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; ON THE CHARACTERIZATION
OF MULTIPATH ERRORS
IN
lIr
SATELLITE-BASED PRECISION APPROACH AND LANDING SYSTEMS
A Dissertation Presented to The Faculty of the College of Engineering and Technology Ohio University
In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
Michael S. Braasch
/
June, 1992
Table of Contents 1. INTRODUCTION
.....................
2 . SATELLITE-BASED NAVIGATION SYSTEMS IN THE PRECISION APPROACH ENVIRONMENT AND THE ISSUE OF MULTIPATH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Multipath in the Precision Approach Environment . . . . . . . . . . . . . 9 3 . EFFECTS OF MULTIPATH ON THE SATELLITE-BASED NAVIGATION RECEIVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 PRN Ranging and DLL Operation . . . . . . . . . . . . . . . . . . . . . . . 3.2 PRN Modulated Signal Description . . . . . . . . . . . . . . . . . . . . . . . 3.3 Coherent PRN Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Coherent DLL Discriminator Curve in the Absence of Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Coherent DLL Discriminator Curve in the Presence of Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 PLL Operation In The Presence Of Multipath . . . . . . . . 3.4 Non-coherent PRN Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Non-coherent DLL Discriminator Curve in the Absence of Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Non-coherent DLL Discriminator Curve in the Presence of Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 PLL Operation in the Presence of Multipath . . . . . . . . . 3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 CIA-Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 CIA-Code with Narrow Correlator Spacing . . . . . . . . . . . 3.5.3 P-Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21 26 31
4 . AIRBORNE ANTENNA PATTERN CHARACTERIZATION . . . . . . . . 4.1 Antenna Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Theoretical Background for Antenna Modeling . . . . . . . . 4.1.2 Application to GPS Antennas . . . . . . . . . . . . . . . . . . . . . 4.2 Airborne Antenna Pattern Modeling . . . . . . . . . . . . . . . . . . . . . 4.2.1 Piper PA-32 (Saratoga) . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Boeing B-737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88 90 90 91 103 107 111
5 . AIRFRAME-BASED MULTIPATH AT THE AIRBORNE RECEIVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 PA-32 Roll Plane Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Structural Flexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123 124 124
11 11 16 17 18
31 32 35 42 42 56 73
6 . OBSTACLE-BASED MULTXPATH AT THE AIRBORNE RECEIVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.1 Obstacle Clearance Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.2 Earth-Surface-Based Multipath . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 . OBSTACLE-BASED MULTIPATH AT THE GROUND REFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Antenna Architecture and Siting Considerations . . . . . . . . . . . . 7.2 Multipath Attenuation Via Earth Blockage . . . . . . . . . . . . . . . . 7.3 Multipath Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 P-Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157 157 159 162 163
8. DATA COLLECTION AND ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . 8.1 Theoretical Basis for Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Ionospheric Correction . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Equipment Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164 164 168 177 180 180 183
9 . CONCLUSIONS AND RECOMMENDATIONS 10. ACKNOWLEDGEMENTS 11. REFERENCES
. . . . . . . . . . . . . . . . . . 194
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197
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199
1. INTRODUCTION
Satellite-based navigation systems, such as the Navigation Satellite Timing And Ranging Global Positioning System (NAVSTAR GPS), are showing potential for a wide variety of new and exciting applications. Real-time attitude and heading determination of dynamic platforms, measurement of flexing in large space structures, and precision approach and landing are just a few of these. Such applications require high accuracy in real time. Precision approach aids, for example, must provide sub-meter positioning accuracy with a data renewal rate on the order of 10 samples per second. However, before this technology reaches full maturity, a major stumbling block remains. It is the effect of multipath. Multipath is the phenomenon whereby a signal amves at a receiver via multiple paths due to reflection and diffraction. Multipath represents the dominant error source in satellite-based precision guidance systems. A review of the literature reveals a multitude of articles discussing the effects of multipath, yet most have very little depth. Some effort has been applied to developing multipath mitigation techniques, yet all of those put forth thus far have limited application. As a result, a comprehensive treatment of the multipath problem is needed. Both characterization and mitigation of multipath must be addressed. Characterization is the requirement for a full understanding of the problem. Mitigation is then possible and is necessary to exploit the full potential of the system. Both are needed for the determination of full system performance and
the development of enor budgets. The work performed for this dissertation addresses the multipath characterization problem in the context of the precision approach and landing operation. This is considered one of the most demanding applications for any navigation system. High accuracy position information must be provided continuously throughout the operation. For obvious reasons, system accuracy requirements are most stringent near the runway. However, this is also the region which brings the aircraft closest to multipath-causing obstacles. As a result, multipath error can be aggravated while the error tolerances become tighter. Furthermore, satellite-based navigation systems must deal with a relatively uncontrolled transmitter environment. Unlike the Instrument Landing System (ILS) and the Microwave Landing System (MU), each of which involves a set of
stationary transmitters, satellite systems use four or more transmitters all of which are constantly moving. This increases the level of complexity in characterizing the multipath environment. In addition, multipath arising from the airframe itself is also possible and can be of sufficient magnitude to be a concern. As will be discussed later, satellite-based precision approaches will require a ground reference station with a satellite receiver. Being at a known location, this receiver will determine the magnitude of errors in the satellite signals and will transmit corrections for these errors to the airborne user. These transmitted corrections will be erroneous, however, if the signals at the ground station are corrupted by multipath. This stems from the fact that multipath, unlike most
3
other error sources such as atmospheric delays and satellite clock errors, is not common to both the ground and airborne receivers. The remainder of this dissertation addresses these issues. A brief background on satellite-based navigation systems and the issue of multipath is presented first. This section includes an overview of the previous research dealing with multipath in the GPS and shows how the unique issues occurring in the precision approach environment have not been covered. This is followed by a derivation of the effects of multipath in the satellite navigation receiver. It will be shown that multipath errors are a function of the relative multipath strength, delay, phase, and phase rate-of-change. Once this has been determined, the remainder of the problem consists of characterizing these parameters in the precision approach environment. As the results will show, consideration of the multipath environment at each airport will be required to ensure proper ground equipment siting and approach procedure design. It should be noted that when this investigation began, two satellite-based navigation systems were considered to be likely candidates for the precision approach and landing application. The first, as was mentioned earlier, was the GPS. The second was the Global Navigation Satellite System (GLONASS) being fielded by the former Soviet Union. Although the success of the GPS in the Persian Gulf conflict seems to have assured its future, the uncertain economy of the Commonwealth of Independent States casts a shadow on the future of GLONASS. As a result, the discussions which follow will be focussed on the
4
GPS . However, the results will be applicable to other satellite-based navigation systems as well.
2. SATELLITE-BASED NAVIGATION SYSTEMS IN THE PRECISION APPROACH ENVIRONMENT AND THE ISSUE OF MULTIPATH
Satellite-based navigation systems, such as the GPS, provide guidance through signals broadcast from the satellite to the user. These signals are formed by digital pseudorandom noise @m) codes which bi-phase modulate a radio frequency (RF) carrier. If the receiver clock is synchronized with GPS time and the prn codes are tracked by the receiver, it can then determine the user's range to the satellites (this process is also known as code-phase tracking). The carrier is also modulated by navigation data which contain information regarding satellite locations. Conceptually then, position determination is a simple matter of trilateration. If the receiver clock is not synchronized with GPS time or if the receiver clock is of low stability, then each of the range measurements will contain an unknown receiver clock bias. The resulting measurement is referred to as a pseudorange measurement:
where:
pi is the pseudorange to the i" satellite
xi, yi, zi are the coordinates of the i" satellite
x, y , z are the coordinates of the receiver
B is the receiver clock bias
c is the speed of light "errors" represent the sum of propagation and receiver errors The receiver must solve for its coordinates and the common clock bias simultaneously and thus requires a minimum of four pseudorange measurements. Error sources include: signal delays induced by the troposphere and ionosphere; satellite clock errors and errors in the broadcast orbital information; intentional satellite clock and orbit information errors (known as Selective Availability, employed by the Department of Defense for security purposes); receiver hardware delays and measurement errors; thermal noise; multipath. Errors that are correlated between two receivers can be reduced significantly through a differential technique. This technique uses a ground reference station at a known location with a satellite receiver. The output of the receiver is compared with the known location of the station and an error is computed. This error represents a correction factor which is broadcast to users in the area. The satellite systems in non-differential form typically provide position information with accuracy on the order of tens of meters. The use of differential corrections provides accuracy on the order of meters. Accuracy on the order of centimeters, however, is possible with a technique known as interferometry. Interferometry is based on tracking the carrier of the signal (rather than the code) and making phase measurements at multiple antennas (this technique is also known as differential carrier-phase tracking). Determination of the threedimensional vectors (baselines) joining the antennas can be accomplished with
7 accuracy on the order of a fraction of a wavelength (the NAVSTAR GPS wavelength is approximately nineteen centimeters). For closely separated users, most error sources are highly correlated except for multipath and receiver errors. Multipath has been cited as a major error source both in differential satellite systems as well as interferometry bchapelle, 1989 and Cohen, 19911. Multipath distorts the signal modulation and degrades accuracy in conventional and differential systems. Multipath also distorts the phase of the carrier and hence degrades the accuracy of the interferometric systems. Furthermore, interferometric systems often employ pseudorange measurements for initialization purposes. Multipath contamination of the pseudorange can increase the time required for initialization. For standard differential systems, signal derogation due to multipath can be severe. This stems from the fact that multipath is a highly localized phenomenon. Multipath sources which affect the ground reference station receiver do not necessarily cause errors in the airborne receiver. Likewise, multipath sources which affect the airborne receiver do not necessarily affect the ground reference station. As a result, the differential corrections which are uplinked from the ground may correct for multipath errors experienced on the ground which are not present in the air. Similarly, the correction may not account for multipath errors which the airborne receiver is experiencing. Multipath effects in pm ranging have been studied for almost two decades. Hagerman (1973) derived relationships involving multipath and pm code-tracking
8 error. This fundamental work formed the basis for the analysis of GPS code and carrier multipath errors during the field tests at the Yuma Proving Ground [General Dynamics, 19791. In the early 1980's, the effect of multipath on short baseline interferometry was studied at the Massachusetts Institute of Technology [Counselman and Gourevitch, 1981, Counselman, 19811 and at the Charles Stark Draper Laboratory [Greenspan, et al, 19821. The studies concluded that the effects of multipath could be reduced to a few centimeters of error over short baselines if the signals could be averaged over a period of an hour or more. Bletzacker (1985) also considered multipath errors in geodetic applications. Performance improvements were obtained by mounting the antenna on RF absorbing material and thereby improving the characteristics of the antenna pattern. Tranquilla and Carr (1990-91) confirmed this by collecting data in stressful environments using a geodetic antenna with and without an R F absorbing ground plane. Falkenberg, et a1 (1988) and Lachapelle, et al (1989) describe marine differential GPS experiments in which multipath was mitigated through the use of
RF absorbing ground planes and filtering schemes. Evans (1986) demonstrated multipath effects on ionospherically corrected code and carrier measurements from a geodetic GPS receiver. Georgiadou and Kleusberg (1988) considered multiple reflections and showed that multipath on short baselines could be detected using dual frequency measurements.
