Location Awareness by Implementing Time of Arrival
Since the emergence of GPS systems, of the various
techniques available to identify location, TOA has
been of least interest. But its importance will be
on the rise again with the development of miniature
and low-power sensing and communications platforms, especially in tactical combat, homeland security, and
crisis management scenarios.
By Bar-Giora Goldberg
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Figure 1. Simplified multipath modeling. |
Location awareness is an issue of great interest in everyday orientation, navigation, and security, as well as special crisis and reconnaissance situations. From flight control, to security and emergency, the ability to accurately identify location of an emitting source is of great value. In certain ways, even location awareness of the cellular network, or 802.11, has been described as one of its challenges. For all practical purposes, cellular phones, laptops, or PDAs have no idea where they are or what is nearby. This is going to change soon.
The installation of the GPS satellites and their release for use in civilian applications, coupled with the development of low cost GPS chipsets, algorithms, and portable systems and displays, created business opportunities. But there are many applications that cannot use GPS, mainly due to their miniature size, and low-power platforms. Such systems lack not only sufficient power sources and size, but also have no direct view to the satellite. Since these small, low-power systems will be the mainstream arena for many future systems, alternative methods are being investigated.
This article serves as a mathematical and practical introduction for the statement of the problems, and the optimal solutions and limits of accuracy in location awareness by TOA.
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Figure 2. Phase shifted baseband signals. |
Physically, TOA requires at least three stations (which can transmit their signals to a central processing unit), but is significantly less sensitive to multipath and requires only complex processing, which is ideal in today's environment of high computational power at low cost and power. Therefore it fits well in applications that have limited size and limited power platforms. Furthermore, it is of extreme interest for military, security, seismic, crisis management, and Homeland Security applications.
Location Basics
Location awareness, the ability to accurately locate the position of an electromagnetic emitter, has always been an important parameter and one that has become ubiquitously possible only of late. From military and security, crisis management to commercial vehicular or emergency situations, location is becoming both mandatory for saving human life and property, and a viable market opportunity.
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Figure 3. Delay estimator block diagram. |
Miniature low-power and battery powered platforms, designed for tactical combat, homeland security, remote sensing, container security, and local identification, can not yet accommodate the extra electronics, power, and antenna requirements necessary to include GPS on board. These systems usually have no clear path to the satellite either. Such systems require a support network in the field to identify their location.
DF systems have been installed in the last few years to identify location of stolen vehicles. DF also serves as an excellent solution in many other applications (like airports and aerial navigation). While DF requires only two stations for triangulation, it usually requires cumbersome antenna systems, and can be very sensitive to multipath. A reflected path can easily be mistaken as the source direction, creating large angular error. It is therefore not a fitting solution for small communications platforms, especially portable types incapable of accommodating a large antenna array and limited by power sources.
DF systems measure the direction of the arriving signal's wave-front. In comparison, TOA systems measure time delay and time of arrival and are much less sensitive to multipath. Furthermore, they require no special antenna systems (See Figure 1). As seen, DF systems can make a serious angular error while TOA will just measure a slightly longer delay.
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Figure 3a. Representation of phase error vectors P1 and P2: (I1+jQ1 and I2+jQ2). |
TOA is an excellent solution for small platforms. Given three receivers in the plane, simple calculation can be developed to triangulate the location of the emitters by solving simple 2nd order polynomial equations.
Statement of Estimation Challenge
The fundamental challenge of differential TOA estimation systems can be stated as follows: A transmitter emits a modulated signal. Two independent receiving stations receive this signal. The challenge is to make an accurate measurement of the time delay difference between these two signals.
If:
S1= cos(Wot+P(t)+Φ1(t))+n1(t)
and
S2= cos(ωo(t+Γ)+P(t+Γ)+Φ2(t)+n2(t)
ωo is the center frequency, Γ is the time delay between the two stations (due to the difference in distance from the emitter), P(t) is the modulation signal (phase or frequency), Φ(t) a slowly varying phase process due to environment (multipath or fading) or receiver components, and n(t) is channel noise. The challenge is to estimate time delay, Γ.
Φ
A Short Math Background
• Basics
Assume that phase (or frequency) modulation is being used. The challenge is clearly a two dimensional problem, (phase and delay); one that requires the accurate estimation of both the signals' relative phase and delay. Readers might confuse delay with phase, and indeed, for a narrow band sin signal they are indistinguishable. Hence sin(ωo(t+Γ)) = sin(ωot+ζ), ζ = ωoΓ. In such case, phase and delay are totally correlated.
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Figure 4. Delay loop. |
But the signals we deal with are not sin waves, they are modulated and that, under such conditions, delay and phase can be completely uncorrelated. In fact this will be very desirable because each can be estimated independently without affecting the other. The purpose is, therefore, to create a processing system in which each can be processed separately, or in other words, make them orthogonal coordinates for the estimation process.
Note the effect of phase shift on the following baseband signals; both are sin(3cos(t) + 1.5sin(2t) + Φ), one with Φ = 0 while the second Φ = 90; clearly no time shift can adjust for the phase difference and correlate them. Similar exercise can be applied to time delay to show that no phase shift can adjust it either. Clearly, these two parameters merge only when the signals are narrow band until they become undistinguishable for a pure sin.
Indeed, the distinction between phase and delay grows as the signal becomes wider in bandwidth (i.e., lowering center frequency of processing); this process achieves total independence (de-correlation of the two parameters) when the estimation process is performed at baseband (ω1=0). This is desirable anyway as we wish to process using baseband DSP.
Mathematical Analysis
Given the complex autocorrelation function:
X(Γ, Φ) =ƒSxS*(dt),
then
X(Γ, Φ) = R(Γ)•ej(ωt + Φ)
R(Γ) is the autocorrelation function of the modulated signal.
