By Ahmed M. Kamel, Daniele Borio, John Nielsen, and Gérard Lachapelle, University of Calgary
A fuzzy tracking system performs as a narrow bandwidth tracking system in terms of noise reduction, and a wide bandwidth tracking system in terms of dynamic response, overcoming the contradiction between receiver bandwidth requirements using classical tracking techniques for either noise reduction or dynamic tracking.
Autonomous navigation systems onboard precision guided missiles or fighter planes depend on GNSS and its very weak signals for positioning and navigation. Performance of a GPS receiver usually depends on the phase-lock loops (PLLs) used to down-convert these weak signals and track their carrier phase and frequency. A PLL can properly work only if its bandwidth is wide enough to track the signal dynamics, which can be significantly high, given the extremely rapid movements, accelerations, and direction changes of a missile or plane. On the other hand, wide-loop bandwidths allow larger portions of noise and interference to enter the tracking loops and disturb the signal tracking process. Excessive noise and interference can lead to loss of lock.
Aiding from a frequency lock loop (FLL) allows reducing the PLL bandwidth. This cannot prevent, however, frequent loss of lock and can be strongly affected by interference. The tradeoff between bandwidth requirements motivates design of alternative tracking systems replacing conventional FLL-assisted-PLLs.
We used fuzzy systems to design and test an innovative FLL-assisted-PLL. The output of a fuzzy controller that replaced standard loop filters drives the numerically controlled oscillator (NCO). The proposed fuzzy frequency phase lock loop (FFPLL) uses both frequency and phase discriminator outputs to generate the required frequency changes to tune the NCO, which in turn generates the local carrier for signal down-conversion.
The main core of any fuzzy system is its fuzzy sets or membership functions (MFs) that map input/output parameters into defined linguistic variables describing the input/output states. Loop discriminator outputs mainly depend on the incoming signal carrier-to-noise power density ratio (C/N0) and have a probability density function (PDF) that, under lock conditions, can be accurately approximated by a Gaussian distribution. Although the mean of this Gaussian distribution is zero under normal tracking conditions, it can be affected by sudden changes in the presence of dynamics that can cause cycle slips and other phase errors. The standard deviation of this distribution is also dependent on the signal quality and hence on the interference level. For these reasons, the discriminator output values have been clustered into several overlapped Gaussian MFs that can linguistically describe their state. The variance of the Gaussian MFs assigned to the phase and frequency discriminator outputs are adaptively tuned according to the incoming signal quality. So any change in the interference power level leads to variations in the Gaussian MF variance to ensure accurate linguistic description of the discriminator output signal. The fuzzy rules are selected to tune the NCO and ensure accurate and robust signal tracking.
We assess performance of the fuzzy tracking system in the presence of different power levels of interference. To generate GPS signals corrupted by radio frequency (RF) interference, we used a hardware GPS signal simulator combined with two external signal generators, and applied different interference levels combined with missile harsh dynamics to test the proposed system. Results show that the fuzzy tracking system significantly improves system robustness and accuracy such that it is able to track very high dynamics with reduced tracking jitter. The system shows resilience against strong interference up to a certain extent where increasing jamming levels are compensated by the online adaptation of the MF distribution on the basis of a small amount of data or C/N0 information.
The system performs favorably against standard tracking loops that cannot sustain the same level of dynamics and interference. The adaptive FFPLL can sustain interference power levels up to J/S = 40 dB. Even when the algorithm loses lock, a fast, reliable reacquisition is obtained when the interference power is reduced.
Most physical processes are nonlinear in nature. Linear approximations and models are employed because linear systems are simple, understandable, and can provide acceptable approx-imations of the actual processes. Unfortunately, most tracking problems are too complex, and their linear approximation does not provide sufficient insight on the system in all environmental conditions.
Standard tracking loop filters are obtained by solving an optimization problem where the noise characteristics and the order of the signal dynamics are known. Different loop orders are obtained for different orders of dynamics. Moreover, the optimization problem is usually solved by considering a linear approximation of the loop. These assumptions are strong, but the standard solution can fail to provide satisfactory performance when the loop is no longer working in its linearity region, or the noise characteristics are not completely known. In such conditions, an approach based on a linguistic description of the system variables may be preferable. In that sense, fuzzy control systems provide sufficient tools for designing a robust alternative to standard loop filter.
In previous cases where researchers tried to use fuzzy techniques for PLL design, they used fuzzy logic controllers (FLCs) in parallel with a classic PLL architecture. We take a different approach, designing a new fuzzy rule-based tracking system to replace the standard FLL-assisted-PLL. The new system uses the noisy phase and frequency discriminator outputs and directly produces a control signal that represents the frequency change required by the NCO to maintain phase lock.