Abidin (1990) examined the effects
of multipath in dual-frequency-measurement-based
ambiguity resolution.
The effect of multipath on ionospheric measurements using GPS was presented by Bishop, et a1 (1985). Their work verified the theoretical multipath relations derived by Hagerman (1973) and considered various mitigation schemes for static applications. S e ~ o t tand Pietraszewski (1987) and Sennot and Spalding (1990) have developed state variable models for the estimation and mitigation of multipath in differential GPS ground reference stations. More recently, Van Nee (1991) has shown that code-phase multipath error traces tend not to be zero mean and can have periods on the order of an hour. This contradicts the popular notion that code-phase multipath can be eliminated in static applications simply through averaging.
2.1 Multipath in the Precision Approach Environment
The majority of the theoretical multipath work performed to date has used only a Geometrical Optics (GO) model for the electromagnetic environment. The GO theory assumes that the received field is comprised of direct, reflected and refracted rays. However, this model is only valid when diffraction effects are negligible. Clearly, this is not the case for airborne applications. Edges on the wings, elevator and rudder are classic sources of diffraction. Theoretical development incorporating the phenomenon of diffraction is thus required. Furthermore, as was discussed in the introduction, the effects of multipath in the final approach and landing environment have not been investigated. In this
application, multipath effects may be divided into three categories: 1) Obstacle-based multipath at the airborne receiver 2) Airframe-based multipath at the airborne receiver
3) Obstacle-based multipath at the ground reference station receiver Characterization and mitigation of multipath in each of these categories must be achieved before satellite-based precision approaches can become a viable technology. Prior to characterizing the errors associated with each of the above categories, the effects of multipath in the receiver must be determined.
11 3. EFFECTS OF MULTIPATH ON THE SATELLITE-BASED NAVIGATION RECEIVER
Fundamental to the understanding of the effects of multipath in the final approach environment is an understanding of pm ranging receivers and how multipath distortion results in ranging errors. This section derives closed form expressions for code-phase and carrier-phase multipath errors resulting from a single multipath ray entering' a stationary receiver. Although most multipath scenarios involve multiple rays, much insight results from the analysis of the single-ray case. It also serves as the starting point from which the multiple-ray case can be considered. The analysis considers the two most prevalent types of receivers: the coherent delay-lock-loop (DLL) and the non-coherent DLL. This section extends the theoretical developments documented by Hagerman (1973) and Van Dierendonck, et al (1992) to include analysis of narrow-correlator spacing in coherent DLL's.
3.1 PRN Ranging and DLL Operation
PRN ranging from satellites involves determining the time-of-transmission of a signal. This is accomplished by tracking the pm code which the satellite places on the broadcast signal. By tracking the pm code the receiver is able to determine the time at which the signal was broadcast from the satellite. By
12 subtracting the signal reception time from the time-of-transmission, the resulting quantity may be converted into a pseudorange measurement through multiplication by the speed of light. The heart of the receiver, then, is the DLL which tracks the incoming pm code (Figure 1). The fundamentals of DLL operation may be found in Spilker (1977, 1978). The results are summarized here. Since the received signal power (-160 dBW) is well below the noise floor, the incoming pm code must be extracted and tracked by correlating it with a copy of the code which is generated in the receiver. Code-correlation involves multiplication of the two codes and integration of the result (hardware implementations usually involve a mixer followed by either a bandpass or lowpass filter). Formally, the autocorrelation function of the pm code is given by [Spilker, 19781:
where:
R is the autocorrelation function T
is the lag
p(t) is the value of the pm code at time t However, since the pm code is a discrete sequence, the integral may be reduce to a sum:
An example is given in Figure 2. When the locally-generated code has been advanced or delayed such that it is aligned with the incoming code, the resulting correlation is maximized. In principle, the incoming code could be tracked by advancing or delaying the locally-generated code such that maximum correlation is achieved. Implementation of this concept, however, is complicated by the fact that the correlation function is symmetrical about the maximum. As a result, the correlation function yields the same value regardless of whether the locallygenerated code needs to be advanced by a certain amount or delayed by the same amount. The DLL solves this problem by implementing three correlators. The incoming code is correlated simultaneously with three versions of the locallygenerated code. One of the locally-generated codes is generated "early"in time by a specified amount, the second is generated "late" by the same amount and the third is "on-time" or punctual. The output of the early and late correlators are differenced to form an error signal. The error signal, also known as a discriminator function, is non-ambiguous since it changes sign depending upon whether the on-time code needs to be advanced or delayed. This error signal drives a voltage-controlled oscillator (VCO) which in turn drives the local pm code generator.
3.2 PRN Modulated Signal Description
The signal broadcast from the satellite in a pm ranging system may be expressed as:
where:
0 . 5 * ~ ~ is the average signal power into the receiver "0
is the frequency of the received signal in radians per second (carrier frequency plus Doppler shift)
~(t)
pm code (either
+ 1 or - 1)
Note that the actual GPS signal is considerably more complicated. The GPS carrier is modulated by two pm codes (the coarse acquisition, CIA code and the precision, P code) in addition to navigation data. For the present purpose of multipath analysis, however, the model given by equation (4) is adequate. By applying trigonometric identities, equation (4) may be rewritten:
Multipath is characterized by four parameters (all of which are relative to the direct signal): 1) amplitude; 2) time delay; 3) phase; 4) phase rate-of-change. For the present discussion, a stable multipath scenario is assumed and thus the
relative phase rate-of-change is assumed to be zero. Relative phase of the multipath is a function of the relative time delay and the reflection coefficient of the reflecting object. If the received signal is composed of the direct signal plus a single multipath ray, it may be expressed:
where:
a
is the multipath relative amplitude
6
is the multipath relative time delay (note: must be negative given the convention used in the equation)
Note that the relative phase of the multipath is not shown. Substitution of equation (5) into equation (6) and inclusion of relative phase yields:
where:
? ' I,
is the multipath relative phase
3.3 Coherent PRN Receiver
This section derives expressions for the coherent DLL discriminator curve in the absence and presence of multipath. Also derived is the expression for the composite phase of the multipath corrupted signal as it is tracked by the carrier-
tracking loop.
3.3.1 Coherent DLL Discriminator Curve in the Absence of Multipath
Following the signal flow depicted in Figure 1, the incoming signal, S ,(t), is mixed with early and late versions of the pm code modulated onto an intermediate frequency (IF):
where:
r
is the DLL tracking error
r,
is the time advance of the early code or the time delay of the late code (relative to the on-time code)
8
is the tracking error of the phase-lock-loop
o,
is the intermediate frequency in radians per second
The output of the mixers is then:
These signals are then passed through bandpass filters (BPF). It is assumed that the passband of the BPF's is narrow enough to reject the sum-frequency terms. The filters also serve to integrate the difference-frequency terms (or, equivalently, to average and scale them) and thus complete the correlation process. The output of the filters is then:
s,(t)sAt)
where:
=
1 - A R(r - sd)cos(o ,t - 8) 2
03
o,
is the difference-frequency (a,= o, -
R(r)
is the correlation function of the pm code
A sufficient approximation of the pm code correlation function is given by:
20 where:
T
is the pm code bit period (note: a pm code bit is also known as a "chip." Accordingly, the pm code bit-rate is also known as the "chipping-rate")
Correlation sidelobes are ignored and infinite bandwidth is assumed. The finite bandwidth of the BPF's distorts the shape of the pm code bits and results in a smoothing of the correlation function penton, et al, 19911. Comparison of the results of this analysis (section 3.5) with those obtained when smoothing the correlation function Fan Dierendonck, et al, 19921 reveals that the smoothing slightly reduces the maximum range error due to a given set of multipath parameters.
As a result, the above representation yields conservative results.
The signals out of the BPF's are thus the early and late correlator functions. The low-pass filters (LPF) perform the conversion to baseband. Assuming a PLL tracking error of zero then yields:
Finally, the discriminator function is formed by differencing the outputs of the LPF's. The normalized form of the discriminator function is thus:
Where the subscript c denotes coherent DLL operation. Plots of the correlation function and the discriminator function are given in Figure 3. Note that the DLL tracks the peak of the correlation function by tracking the zero-crossing of the discriminator function since both occur for
7
=
0. As will be derived in next section, multipath distorts the discriminator curve
such that the zero-crossing occurs for some non-zero
7.
Thus, the
7
corresponding to the zero-crossing is the DLL tracking error due to the multipath. The ranging error is then given by the opposite of the tracking error.
3.3.2 Coherent DLL Discriminator Curve in the Presence of Multipath
In the presence of multipath, the incoming signal is given by equation (7). In the case of the coherent DLL, the early and late codes are modulated onto an IF which is phase-locked to the downconverted incoming signal. However, the incoming signal has been perturbed due to the presence of the multipath. Thus, the phase lock loop tracks the phase of the composite signal and not that of the direct. The early and late signals are now given by:
where:
0;
is the composite phase of the direct plus multipath signal
8
is the PLL tracking error
The relations governing the composite phase of the direct plus multipath signal will be derived in the next section. The outputs of the mixers are given by multiplying (7) by (18) and (19):
Passing these signals through the BPF's completes the correlation process and removes the sum frequency terms. The outputs of the filters are then:
The low-pass filters convert the signals to baseband yielding:
Assuming a PLL tracking error of zero and taking advantage of the fact that the cosine is an even function yields:
Again, the discriminator function is formed by differencing the outputs of the LPF's. The normalized form of the discriminator function is thus:
Where the subscript cm denotes coherent DLL operation in the presence of multipath .
3.3.3 PLL Operation In The Presence Of Multipath
Prior to entering the phase-lock loop, the incoming signal is mixed with the on-time code modulated onto the local oscillator frequency. The incoming signal in the presence of a single multipath ray was given by equation (7). The on-time signal is given by:
After mixing the two signals,
Passing through the BPF yields:
Assuming a PLL tracking error of zero yields:
Recall that 8: is the composite phase of the direct plus multipath signal. Talung that as the reference, equation (32) may be rewritten:
Where o,= o, - a,. The composite phase, 8:,
may now be determined by
considering two operating regions.
Region I: Absolute value of the DLL tracking error is less than one chip.
In this region, the correlation function, R(r), is non-zero. Using the trigonometric identity: cos(u+v) = cos(u)cos(v) - sin(u)sin(v), equation (33) may be rewritten:
This expression may be condensed further yielding:
Using the same trigonometric identity mentioned above, the following form may be derived:
This form is used since an expression for the composite phase is desired. By directly applying the identity, one would obtain:
However, these are not valid expressions. Upon examination of equation (37), one notices that C O S (must ~ ~be negative. This follows from the fact that a is always positive and R(r) is always positive in region I. Thus, for the absolute value of cos(0;) to be less than or equal to 1, cos(0,J would have to be negative. This, of course, is not true in general. As a result, an alternate expression for 0; must be determined. This can be accomplished by forming the expression for the tangent:
This expression for the tangent is valid since the tangent function ranges from positive to negative infinity. For region I operation then, the composite phase of the signal entering the phase-lock loop is given by the arctangent of the right-hand side of equation (39). The interdependency of the code- and carrier-
30 tracking loops may also be observed in equation (39) by noting the presence of the code-correlation function.
