Here's a short reply to the comments of Bob, Attila, Magnus, and others. Thanks for reading the paper carefully. I appreciate it. Some of the comments are quite interesting, other seem off the mark. Let's start with an interesting one:
The issue I intended to raise, but which I'm not sure I stated clearly enough, is a conjecture: Is least-square fitting as efficient as any of the other direct-digital or SDR techniques? Is the resolution of any direct-digital system limited by (a) the effective number of bits of the ADC and (b) the number of samples averaged? Thanks to Attila for reminding me of the Sherman and Joerdens paper, which I have not read carefully before. In their appendix Eq. A6 they derive a result which may or may not be related to Eq. 6 in my paper. If the conjecture is true then the SDR technique must be viewed as one several equivalent algorithms for estimating phase. Note that the time deviation for a single ADC channel in the Sherman and Joerdens paper in Fig. 3c is about the same as my value. This suggests that the conjecture is true. Other criticisms seem off the mark: Several people raised the question of the filter factor of the least-square fit. First, if there is a filtering bias due to the fit, it would be the same for signal and reference channels and should cancel. Second, even if there is a bias, it would have to fluctuate from second to second to cause a frequency error. Third, the Monte Carlo results show no bias. The output of the Monte Carlo system is the difference between the fit result and the known MC input. Any fitting bias would show up in the difference, but there is none. Attila says that I exaggerate the difficulty of programming an FPGA. Not so. At work we give experts 1-6 months for a new FPGA design. We recently ported some code from a Spartan 3 to a Spartan 6. Months of debugging followed. FPGA's will always be faster and more computationally efficient than Python, but Python is fast enough. The motivation for this experiment was to use a high-level language (Python) and preexisting firmware and software (Digilent) so that the device could be set up and reconfigured easily, leaving more time to think about the important issues. Attila has about a dozen criticisms of the theory section, mostly that it is not rigorous enough and there are many assumptions. But it is not intended to be rigorous. This is primarily an experimental paper and the purpose of the theory is to give a simple physical picture of the surprizingly good results. It does that, and the experimental results support the conjecture above. The limitations of the theory are discussed in detail on p. 6 where it is called "... a convenient approximation.." Despite this the theory agrees with the Monte Carlo over most of parameter space, and where it does not is discussed in the text. About units: I'm a physicist and normally use c.g.s units for electromagnetic calculations. The paper was submitted to Rev. Sci. Instr. which is an APS journal. The APS has no restrictions on units at all. Obviously for clarity I should put them in SI units when possible. Ralph KM6IYN _______________________________________________ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.