Below are calculations showing the time for direct Y-code acquisition with older application specific integrated circuit (ASIC) technology and current ASIC technology. In the analysis, the following assumptions were made:
(1) |
Receivers have limited knowledge of their current position. |
(2) |
Receivers are using the latest satellite ephemerides. |
(3) |
Time is known to ± 1 second. |
The Y-code has 10^{7} chips to search, given a 1-second uncertainty in clock offset (10.23 million chips per second). A well-designed receiver can obtain a signal-to-noise ratio of 12.6 dB in 0.001 seconds, based on the following derivation:
Noise power = kTB,
where k is Boltzman's constant, k = -198.6 dBm/Hz/Kelvin or 10^{-19.86} milliwatts/Hz/K. Assume the system temperature, T, is 100 Kelvin, then B, the noise bandwidth, is taken to be 1/0.001 seconds, or 1,000 Hz. Thus:
Noise power = (10.^{-19.86} milliwatts/Hz/K)(1,000 Hz)(100 K)
= 10-^{14.86} milliwatts or -148.6 dBm
Given the minimum received power level for the L_{2} signal, which is -136 dBm, the ratio of signal-to-noise can be calculated:
Signal-to-noise = received power - noise power
= -136 dBm -(-148.6 dBm)
= 12.6 dB.
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Appendix K
Direct Y-Code Acquisition
Below are calculations showing the time for direct Y-code acquisition with older application specific integrated circuit (ASIC) technology and current ASIC technology. In the analysis, the following assumptions were made:
(1)
Receivers have limited knowledge of their current position.
(2)
Receivers are using the latest satellite ephemerides.
(3)
Time is known to ± 1 second.
Old Technology (100,000 Gate ASIC)
The Y-code has 107 chips to search, given a 1-second uncertainty in clock offset (10.23 million chips per second). A well-designed receiver can obtain a signal-to-noise ratio of 12.6 dB in 0.001 seconds, based on the following derivation:
Noise power = kTB,
where k is Boltzman's constant, k = -198.6 dBm/Hz/Kelvin or 10-19.86 milliwatts/Hz/K. Assume the system temperature, T, is 100 Kelvin, then B, the noise bandwidth, is taken to be 1/0.001 seconds, or 1,000 Hz. Thus:
Noise power = (10.-19.86 milliwatts/Hz/K)(1,000 Hz)(100 K)
= 10-14.86 milliwatts or -148.6 dBm
Given the minimum received power level for the L2 signal, which is -136 dBm, the ratio of signal-to-noise can be calculated:
Signal-to-noise = received power - noise power
= -136 dBm -(-148.6 dBm)
= 12.6 dB.
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12.6 dB is more than adequate for detection, which means that the ratio of signal voltage-to-noise is 4.3. If the detection threshold were conservatively set at three times the noise there would only be a 1-three sigma, or about 1 percent probability of false detection.
If a receiver is implemented with a parallel search capability of 1,000 correlation channels, a full search over 1 second of delay could be accomplished in 10 seconds based on the equation below.1
(107 chips)(0.001 correlation channel sec/chip search)/(1,000 correlation channel)
= 10 seconds.
This assumes that the signal Doppler is known to about 1,000 Hz, which corresponds to about 200 m/second, or 720 km/hr.
Current Technology (500,000 Gate ASIC)
The search time would be reduced by a factor of 5, to 2 seconds. Using the same procedure as above, if a receiver is implemented with a parallel search capability of 5,000 correlation channels, a full search over 1 second of delay could be accomplished in
(107 chips)(0.001 correlation channels sec/chip search)/(5,000 correlation channel)
=2 seconds.
Again, this assumes that the signal Doppler is known to 1,000 Hz, which corresponds to about 200 m/second, or 720 km/hr.
Discussion
For both cases, modest assumptions about receiver capabilities have been made. Time keeping accurate to 1 second is within the range of a wristwatch-level oscillator over a day or so. Most platforms can estimate their velocity to 720 km/hr. If the velocity and time are not known to this level, additional multiples of the 10- or 2-second search would be required. Once the first satellite is acquired, the receiver clock can be fixed to about 0.01 second, so searches for additional satellites can be done sequentially taking about 0.1 second each. We have also assumed that the receiver has on-board ephemerides for the satellites to allow position solutions immediately following acquisition of the first four satellites. If there are no on board ephemerides, it takes about 30 seconds to receive all five ephemeris subframes, so 30 seconds should be added to obtain a time-to-first-fix.
1
A chip to perform the parallel search would require about 100,000 gates if implemented in a gate array, and these have been available for many years. (For comparison, 500,000 gate arrays are now available.) About 50,000 gates would be required to implement 1,000 correlation channels in a more efficient full-custom ASIC.