Region 11: Absolute value of the DLL tracking error is greater than or equal to one chip.
In this region, the correlation function, R(r), is assumed to be zero. Although this is not strictly true, it is a reasonable approximation in light of its proximity to zero relative to the peak value of the correlation function. Using this assumption, equation (33) simplifies to:
Since the equation is already in the form of equation (36), by inspection:
For region I1 operation then, the composite phase of the signal entering the phase-lock loop is simply the multipath phase relative to the direct signal. The direct signal phase is taken to be zero. Conceptually, region I1 operation involves the DLL tracking the multipath rather than the direct signal. In either operating region then, the multipath-induced carrier-phase measurement error is given by 0;.
3.4 Non-coherent PRN Receiver
This section derives expressions for the non-coherent DLL discriminator curve in the absence and presence of multipath. Also derived is the expression for the composite phase of the multipath corrupted signal.
3.4.1 Non-coherent DLL Discriminator Curve in the Absence of Multipath
As shown in Figure 1, the outputs of the early and late correlators (mixers plus bandpass filters) are squared prior to being low-pass filtered and differenced. The signals entering the LPF's are given by squaring the expressions in equations (12) and (13):
Note that since the early and late signals are being generated non-coherently, a PLL is not employed. As a result, 0 is simply a phase offset rather than being a PLL tracking error. Equations (42) and (43) may be expanded yielding:
The LPF's reject the double-frequency terms leaving:
Again, the discriminator function is formed by differencing the outputs of the LPF's. The normalized form of the non-coherent discriminator function is thus:
Where the subscript n denotes non-coherent DLL operation.
3.4.2 Non-coherent DLL Discriminator Curve in the Presence of Multipath
In the presence of multipath, the incoming signal is given by equation (7). In the case of the non-coherent DLL, the early and late codes are modulated onto
33 an IF which is frequency-locked (but not phase-locked) to the downconverted incoming signal. The early and late signals are then given by:
where:
0 is the phase offset between the locally generated IF and the down-
converted incoming signal
Note that these expressions are similar to those for the coherent DLL (see equations (18) and (19)). The outputs of the early and late correlators may be obtained by ignoring the composite phase term in equations (22) and (23):
These signals are then passed through the squaring devices yielding:
The LPF's reject the double-frequency terms yielding:
Finally, the discriminator function is formed by differencing the outputs of the LPF's. The normalized form of the non-coherent discriminator function is thus:
Where the subscript nm denotes non-coherent DLL operation in the presence of a single multipath ray.
3.4.3 PLL Operation in the Presence of Multipath
Normal operation of a non-coherent DLL involves the code-tracking loop working in parallel with a frequency-tracking loop. As such, the hardware in the basic DLL does not support carrier-phase measurements. In such implementations, carrier-phase tracking is accomplished with a separate PLL. Prior to entering the PLL, the incoming signal is mixed with the output of the on-time modulator and the result is bandpass filtered. Since the DLL is non-coherent, the on-time signal will be frequency-locked but not phase-locked to the incoming signal. The
36 resulting mixed and filtered signal will therefore not be completely stripped of the prn code. However, the remaining portion of the pm code will have a frequency content in the megahertz range (CIA code chipping rate: 1.023 MHz;P-code chipping rate: 10.23 MHz). Since the tracking loop bandwidth of typical PLL's is less than 20 Hz, the code will not affect the carrier-tracking. As with coherent DLL operation, the phase of the incoming carrier is perturbed by the presence of the multipath. However, the carrier-phase error is not dependent upon the behavior of the code-tracking loop. Assuming ideal PLL operation, the composite phase of the direct plus multipath signal is tracked. To aid in the derivation of the composite phase, the prn code modulation in equation (4) is ignored and the result is rewritten in phasor form @lane wave propagation is assumed):
where:
0 . 5 * is~ the ~ average signal power into the receiver
k, = (2711 A)uD and is the wave number vector (u, is the unit vector from the satellite to the receiver)
r, is the position vector from the origin to the receiver rZ is the carrier wavelength o, is the frequency of the received signal (carrier plus Doppler shift)
t is time Assume A = 1 and let
Equation (58) may now be simplified:
sl(t) =
dh
The multipath signal differs from the direct by a magnitude and phase factor and
may be written:
where:
a is the relative amplitude of the multipath Q~~
is the absolute phase of the multipath
8, is the relative phase of the multipath
The received signal is the sum of the direct plus multipath:
Taking the phase of the direct signal as the reference, equation (62) may be rewritten:
38 By employing Euler's identity, equation (63) may be rewritten in terms of its real and imaginary components:
The composite phase then is the arctangent of the ratio of the components:
Again, since the composite phase was derived relative to the direct signal phase, the composite phase is thus the carrier-phase error. By comparing equation (65) and equation (39), one notices that the composite phase tracked by the PLL in the coherent DLL is distorted by the code-tracking loop. However, this is not the
case in a non-coherent DLL receiver architecture since carrier-phase tracking is accomplished separately in hardware. To derive the expression for the maximum carrier-phase error, the derivative of the composite phase is taken with respect to the phase of the multipath. Employing the formula for the derivative of the arctangent:
Carrying out the derivative yields:
Expanding this expression:
Canceling the numerator of the first term with the denominator of the second term and using the trigonometric identity, cos2u
+ sin2u = 1, yields:
40 By setting the derivative equal to zero, the multipath relative phase giving rise to the maximum carrier-phase error may be determined:
Substitution of equation (70) into equation (65) yields the expression for the maximum carrier-phase error:
-a)) 0L= tan-'{ 1 +aas i n ( * ~ o s - ~ (a>) COS(~COS-~(-
=
tan-' { a sin(*us-'(- a)) 1- a 2
Recalling the trigonometric identity,
sin(--'(u))
Equation (71) may now be simplified:
Recalling a second trigonometric identity,
=
dl - u2
Use of the above equation yields the final form for the maximum carrier-phase error:
For alternative derivations see Georgiadou (1988) and Bishop (1985). Assuming that the multipath strength is always less than or equal to the direct, it follows that the carrier-phase error can be no more than 90 degrees. At the GPS L1 band (1575.42 MHz), this corresponds approximately to 4.8 centimeters. As will be shown in the next section, code-phase errors can exceed 100 meters. Carrierphase tracking is thus an attractive technique. It should also be noted that the 90 degree phase error only occurs for a = 1 (and therefore 8, = 180 degrees; see equation (70)). In most instances the multipath will not be that strong and the carrier-phase error will then be less than 90 degrees. In fact, the case where a = 1 and Om= 180 degrees is undefined. This corresponds to having the multipath signal being equal in amplitude but opposite in phase with respxt to the direct. As a result, the total signal entering the receiver is null.
3.5 Simulation Results
Having derived the multipath error equations, a parametric study was performed to quantify the error encountered under various multipath conditions. The error equations were implemented in FORTRAN routines and the simulations were performed on an 80386-based personal computer.
Pseudorange errors as a function of relative multipath amplitude and delay are given in Figure 4 through Figure 11 for the standard coherent and noncoherent DLL's. Note that the standard separation of 1 pm bit period between the early and late correlators is assumed. In each figure, the relative multipath amplitude is constant and the relative phase is taken to be a function only of the relative time delay. The rate of oscillation of the error curves is high and thus only the envelope may be easily distinguished. Examination of the envelopes reveals that peak multipath error increases with increasing multipath strength. It is also important to note that in all cases the error drops to zero at a delay of approximately 1500 nanoseconds. Since one CIA code chip is roughly 1000 nanoseconds (300 meters), it may be concluded that the receiver will reject multipath with delays of greater than 1.5 chips.
pseudorange error in nanoseconds
51 For a better understanding of the behavior of the error curve within the envelope, Figure 12 and Figure 13 show pseudorange error, and the corresponding error spectrum, over a small range of time delay. When the relative multipath amplitude is small, the error varies sinusoidally as a function of relative path delay and is thus narrow-band. When the relative amplitude is large, however, the error is not sinusoidal and in fact contains sharp discontinuities. Not surprisingly, the signal is thus wide-band. This disproves the popular myth that every peak in a multipath error spectrum corresponds to a separate multipath ray. The nonsinusoidal behavior is a result of the non-linear signal processing which the DLL employs. The effects reach farther than simply causing an increased error signal bandwidth. As was first shown by Hagerman (1973) and later by Van Nee (1991), and Braasch and Van Graas (1992), the error signal is not zero-mean. In situations where the multipath relative phase is fluctuating (non-zero relative phase rate-of-change), the errors will not average out to zero. Plots of average error versus time delay are given in Figure 14 and Figure 15. Since relative-phase is a complex function of the electromagnetic properties of the reflecting object in addition to the time delay, error may be computed by holding relative amplitude and phase constant, and varying the time delay. For each of the plots shown, error was averaged over ten relative-phase values evenly spaced between 0 and 180 degrees. As can be seen in the plots, not only are the error traces non-zero mean, the average value can easily be several tens of nanoseconds. For example, a multipath-to-direct ratio of -8dB yields a
(su) Joua apo:,
(su) l o u a apo:,
(su) lorla a8uelopnasd a a e ~ a ~ e
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56 peak average error of 30 nanoseconds for the coherent DLL. This corresponds to a ranging error of approximately ten meters.
3.5.2 C/A-Code with Narrow Correlator Spacing
Fenton, et al (1991) and Van Dierendonck, et a1 (1992) describe a recent (patented) development in GPS receiver design which lessens the effect of multipath by narrowing the spacing of the early and late correlators. By using a small portion of the correlation function (around the peak) to form the discriminator, maximum multipath error is reduced and multipath with relative delays of approximately 1 chip or greater is rejected entirely. Current hardware is capable of 0.1 chip correlator spacing. With wideband processing of the signal, it is anticipated that future implementations could reduce the correlator spacing to 0.05 chip. Since the error equations were derived with early and late correlator offsets as a variable, analysis of the narrow correlator spacing technique is possible with the previously developed FORTRAN routines. The results are shown in Figure 16 through Figure 31. Note that the peak errors are reduced by approximately a factor of ten from the 1 chip correlator spacing results. It should be noted that although Hagerman (1973) and Van Dierendonck, et al (1992) indicate the narrow correlator concept only applies to the non-coherent DLL, the results clearly show this is not the case. Coherent designs benefit as well. More
CIA-Code; M/D= -20tlB; Coherent DL14 0.1 C h ~ pEarly-Late S p a c ~ n g
I
7- -
7 -
--
-
1
-
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-
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-
-
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rrlultipatll rel;itiw time delay in nanoseconds
~ i g u r e16. C I A - C o d e M u l t i p a t h E r r o r ; M I D
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C o h e r e n t DLL;
0 . 1 C h i p Spacing
pseudorange error in nanoseconds
pseudorange error in nanoseconds
I
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multipath relative time deli~yin n a ~ ~ o s e c o n d s
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Figure 24. CIA-Code Multipath Error; M / D
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Figure 25. C/A-Code Multipath Error; M/D
1
('/A ('ode;
27. CIA-Code Multipath Error; MID = -4.4dB; Coherent DLL; 0.05 Chip Spacing
multipath relative time delay in nanoseconds
CIA-Code; MID=-4.4dB; Coherent DLL; 0.05 Chip Early-Late Spacing
pseudorange error in nanoseconds
73
recently, Van Dierendonck (1992) has confirmed this. Plots of average error versus time delay are given in Figure 32 through Figure 35. Again, an improvement of approximately a factor of ten is gained. Careful examination also reveals that the coherent DLL performs slightly better than the non-coherent. This is not surprising in light of the 1 chip spacing results.
