Case study: test of NG chip sequence

In the Caltech RMPI, the PRBS for each channel is loaded from a 128-bit shift register, so this can be programmed to an arbitrary sequence. However, this adds significant power cost to the design. Since this is the first working RMPI chip, it was considered worth the sacrifice in order to allow the extra flexibility: for example, it is possible to implement the run-length-limited ideas.

The Northrop Grumman chip saves power by avoiding shift registers, and instead generates the PRBS sequence from a LFSR. Initially, the direct outputs of a LFSRs were used, but after this author’s analysis in January 2009, it was identified that this caused significant performance degradation, partially due to the occasional very long runs of +1 or₋1. For a system with many channels, this is not an issue, but the NG chip only has 4 channels. In addition, the sequences sent to each channel were correlated, which added to the problems. Offsetting (aka staggering) the channels helps the situation, but not enough to match the performance of the Caltech RMPI.

Following the analysis, the PRBS was modified so that it was generated by a Gold code, which can be created by mixing the outputs of LFSR, so the additional hardware complication was minimal. The output of each Gold code repeated every 63 periods, but only the first 52 periods were used. Thus each of the four channels used a PRBS with lengthNchip= 52. Because the NG design uses

four channels, the ADC rate is greater than in the Caltech design, and operates just below 100 MHz, so thatNint= 52. ThusNchip=Nint.

This design underperformed, so we analyzed the system. An empirical test was conducted as follows, which consisted of 100 independent trials. For each trial, a carrier frequency was picked uniformly at random from between 300 MHz and 2.5 GHz, which was modulated by a simple trapezoidal pulse that spanned almost all the signal; the Nyquist length wasN = 1024 so this was about 200 ns. To be realistic, synthetic noise was added both to the signal and to the measurements, calculated to give 60 dB SNR separately.

Reconstruction was done via two methods. The first method used the over-complete Gabor dictionary with about 8 reweightings using the NESTA algorithm. The second method was more complicated, and consisted of first reconstructing the signal with a DCT orthobasis and reweighting a few times, followed by using the over-complete Gabor dictionary with reweightings. The second method almost always worked better. These techniques will be discussed in_§2.7.

Reconstructions were declared either a success or a failure. To fail, two criteria had to be met (in practice, criterion 1 was almost never met without criterion 2 also being met). Both criteria were based on the frequency only, using Welch’s method to estimate the PSD. The criteria were

1. Failure Criterion 1: the peak of the spectrum was not close to the actual carrier frequency. The allowable tolerance was set to the the width of the main lobe at -20 dB from the peak.

Failure rates per 100 tests

Design Method 1 Method 2

Nchip= 52, stagger 1 46 33 Nchip= 52, stagger 2 44 31 Nchip= 52, stagger 10 48 35 Nchip= 52, seed 1 57 43 Nchip= 52, seed 2 54 39 Nchip= 52, seed 3 50 35 Nchip= 51, seed 1 25 13 Nchip= 104, seed 1 13 7 Nchip= 104, seed 2 20 7 Nchip= 104, seed 3 20 9 Nchip= 3276 0 0

Nchip= 312, random signed Bernoulli 0 0

Entire channel a Gaussian matrix 0 0 Caltech design,Nchip= 128 0 0

Figure 2.31: June 2009 tests of the Northrop Grumman chipping sequence. The first half of the table show various NG designs; Nint = 52. When Nchip = Nint or Nchip is short, the performance suffers. The NG design was

modified to useNchip= 3276 which results in much better performance. The second half of the table show some

other models for comparison; the Caltech design seems to work well.

than the main peak.

By inspection, most signals that passed these frequency criteria also looked good in the time domain. The results are presented in Table 2.31. The LFSR depends on an initial state, referred to as a “seed,” and the test was run using various seeds in order to determine if the poor performance was due to an unlucky choice of the seed.

This alignment of Nchip = Nint proved to be disastrous, and shifting the channels relative to

one-another did nothing to improve the results, nor did changing the seeds of the Gold codes. These results are reported in the first 6 rows of the table.

Initially attempts tried two fixes that yielded only modest improvement. The first fix is to take 51 samples of the Gold code to use for the periodic PRBS, with the idea that the mis-alignment

Nchip 6= Nint will improve recovery. This appears to have helped (see Table 2.31). A similar

modification is to take 104 samples from the Gold code (which means the last 41 sample are duplicates of the first 41), and this also helps. However, neither modification matches the performance of the Caltech design or other baseline tests shown at the bottom of the table.

In light of these results that show poor performance compared to the Caltech Φ matrix, the NG team found a way to devise a PRBS of lengthNchip= 52×63 = 3276 by adjusting the LFSR inputs

to the Gold code. According to the above criteria, this adjustment is extremely beneficial and results in no errors per 100 tests.

Note that in order to fully characterize the system, Φ must have 3276 rows. For the Caltech system,Nchip= 128 andNint= 100, so their LCM is 3200; thus Φ has 3200 rows, so calibration of