law, then, relates the two frequencies fv and fob reflected from the same true height (the distance BD in Fig. 6).

In order to determine sec ф and fob values from vertical – incidence soundings (which measure the virtual height h’), we need two more theorems. Breit and Tuve’s theorem states that the time taken to traverse the actual curved path TABCR in Fig. 6 at the group velocity vg equals the time necessary to travel over the straight-line path TER at the free-

E Figure 6. Plane geometry describing vertical and oblique ionospheric propagation. |

n2 = 1 – |

(1 – jZ) – |

2(1 – X – jZ) |

1/2 |

Y 4 IT |

+ Yl2 |

± |

4(1 – X – jZ)2 |

where |

n = complex refractive index = (u _ jx) ш = angular frequency of the exploring wave (rad/s) |

700

Sec ^(corrected) 3.0 2.0 |

16 20 24 28 32 36 40 44 48 52 56 60 70 80 Angle of departure (deg) |

Figure 7. Logarithmic transmission curves for curved earth and ionosphere, parametric in distance between transmitter and receiver. |

1.0 |

600

is 500 ra

<d 400 sz

я

3 300 200 100 0

1 |

+ Yl2 |

=F I 7 |

2 |

(15) |

wN = angular plasma frequency wH = angular gyrofrequency = B0|e|/m (rad/s) wL = longitudinal angular gyrofrequency = (B0|e|/m) cos в wT = transverse angular gyrofrequency = (B0|e|/m) sin в X = wN/w2 Y = wH/w Yl = wL/w YT = wT/w Z = v/w в = angle between the wave-normal and the magnetic field inclination The Polarization Relation. We begin by defining the polarization ratio R as R = —Hy/Hx = Ex/Ey (13) Then we can write the double-valued polarization equation as |

In the upper F region of the ionosphere where the electron – ion collision frequency is very low, we may simplify the dispersion and polarization equations by dropping the Z term (since v « 0). Equations (12) and (14) then become (for no absorption) |

If we further simplify Eq. (12) by dropping the Y terms (no magnetic field), then we obtain n2 = 1 — X, which is equivalent to Eq. (1). |

2 1 — X — jZ I 4 (1 — X — jZ) |

2X (1 — X) |

n2 = 1 — |

n — 1 |

(14) |

(16) |

and |

Y 4 YT |

1/2- |

X |

2(1 — X) — YT2 ± [YT4 + 4YT2(1 — X)2]1/2 |

According to magnetoionic theory, a plane-polarized EM wave traveling in a medium like the terrestrial ionosphere will be split into two characteristic waves. The wave that most closely approximates the behavior of a signal propagating in this medium, without an imposed magnetic field, is called the ordinary wave, and the other is called the extraordinary wave. These terms are taken from the nomenclature for double refraction in optics, although the magnetoionic phenomena are more complicated than the optical ones. The ordinary wave is represented by the upper sign in the polarization Eq. (14), except when the wave-normal is exactly along the direction of the magnetic field. Anomalous absorption occurs for the extraordinary wave when its frequency equals the electron gyrofrequency (fH = |B| e/me « 0.8 to 1.6 MHz). These frequencies lie in the medium-frequency (MF) band; consequently the absorption of the extraordinary wave [A « (f — fH)2] is large and the polarization of the transmitted wave is important in the determination of the fraction of the incident power that goes into the extraordinary wave. This is especially true near the dip equator, where the magnetic field is nearly horizontal and the field is usually vertical. In addition to anomalous absorption effects near the electron gyrofrequency, the wave may also experience significant lateral deviation. This is illustrated for vertical and oblique propagation in Sections 11.2.2 through 11.2.4 of Ref. 5. If Eq. (16) is recast as a funtion of w and we define f(e) = j(sin2 e)/cos в and wc = (B0|e|/m) f(e), then it will be seen to describe an ellipse. The quantities f(e) and wc play an important part in the description of the polarization behavior of waves in magnetoionic theory. The magnitude of wc is independent of frequency, but varies with the angle between the wave normal and the magnetic field, в, whereas the sign of wc depends on the sign of the charge e and on the direction of the magnetic field. For longitudinal propagation wc = 0, and for transverse propagation wc ^ “.In the case where X = 1, the quantity wc primarily determines the polarization of the wave. A very complete discussion of R as a function of X and of the variation of the polarization ellipse is given in Ref. 21. A more complete understanding of the behavior of EM waves in the terrestrial ionosphere may be obtained by em- |

