## Monthly Archives: June 2014

## 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.

## Radio Propagation in a Magnetized Plasma

Before proceeding with a discussion of the Appleton (mag – netoionic) equations, we need to define two quantities contained explicitly in the equations. The first is v, the number of collisions per second (collision frequency) between electrons and heavier particles (ions and neutrals). Another quantity, the gyromagnetic frequency or gyrofrequency, is the natural frequency (Hz) of gyration of an ion or electron in a magnetic field of strength B0 (Wb/m2) and is given by

f

J0

dz

max

h (f) —

(6)

l(f, z)

where z is the true height, Zmax is the maximum height reached by the frequency f, and n is the refractive index at Zmax for the frequency f. A good discussion of the relation between true height and virtual height is given in Ref. 19.

## Comparing Open-Loop Experiments with Simulations

Figure 8. The general form of the TSAC model. [From A. T. Bahill and T. M. Hamm, Using open-loop experiments to study physiological systems with examples from the human eye movement systems, News Physiol. Sci., 4:107,1989 (56).] |

Insight into the behavior of the smooth-pursuit system under open-loop conditions was sought by Harvey (30) through a comparison of experimental results with those from simulations. The simulations were performed using the target-selective adaptive control (TSAC) model (19) shown in Fig. 8. This model has three branches. The top branch, the saccadic branch, generates a saccade after a short delay whenever the disparity between target and eye position is too great. The middle branch, the smooth – pursuit branch, produces smooth tracking of moving targets. The input to the smooth-pursuit branch is velocity, so the first box (labeled smooth-pursuit processing) contains a differentiator and a limiter. The box labeled smooth – pursuit controller and dynamics contains a first-order lag (called a leaky integrator), a gain element, a time delay, a saturation element, and an integrator to change the velocity signals into the position signals used by the extraocular motor system. The bottom branch contains the target – selective adaptive controller that identifies and evaluates target motion and synthesizes an adaptive signal, Rc, that is fed to the smooth-pursuit branch. This signal permits zero-latency tracking of predictable visual targets, which the human subject can do, despite the time delays present in the oculomotor system. The adaptive controller must be able to predict future target velocity, and it must know and compensate for the dynamics of the rest of the system. The adaptive controller is separate from the smooth-pursuit system in the model and also in the brain (11). The adaptive controller sends signals to the smooth-pursuit system and also other movement systems (34). All of these branches send their signals to the extraocular motor system, consisting of motoneurons, muscles, the globe, ligaments, and orbital tissues. And of course, the final component of the model is a unity gain feedback loop that subtracts eye position from target position to provide the error signals that drive the system. The solid lines in this figure are signal pathways, while the dashed lines are control pathways. For instance, the dashed line between the saccadic controller and the smooth-pursuit controller carries the command to turn off integration of retinal error velocity during a saccade.

In the experiments, many different target waveforms are used. The step target was presented to the subject to verify that the technique of opening the loop using electronic feedback was working. Because the step target introduced a position error rather than a velocity error, this experiment opened the loop on the saccadic system rather than the pursuit system. A position error with the feedback loop opened should have elicited a staircase of saccades. If this expected open-loop response to the step target was seen, then the electronic feedback was opening the loop correctly, as between 1.5 s and 2.5 s of Fig. 7.

There was difficulty in getting consistent results for sinusoids with the loop opened. The most consistent results obtained for such presentations came from the first few seconds after the loop has been opened. This finding suggests that the difficulties with open-loop sinusoids were probably due to the involvement of high-level processes such as adaptation. Once the loop was opened, the behavior of the target changed. Often the subjects would appear to respond to this change in target behavior by changing their tracking strategies. Figure 7 shows a presumed example of such a change in human tracking strategy. Between 1.5 s and 2.5 s of this record the subject behaved as one would expect for a subject tracking an open-loop target; there is a saccade every 200 ms (approximately the time delay before the saccadic system responds to a position error). However, in the middle of the record, the saccades cease; it seems that the subject turned off the saccadic system. Such saccade free tracking was common in these experiments and in other open-loop experiments (16,28,29,32,33,35). The records are strikingly devoid of saccades in spite of the large position errors, a finding that, oddly, received little comment by previous investigators (except for 33), although it is often seen in the data.

By way of comparison, the model is shown tracking a sinusoid under open-loop conditions in Fig. 9. To simulate the changes in strategy that are apparent in the human data of Fig. 7, the model characteristics were changed at intervals. From 2 to 4.25 s there is normal closed-loop tracking. At 4.25 s the loop was opened, the adaptive controller was turned off, and the smooth-pursuit gain was reduced to 0.7, thus producing a staircase of saccades similar to those shown in Fig. 7. At 7.25 s the saccadic system was turned off, the adaptive controller was turned back on, and the gain of the smooth-pursuit system was returned to its normal value; the model tracked with an offset similar to that of Fig. 7. This type of position offset was often noticed in human subjects during open-loop tracking. Finally, at 10.5 s the adaptive controller was turned off and the model tracked without an offset but with a time delay as was seen in some subjects.

