Click here for the enhanced PDF version of this article

Developing specifications for new military handheld receivers requires a better understanding of receiver dynamics. While some are easy to assess such as weight, size and power requirements (based on available battery size), others may not be as simple and are usually addressed in technical information meetings (TIMs) between representatives of the military and manufacturers/developers. Even within this group there remains a set of requirements considered as legacy hand-me-downs from previous specifications.

This article attempts to provide a better understanding and description of what receiver dynamics are about and shows how simulation can be of value in the development of military receiver specifications. It uses as an example the problem of determining a reasonable value of dynamic jerk stress via first-order simulation with such engineering tools as MathCAD, MATLAB and Simulink.

Jerk is the time rate of change of acceleration. Also, acceleration is the time rate of change of velocity and velocity is the time rate change of distance. Dropping a receiver from any height will not generate a jerk force prior to hitting the ground. This is because it has weight resulting from acceleration due to gravity, which can be considered a constant value that does not change.

To make this work you would need to add an additional measurable force that changes with time to the force of gravity, while monitoring its performance on its path toward impact. An example is an airplane rolling into a turn or a step input applied to its elevator controls such that a maneuver may be modeled as a ramp in acceleration, starting at zero acceleration at time t_o and leveling off at constant value acceleration A at time t₁.

Prior to getting into modeling, a description of the tracking loops would be beneficial in understanding what is meant by “the weakest link” for better insight into the modeling process.

A typical GPS receiver contains two tracking loops, a carrier tracking loop to track the carrier phase and a code-tracking loop to track the signal code to within a small fraction of the chip duration. Most GPS receiver designs have a mode of operation employing a non-coherent delay-locked loop (DLL) for code tracking and a Costas phase-locked loop (PLL) for tracking the Doppler-shifted carrier resulting from the relative dynamics of the transmitting (satellite) source and receiver.

The pseudo ranges obtained from code tracking provides a position fix, while the pseudo-range rate (or delta range) estimates from the Costas-loop provide a velocity fix accomplished by counting the number of Doppler frequency shifts of carrier cycles that occur over a finite time interval.

At this juncture, note that the receiver is designed to handle the Doppler associated with satellite motion by estimating its value and then eliminating the effect. The Doppler dealt within this article can be considered as being over and above those associated with satellite motion.

The carrier-tracking loop is more sensitive to dynamics because it tracks a much higher frequency than the code-tracking loop. And generally in a conventional GPS receiver if the carrier-tracking loop loses lock during an abrupt maneuver the other loops will subsequently also lose lock. To help overcome this problem, the other loops are typically aided either from the carrier-tracking loops or from an external navigation source

When a receiver is in a low jamming/interference environment, the carrier loop can provide aiding to the code loop. However, when a receiver is in a medium or high jamming/interference environment the carrier loop may not be able to maintain lock and alternate approaches, such as frequency lock loop (FLL) or inertial INS aiding will be required.

In a military receiver, requirements associated with dynamics and noise suppression as jamming are two of the most difficult requirements to meet in the design of a GPS code-tracking loop.

For example, if an appreciable amount of white Gaussian noise exists on the input signal to the PLL then you will probably want to use a PLL with a narrow or small equivalent noise bandwidth to filter the noise. However, if you want to track a signal by reproducing its waveform as close as possible then you need a bandwidth that is relatively high.

A way of dealing with this is to use an adaptive approach where the receiver automatically adjusts the bandwidth to compensate for added noise or dynamics at the input of the receiver. A common practice dealing with the above is to limit the lower bound of the equivalent loop noise bandwidth to between 5 Hz and 15 Hz to allow in-band noises as clock jitter and VCO 1/f flicker noise to be tracked. This practice greatly reduces the risk of PLL cycle slip. It should also be noted that during conditions of jamming a military receiver equipped with adaptive capabilities may temporarily exceed these bounds.

Since the goal is to evaluate dynamic performance with special regard to jerk performance, we need a tool that simulates the effect of velocity, acceleration and jerk on the input signal to the Costa loop.

This tool is in the form of the following equations.

The first being the “distance” equation relating velocity acceleration and jerk to distance traveled, the second to the Doppler equation relating the carrier frequency, speed of light and instantaneous velocity to the instantaneous Doppler frequency.

Velocity and frequency are measures of distance covered per unit time, with velocity being in terms of meters or feet and the frequency in cycles (or Hertz), which can be converted to wavelengths in terms of meters or degrees. It is also a measure of distance as shown in Table 1 where lambda (λ) is the wavelength of the L₂ carrier at a frequency of 1.2276 GHz.

If one were to view the effects of velocity, acceleration and jerk on the incoming Doppler it would appear as shown in Figure 1 where the green waveform is the actual instantaneous waveform incorporating not just the basic Doppler frequency but also the effects of acceleration and jerk as a form of frequency modulation.