(23) |

X |

(24) |

ploying two approximations. The quasilongitudinal (QL) approximation applies when the wave is propagating nearly parallel to the geomagnetic field, and the quasitransverse (QT) approximation applies when the wave propagates in a direction nearly normal to the geomagnetic field. References 21 and 22 contain extended discussions of the QL and QT approximations:

QT: YT4 » 4(1 – X)2Yl2

QL: YT4 << 4(1 – X)2Y2

## ILS CHARACTERISTICS

An instrument landing system (ILS) consists of ground-based transmitters and airborne equipment that provides lateral, along-track, and vertical guidance. The lateral signal is provided by a localizer transmitter and the vertical signal is provided by the glideslope transmitter. The airborne ILS receiver is capable of receiving and processing both the localizer and the glideslope signals. The along-track information is provided by marker beacons (transmitters located along the descent path that provide a narrow vertical radio signal) or distance measuring equipment (DME). The marker beacon receiver can be part of the ILS receiver or it can be a separate receiver (see Fig. 5).

The localizer beam is almost always aligned to guide the aircraft directly over the runway threshold. In certain situa-

tions, only a localizer signal is provided and no electronic glideslope signal is provided. In some localizer-only situations, the localizer signal is not aligned to the runway but instead provides guidance to some location from which the pilot has other means to complete the landing.

In some cases, the ILS DME transponder delay is adjusted so that the sensed DME distance is zero at the runway threshold instead of at the DME antenna. Such DMEs are known as biased DMEs and the bias is indicated on the approach chart. There are two general arrangements of biased DMEs (see Fig. 6). In one case the DME antenna is located at the glideslope antenna and adjusted to read zero at the corresponding runway threshold. In the other case, the DME antenna is located midway between the two runway thresholds at opposite ends of the runway. In this case the single DME will support approaches from either direction and will read zero at both runway thresholds.

The localizer signal is transmitted on assigned frequencies between 108.1 MHz to 111.95 MHz. As shown in Fig. 5, the localizer antenna is usually located past the far end of the runway very near the extended runway centerline. The signal pattern are two main lobes on each side of the center line. The left lobe is predominantly modulated at 90 Hz and the right lobe is predominately modulated at 150 Hz. Along the center line, the two modulated signals are equal (see Fig. 7).

The localizer signal also extends backwards and is called the back course. This signal can be used for guidance but the modulation convention is reversed. There is no glideslope signal provided on the backcourse region. When flying the back – course, either the equipment must reverse the localizer indications or the pilot must recognize and fly the reversed indications.

The glideslope signal is transmitted on assigned frequencies between 328.6 MHz and 335.4 MHz. The glideslope antenna is located on the side of the approach end of the runway. The glideslope signal consists of two main lobes on each

Glideslope and DME antenna • |

Antenna beam pattern |

Back course |

Localizer course |

Localizer antenna |

Localizer beam centerline |

Runway |

Glideslope |

An omnidirectional antenna is used to receive the signal to aid in tuning the receiver. The ADF receiver determined the bearing of the station using the directional sensitivity of loop antenna. The loop antenna may be physically rotated to determine the bearing to the signal or the bearing may be determined using electronic sensing of the signal strength from more than one loop.

400 Hz |

Inner marker |

Middle marker |

Outer marker |

side of the desired glideslope path. The glideslope descent angle is usually three degrees. To provide obstacle clearance or to reduce noise, steeper glideslope paths are used. The upper glideslope lobe is predominantly modulated at 90 Hz and the lower lobe is predominantly modulated at 150 Hz (see Fig. 8).