The P-Code is the second pm code which is modulated onto the GPS carrier. Although it is available to the public during the current constellation buildup, Defense Department policy indicates that this code will be encrypted (forming the so-called Y-Code) so as to be accessible only by authorized users. For those users having access to it, however, considerable multipath reduction or rejection is gained. Since the P-Code (10.23 MHz) is modulated at a rate 10 ten times higher than the CIA-Code (1.023 MHz), its chips are thus one-tenth the length. The P-Code therefore is much less sensitive since it is affected only by multipath with relative time delays less than 1.5 times its own chip. Figure 36 through Figure 45 quantitatively confirm these qualitative results. Maximum error has been reduced by a factor of ten. The average error plots show the P-code error to have the same shape as the CIA-code with just the scale being changed.
Figure 32. CIA-Code Average Multipath Error; Coherent DLL; 0.1 Chip Spacing
multipath relati% time delay (IS)
CIA-Code; Coherent DLL; 0.1 Chip Early-Late Spacing
(su) Jolla aBue1opnasd a a e ~ a ~ e
Figure 34. C/A-Code Average Multipath Error; Coherent DLL; 0.05 Chip Spacing
multipath relative time delay (ns)
CIA-Code; Coherent DLL; 0.05 Chip Early-Late Spacing
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200
400
800
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1200
1400
-20dB; Non-coherent DLL; 1 Chip Spacing
multipath relative time delay in nanoseconds
600
1
I
-30 I
1
-
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-20 -
-10 -
0
10-
I
Figure 40. P-Code Multipath Error; MID
2 a
9
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E
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P-Code; M/D=-20dB; Non-coherent Dm 1 Chip Early-Late Spacing
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200
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400
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800
I
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1200
I
1400
-8dB; Non-coherent DLL; 1 Chip Spacing
multipath relative time delay in nanoseconds
600
-
-30 -
0
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-
-20 -
- 1 1(11
0
10
Figure 41. P-Code Multipath Error; M/D
0 iB %
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400
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800
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600
I
1200
I
1400
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-10
0
10
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20
~ i g u r e43. P-Code Multipath Error; MID = -4.4dB; Non-coherent DLL; 1 Chip Spacing
%
a =r
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4. AIRBORNE ANTENNA PATI'ERN CHARACTERIZATION
In chapter 3, the relationships governing the response of the receiver to multipath were derived. Multipath was parameterized in terms of amplitude, time delay, phase and phase-rate relative to the direct signal. Prior to arriving at the receiver tracking loops, multipath must pass through the antenna. Realizable antennas (as opposed to theoretical isotropic radiators) do not receive signals equally from all directions. In fact, partial multipath rejection is built-in to some antenna designs by shaping the gain pattern. Since most multipath arrives from angles near the horizon, multipath may be rejected by shaping the pattern to have low gain in these directions. However, extensive shaping of the pattern requires either a large aperture or multiple elements and signal processing. For the ground reference system this is not a problem. A high profile antenna with low gain at the horizon is acceptable. Such is not the case for the airborne user. High profile antennas are not desirable for two reasons. One is the additional drag they induce on the aircraft. Secondly, unlike the ground reference station, airborne antennas must continue to receive the satellite signals while undergoing dynamics. A sharp null in the pattern could result in the loss of a signal if the aircraft happens to be banked appropriately. Accordingly, most airborne installations involve conformal microstrip antennas. In addition to the benefits just mentioned, microstrip antennas offer small size, low weight and cost and relative ease of installation palanis, 19821.
89 Characterization of the airbome antenna pattern involves determining how the presence of the airframe distorts the free-space pattern. The basic problem is one of scattering by a complex three-dimensional object. Since closed-form solutions do not exist for realistic airplane models, computational techniques must be employed. The leading candidates for such a problem are the (Uniform) Geometrical Theory of Diffraction (GTD) and the Method of Moments pansen, 1981, Harrington, 1968, Balanis, 1982, Balanis, 19891. The GTD models scattering problems by computing reflected and diffracted field components. Since it is a high-frequency asymptotic technique it is therefore applicable only to those problems involving objects whose dimensions are large compared to a wavelength. The Method of Moments, on the other hand, is a general technique applicable to all problems. The Method of Moments solves for the scattered field by determining the currents induced on the object and then computing the standard radiation integrals. However, implementation of the Method of Moments is restricted to problems whose objects have dimensions on the order of a wavelength. This stems from limitations in current computer memory size, finite word length and processing speed. Since even the smallest airplane is large compared to the GPS wavelength (approximately 19 centimeters), the GTD is the choice for solving the airbome antenna pattern problem.
4.1 Antenna Model
Prior to implementing the GTD to solve the problem, a model for the antenna itself must be derived. Microstrip antennas are composed of two parallel conducting plates separated by a thin slab of dielectric material. The bottom plate may be thought of as a ground plane. The top plate, also called the patch, is attached to the feedline. Closed form expressions describing the behavior of microstrip antennas have been derived by assuming the antenna to be an ideal cavity
m, 19881. Closed-form expressions may only be obtained
with antennas
whose patches are in geometrically simple shapes. More complex structures giving rise to circular polarization may be modeled as cavities but solution requires computational techniques [Carver, 19811. The method of finite elements is one such technique but accurate results require an extremely computationally intensive procedure [Coffey, 19921. In any case, these models are ideal and therefore cannot represent the irregularities in the gain pattern due to imperfect manufacturing technology [Lo, 19921.
4.1.1 Theoretical Background for Antenna Modeling
An alternative approach is to find an equivalent model which has the same gain pattern as that measured from the antenna of interest. This approach is attractive in that the peculiarities associated with the physical device may be
91 circumvented. One need only determine the antenna aperture current distribution which gives rise to the measured gain pattern. A computer model developed at the Ohio State University ElectroScience
Laboratory performs this function [Pelton, 19781. The model is based on an antenna synthesis technique described in Mautz (1975). The technique is founded upon the relationship between the source and its associated field:
where:
T is a linear operator f is the source g is the field radiated by the source
In the synthesis problem, a desired field, go, is specified. The goal is to find the optimum f such that the difference between go and g is minimized. The source is assumed to be a discrete set of elemental radiators. The desired radiation pattern is described by a discrete set of field points. Typically the required number of elemental radiators is small compared to the number of specified field points. The problem is thus an overdetermined set of simultaneous equations and the least squares solution is therefore:
However, in order to perform this computation, go must be specified in magnitude
92 and phase. If the phase pattern has not been specified, an iterative procedure is required. The procedure starts by assuming a set of phase values for go. The least-squares solution for f is formed and g is calculated. Pattern error is formed by calculating the norm of the difference between g and go. In general, the individual elements of g and go will not be in phase. For minimum pattern error, however, they must be in phase. The phase terms in go are therefore updated to be in phase with g. At this point the procedure repeats by forming the leastsquares solution for f with the updated go. This procedure iterates until the pattern error has converged to a minimum value. In the synthesis problem, a desired gain pattern is specified. In the antenna modeling problem, actual antenna pattern measurements are used. Since measured radiation patterns are typically given in magnitude only, the iterative procedure described above may be applied. The result is a set of current elements which, in terms of radiation patterns, is equivalent to the actual antenna. Pelton (1978) describes an adaptation of the iterative procedure wherein the rate of convergence was increased by employing a gradient search in the minimization process. It is interesting to note that the synthesis results of the procedure are far from spectacular (Mautz, 1975). However, as will be shown later, the antenna modeling results match closely with the measured data.
4.1.2 Application to GPS Antennas
The computation in the iterative procedure primarily involves matrix operations. As such, the FORTRAN code provided in Pelton (1978) is not optimum in terms of run time. Accordingly, the author translated the FORTRAN code into MATLAB. In order to verify the proper operation of the MATLAB code, test data provided in Pelton (1978) was used. The results matched those which were published. Having gained confidence in the code, GPS antenna patterns were obtained for modeling. Since the GPS signal is circularly polarized, measured data is given in the form of axial ratio patterns D,19881. The axial ratio patterns for Sensor Systems antennas S67-1575-16 and S67-1575-2 mounted in the center of circular ground planes (4 feet in diameter) are given in Figure 46 and Figure 47. The patterns are formed using a linearly polarized source antenna which rotates as the pattern is being taken. The outer envelope of the pattern is the response of antenna to a vertically polarized source. The inner envelope is the response to a horizontally polarized source. The difference between the two gives the axial ratio pattern. The discretized source was formed by two sets of magnetic current elements. One set was placed perpendicular to the pattern cut. The iterative procedure was used to determine the magnitude and phasing of each of the elements for the vertical polarization pattern (Figure 48 and Figure 49). Circular
Figure 47. Axial Ratio Pattern for GPS Antenna S67-1575-2
98
polarization was then obtained by placing a second set orthogonal to the first set and feeding the elements with the same magnitude but 90 degrees shifted in phase. The synthesis results in Mautz (1975) and the antenna modeling results in Pelton (1978) showed that element spacing should be no greater than one-quarter wavelength. Since the GPS antennas are approximately one-half wavelength in diameter, three elements were used in each set. The pattern error as a function of iteration number is given in Figure 50 and Figure 51. Modeled versus measured results are given in Figure 52 and Figure 53. The magnitude and phasing for the elements modeled after the vertical patterns are given in Table 1.
TABLE 1
S67-1575-16
Magnitude
Phase (deg .)
S67-1575-2
Magnitude
Phase (deg .)
1
0.5262
145.9153
1.0000
0.0000
2
1.0000
0.0000
0.6835
-176.1413
3
0.5262
145.9010
0.9747
0.1009
The driving force behind finding the maximum allowable spacing was the desire to minimize the number of modeled elements and thus minimize run time. However, no mention was made in the literature with regard to decreasing spacing in an effort to improve results. In an attempt to improve performance, the author tried additional elements with a closer spacing. Spacing the elements
Figure 50.
10.'
