For instance, 100 data-points is collected at a sampling time of 'dt,' which is followed by another 100 data-points collected at '2*dt', followed by 100 data-points more sampled at '4*dt' and so on. Modern commercial correlators used in DLS, DWS, FCS use the so-called multi-tau scheme, and I'd like to do the same in post-processing. Does anyone have any experience with implementing a multi-tau scheme?
I think to know which is the basical algorithm. Let's suppose, as you say, that you have a data string D(t1), D(t2), D(t3)...etc. First point of the correlation function (at 1 tau) is the average of the product of consecutive data (e.g. average of D(t1)*D(t2), D(t2)*D(t3), D(t3)*D(t4)...); second point (at two tau) is the average of D(t1)*D(t3), D(t2)*D(t4), D(t3)*D(t5)...etc.
Of course, the statistics at high tau is low and you need to acquire a long string of data. So you need to acquire only one data string at the time steps of the minimum tau that you wish to obtain in the correlation function.
The method and algorithm have been described in this book "Dynamic light scattering : the method and some applications / edited by Wyn Brown" in the chapter "Single-photon correlation techniques" and "Noise on photon correlation functions and its effect on data reduction algorithms" by the late Klaus Schaetzel and Rainer Peters respectively.
These chapters describe how it works and how to normalize the auto-correlation function.
Klaus Schätzel invented the multiple-tau scheme and published his ideas in earlier references if you do not have access to Wyn Brown's book. Here are a few of those, in case you want to dive into the details:
K. Schätzel; "Noise in Photon Correlation and Photon Structure Functions";
Optica Acta, 30, 155-166 (1983).
K. Schätzel; "Correlation techniques in dynamic light scattering";Appl. Phys. B42, 193-213 (1987).
K. Schätzel, M. Drewel, S. Stimac; "Photon Correlation at Large Lag Times: Improving statistical accuracy"; J. Modern Optics 35, 711-718 (1988).
K. Schätzel; "Noise on Photon Correlation Data"; Quantum Optics 2, 287-305 (1990).
Alternatively, check the list at http://scholar.google.com/scholar?hl=en&q=schaetzel+photon+correlation for hits on more recent publications that have built on and referenced the concept.
Thank you for the suggested papers and book chapters. I will post here again if I have additional questions after going through these references. I appreciate your responses.
Multi-tau machines have been in use since the early 1970s (Benedek lab). An important issue not widely appreciated is that you have to be careful to calculate the delay times correctly. This is sometimes (incorrectly) described as the tau/2 correction. The correct result, with full calculation, is in my paper G. D. J. Phillies, "On the Temporal Resolution of Multitau Digital Correlators", Review of Scienti fic Instruments, 67, 3423-3427 (1996)
This can be done with some ease in software, however even with an i7 processor, it is difficult to perform simple multi-tau, while collecting data rates > 100kHz, in real time with resolutions exceeding 1 us. Each additional decade of timing resolution requires tremendous computing resources and it quickly becomes impractical to replace a hardware correlator with a straight multi-tau multiplication scheme for fast timing resolutions and high data rates. One method to mitigate this was published by E. Schaub in 2012: http://www.ncbi.nlm.nih.gov/pubmed/22330459.
If you want to start with very narrow channels, look for something like an Nvidia Tesla board (400+ processors, effective maximum speed 1.15 teraflop when the problem is highly parallelizable; my numbers are for an older model), which this one is. You can find these second-hand on ebay for around $1000. You will need a good power supply.
I don't know after 4 years weather it helps or not,
Multiple-tau autocorrelation (M-tau ACF) estimates the ACF over a quasi logarithmic time scale.
In the article of "FPGA implementation of a 32x32 autocorrelator array for analysis of fast image series." (available at: http://arxiv.org/pdf/1112.1619.pdf). 1024 ACF at the same time are estimated.
Shortly before your question arised a resource optimized version of M-tau calculation was published in "Multiplexed multiple-tau auto- and cross-correlators on a single field programmable gate array" (published in Review of Scientific Instruments, but detailed version is available at http://arxiv.org/pdf/1112.1616v1.pdf).
And I could have helped you.
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