These simulations help explain some confusing data in the literature by allowing us to suggest that when the loop on the human smooth-pursuit system is opened, subjects alter their tracking strategy to cope with altered target behavior. Some subjects continue to track with all systems (producing a staircase of saccades), some turn off the saccadic system (producing smooth tracking with an offset), some also turn off the adaptive controller (producing smooth tracking without an offset), and some change the gain on the smooth-pursuit system. Thus, each subject ap-

Figure 9. Position of the target (dashed) and model (solid) as functions of time under a variety of conditions. At the first arrow, the loop was opened, at the second arrow the saccadic system was turned off, and at the third arrow the adaptive controller was turned off. Tracking patterns similar to each of these are common in human records. [From A. T. Bahill and D. R. Harvey, Open-loop experiments for modeling the human eye movement system, IEEE Trans. Syst. Man Cybern., SMC-15: 249, © 1986 IEEE (30).]

pears to adapt to the novel tracking task created by opening the loop by selecting subcomponents of the smooth-pursuit system and/or changing parameters within those subsystems. All these strategy changes are within the possibilities provided by the model.

To eliminate these changes in strategy, recent studies of open-loop smooth pursuit tracking only use the first 140 msec of target movement (35a).

Such plasticity is common in physiological systems. Repetitive stimulation of the vergence eye movement system indicates that that the speed of an individual movement depends on the size of the preceding target movement (37a). Multifaceted control is also common in other physiological systems (see, for example, 36 and 37). Thus, the potential exists in other physiological control systems for changes in strategy—that is, a change in the balance of control subsystems in different physiological states whether these states occur “naturally” or are imposed by an investigator. Such changes may occur in different behavioral states as observed, for example, for respiratory control in the newborn (38). Consequently, it should not be surprising that when an investigator attempts to open the loop on a control system the control strategy changes. This section demonstrates this principle for the eye movement system.

The technique of opening the loop on a physiological system in order to better understand its behavior is very powerful as long as care is taken to acknowledge that the human is a complex organism and is likely to change its behavior when the input changes its behavior.

## The Virtual Height Concept

If we consider an RF pulse traveling vertically upward into the ionosphere at the speed of light, v = c, it will be reflected

at the virtual height, h’. The time required for the pulse to be reflected from an ionospheric layer and return to the earth is space velocity c. Referring to the geometry shown in Fig. 6, we can write the expression

2 fh dz C Jo M |

t ‘ ^X c JtER sin Фо |

4 c Jti |

f — (4)

J0

then the virtual height can be found from h'( f) = і ct, or

, h

h'(f) = -=

J0 J1

D. = — sin ф0 c TE + ER |

(8) |

dz |

(5) |

л/1 – fn/f2 |

Martyns theorem may be written concisely as |

Since the pulse always travels more slowly in the layer than in free space, the virtual height of a layer is always greater than the true height. The true height and virtual height are related by the integral equation

h’ob — К |

(9) |

Smith (20) devised a set of logarithmic transmission curves, parametric in range, for the curved earth and ionosphere. They are shown in Fig. 7 and are sufficiently accurate for the distances shown.

## VOR EQUIPMENT CHARACTERISTICS

VHF Omnidirectional Range (VOR) equipment consists of a ground station (transmitter) and an airborne receiver. The VOR ground station continuously transmits a signal that may be used by all aircraft within reception range of the signal. Using the VOR signal, the receiver determines the bearing from the ground station to the receiver. VOR stations are often colocated with other navigational aids such as DME and TACAN stations. The frequency of VOR stations ranges from

108.0 MHz to 117.95 MHz.

There are two types of VOR transmitters, Doppler VOR and conventional VOR. Doppler VOR has limited usage. The details of conventional VOR follow.

The transmitted signal from the VOR station consists of a VHF carrier and a 9960 Hz subcarrier. The VHF carrier is amplitude modulated by a variable 30 Hz signal whose phase is dependent upon the bearing with respect to the ground station. The 9960 subcarrier is frequency modulated by a 30 Hz reference signal. The subcarrier is modulated between 10440 Hz and 9480 Hz (see Fig. 3).

Figure 4 illustrates the phase-bearing relationship of the two components of the VOR signal. At 0° bearing, the phase of the variable signal is the same as the phase of the reference signal. At 90° bearing from the ground station, the variable signal is 90° out of phase with respect to the reference signal. The phase difference between the two signals is proportional to the bearing from the ground station to the receiver.

The phase difference between the variable signal and the reference signal is used by the receiver to determine the bearing from the ground station to the receiver. Essentially, the receiver separates the subcarrier from the VHF carrier, detects the phase of the 30 Hz signal in each and then determines the phase relationship.

In a conical area above the VOR station, the phase difference between the two signals cannot be detected reliably. This area is called the VOR cone of confusion. Receivers have monitors that detect this condition and provide alerts to the pilot that the signal is unreliable.

In general, the VOR ground station antenna is physically aligned so that the VOR signal indicates 0° bearing when the receiver is magnetically north of the ground station. That is, in general, the VOR bearing from the ground station to the receiver is the same as the magnetic bearing to the receiver. However, the magnetic field of the earth is constantly changing. In Europe and the United States there are regions where the magnetic north is changing 1° every seven years. Therefore even if the VOR signal was aligned with magnetic north at one instant in time, after some period of time there can be a significant difference. When the difference becomes too large (2° or so), the VOR antenna is usually realigned.

The accuracy of VOR signals is degraded by range and terrain. Generally, the VOR signal error is less than 1°. As the range from the VOR station is increased, the VOR signal bearing often oscillates around the true bearing. This effect is known as scalloping.

There are several types of VOR indicators that are in common use. Most of the indicators provide bearing as a rotating arrow pointing to the bearing angle on an azimuth card. The azimuth card is slaved to the heading sensor so that the current heading is at the top directly under the lubber line. The indicators often include a distance indication that is connected to the DME.