When the signal is sparse (in time, or frequency or some other basis), i.e. it has spectral or temporal "gaps" wherein it is zero, with locations known or even unknown, then it is possible to recover it "beyond Nyquist", i.e. with sampling rate lower than the two-sided bandwidth. There are lots of papers about it in the compressed sensing literature.

Look up for example the papers of Prof. Yonina Eldar from my school and references therein.

Hi Subhasish,

I worked on this topic during my PhD. You'll find some results in the following paper.

Sarasij Das and Tarlochan Sidhu. 'Reconstruction of Phasor Dynamics at Higher Sampling Rates using Synchrophasors Reported at Sub-Nyquist Rate.' Innovative Smart Grid Technologies (ISGT), 2013 IEEE PES, 24-27 Feb 2013, Washington, D.C

Subhashish,

Analog Devices has many Application Notes related to Signal Sampling and particularly under-sampling. Visit the website for High-speed ADCs (>1MSPS sampling rate) and you will find some useful information related to your question.

Thanks everyone , I'm going through the suggested materials.

Dear Friend Victor Maneul (UNAM)

Thanks for your insightful inputs on the actual Nyquist's Criterion and Shannon's theory. Interesting information!

Thanks once again!

Especially in relation to communication, you might want to have a look at these papers of ours:

• http://vbn.aau.dk/en/publications/downsampling-of-dft-precoded-signals-for-the-awgn-channel%2818feb133-b4d2-4e62-867b-1714dae6c931%29.html

• http://vbn.aau.dk/en/publications/demodulating-subsampled-direct-sequence-spread-spectrum-signals-using-compressive-signal-processing%280a9dc47a-e849-4321-9b0d-8bbac6d3159e%29.html

@Subhasish Can You elaborate a bit on how You propose to correlate energy content (RMS) of the signal to the ensamble of possible DFT spectras?

It seems to me that You need to have at least some temporal information on the nature of the signal (such as sparseness, waveform shapes, phase etc. as in the previous interesting papers) in order to improve on Nyquist. Then adaptive estimators, correlation etc. can improve decoding.

Assuming a linear transmission media with known attenuation, the reciever can know the transmitted energy of the signal. Or the energy value could be transmitted digitally. But just this seems in itself to be of no help in decoding the signal translated to different energy levels since the the waveform will be the same. (Assuming You are well above the noise floor) There has to be a correlation between energy and waveform, which does not seem to exist in the simple linear case.

As per my understanding that if we have a bandlimited baseband signal and we assume that there may be undersampling cases also, so when we estimate the DTFT of the received signal ; there could be few possible solutions whose decomposition may lead to the actual spectrum( assumed to be derived form the DTFT of the received one), given the possible spectrums there are spectrums for which the mean square errors in terms of L2 equivalency w.r.t the transmitted(value received via digital coding) be within some delta.

Could it be possible to find the transmitted signal in terms of equivalent signal.

IIR, if all you want is a measure of signal energy per N samples, undersampling and consequent aliasing doesn't matter at all. Just take the average (1/N) * Sum (x[i]; i=0 to N-1), or do it as a running average, adding new sample, subtracting oldest; repeat. EDIT (Correction, 3/28/2018): A LPF ahead of the sampling, with cutoff frequency equal to the bandwidth of the (baseband) signal is desirable, but not necessary. In my case I was only interested in measuring the *energy* of a 3 kHz pink noise signal, not its *spectrum*. The maximum sampling rate I had in the uP was about 1.5 kHz (whereas at least 6 kHz would have been necessary to prevent aliasing or spectral folding). So I was stuck with spectral folding, but it did not matter because the total signal energy was unaffected. There is one potential benefit spectral folding: It turns out that if you use a sampling rate of half or a quarter of the 6 kHz (2 x fmax), the spectral folding flattens the spectral energy. In any case, all you have to do to estimate the energy of the whole 3 kHz band is sum the data using the boxcar average mentioned above in my original post. None of the energy gets lost in the spectral folding. (I used this undersampling approach in the EAD TheaterMaster home theater processor back in 1995, and we got very accurate noise measurements on the built-in SPL meter, although most people would just use the automated noise setup, which in fact did not need accuracy because it merely adjusted channel volumes until a calibration level was achieved.)

@subhasish, I recomend you to see my paper which deals with the problem of trying to recover an under-sampled signal. the paper presents basic information.

Article A Simplification to the Fast FIR-FFT Filtering Technique in ...

Hi Subhasish,

you have to look to the notion of Compressive Sensing (http://dsp.rice.edu/cs ).. norm l1 etc.. developped by Candès, Tao, Donoho.. if you know your signal is sparse you can sample much under the Nyquist limit.. it is very very powerful

I'm not very clear on what you're trying to accomplish here. So I will throw out some ideas that may or may not help in thinking this through:

1) Go outside the box and consider this: the samples represent a population of values and there is surely a distribution of values which may be well-represented by a set of sparse samples. Obviously the more samples the better unless the underlying distribution has important dynamics. If knowing the distribution of values is helpful then this may be of great help.

2) I've not thought of this before but someone must have:

Consider a transmitter that has the capability to transmit up to a bandwidth of N Hz and we'll assert that we don't care what N is.

Also, consider a receiver that has the capability to receive the same bandwidth N.

Also, consider a sampler which has sample rate S

IMHO that quite often we are able to extract information with under sampling with additional prior knowledge for example we are able to fill vowels in most words in text.. this leads us to say

cryptography ..( a famous Sherlock Holmes story about deciphering where

as well known likely letter is 'e' ) . So if we interpret energy as length of

vector^2 in n dimension then in n space we could locate possible

usually meaningful set of points with some probability ..an nball sphere..

.. so the minimum distance measure to these significant points would perhaps

lead us to an answer..

There is a fairly recent technique called L1 minimization or l1magic

which might answer your needs. you may like to visit ..

http://users.ece.gatech.edu/~justin/l1magic/#code

If you start experimenting with L1-minimisation reconstruction in Matlab (cf. l1magic above), I have found that YALL1 and SPGL1 are some of the fastest implementations currently around: http://yall1.blogs.rice.edu/, http://www.cs.ubc.ca/~mpf/spgl1/.

Also, note that the family of greedy iterative reconstruction algorithms are usually much faster but may not be able to reconstruct for as low undersampling ratios as L1-based solvers. Look up CoSaMP and IHT. See also http://sparselab.stanford.edu/OptimalTuning/sp2009.pdf