-
10-2--
iteration number
15
20
25
30
Mean-Squared Pattern Error For Modeled GPS Antenna S67-1575-16
I
10
I
5
-
-
-
-
-
0
-
-
-
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-
-
-
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-
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S67-1575-16; Mean-Squared Pattern Error 7
-
-
-
-
-
100.-
103 one-eighth wavelength improved the results only slightly and did not justify the additional computational burden. Spacings of closer than one-eighth wavelength yielded near-singular matrices and disastrous results. The one-quarter wavelength spacing was thus retained.
4.2 Airborne Antenna Pattern Modeling
Having obtained models for the antennas, the airborne antenna patterns could be determined. The GTD modeling was performed using the program NEWAIR3 developed by the Ohio State University ElectroScience Laboratory [Kim, 1984 and Burnside, 19851, The code models the aircraft fuselage with a composite ellipsoid. Wings and stabilizers are modeled with flat plates. The code
can model direct, reflected and diffracted rays as well as second order interactions. The magnetic current elements, derived above, are mounted on the surface of the composite ellipsoid. The code can perform arbitrary pattern cuts in addition to the principal planes. Other capabilities include the generation of field patterns due to particular interactions. For example, one can obtain that portion of the pattern due only to diffraction. In order to determine the sensitivity of the patterns to airframe type, a Piper PA-32 (Saratoga) and a Boeing B-737 were modeled. To determine the sensitivity to antenna type, the PA-32 was modeled with the S67-1575-16 and S671575-2 antennas. Principal plane patterns (Figure 54 through Figure 56) were
Figure 54. Roll Plane Pattern Cut
Figure 55. Elevation Plane Pattern Cut
Figure 56. Azimuth Plane Pattern Cut
107 calculated using only first-order interactions. This is the recommended procedure since the second-order terms are typically negligible and run-time is considerably increased. In addition, if the second-order terms are not negligible, their absence
can be noted as minor discontinuities in the patterns [Burnside, 19851.
4.2.1 Piper PA-32 (Saratoga)
The Piper PA-32 is a single-engine, six-passenger general aviation aircraft (Figure 57). The antenna was mounted centered on top of the fuselage just behind the cockpit. The fuselage was modeled by a composite ellipsoid and the wings and stabilizers were modeled by flat plates. Although true radiation pattern data was not available, a reasonableness check was accomplished through GPS data collection. Signal-to-noise ratio (SNR) data were collected from an Ashtech P12 GPS receiver mounted in the Avionics Engineering Center's PA-32, N8238C. The aircraft was parked on the ramp at the Ohio University airport, UNI. Data were collected from satellite 21 on Julian day 106, 1992 using Sensor Systems antenna S67-1575-14. Although an accurate axial ratio pattern for S67-1575-14 was not available, the pattern for S67-1575-16 is a close match and hence was employed. SNR as a function of satellite elevation angle is given in Figure 58 and the corresponding computed antenna gain pattern is given in Figure 59. At first glance the two would appear not to compare well. However, the additional 4 dB roll-off in the measured SNR is due to atmospheric
Figure 57. Piper PA-32 Saratoga
111 attenuation of the satellite signal which is typical for low elevation angles [Spilker, 1977 and Braasch and Van Graas, 19911. Taking this into account, the measured and modeled data compared quite favorably. The full radiation patterns are given in Figure 60 through Figure 62. The results satisfy intuition. The patterns are well-behaved above the local horizontal plane and show significant attenuation below. The results for antenna S67-1575-2 are given in Figure 63 through Figure 65. The patterns are quite similar to those for the previous antenna. The effects of the sharper pattern roll-off, recorded in the measured data, may be observed.
4.2.2 Boeing B-737
To contrast the general aviation aircraft, the Boeing B-737 represents typical transport aircraft (Figure 66). The GPS antenna, S67- 1575-16, was mounted on top of the fuselage approximately 2 meters forward of the wings. Again, the fuselage was modeled by a composite ellipsoid and the wings and stabilizers were modeled by flat plates. The model, and verification of its accuracy, is given in [Burnside, 19851. The principal plane patterns are given in Figure 67 through Figure 69. The elevation and azimuth plane patterns are quite similar to those for the PA-32. However, the roll plane pattern is significantly different. Although the general
Figure 61.
PA-32; Elevation Plane Power Pattern; S67-1575-16
pattern angle (degrees)
PA-32; Elevation Plane Pattern; S67-1575-16
(a)
SIDE V l E W
FRONT V
( c ) TOP V l E W
Figure 66. Boeing B-737 (from [Burnside, 19851)
122 trend of the pattern roll-off is similar, the B-737 pattern is much smoother. This is a direct result of the antenna being mounted forward of the wings. In the case of the PA-32, the antenna was mounted above the wings. The wings caused diffraction which revealed itself in the form of rapidly oscillating peaks and nulls in the pattern.
123 5. AIRFRAME-BASED MULTIPATH AT THE AIRBORNE RECEIVER
Having determined the characteristics of the airborne antenna pattern, it is now possible to extend those results to determine the multipath environment at the airborne receiver. The airborne pattern directly describes the receptivity of the antenna to obstacle-based multipath. This will be discussed further in the next chapter. By examining the various components of the airborne pattern, the airframe-based multipath environment may be characterized as well. Any signals other than the direct one from the satellite are multipath signals. Because of an increased path length they are delayed in reaching the receiver and therefore corrupt the direct signal. The frrst-order GTD interactions are the source, reflected and diffracted rays. The source field represents the direct signal. The reflected and diffracted fields represent airframe-based multipath. The errors induced by this multipath are a function of the relative amplitude, time delay, phase and phase-rate. Characterization of the reflected and diffracted magnitude patterns (relative to the direct signal magnitude) provides the relative amplitude parameter. Reflected and diffracted phase patterns, relative to the direct, provide the relative phase parameter. Phase-rate is a function of satellite-scatterer-receiver geometry and dynamics. Relative time delay is also a function of geometry but is bounded by the dimensions of the aircraft. The wingspan, typically, is the largest dimension on a civilian aircraft. For antennas mounted on the fuselage, then, the largest relative time delay
124 corresponds to the incident wave just grazing the antenna, passing on to the wingtip and then diffracting back. The maximum relative time delay is thus twice the distance from the antenna to the wingtip (scaled by the speed of light).
5.1 PA-32 Roll Plane Patterns
To illustrate these concepts, reflected and diffracted roll plane patterns were computed for the PA-32. Surprisingly, no reflected field exists. Upon further investigation it was discovered that the fuselage actually prevents the antenna from directly impinging upon the wings. The diffracted field patterns are given in Figure 70 and Figure 71. The magnitude pattern is in dB relative to the peak value of the total field pattern. Although small for the most part, the field does come to within 10 dB of the direct signal.
5.2 Structural Flexing
Preliminary investigation in the area of airframe-based multipath error revealed a phenomenon referred to as "multipath randomization" [Braasch, 19911. It was observed that airframe-based multipath errors are noise-like while in-flight. This allows for error variance reduction by smoothing the pseudorange against the camer measurements. Figure 72 through Figure 75 show how this can occur.
131 The roll plane diffraction pattern for the PA-32 was modeled again but with the dihedral of the wings removed. This corresponds to the flexing of the wings while in-flight. Comparison of the dihedral and no-dihedral plots reveals a shift in the magnitude pattern dependent upon the position of the wings. The phase patterns differ as well. Thus, as the wings flex and bend, the phase and magnitude of the multipath oscillates yielding the observed results. It should be noted that although this phenomenon can be exploited to reduce error variance, a bias will remain. As shown in chapter 3, average multipath error will be close to zero if the multipath relative amplitude is weak. However, average error can be several tens of nanoseconds even for moderate relative amplitude levels. Accordingly, full exploitation of this phenomenon will require a linear receiver architecture yielding zero-mean multipath error.
132 6. OBSTACLE-BASED MULTIPATH AT THE AIRBORNE RECEIVER
This environment may be divided into two categories: 1) Obstacles located below the aircraft; 2) Obstacles located above the aircraft. For any given obstacle, its relative height will depend upon the location of the aircraft on the approach. An obstacle may be below the aircraft at the beginning of the approach but above at the end. The airborne antenna patterns from chapter 4 reveal the response to multipath from above and below the aircraft. Although greatly attenuated, multipath signals from below the aircraft maintain sufficient strength to cause measurable errors. For multipath signals from above the aircraft, little or no attenuation occurs. In this case the antenna achieves little multipath rejection and thus the environment around the aircraft must be examined.
6.1 Obstacle Clearance Surfaces
At the start of this investigation, it was thought that the minimum physical separation of objects from a precision approach path would yield minimum multipath relative time delays. If the minimum delays were greater than 1500 nanoseconds, multipath immunity would have been achieved. Unfortunately, this is not the case. The international requirements governing the minimum physical separation of objects from approach paths are
133 known as obstacle clearance surfaces (OCS) [ICAO, 19831. A careful study of UCAO, 19831 reveals that obstacles of any height can be as close as 685 meters from an aircraft on a nominal approach. The receiver-object separation primarily affects the relative amplitude of the multipath signal. As will be demonstrated next, near-zero relative time delays can be obtained regardless of the receiver-object separation. Consider the two-
dimensional satellite-object-receiver geometries depicted in Figure 76 through Figure 78. Since the GPS satellites are at a distance of approximately 2x10' meters from users near the surface of the Earth, plane wave propagation may be assumed at the receiver and object. Accordingly, in Figure 76 with the satellite at the left of the receiver, the minimum multipath relative time delay is twice the distance from the object to the receiver, 2d. With the satellite above (Figure 77), the minimum relative delay has been reduced to d. However, with the satellite on the right (Figure 78), the relative time delay approaches zero. It can be concluded that an angular region exists around the receiver to object line-of-sight in which a satellite could produce multipath with a relative time delay of less than 1.5 prn code chips. To derive the bounds on this region, consider Figure 79. The multipath relative time delay is given by (d-dcose). The bounds on the critical region are given by the 0 for which (d-dcose) is equal to 1.5 chips. Solving for 8 yields:
where r,,, is maximum multipath relative time delay for which the receiver is sensitive. For the case of standard CIA-Code receivers,
I,,,
is 1.5 chips (1500
nanoseconds). For the extreme case where d is 685 meters, 8,
evaluates to 70
degrees. Thus an extremely wide multipath region exists. It should be noted, however, that the derivation 'assumed the receiver, object and satellites all to be at the same height. Therefore, this multipath region is only associated with lowelevation angle satellites. As mentioned earlier, object-receiver separation primarily affects multipath signal strength. Since relative time delay was shown not to have a non-zero minimum, the investigation was focused on characterizing multipath relative amplitude. From a conceptual viewpoint for finite-sized objects, increased objectreceiver separation will result in decreased multipath relative amplitude. Since the minimum separation has been determined to be 685 meters, what remains to be shown is the size of an object which produces significant multipath given this separation. Following the theory of optics, significant signal reflection does not occur for objects whose dimensions are much less than the first Fresnel zone [Beckrnann, 19631. The first Fresnel zone is thus a rule-of-thumb in determining minimum object size to yield non-negligible multipath signals. In the optics sense,
139
the first Fresnel zone is an ellipsoid about the line-of-sight consisting of all points whose combined path length to the receiver and transmitter is one-quarter wavelength longer than the line-of-sight distance. Often the term Fresnel zone is used to indicate the cross-section of the ellipsoid (a circle) at a given point along the line of sight. The radius of the circle is a function of wavelength and the receiver-object-transmitter separation and is referred to as the Fresnel zone radius. The ellipsoid represents the volume of space around the line-of-sight which must be free of obstacles to insure clear transmission. The same theory may be applied to the case of reflection by considering propagation from the transmitter to the image receiver. The intersection of the ellipsoid with the reflecting object defines that portion of the object which is significant in producing the reflected field. Accordingly, if an object is small in terms of the Fresnel zone radius, the signal it reflects is of negligible magnitude. For cases in which the receiver and object are closely spaced relative to their distance to the transmitter, the expression for the first Fresnel zone radius is given by [Shibuya, 1983:
where d is the distance between the object and receiver. A plot of the first Fresnel zone radius versus range (for the GPS wavelength) is
given in Figure 80. As the plot shows, the radius is well under 100 meters even for receiver-object separations of 10 kilometers. For a separation of 685 meters,
141 the radius is 11.4 meters. Thus relatively small objects can produce significant multipath signals. These general results may be confirmed by a parametric bistatic scattering study. Bistatic scattering involves determining the fields scattered by a given object under illumination from a fixed source. For this study, the model, GEMACS (General Electromagnetic Model for the Analysis of Complex Systems) developed by Dr. Edgar Coffey was employed [Coffey, 19901. For computational efficiency and ease of analysis, a two-dimensional perfectly conducting strip illuminated by a vertically polarized source was modeled (a two-dimensional strip is modeled in GEMACS by making the strip very long in the vertical dimension). Three strip widths were modeled: 10 meters, 50 meters and 100 meters. The source was located at
t$
= 305 degrees and the strip was
placed in the Y-Zplane. The nominal specular reflection angle was thus @ = 55 degrees. The scattered field strength (relative to the direct signal) was computed for angles around the nominal reflection angle at a distance of 685 meters from the centerline of the strip. The results are given in Figure 81 through Figure 83. As expected, multipath relative amplitude is high for the 50 and 100 meter strips. The 10 meter strip, occupying less than half of the first Fresnel zone diameter, is on the borderline of producing a negligible scattered field.