Marker beacons signals are transmitted at 75 MHz and are modulated at 400 Hz, 1300 Hz, or 3000 Hz. The transmitters are located along the descent path of to the runway. Figure 9 shows the general arrangement of the beacons. When the aircraft passes over the beacons, the marker beacon receiver detects the signal and provides an indication to the pilot of the passage. The exact location of the marker beacons is given on the approach procedure chart. Inner marker beacons are installed at runways with Category II and Category III operations.

In typical operation, the pilot maneuvers the aircraft to cross the localizer signal centerline. At this time, the localizer receiver provides an indication that the localizer signal is being received and provides a lateral deviation indication showing the aircraft displacement from the centerline. Using the lateral deviation indication, the pilot steers to the localizer centerline until the glideslope receiver indicates reception of the glideslope signal. At that time the pilot has both lateral and vertical indications to guide the aircraft on the desired glideslope path. The marker beacons or DME indications provide along-track indications of the progress of the descent.

## MAKE THE MODEL I

The human can overcome a time delay and track visual targets with zero latency. This is nicely demonstrated by the smooth-pursuit eye movement system. We found that if our model was to emulate the human, it had to predict target velocity and compensate for system dynamics. The model accomplished this using a prediction algorithm. To help validate the model, a sensitivity analysis and a parameter estimation study were performed.

Figure 10 shows our model for the human eye movement systems. Like the human, this model can overcome the time delay and track a target without latency. To do this, the model must be able to predict future target velocity and compensate for system dynamics. In this section, a least-mean-square technique for predicting target velocity is described. After incorporating this prediction algorithm into the model, the model was studied to learn more about the model, and hopefully about the human. In particular, we performed a sensitivity and analysis of the predictor and then investigated how parameter variations affected the MSE between the predicted output and the actual target waveform.

This section primarily examines the smooth-pursuit eye movement system. The earliest model for the smooth – pursuit system is the sampled data model developed by Young and Stark (27). Because of later evidence presented by Robinson (39) and Brodkey and Stark (40), the pursuit branch is no longer viewed as a sampled data system, but rather as a continuous one.

There is one physically realizable model capable of overcoming the time delay in the smooth-pursuit branch, the TSAC model developed by McDonald (18, 19). This model with the saccadic and smooth-pursuit branches and their interactions is shown in Fig. 10. The computer simulation that implements this model was written in the C language on a Unix computer system.

Referring to Fig. 10, the input to the smooth-pursuit branch is retinal error, which is converted to velocity by the differentiator. The limiter prevents any velocities greater than 70°/s from going through this branch. [The numbers given in this section are only typical values, and the standard deviations are large—for example, and LaRitz (41) showed smooth-pursuit velocities of 130°/s for a baseball player.] The leaky integrator K/(ts + 1) is suggested from (a) experimental results showing that humans can track ramps with zero steady-state error (7) and (b) open-loop experiments that showed a slope of — 20 dB per decade for the pursuit branch’s frequency response (30). The gain, K, for the pursuit branch must be greater than unity, since the closed-loop gain is almost unity. Currently used values for the gain are between 2 and 4 (30, 42). The e~sT term represents the time delay, or latency, between the start of the target movement and the beginning of pursuit movement by the subject. A time delay of 150 ms is currently accepted (30,33). The saturation element prevents the output of any velocities greater than 60°/s, the maximum

velocity produced by most human smooth-pursuit systems.

The model must be able to overcome the 150 ms time delay and track with zero latency. Because the smooth- pursuit system is a closed-loop system, the model’s time delay appears in the numerator and the denominator of the closed-loop transfer function,

An adaptive predictor using adaptive filters was designed to overcome the time delay in the numerator. Compensation for the model’s dynamics overcomes the time delay in the denominator.

