I have more or less convinced myself that the optimal filtering for this signal is carried out as follows:

- Multiply by a replica of the signal, with a time offset tau.
- Multiply by a replica of the signal shifted in phase by pi/2.
- Square and add preceeding results.
- Vary time offset tau to find maximum.
I also think that almost the same result is obtained by
frequency-shifting the signal from a reference frequency, and averaging
to effect low-pass filtering (down-sampling). This seems mathematically
to simply shift the frequency by an amount equal to the frequency of the
carrier. Then quadrature replicas are constructed by subtracting the
frequency of the carrier, and the detection carried out in the same
way.
I'm going to try this out now. Then think more later about its
validity.
**December 27.**Get to work. I will use data in file s1245b.dat, 90 sec of 1000-Hz data: npt (long) = 90,000, yd(npt) (double), websig(npt) (float). (See November 25.)21:30 I have a nice first result. I multiplied the data of s1245b.dat by sin(260 Hz * 2 p t) and averaged in blocks of 128 points, plotted here.

Next two replicas are constructed. Replica q1raf is a swept frequency, starting at f1 = 259.375 Hz and sweeping over 1.523 Hz in 80 seconds. A constant 260 Hz is subtracted from this frequency. The quadrature replica, q2raf, is 90 degrees behind q1raf in phase. The two replicas are shown here.

The detection now proceeds by multiplying the signal by the two replicas, summing the arrays, and adding in quadrature. Here is the result.

That's all for today.

**Friday December 28, 2001.**To proceed with the analysis of the Webb signals, I need to do some mass data analysis:- Carry out the cross-correlation with the RAFOS quadrature replicas for all of the beam-formed, detected and down-sampled files. From each one, create a file of the cross-correlation output.
- Scan the files of cross-correlation output for peaks. See how effectively the peaks are detected. Make a file of detected peaks.
- Identify triple detections and try to determine the current speed. (We expect the resolution to be crappy.)