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Looks like Mathematica 9 is released. I haven’t yet had a chance to take a look. Glancing through their release notes, a few interesting things I hope to try at some point are

– Signal Processing, which for some reason was fairly week on Mathematics till date, compared to Matlab for instance.
-The (random and social) network analysis tool is something I hope they made powerful.
-Integration with R.
-Time series, random process analysis new features and may be more.

Windowing techniques have always offered confusion to me. The hardest one was to remember what is what and how they looked in frequency domain (the time domain view was something thankfully I recollect from the names). I hit upon this yet again while trying to map the theoretical spectrum to spectrum computed from discrete samples (for an OFDM modulated signal) and then to analog spectrum measured by an instrument (By the way, I figured out that, the closest discrete technique which maps to the spectrum computation in spectrum analyzer is the Danielle method, which for some reason is not there in Matlab!) . For ages, I was using the pwelch (now spectrum.pwelch) function (pwelch method to estimate the spectrum) to compute/plot the spectrum in Matlab. Usually, some windowing as well is used to adjust the mean square versus resolution.  What window function is more suitable is something that I’ve never mastered. These days, doing a Wikipedia search to find a temporary fix and then move on is the adopted, yet not entirely satisfying strategy. The frequency domain characteristic of the popular window functions are now in here for reference, thanks to Marcel Müller. Since I may need it once in a while, I’ll keep a copy here for my own selfish quick reference. If you ever need to grab a copy, please be aware that, the ownership of this is with them.

It was in 2006, I guess, I had written a mail to Mathworks on this topic. I did get an acknowledgment from them saying that they will incorporate this into the future version. I don’t see it yet. I wonder why! Anyway, here is the Email I had sent to them. Unfortunately, I couldn’t trace their Email reply since it was sent to my earlier official ID, which has changed! Hopefully, I will ping Mathworks  again and get this up.

I believe the theoretical BER under fading scenarios, provided by the ‘berfading’ function in Communication toolbox is not quite accurate, for frequency flat fading, when Doppler shift to be considered. Currently, for example, the berfading gives out the BER for Rayleigh (or Rice) fading channel, without considering the impact of Doppler spread. In the presence of  Doppler, there may be an irreducible error floor for certain modulation schemes, especially differential modulation schemes. There are theoretical results (not closed form) to reflect such impacts. Closer approximations of the average bit error probability are derived in closed form for some of the differential schemes (say DBPSK, DQPSK etc).

Just to illustrate the point, I am enclosing here, a snapshot of the comparative BER results. For DBPSK, the exact BER, considering maximum Doppler frequency, for a uniform scattering model (Jakes spectrum) can be better approximated to Pb=(0.5./(1+EbN0) ) .*(1+ (EbN0.*(1-besselj(0,2*pi*Ts*fd)) )); The simulation results closely match this result, as against the berfading  result “berfading(SNR,’dpsk’,M,1)”

The BER curves are shown in the figure, attached with. Blue curve is what Matlab ‘berfading’ gives. The pink (Pb_doppler_theory) is the approximate and more correct bound, for Jakes spectrum (See [1], [2], [3]). The latter result is in close unison with the simulation as well.It would be nice if Mathworks can modify (may be in the next release) berfading such that, the  theoretical BER expected take care of such fading characteristics (in this case Doppler). I am  using 2006b version of matlab. In this only Jakes spectrum can be compared. For other  spectrum (I read in the Mathworks website that other spectrums can be configured in later
versions of Matlab/toolboxes), appropriate modifications to be done.

[1] P. Y. Kam, Tight bounds on the bit-error probabilities of 2DPSK and 4DPSK in nonselective Rician fading,IEEE Trans. Commun., pp. 860-862, July 1998.
[2] P. Y. Kam, Bit error probabilities of MDPSK over the nonselective Rayleigh fading channel with diversity reception,IEEE Trans. Commun., pp. 220-224, Feb. 1991.
[3] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels A Unified Approach to Performance Analysis, Wiley 2000.


June 2018
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