We used several different techniques for predicting target velocity. There are many more to choose from (see the “Adaptive filters’” and “Filtering theory” sections of this encyclopedia). Now we will make a detailed presentation ofone of these prediction techniques. The nonmathematical reader may skip this section (all the way to “VALIDATE THE MODEL”) without loss of continuity.

Figure 11. Implementation of the weight adjustment algorithm. [From D. R. Harvey and A. T. Bahill, Development and sensitivity analysis of adaptive predictor for human eye movement model, Transaction of the Society for Computer Simulation, December 1985. © 1985 by Simulation Councils, Inc., San Diego, CA. Reprinted by permission. (20).] |

Figure 10. McDonald’s TSAC model has three branches: smooth pursuit, saccadic, and the adaptive predictor. |

To solve the Wiener-Hopf equation it is necessary to compute the correlation matrices. However, this would require a lot of computer time; furthermore, these matrices cannot be computed in advance, because this would require a priori knowledge of the statistics of the input signal.

Because it is difficult to compute the true gradient, we use an estimate of the gradient, which is equal to -2E(j)X(j). Our algorithm is a form of the method of steepest descent using estimated gradients instead of measured gradients. Using this estimated gradient, the adjustment algorithm can be written as

1) = W{j) – j – 2ksE{j)X{ j)

Figure 11 illustrates the implementation of this weight adjustment algorithm. If the input signals are uncorrelated, then the expected value of the estimated gradient converges to the true gradient without any knowledge of the input signal’s statistics.

During the operation process of the LMS filter, illustrated in Fig. 12, the tapped delay-line input signals are weighted, using the gains from the adaptation process and summed to form the output signal. The difference between the desired output signal and the actual output of the filter is the error that is fed back to the weight adjustment algorithm.

The speed and accuracy of the filter while converging to the optimal solution depends on several factors. Because noise is introduced into the weight vector from the gradient estimates, it follows that if the filter is allowed to

Figure 12. The LMS adaptive filter. The boxes labeled “Weight adjustment” contain systems like Fig. 11. [From D. R. Harvey and A. T. Bahill, Development and sensitivity analysis of adaptive predictor for human eye movement model, Transaction of the Society for Computer Simulation, December 1985. © 1985 by Simulation Councils, Inc., San Diego, CA. Reprinted by permission. (20).] |

converge slowly, less noise will be introduced during each adaptation cycle and the convergence will be smoother. Regardless of the speed with which the filter converges, some noise will be introduced. This noise prevents the filter from converging to the minimum MSE. The ratio of the excess MSE to the minimum MSE gives a measure of the misad – justment of the filter compared to the optimum system. The misadjustment depends on the time constant of the filter’s weights, where the time constant is defined as the time it takes for the weights to fall within 2% of their converged value. A good approximate formula for the misadjustment, M, is

This algorithm shows that M is proportional to the number of weights, n, and inversely proportional to the time constant, tMSE. The time constant тMSE can be measured experimentally for each simulation. However, we would prefer to find an analytical way to estimate it. We can do that as follows.

To ensure convergence the proportionality constant, ks, in the weight adjustment algorithm must be within the following bounds:

1

0 < ks <

where E[X2j] is the expected value of the square of the jth input. For slow and precise convergence, ks should be within the following more restrictive bounds:

і

0 < ks «

According to Widrow (43,44), for a filter using tapped delay – line input signals, the time constant is related to the proportionality constant by

Figure 13. The adaptive predictor. The boxes labeled “Adaptive filter” and “Slave filter” contain systems similar to those in Fig. 12. [From D. R. Harvey and A. T. Bahill, Development and sensitivity analysis of adaptive predictor for human eye movement model, Transaction of the Society for Computer Simulation, December 1985. © 1985 by Simulation Councils, Inc., San Diego, CA. Reprinted by permission. (20).] |

In summary, an adaptive filter is made up of a tapped delay line, variable weights, a summing junction, and the weight adjustment algorithm. The filter adjusts its own internal settings to converge to the optimal solution. Due to noise from the gradient estimate, the accuracy and speed of convergence depends on the number of weights and the proportionality constant, ks.