New Signal-Tracking Approach
GPS L1 signals consist of carrier, spreading code, and navigation data. To successfully demodulate the navigation data from the received signal, an exact carrier wave replica must be generated, generally using PLLs and FLLs. Figure 1 shows the basic block diagram of a standard PLL. The two first multiplication stages are required to wipe off the input signal carrier and pseudorandom noise (PRN) code required for any CDMA communication system. A local replica of the PRN code is provided by the delay lock loop (DLL) and is used to remove the PRN sequence from the incoming signal. The carrier loop discriminator is used to estimate the phase error between local and incoming carrier. The discriminator output, which represents the phase error, is then filtered and used to tune the NCO, which adjusts the frequency of the local carrier wave. Thus, the local carrier wave tends to be a precise replica of the input signal carrier.
PLL design is a challenging task, particularly if the receiver is affected by high dynamics, or if the input signal power is low due to signal interference or degraded environments. It is therefore desirable to provide robust algorithms for the PLL design.
FLLs are more resilient against signal dynamics and produce accurate velocity measurements. PLLs however also provide signal-phase information, leading to a simplified data demod-ulation process as compared to FLLs. Several attempts to combine the benefits of both loops have been done in the past, leading to various FLL-assisted-PLL schemes where the joint use of the two loops becomes an effective way to accomodate high signal dynamics. The ability of a tracking loop to track signal dynamics is also determined by the loop order. For high dynamic
scenarios, a 3rd order PLL is usually used as it is only sensitive to acceleration jerks. Higher-order PLLs can produce system instability and greater noise level. Figure 2 shows the loop filter of a typical 2nd order FLL-assisted 3rd order PLL, where T is the update period of the loop. All the gains shown in the figure are design parameters and function of loop bandwidths, Bnp and Bnf , as reported in Table 1.
The response of a GPS receiver to different signal-to-noise levels depends mainly on the code and carrier (phase/ frequency) tracking loop bandwidths. However, there is a trade-off between noise resistance and response to dynamics. Narrow bandwidth track-ing loops are more resistant to noise, which makes them suitable for moderate jamming environments. Wide bandwidth tracking loops are more responsive to dynamics. Thus, tracking loop bandwidth requirements for GPS receivers are conflicting. One solution is to adapt the tracking loop bandwidth to the receiver measured carrier-power-to-noise density ratio (C/N0) and receiver dynamics. However, this approach can hardly solve for both concerns at the same time; trade-off must be found.
Automatic control methods based on artificial intelligence approaches (for example, fuzzy systems, neural networks, and genetic algorithms) have emerged as an alternative model to analytic control theory. One of the greatest advantages of fuzzy controllers is the simple and intuitive design. On the other hand, this simplicity is perhaps the primary cause of their initial slow acceptance among the control community.
Figure 3 shows the structure of the system design, where the standard loop filter is replaced by the proposed FFPLL controller. The fuzzy controller is composed of three consecutive layers named as fuzzification, fuzzy associative memories (FAMs, or fuzzy rules or fuzzy associations), and defuzzification layers.
The fuzzification layer is composed of a number of fuzzy sets characterized by MFs determined by the designer. These MFs are responsible for converting the crisp input values into linguistic values. The defuzzification layer is related to the fuzzification layer through the FAM rules that compose the second layer. FAM rules operate in parallel and to different degrees. Each is a set-level implication and represents ambiguous expert knowledge or learned input-output transformations. The system nonlinearly transforms exact or fuzzy state inputs to a fuzzy set output. This output is defuzzified with a centroid operation to generate an exact numerical output.
The fuzzy frequency/phase tracking system is designed to rapidly recover the signal frequency in the presence of large frequency errors, that is, after acquisition/reacquisition, and to behave as a PLL, with precise phase recovery, in the case of small frequency errors. The fuzziness of the system inputs is mainly due to the low power of GPS signals with respect to thermal noise, the main source of phase/frequency jitter. Noise distribution then plays a major role in the system design. This is why an a priori knowledge of expected signal parameters such as C/N0 is essential. This knowledge can be achieved during signal acquisition or in the first stages of signal tracking. For example; a signal with a C/N0 equals to 39 dB-Hz, in static condition and in an interference-free environment, is characterized by a phase discriminator output with a distribution approximately Gaussian as shown in Figure 4. The standard deviation of this signal, when using a standard PLL, can be theoretically calculated as follows:
where (rad) is the standard deviation the dot-product discriminator, which also suits well the arctangent discriminator used in this research, T (s) is the predetection integration time and c / n0 carrier to noise power expressed as a ratio (Hz).