6.2 Earth-Surface-Based Multipath
From the discussion of the previous section, it follows that at low altitudes, the surface of the Earth (both land and water) represents an "object" which gives rise to a large multipath region. The satellite elevation angle which gives an upper bound to the multipath region is dependent upon the altitude of the aircraft. To evaluate this bound definitively, consider the ground reflection depicted in Figure 84. The receiver is located at an altitude of h above the surface of the Earth. The Earth surface is assumed to be a perfectly conducting infinite plate. As such, geometrical optics and image theory may be employed @3alanis, 19891. The "bent" path of the reflected signal may be thought of as a "straight" path to an image receiver located a distance h below the plate. Determination of multipath relative time delay, 6, may be accomplished as shown in Figure 85 and is given by:
The corresponding satellite elevation angle, I#, is the complement of @:
The upper bound on the multipath region is given by the relative time delay of r,,,.
@
which gives rise to a
This may be determined by setting the relative time
delay equal to r,,,, solving for I$ and then computing I#:
As expected, the upper bound on the multipath region is thus a function of receiver altitude. Figure 86 and Figure 87 plot this bound for standard CIA- and P-Code receivers. The upper bound levels off at zenith when altitude is less than or equal to one-half of r,,. Note that r,,, is 440 meters for standard CIA-Code receivers and 44 meters for P-Code receivers. The order-of-magnitude decrease in r,,, for P-Code over CIA-Code yields a significantly smaller multipath region. For altitudes of 169 meters or above, the multipath region is below 7.5 degrees.
Typical civilian receivers do not track satellites below 7.5 degrees (also known as the mask angle) due to unacceptable atmospheric signal derogation. However, precision approach guidance below this altitude may require tracking of satellites in the multipath region. Having characterized the range of Earth's surface multipath relative time delay, the strength of the multipath may be determined by examining the reflection coefficient. For electromagnetic wave incidence from air onto nonferromagnetic, lossy dielectric media, the reflection coefficients for vertical and horizontal polarizations are given by [ S tutzman and Thiele, 19811:
where:
and where: 0, is the angle of incidence (relative the surface normal) E,
is the relative permittivity of the lossy media
E,
is the absolute permittivity of free space
a is the conductivity of the loss media
~ ) sea water ( ~ , = 8 1 ,a=4) at the GPS L1 Results for dry soil ( ~ , = 2 . 8 , 0 = 1 0 and frequency (1575.42 MHz) are given in Figure 88 through Figure 91. Although dependent upon incidence angle, the results show that significant multipath can originate from the Earth's surface. This is particularly true for seawater. Although data have not been collected, it is likely that multipath from
10
50 angle of incidence (degrees)
40
70
T~---
Reflection Coefficient Magnitude For Dry Soil
30
.--
__-__--__--
.._------- ..-
horizontal polarization: dashed
vertical polarization: solid
20
--
Reflection Coefficients for Dry Soil
7-
__.__-----_._._..._-------
Figure 88.
OO
0.1 -
0.2 -
0.3 -
0.4 -
0.5 -
0.6 -
0.7 .-
0.8 -
0.9 -
1 r---
----------
---------
80
-
-
-
(suepe~)aseqd ~uapgao:, u o p ~ q a ~
156 the ocean surface could severely corrupt precision guidance of aircraft to ships
and offshore oil platforms.
157 7. OBSTACLE-BASED MULTIPATH AT THE GROUND REFERENCE STATION
The multipath error equations derived in Chapter 3 give general guidance regarding the siting of the ground reference station to avoid multipath. These equations essentially relate two principles for the avoidance of multipath. The frrst is to site the station so as to maximize relative time delay of multipath from nearby objects. Secondly, the station should be located such that multipath arrives from directions in which the antenna has low gain. As was shown in Chapter 6 , no minimum receiver-object separation exists such that multipath relative time delay is kept above a given critical value. Furthermore, the size of the first Fresnel zone is relatively small even for receiver-object separations of several kilometers.
7.1 Antenna Architecture and Siting Considerations
As mentioned in Chapter 4, multipath generally arrives from the horizon. This is true in the case of ground reflection as well as for other objects in and around the airport. Since a conformal antenna is not a requirement of the ground reference station, antennas with larger apertures may be used. The increased aperture size allows for shaping of the gain pattern to attenuate multipath arriving at or below the horizon. The desired gain pattern is then nearly omnidirectional
158
in the upper hemisphere and attenuated to whatever extent possible at the
horizon and below. The ideal siting would be such that the antenna is above all objects in the vicinity. Note that siting above all nearby objects is not a trivial
task. The tallest object on an airport typically is the control tower. Although being above all nearby objects, the top of the tower is not an ideal candidate for ground reference station siting. This follows since control towers often are near other buildings on the airport surface. By virtue of their height, the top of the tower also is usually the location for many communication antennas which would further distort the pattern of an antenna for a satellite navigation receiver. Even if an "ideal" location is available, the non-zero gain of the antenna below the horizon must be accommodated, Furthermore, depending upon the type of antenna used, a ground plane may or may not be required. Mounting on a ground plane has two effects. The first is to distort the antenna free-space pattern such that gain near the horizon is increased. Secondly, the presence of a ground plane results in significant diffracted energy entering the antenna [Balanis, 19821. Even energy originating from below the ground plane will enter the
antenna via edge diffraction. Furthermore, as indicated in the name, the ground plane acts as the "ground" or reference for the voltage induced in the antenna by the incident field. Multipath illuminating the ground plane from below will distort the desired signal entering from above. In this sense, even the antenna feed line acts as part of the ground plane. As a result, placing the antenna above all objects could actually increase the level of multipath distortion.
159 Possible techniques to overcome the diffraction problem include the use of optical data links between the antenna and receiver. This reduces the effect of multipath incident upon the feedline. Bletzacker (1985) discusses the use of RF absorber material to alleviate the distortion of the pattern above the horizon. By mounting RF absorber material between the ground plane and the antenna, the effect of edge diffraction is reduced. The antenna pattern above the horizon then behaves more closely to the desired free-space pattern. RF absorber could also be used to reduce the effect of multipath incident from below the ground plane. By mounting RF absorber material on the bottom of the ground plane (and around the feedline), the signal ground or reference will experience reduced distortion. These techniques are in their infancy as applied to GPS, however, and further research is needed to refine them.
7.2 Multipath Attenuation Via Earth Blockage
A unique alternative for reference station siting has been developed both in Germany and the United States [Van Willigen, 1992 and Van Graas, 19921. Instead of searching for a high point, the antenna is placed at the bottom of a hole dug in the ground. The goal is to locate and dig the hole is such a way as to shadow the antenna from all objects. Multipath from "below"is non-existent. Multipath from objects impinges on the ground at grazing angles and is thus almost entirely reflected (Figure 88 and Figure 89). The remaining energy which
160 enters the ground is severely attenuated by the ground prior to entering the antenna. For a more quantitative evaluation of the effect of the ground, the skin depth may be calculated. The concept of skin depth arises from the general expression for wave propagation in lossy media. The equation for a uniform plane wave, linearly polarized in the x direction, propagating in the positive z direction in an unbounded lossy medium is given by [Balanis, 19891:
where:
2, is a unit vector in the x direction E: is the magnitude of the field at the origin y is the propagation constant (y =a +jB) a is the attenuation constant
B is the phase constant The attenuation constant is given by w a n i s , 19891:
The units of a are Nepers per meter. The dimensionless unit, Nepers, is used since a represents attenuation [Cheng, 19831. Referring back to equation (86), a is observed to control the rate of decay of the wave. When az is equal to 1, the
161 wave has decayed to e-' = 0.368 = 36.8% of its original magnitude. Note that this represents the electromagnetic field spatial domain analog of the time constant in electrical networks [Johnson, 19841. Accordingly, the skin depth is defined as the distance traveled through the lossy medium during which the wave is attenuated to 36.8% of its original value. Denoting skin depth by 6:
To determine typical values of skin depth, o is taken at the GPS L1 frequency, 2~(1575.42MHz). For soi1;the permeability is that of free space:
p = po =
4~x10-'. Conductivity of soil is dependent upon moisture content but ranges from a = 10'~for wet soil and a = 10' for dry soil. A typical value for soil permittivity
is 2.8
E,,
where
E,
is the permittivity of free space
(E,
= 8.854~10-12).