This change in the Doppler frequency can be plotted, as illustrated in Figure 2, for jerk values of 1, 2 and 10 m/s³, combined with an acceleration of 1m/s² and initial velocity of 1.57 m/s — where the velocity of 1.57 m/s is the average velocity of a person carrying a 0 to 70-pound backpack.

The ordinate (frequency) data associated with each plot illustrated in Figure 2 can be normalized and converted to a voltage versus time plot by multiplying the ordinate value by 1 V and then dividing by the frequency deviation at time T to represent a maximum voltage of 1 from time 0 to a time T as illustrated in the signal builder profile shown in Figure 3.

The profile can then be used in conjunction with the Simulink signal shaper block and continuous time VCO block shown in the left section of Figure 4 by adjusting the gain sensitivity of the VCO in terms of Hertz/Volt to match the deviation in frequency at time T.

By doing so, one is essentially converting the plots into a Doppler frequency incorporating effects of acceleration and jerk.

Since the carrier is suppressed to the point where conventional signal-detection methods no longer work, it must consequently be regenerated via a Costas carrier/data recovery tracking loop as illustrated in the right section of Figure 4 and is an essential element on the receiving end of a suppressed carrier system.

In any case, when phase lock is obtained the upper branch of the system is in phase quadrature with the VCO output phase and can be modeled as an equivalent PLL.

The function of the lower in-phase branch is used in the recovery (or prediction) of the 50 Hz data signal, which is then normally passed on to the next phase of processing However, since we are focusing more on the tracking loops' ability to maintain lock and not on data recovery, only the equivalent residual noise effects generated by the modulated BPSK data are incorporated in the simulation.

The left section of the test setup includes a 25 Hz square wave bi-phase modulated onto the VCO's output, which in turn, is combined with band-limited white noise simulating the effect of suppressed carrier downconverted sideband modulated data, and the BPSK data is then essentially removed via the second multiplier.

The frequency effects of the simulated Doppler frequency are incorporated onto the modulated waveform via the above process along with the added band-limited white and residual BPSK-generated noise to effect an equivalent value of carrier-to-noise density ratio (C/N_o) of approximately 37 dB.

The freedom of being able to select a VCO center frequency compatible with timing and delay requirements is a big advantage in the use of this equivalent baseband approach, which takes advantage of the transparent nature of Doppler propagation through an IF channel.

Second- and third-order loop filters with equivalent loop noise bandwidths ranging from 3 Hz to 15 Hz were simulated via the above setup using the input Doppler profile illustrated in Figure 3. The phase relationship between the input (Doppler modulated) VCO and output of the Costas tracking loop was recorded via the scope output as illustrated in Figure 4.

Various values of predition filter time constants ranging from 4 ms to 20 ms were also experimented with and are included in the descriptions of the second- and third-order simulation results.

The threshold of interest is the maximum dynamic response for an allowable 15° deviation between the input and output VCOs constrained by an added C/N_o requirement of approximately 37 dB.

Associated with the decision is the trade between a third-order and a second-order system, where the second-order system requires fewer parts, is less complex and can be more cost effective.

The goal was to determine a reasonable first-order lower bound of dynamic jerk (in m/s³) for incorporation into a specification.

In this case, a dynamic jerk of 10 m/s³ associated with a velocity of 1.57 m/s, acceleration of 1m/s², and time constraint of 0.6 seconds can be specified with a degree of reasonable confidence — provided a third-order system is used.

1. Passmore, R, “Human Energy Expenditure,” Physiological Reviews, 1955, vol. 35, pp. 801-875.

2. Floyd M. Gardner, “Phase Lock Techniques,” second edition, 1979.

3. Dah-Jing Jwo, “GPS Receiver Performance Enhancement Via Inertial Velocity Aiding.” The Journal of Navigation, vol. l 5, 2001, pp. 115-117.

4. Elliot D. Kaplan, “Understanding GPS Principles and Applications,” Artech House Publishers, 1995.

5. Dan H. Wolaver, “Phase-Locked Loop Circuit Design,” Prentice Hall, 1991.

6. Katsuhiko Ogata, “Modern Control Engineering,” fourth edition, Prentice Hall Inc., 1997.

7. Pratap Misra and Per Enge, “Global Positioning Position Signals, Measurements and Performance,” Ganga-Jamuna Press, 2001.

Paul P. Wollam is a consultant and founder of WEDAS. He received a B.S. in Engineering from Los Angeles State College in 1963 and a M.S. with a major in Communications Engineering from Loyola Marymount University in 1975. Wollam has more than 35 years of engineering design/development experience spanning the frequency range of dc to lightwave and can be reached via e-mail at pwollam@sbcglobal.net.

RSS    Save to Del.icio.us  Digg This