Figure 4 shows the time-domain representation for the phase-discriminator output during tracking the incoming signal received from PRN 5 using a 4 Hz 3rd-order PLL in 1-millisecond coherent integration time and its histogram with the Gaussian function approximation. The corresponding Gaussian probability density function (PDF) in this case covers the signal expected values in standard tracking conditions at certain C/N0 levels, and it can be linguistically described as zero-state if compared to the ideal phase discriminator output. The mean and standard deviation, which are the two main parameters that govern the Gaussian distribution function, are directly related to the signal dynamics and signal quality respectively.
Receiver dynamics can cause phase tracking errors, and hence the distribution mean will be shifted from zero. On the other hand, the changes in signal quality will produce changes in the standard deviation as illustrated in Equation (1). An appropriate mapping between the signal PDF and fuzzy MFs can be made, and in this case, the probability of occurrence described by the PDF will be replaced by a degree of occurrence sensed by a number of overlapped Gaussian MFs as shown in Figure 4(c).
Using this approach, both phase and frequency-error inputs in addition to the NCO tuning-frequency output domains are clustered into several overlapping Gaussian fuzzy sets, each of them describing a certain linguistic definition of input or output value (big, medium, small, zero, and so on). The number of MFs adopted for the fuzzy controller is reported in Table 2.
The number of fuzzy sets associated with each fuzzy variable is a design parameter selected according to the required tracking accuracy. How much these contiguous sets should overlap is also a design issue depending on the problem at hand; too much overlap blurs the distinction between the fuzzy set values, whereas too little overlap can produce excessive overshoot and undershoot.
The fuzzy rules that relate all the linguistic variables can be expressed as:
Ri : if x1 is Ai1 and x2 is Ai2,
then y is Bi. i = 1, 2 . . . 81
where x1, x2, and y are linguistic variables, and Ai1, Ai2 and Bi are linguistic labels (or fuzzy sets) characterized by an MF. A defuzzification process is used to determine a crisp value according to the fuzzy output from the inference mechanism. The fuzzy centroid method, which calculates the center of the area of the infer
ence mechanism output possibility distribution, is used as defuzzification strategy in the FFPLL. The output y is obtained as
where n is the number of fuzzy output sets, yi is the numerical value of the ith output membership function, and u(yi) represents its membership value at the ith quantization level. Table 3 shows the fuzzy rule table providing the human knowledge base of the controller.
Gaussian MFs ended by trapezoidal rules were chosen as shown in Figure 5, Figure 6, and Figure 7, respectively. The variance of each Gaussian function can be changed according to signal noise level as described earlier, and online adaptation can be performed as described in a later paragraph. The FAMs are designed to act like an FLL for fast frequency tracking recovery in case of large frequency error indicated by the frequency discriminator. That can be seen in Table 3 in all the rules except when the frequency error is in the zero region. In this case it starts to look for the phase error, which is indicated by the phase discriminator for accurate phase tracking, and to extract the required data message.
As shown in Equation (1), higher C/N0 values ensure a small noise standard deviation, hence accurate and stable tracking. Increasing signal interference level will decrease the C/N0 level.
Interference signal power usually changes according to the receiver maneuver by approaching or moving away from a jammer, jammer motion, or to the jammer power changes. These changes affect the effective C/N0 on the receiver side. The analogy between Gaussian noise distribution and fuzzy MFs as shown in Figure 4 still holds, but a continuous change of the MF parameters — particularly the standard deviation — is required to cope with the C/N0 variations.
For online adaptation of the MFs, the noise standard deviation associated with the phase and frequency discriminator outputs must be continuously estimated. This can be done using past samples from the phase and frequency discriminators. Small analysis windows, used for collecting past phase and frequency discriminator samples, should be used to properly follow rapid changes due to the interfering signal. A tradeoff between sensitivity and accuracy must be taken into consideration. For this research, we found a small analysis window with a width of 1 second to be enough for good sensitivity at high dynamics. Figure 8 shows the modified FFPLL system with the standard deviation estimation. This information is used for the online adaptation of the Gaussian fuzzy MFs.
Test and Simulation
The primary equipment used for testing the proposed algorithm is a hardware simulator. The hardware configuration is capable of producing GPS signals in the L1, L2 and L5 frequencies in addition to adjustable additive interference through two separate signal generators. Several custom scenarios representing typical missile motion in space have been designed and tested. The radio frequency (RF) signals are collected through a front end after passing through an external low noise amplifier (LNA) using sampling frequency of 10 MHz, and saved for post-processing.
To assess performance of the tracking algorithm under interference and dynamic effects, we designed two categories of simulation scenarios. The first category is designed to test interference effects where a static receiver with gradually increasing interference level has been used. Both the interference and high dynamic effects are examined in the second category, in which scenarios of a missile that maneuvers near an interference source are designed. Four different tracking schemes are used for GPS signal tracking. They include the usage of a standard PLL with narrow and wide bandwidths (4 Hz and 14 Hz, respectively), FLL-assisted-PLL using narrow bandwidths (3/4 Hz), and finally the new FFPLL. The performance of each algorithm is evaluated by assessing the continuity of tracking during high dynamics, that is, the ability of the receiver to maintain lock, and the noise standard deviation of the estimated Doppler.