Substitution into equations (87) and (88) yields: 6 = 0.889 meters (wet 6 = 88.8 meters (dry
soil: a =
soil: D = 1 0 3
Note that after a distance of three skin depths, the wave has attenuated to 5% of its original amplitude. Thus, if the ground reference station can be sited such that multipath must travel several skin depths through the earth prior to entering the antenna, the multipath relative amplitude will essentially be negligible. Best system performance is therefore obtained when the soil has been saturated with moisture.
162 Although theoretically appealmg, this technique has several drawbacks. Being highly dependent upon ground moisture, the system performance will fluctuate extensively over time. Secondly, it is difficult to find a suitable location which is removed from nearby buildings. For example, the locations which are appropriate for the ILS or the MLS ground system antennas are removed from buildings but are not suitable for satellite antenna siting. This stems from the fact that aircraft flying over or past these locations would block visibility.
7.3 Multipath Randomization
Chapter 5 introduced the theoretical basis for airframe-based multipath randomization. Braasch and Van Graas (1992) demonstrated how this could be achieved at the ground reference station. Although the technique was demonstrated by physically moving a GPS antenna in the presence of a reflector, an electronic implementation would involve some form of a phased-array. Instead of physically moving a single antenna, multiple antennas would be sampled at random to simulate motion. However, use of the standard non-linear receiver architecture again limits this scheme to error variance reduction only. No bias reduction is achieved. Exploitation of this technique will remain limited until linear receiver architectures are developed.
Due to their tremendous multipath performance, P-Code receivers would appear to be the leading receiver candidate for the ground reference station. The narrow-correlator spacing CIA-Code receiver is not chosen since it does not completely reject multipath unless the relative delay is 1000 nanoseconds or greater. P-Code receivers, however, reject multipath with relative delays of 150 nanoseconds or greater. In the event of encryption of the P-Code into the YCode, the Federal Aviation Administration could operate ground reference stations with keyed receivers. Security could be maintained by having the Department of Defense control the operation of the receivers. The use of RF absorber material above and below the ground plane would also be appropriate. This will improve the characteristics of the installed antenna
pattern, attenuate short-delay multipath and prevent disturbance of the signal reference.
8. DATA COLLECTION AND ANALYSIS
Having laid the theoretical foundation for multipath error evaluation, data collection and analysis was performed to determine what behavior is typical "in the field. " Accordingly, a data collection effort was carried out to gather multipath data at ground reference station sites as well as airborne installations. Data were collected on the rooftop of the Ohio University Avionics Engineering Center hangar and at a geodetic marker in a nearby field. These sites were used as typical locations for ground reference stations. For airborne installations, data were collected from antennas mounted on a Piper PA-32 Saratoga and a Douglas DC-3. Ramp and flight data were collected from each aircraft.
8.1 Theoretical Basis for Analysis
In order to determine the errors in a given measurement, a truth reference is needed. As the name implies, an ideal truth reference provides the true value which the measurement is attempting to determine. In most cases such a reference is not available. An acceptable alternative is to use a second measurement which is at least an order of magnitude better in accuracy than the measurement being analyzed. A further complication arises when trying to separate error sources from a given total measurement error. Even if the total error can be determined, it is often difficult to distinguish among the various error
sources. Although an absolute truth reference was not available for the data collection, the GPS signal itself may be exploited to isolate the combination of multipath error plus receiver error [General Dynamics, 1979 and Evans, 19861. The GPS code and camer-phase (integrated Doppler) measurements may be expressed as follows praasch, 1990-911: dcd = ds + +
+
dHw
-- ds + ct-
d+
where:
SA
ctrW
SA
+
dm
+
dmp + dbm + LIRE dcd-,, + code-noise + dd-mp ct,
+
- ctm + dmp - dbm +
URE + phase-noise + dpb-,
dpk--
+
+
A
d,,
is the code measurement
d,,,,
is the carrier-phase (integrated Doppler) measurement
d, is the true line-of-sight range from the satellite to the user c is the speed of light t,
is the receiver clock offset from system time
t,, is the satellite clock offset from system time
&, is the apparent
signal path length increase due to propagation
through the troposphere dio, is the apparent signal path length increase due to propagation
through the ionosphere
166 URE (User Range Error) is the range error due to satellite clock and orbit uncertainty SA (Selective Availability) is the intentional degradation of the
satellite clock and orbit information by the Department of Defense (used for security purposes) d,
is receiver hardware delays
" meas" represents receiver measurement errors
noise is a combination of receiver noise and diffuse multipath " mp" represents non-diffuse multipath
A is a range difference between the code and integrated Doppler
measurements due to an integer wavelength ambiguity
Diffuse multipath arises from reflection and diffraction from a group of electrically small objects. Each of these objects individually produces a negligible multipath field but the sum effect can be on the order of receiver noise values. This effect is lumped in with receiver noise since it is generally uncorrelated over time and therefore noise-like in behavior. Note that the ionosphere term is equal in magnitude but opposite in sign for the two measurements. This will be derived later. The integer wavelength ambiguity arises in the carrier measurement since the basic measurement is that of fractional phase only. For the moment, consider the observable obtained by differencing the code and carrier-phase measurements:
d , - d p h = 2dh dd-- a'M-, - phase-noise + dcd, - dph-w +
+
code-noise A
-
As was proven in Chapter 3, carrier-phase multipath will not exceed 4.8 centimeters as long as the strength of the multipath is less than the direct signal. In addition, state-of-the-art receivers exhibit phase-noise values on the order of
0.1 millimeter (1-sigma) Ferguson, 19911 and basic receiver phase measurement errors typically are also negligible praasch and Van Graas, 19911. Since codemultipath errors are typically on the order of meters, carrier-phase multipath and noise and receiver phase measurement errors may be neglected. The typical application of this process involves differencing code and carrier data collected over a given period of time. The integer ambiguity may thus be removed by subtracting out the bias. What remains is a combination of multipath, receiver code measurement error, noise and an ionosphere term:
(dd-dPk)@
-- 2dh
+
dC&,
+
code-noise
+
drod.-,,p
(92)
Although receiver code measurement errors can be correlated over time, receiver noise is not and therefore may be reduced through filtering. The filtering methodology will be outlined later. The remaining term besides multipath is then the effect due to the ionosphere. However, as will be shown next, measurements from two different carrier frequencies can be used to eliminate the ionosphere term.
8.1.1 Ionospheric Correction
GPS satellites broadcast three pm codes for navigation purposes. The CIA-Code and one P-Code are broadcast at the link 1 (Ll) frequency of 1575.42 MHz. A second P-Code is broadcast at the link 2 (L2) frequency of 1227.6 MHz. By making P-Code measurements on both the L1 and L2 frequencies, it is possible to remove the added delay due to the ionosphere. First, the effect of the ionosphere on code and carrier (group and phase) propagation velocities will be derived. In general the ionosphere is an inhomogeneous, time-varying magnetoplasma [Chen, 1983 and Yeh and Liu, 19721. However, an acceptable first order approximation may be obtained by modeling the ionosphere as a homogeneous non-time-varying isotropic plasma [Spilker, 19781. The wave equation in such a media is given by [Chen, 19831:
where k is the wave number, k, = 2rl A and
where:
o, is the plasma frequency given by:
where:
Nois the density of free electrons e is the charge of an electron (1.602x10-l9coulombs) m is the mass of an electron (9.1 1x10"l kilograms) E,
is the permittivity of free-space (8. 854x10-l2coulombs per
~ewton-mete?)
Conditions for non-trivial electric fields may be determined by setting the coefficient in equation (93) equal to zero and thus deriving the dispersion equation:
Since at GPS carrier frequencies, o > o,, solving equation (96) for the wave number yields:
The index of refraction is given by the ratio of k and k,,:
The phase velocity of waves propagating through this media is given by the ratio of the carrier frequency to the wave number [Chen, 19831:
where c is the propagation velocity of free-space (i.e. the speed of light, c = 3x lo8 mls). The phase velocity is thus observed to exceed the speed of light. The magnitude of the group velocity is given by the derivative of frequency with respect to the wave number [Chen, 19831:
To evaluate this, the dispersion equation is used:
However, the dispersion equation gives the expression for o2rather than o and thus an identity from calculus is required:
Evaluating the derivatives for the intermediate quantities:
Therefore, the group velocity is given by:
Comparing equation (99) with equation ( l a ) , it follows that for propagation in an isotropic plasma:
The net first-order effect of the ionosphere then is to decrease the propagation velocity of the code (relative to the speed of light) and to increase the velocity of the carrier by the same amount. This explains the opposite effects of the ionosphere on the code and carrier as seen in equations (89) and (90). Having derived the expressions for index of refraction and group velocity, the ionospheric correction may be determined. As was mentioned previously, the ability to remove the effect of the ionosphere is dependent upon making measurements at two separate carrier frequencies. At this point the claim is reasonable since the propagation velocity is observed to be a function of the frequency of the wave. Before deriving the ionospheric correction, the increased propagation delay due to the ionosphere is derived. The ionospheric delay is given by the difference between the propagation delay from the satellite through the ionosphere to the receiver, and the delay
173 which would have been experienced had the propagation path been only through free-space:
The free-space propagation delay is given by the range from the satellite to the receiver divided by the speed-of-light:
More generally, the range from the satellite to the receiver may be expressed as a line integral taken over the propagation path:
The propagation delay including the effects of the ionosphere may be determined by incorporating the group velocity into the above equation:
OL
This expression may simplified considerably by taking advantage of the fact that at
174
GPS frequencies, a2 > > 0;. In this case, it is justified to retain only the first two terms in the binomial expansion of equation (98):
Thus, the expression for the group velocity is simplified:
With this, the propagation delay expression reduces to:
A more convenient form may be obtained as follows:
Since
0 :
< < 02,the term multiplying c in the denominator may be assumed to
be equal to 1. Thus,
The increased path delay is thus:
Which corresponds to a ranging error of:
Substitution of equation (95) into (1 16) and evaluation of the constants yields:
Where TEC stands for total electron content along the path from the satellite to the receiver. However, since TEC is usually an unknown quantity, the ionospheric delay cannot be determined directly. The usual technique is to form a combination of range measurements made at two separate carrier frequencies.
176 This follows from an examination of equation (89). Note that code measurements made at two separate frequencies will be the same except for receiver noise, multipath and ionospheric delay. Neglecting receiver noise and multipath for the moment, the ionospheric observable is formed by differencing range measurements taken at two frequencies:
Substitution of equation (117) into (118) yields:
Converting to a common denominator:
the ionospheric delay may now be computed:
Ideally then, the ionospheric delay can be computed and then removed from the
177 code and carrier measurements. However, it must be recalled that receiver noise and multipath have been neglected. Use of standard code measurements to compute and remove the ionospheric delay will actually increase the residual noise level and confuse the identification of multipath. In this case, carrier measurements can be employed since they have negligible noise and multipath contamination (relative to those of the code). Since the carrier measurements have an unknown integer wavelength ambiguity, they cannot be used to compute absolute ionospheric delay. However, they can be used to eliminate the ionospheric term in the multipath observable (equation (92)) since the unknown ambiguities result in a bias which can be subtracted. To illustrate this technique, differenced code and carrier measurements are given in Figure 92. The upward sloping trend is the divergence of code and carrier due to the ionosphere. The ionospheric correction may be formed using camer measuremenl. By removing this term from the raw code minus carrier
data, the result yields residuals composed of multipath, receiver errors and noise (Figure 93).