The real value of X(Γ, Φ) is equal to R(Γ)(cos(ωiΓ+(Φ). ωi is the processing frequency.
In Gaussian noise, it is well known that the error estimates (fundamentals presented in standard communications theory have been omitted) from the various 2nd derivatives of X(Γ,Φ), will be as follows:
• The standard deviation of the phase estimate is given by: (σΦ2 ~ 1/SNR. As an example, suppose that our signal bandwidth (data rate) is B = 250 kHz, and the receiver SNR is 12 dB. Assume also that the tracking loop that will estimate the phase error has a loop BW of 1 kHz, and therefore will improve overall SNR by 250k/1k = 250 (24 dB). The overall SNR of the processing loop will therefore be 12 + 25 = 37 dB. Hence, the estimated phase error shall have a standard deviation of σΦ = –17.5 dB or 1/57 or radian/57 = 1.
Note: this is, of course, very similar to the well known results derived for phase noise theory in oscillators and PLL results, where SNR = –20log(phase jitter).
• The standard deviation of delay estimate, is given by: σΦ2 ~ 1/SNRxB2. This is also similar to the well-known basic Radar equation. The delay estimate is a function of SNR and signal bandwidth. The higher the signal's bandwidth (the sharper the autocorrelation function), the better the accuracy of the estimator.
Example: suppose that our signal has bandwidth of 250 kHz, channel SNRc is 12 dB, and tracking loop BW is 250 Hz, hence overall loop SNR is SNRc+ 250k/.25, or: 12 + 30 = 42 dB — 16,000.
Then delay estimate is approximately st = 1/BWx√16000 ~ 1/BWx126 = 4/126 μsec or ~ 31 nsec which calculates in free air to ~ 10 meters.
• It is also known that the average of the estimated product of the two parameters (their covariance) is given by: E(ΓxΦ) = ωi/B2xSNR, which shows what we already know intuitively, that if ωi is 0 (baseband processing) the average is 0, meaning that the estimators of the two parameters are uncorrelated. Therefore, by processing only the estimator in baseband, one can de-correlate the two parameters.
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Figure 5. Error signal generation for DLL. |
This is especially convenient because a DSP will be used for processing and therefore down convert the RF signal to its I and Q baseband components. It is both fundamental and practical to process the signals in baseband (or low IF frequency) such that phase and delay can be estimated independently. This will allow the setting of an independent phase estimation loop (similar to PLL) and delay measurement (delay lock loop). The PLL will align the signals' phases so that delay measurement can be done accurately with two signals for which phase difference has been removed, and they are left only with different delay.
System Realization
Based on the above, a simplified block diagram for the system estimator can be drawn. The system shall be realized in baseband, using DSP processing (see Figure 3).
K1 and K2 are the VCO constants. K1 is a real VCO, assuming that the RF signals operate in the >20 MHz range. Most commercial or industrial sensor signals actually operate in the various ISM bands (200, 400, 900, 2500, and 5700 MHz). K2 is mostly a variable delay generator (as is commonly used in timing systems) using 2 to 3 nsec discrete steps, typically (propagation delay of a gate).
Referring to Figure 3, the phase detector can be realized in several ways, depending on the style of the DSP designer. For example, since the signals can be represented by ejΦ1 and ejΦ2 or (I1+jQ1) and (I2+jQ2), a convenient phase error can be calculated by: P2=I1Q2-I2Q1, or more accurately by ej(Φ-Φ2)) which calculates to be P1=atan(I1Q2-I2Q1)/(I1I2+Q1Q2). P1 and P2 are shown in Figure 3a ("j" is the phase error in degrees). There are other various functions used by DSP designers to calculate phase difference.
Delay can be controlled by an element like a FIFO or other types of registers. The clock to the delay element has to be able to resolve the required accuracy. For example, 25 nsec accuracy requires clock that is higher than 40 MHz, etc. ... remember also that measurement resolution will be better than the delay step, as the estimator will average many readings and generate fractions of the delay element. Initial error and practical analysis indicates that while overall system resolution can be very good, absolute accuracy is probably limited to the five to ten feet (or nsec) range. The delay loop will be of the form of most tracking systems with a mechanism to generate a phase error such that loop can be closed, as shown in Figure 4.
The error function for the DLL usually operates on the autocorrelation function of the signal. For a given function (say (sinx/x)2), the error function will resemble that of Figure 5. The two delayed signals and the error generator (green) are shown.
Summary
A statement of TOA systems and a brief mathematical analysis has been presented as an optimal solution of the two dimensional system. The general estimator uses a PLL to remove the phase differences and then proceeds to a delay lock loop to measure time delay. The close loop error estimator can achieve resolution that is generally given by ((SNR)-1xB) )-1 where B is the signal's bandwidth and SNR the received signal to noise ration. The overall process is complicated since three reference points are necessary to perform the triangulation. In addition, multipath and fading effects add to overall system errors, but they are beyond the scope of this article.
TOA systems have been shown to be capable of excellent delay (distance) measurement accuracy, and mainly require excellent processing power, which keeps getting better and cheaper. The system is practical and especially effective for stationary or slow moving emitters, as is the case of interest generally for sensing devices distributed in the field or for security and safety purposes. In overall analysis, however, TOA offers significant advantages in multipath resistance, size, power, portability, and accuracy.
WD&D
About the Author
Bar-Giora Goldberg is CTO of Avaak, a San Diego based hi-tech start up operating in the Sensor-Net space (www.avaak.com). He can be reached at giora@avaak.com.
Wireless Design & Development
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