Interference Effect on Accuracy
The first test category involves studying the interference effect on GPS signal tracking capability and accuracy, using a custom scenario of a static GPS receiver with gradually increasing interference level. A continuous wave (CW) interference signal centered at the L1 frequency is combined with the generated GPS L1 signal and collected by the front end for post processing. Figure 9 shows the increasing interference effect on the signal quality particularly the signal C/N0. In this scenario, the jamming to signal (J/S) interference power is gradually increased every 10 seconds in steps of 10 dB each starting from 0 dB higher than the GPS L1 power.
After reaching an interference power of about 40 dB higher than the GPS power, none of the tracking algorithms was able to track the signal and hence 40 dB is considered the maximum jamming tracking threshold. Figure 10 shows the estimated Doppler standard deviation for PRN 23 using the four tracking schemes described earlier at different interference levels. It is clear that the FFPLL scheme is superior to the other three conventional tracking schemes in terms of Doppler tracking jitter and hence tracking accuracy. The changes in C/N0 level due to the increasing interference level affect the discriminators output noise level as described in equation (1). These effects can be noticed clearly in Figure 10. On the contrary, these changes are almost absorbed by the adaptive FFPLL, and hence the C/N0 changes have a minimum effect on the Doppler tracking accuracy.
Interference and High Dynamics
The second test category assesses the system performance under CW interference and high dynamics. The scenario considered here comprises the effect of missile maneuver near an interference source. Due to this maneuver, the GPS signal C/N0 is changed with the distance from the interference source. The missile velocity in this scenario is increased to reach 300 meters/second performing hard maneuvers with acceleration up to 8 g and jerks up to 50 g/second. The same scenario is repeated five times with different CW interference powers. Due to missile high dynamics narrow bandwidth PLL or FLL/PLL was not able to p
rovide continuous signal tracking and losing lock occurred, that is why only a 14 Hz bandwidth PLL and FFPLL are considered. Interference powers generated are 20, 30, 40, 45, 50 dB respectively above normal GPS signal power. Figure 11 shows the 3D plot of missile trajectory and its maneuver near the jammer, while Figure 12 shows the effect of this maneuver on the signal C/N0 for PRN 3 when a 40 dB interference signal is applied. C/N0 increases and decreases according to the separation from the interference source.
Tracking results show the ability of continuous tracking under interference level up to 40 dB higher than the GPS signal for both PLL 14 Hz and FFPLL. Higher levels of interference lead to tracking loss. FFPLL is able to recover tracking mode and retrieve the signal phase when interference source is disabled due to missile maneuver away from the jamming source whereas the wideband PLL is not able to retrieve back the signal phase in these high dynamics conditions.
Figure 13 shows the effect of adding a 40-dB interference signal on PRN 3 estimated Doppler and Doppler standard deviation respectively, using PLL 14 Hz and FFPLL. Tracking continuity is achieved using both algorithms; the interference signal greatly affects PLL tracking accuracy whereas FFPLL tracking accuracy is much better in both interference and interference free conditions.
The fuzzy tracking system solves the contradiction between receiver bandwidth requirements using classical tracking techniques for either noise reduction or dynamics tracking. It shows better performance in both cases since it performs as a narrow bandwidth tracking system in terms of noise reduction, and a wide bandwidth tracking system in terms of dynamic response.
The fuzzy tracking algorithm FFPLL provided tracking robustness in very high dynamics and signal interference up to 40 dB higher than GPS L1 power. The noise level calculated from the estimated Doppler is small, equivalent to results obtained with a very narrow PLL bandwidth under normal conditions. During high dynamics, tracking continuity is achieved using FFPLL with dynamic performance comparable to a wideband PLL or FLL/PLL. Signal tracking recovery is achieved if the interference power causing signal tracking denial is reduced or turned off.
Spirent GSS7700 simulator, National Instruments PXI 5661 front-end.
Ahmed M. Kamel is a Ph.D. candidate in the Position, Location and Navigation (PLAN) Group at the University of Calgary. He holds an M.Sc. in electrical engineering from Military Technical College (MTC), Cairo, Egypt.
Daniele Borio received a Ph.D. in electrical engineering from Politecnico di Torino, Italy, was a senior research associate in PLAN Group, and is a post-doctoral fellow at the Joint Research Centre of the European Commission.
John Nielsen is an associate professor at the University of Calgary.
Gérard Lachapelle is professor of geomatics engineering at U. of Calgary, Canada Research Chair in wireless location, and head of the PLAN Group.