8.1.2 Noise Reduction
As the figures show, it can be difficult to distinguish multipath errors amongst the noise. Non-diffuse multipath error is conelated over time but can be hidden if its amplitude is comparable to the receiver noise. It is desirable then, to
(m) slsnp~sa~ duaropnd ~bw
raw pseudorange residuals (m) L
Or-
180 reduce the effects of receiver noise and thus make the multipath errors more distinct. This may be accomplished by filtering the code measurements prior to forming the multipath observable. Two common filtering schemes are the Hatch filter patch, 19821 and the complementary Kalman filter [Van Graas and Braasch, 1991-921. Both yield similar results and are based on smoothing the code measurements with the stable carrier measurements. For the data presented in this section, the Hatch filter was utilized. The input to the Hatch filter is the previous N code and carrier measurements and the measurements for the current time. The smoothed code estimate is computed by averaging the current and previous N code measurements. The change in range due to satellite and user dynamics is resolved with the carrier data. For the results presented here, the Hatch filter averaged over the previous 100 seconds of data. An example is shown in Figure 94 and Figure 95. The filter clearly achieves its goal to reduce noise.
8.2 Data Collection
8.2.1 Equipment Set-up
All installations (except the geodetic marker) used the same conformal microstrip antenna. In mounting the antennas for the ground and flight testing,
(m) s[snp!sa~duaropnasd paqawms
183 standard procedures and practices were followed. For the hangar rooftop installation, the antenna was mounted on a circular ground plane approximately two feet in diameter. For the data collection at the geodetic marker, a geodeticantenna was mounted on a standard geodetic-antenna ground plane (circular, smoothed edges, 30 centimeters in diameter). In order to characterize the multipath environment, RF absorbing material was not used in the installations. For the aircraft installations, the antenna was mounted on top of the fuselage slightly aft of the cockpit. For the static data collection, the DC-3 and PA-32 were puked on the ramp approximately 150 meters from the hangar. This was done in order to isolate multipath from the ground and the airframe and to reduce the effects of multipath from other obstacles. In order to demonstrate the effect of multipath from the hangar itself, the PA-32 was also puked on the ramp approximately 30 meters in front of the hangar. The DC-3 flight data was collected while the aircraft was on final approach to runway 7 of the Ohio University Airport (UNI). The PA-32 data was collected during an instrument pattern flight following departure from runway 25.
8.2.2 Results
Figure 96 through Figure 103 give the results of the data analysis. All of the data shown was collected during mid-April of 1992. In all cases both CIA-
*-C /-
/---..
P-Code: dashed
CIA-Code: solid
Satellite 2
I
--
---- -
I
--,
-
Figure 96.
run time in minutes
I
15
Hangar Rooftop Data
I
10
I
5
20
25
I
.F--'
-5 0
F'
-
-
-
-
-
-2 -3
B8a
,--.
1
Hangar Roof; CIA-Code and P-Code Multipath and Receiwr Error
-4 -
-1
n
I
O!fl++
',-/k.,e
1-
2-
a a
7
a
B
E
2
7
a
a
w
-
fi
n
-
3-
4
5,
(m) srsnp!slr a8uaiopnasd paqawms
(m) spnppal a8uelopnasd paqlooms
(m) s~snp!s?rB u s ~ o p n s dpagooms
(m)spnp!su duaropnasd poqooms
192 and P-Code were analyzed. The data taken from the top of the hangar (Figure 96) and the geodetic marker (Figure 97) show CIA-Code multipath levels on the order of 1.5 meters. The P-Code multipath remains below 0.5 meter. The DC-3 (Figure 98) and PA-32 (Figure 99) ramp data show behavior similar to that of the hangar roof and geodetic marker. The PA-32 multipath error (P-Code in particular) is slightly smaller than that for the DC-3. This could be the result of a smaller airframe producing extremely weak short-delay multipath. The influence of the hangar on the PA-32 multipath error is quite evident for satellite 18 (Figure 100). In this case the hangar was located at an azimuth angle of approximately 160 degrees with respect to the aircraft. Satellite 18 was located at an azimuth angle of approximately 210 degrees. Being approximately 30 meters from the front of the hangar, short delay diffractive multipath was incident upon the aircraft antenna. For satellite 19 however, the error is not as severe (Figure 101). This is to be expected since satellite 19 was at an azimuth angle of approximately 300 degrees. This geometry lead to weaker multipath. The results of the flight data are given in Figure 102 and Figure 103. The CIA- and P-Code multipath levels again are similar to those observed while parked on the ramp away from the hangar. As expected, the P-Code performs quite well and represents a significant improvement over the CIA-Code. Due to the similarity in multipath behavior over all of the installations, it
can be concluded that the dominant source of multipath was the Earth's surface. This follows since the ground was the only multipath source which was common to
193 all of the installations. This explains the non-zero variance observed in the flight data. Airframe flexing can be exploited to reduce airframe-based multipath but it does not have a significant impact on ground-based multipath. Finally, it should be noted that no attempt was made to model the data collection scenarios. Modeling was not attempted due to extreme sensitivity to absolute aircraft or ground plane attitude and a lack of multipath source isolation. The antennas were in the presence of ground planes, buildings, undulating terrain and other aircraft. Precise modeling of such scenarios is virtually impossible. The theoretical development discussed in the earlier chapters provides insight into the range and behavior of multipath error. The data collection reveals the types of error which typically can be expected. The data collection also acts as a reasonableness check on the theory.
9. CONCLUSIONS AND RECOMMENDATIONS
Satellite-based navigation systems such as the GPS are currently being considered for use as precision approach aids. However, before these systems can reach full maturity the issue of multipath must be addressed. Multipath represents the dominant error source for the precision approach application. The work presented in this dissertation has provided insight into the range and behavior of multipath errors in satellite-based precision approach and landing systems. Multipath error has been shown to be a function of multipath strength, delay, phase and phase rate-of-change relative to the direct signal. These parameters have been characterized for the precision approach environment and have been shown to be capable of producing severe multipath error. In the absence of pathological multipath-producing obstacles, collected data reveals the Earth's surface to be the major multipath source. Assuming the use of conventional coherent or non-coherent DLL architectures, multipath error has been shown to be a non-sinusoidal and non-zero mean process. The non-sinusoidal error was shown to be a wide-band process when the multipath signal strength was close to that of the direct. This disproves the popular myth that every peak in a multipath error spectrum is the result of a separate reflector. Conventional CIA- and P-Code tracking were examined. In addition, narrow-correlator CIA-Code architectures were studied. The current literature
195 claims that only non-coherent DLL's benefit from the narrow-correlator technique. Work presented in this dissertation showed that coherent designs benefit as well. The results of the study indicate that sensitivity to multipath error is reduced if high elevation angle satellites are used. Obstacles, the Earth's surface and the airframe itself all yield smaller multipath error for high elevation angle satellites. The minimum satellite elevation angle to reduce multipath errors will be dependent upon the environment of the airport under consideration. Given the superior performance of the P-Code over the CIA-Code, it is recommended that P-Code receivers be used in the ground reference station. For security purposes in the event of encryption of the P-Code, the ground station could be operated by the Federal Aviation Administration and controlled by the Department of Defense. Differential corrections from such a system would experience significant reduction in multipath contamination than a corresponding CIA-Code receiver. In addition, antenna pattern shaping and the use of rf absorbing material to reduce multipath error at the ground reference station must be studied indepth. These techniques hold the promise of virtual elimination of the residual PCode multipath error. More generally, siting criteria for ground reference stations need to be developed. The observed phenomenon of airframe-based multipath randomization has been verified theoretically. Multipath mitigation through carrier smoothing is
196 limited to variance reduction since the DLL yields multipath errors with non-zero bias values. Full exploitation of multipath averaging phenomena will not be achieved until linear receiver architectures have been developed. It is recommended that further flight data collection be performed to augment the data presented here. Multipath data needs to be collected at sites for which significant multipath-producing obstacles (i.e. mountains or skyscrapers) are present. In addition, flight data should be collected using narrow-correlator receivers to evaluate their performance. Furthermore, a simulation program needs to be developed to aid in the multipath analysis of these types of sites. Prior to providing precision guidance down to the runway and on the airport surface, multipath errors from parked or taxiing aircraft and other vehicles must be characterized. The goal of such a study would be to determine those regions around the runway and taxiway (also known as critical areas) which must be restricted from aircraft and vehicles to ensure accurate guidance.
10. ACKNOWLEDGEMENTS
The work presented in this dissertation was funded in part by the Federal Aviation Administration under Contract DTRS-57-87-C-00006, 'ITD-50. Additional funding was also provided by the National Aeronautics and Space Administration under the GPS Interferometry Grant NAG-1-1423 and the Joint University Program for Air Transporatation Research Grant NGR 36-009-0 17. Dr. Javad Ashjaee and Mr. Mark Kuhl of Ashtech, Inc. are thanked for the provision of the GPS receiver used in the data collection and for their help in the data analysis. Mr. Mark Goldberg of Sensor Systems, Inc. is thanked for the provision of the axial ratio patterns and for his assistance in the antenna analysis. Dr. Edgar Coffey of Advanced Electromagnetics is thanked for his assistance in the installation and use of the GEMACS code. The author expresses his utmost gratitude to Dr. Frank van Graas, Assistant Professor of Electrical and Computer Engineering, for his unceasing encouragement, support and help throughout this project. The author is indebted to him for his tireless efforts in molding the author into a researcher. Appreciation is also extended to Dr. Robert W. Lilley, Director of the Avionics Engineering Center and Professor of Electrical and Computer Engineering at Ohio University, and Dr. Roger Radcliff, Assistant Chairman and Professor of Electrical and Computer Engineering at Ohio University, for their valuable suggestions and remarks.
198
Dr. Joseph Recktenwald, Assistant Professor of Civil Engineering at Ohio University, and Dr. John Tague, Associate Professor of Electrical and Computer Engineering at Ohio University, are acknowledged for their help in reviewing this document and serving as members of the Dissertation Committee. Mr. Michael F. DiBenedetto, Program Engineer at the Avionics Engineering Center, is thanked for his friendship throughout the years, his encouragement throughout the Ph.D. program, and for all of the helpful comments provided throughout this project. Mr. Jaikishan Rajendran, Graduate Intern at the Avionics Engineering Center, is acknowledged for his help in installing the OSU airborne antenna pattern code. Mr. Edward Breeuwer, exchange student from Delft University, is thanked for his help in implementing the multipath equations in FORTRAN. The author also expresses his sincere appreciation to his parents for their love and support throughout this program. To his wife Soo Yin, all love is extended for her patience, support, encouragement and help. Final thanks is given to the heavenly Father. Without Him, nothing is possible.
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