The newest articles I'm aware of are

http://www.eurekalert.org/pub_releases/2012-05/miot-tec051512.php

http://phys.org/news/2013-01-mind-noise-quantum-entanglement-channel.html

http://web.mit.edu/newsoffice/2010/network-coding-part2.html

Regards,

Joachim

In development of my answer in discussion “Did the digital versus analog game come to end?” initiated by Dr A. Zekry.

In digital communications, bit-rate (i.e. the rate of the bits transmission) equal to the channel capacity is not achievable. Consequently developed methods of quantizing, coding, keying, signal processing algorithms step by step approach the bit rate to this boundary but it will be never attained by digital communication systems (CS).

It looks some incredible, but this is possible in wireless feedback CS (FCS) with the analog AM (DSB SC) transmitters (shortly - AMT) in forward channels. These systems can be realized in practice and may transmit the signals at the Shannon’s limits. Below we explain, how this can be done.

Introductory assumptions:

Both feed-forward channels are stationary, memory- less and channel noises are white Gaussian. The input signals are sampled and each sample is transmitted in n cycles, in the same way and independently from previous samples. Unlike traditional approaches, we consider possible saturation of AMT. If the signal at the input of transmitter falls outside the boundaries of its linearity, transmitter generates pure sine-wave with the amplitude A0 (saturation level). As our researches show, omission of saturation was a reason of impossibility to implement the results of pioneer and later researches in optimal AFCS (of Elias, Shannon, Kailath, Goblick, Omura, Shalkwijk, Butman, Gallager and other authors).

Principle of AFCS functioning (communication scheme).

At the first cycle, AMT transmit the sample as it is. The modulation index M1 is set to the “permissible” value which guarantees that saturation of the transmitter will be excluded at the confidence level mu

Claud Shannon in his famous limiting capacity law in late 40's put the limit but did not give a hint on how such limit can be approached except that coding/error-correcting codes can be found that can set the error rate to arbitrary low values as long as the rate of bits transfer through the channel is less than this capacity limit for the given noise level. The theory is based on the assumption of Gaussian noisy channel.

Research in coding showed that some codes are more efficient in obtaining lower error rates than others therefore approaching more towards the capacity limit. For example, turbo codes are more efficient than block codes. This is on the account of complexity. Therefore complexity of coding schemes is the first part of the answer to how to approach Shannon's limit. Of course, it took long to arrive at Turbo codes in 1990's. The second approach is more redundancy. So, for the same block code adding more redundant bits will reduce the bit error rates. The third approach is repetition where the data is sent again and again in order to improve decision on received bits (e.g. repeating transmission and detection 3 times and taking the decision farming 2 or 3 times out of the 3 trials). Both more redundancy and repetition is used. However, to a limit. In theory, the Shannon limit can be approached by having very long block codes or repeating more and more (with delay not an issue). However, code efficiency decreases more and more so that the net useful data is reduced also more and more. Hence, coding, with repetition, complexity and redundancy is generally used with caution.

Modern wireless systems looking for higher throughput, they resort to some coding, moreover, try to push the ceiling higher by first trying to improve S/N ratio by either improving S, or received signal (e.g. by collecting and constructively collecting scattered and reflected multipaths) or reducing N, the noise (e.g. by avoidance, spreading, cancellation, ...etc). This way throughput increases logarithmically with S/N improvement. A better way is of course to increase the bandwidth, B, which gives linear improvement rather than logarithmic one. Therefore new systems now head to more bandwidth, 4G- LTE uses aggregated 100 MHz bandwidth while its successor 3G-WCDMA uses 5MHz bandwidth. Best of all is to find methods such as the multi-antenna systems (MIMOs) in which the throughput is proportional to the number of javascript:antenna elements involved. I personally like to think of spatial multiplexing using MIMOs to add capacity in wireless channel similar to running more cables in parallel in wired communications to increase capacity. I call this "Wireless Channel Re-Wiring". Thanks. @AlDmour.

Dear professor Platonov,

You explained the modulation technique used and the method of transmission and detection of the samples in a communication system that can satisfy the shanon capacity limit as given in the question. would you please show us how your system satisfies the shanon equation by mapping your system parameters to the parameters in the shanon equation? You speak about sending the same sample n times. Is your system continuous times or discrete time? Sending samples n times would increase the required bandwidth n times since it increases the symbol rate by n times. This would decrease the bandwidth efficiecny n times.

Please professor comment!

Dear Dr Ismat,

Thank you for your answer. The first part of the answer is in the focus while the last part though interesting it is devoted to the trade off the bandwidth and the signal to noise ratio in digital modulation systems and the frequency reuse in the wireless communication systems.

Dear Colleagues, excuse me for a brief breakout in discussion, I was very busy.

Thank Dr Aldmour for interesting comment and Dr. Zekry for a question.

The topic is worth discussion.

Some words on “digital” information theory (IT) (for discrete sources and coding).

All theoretical results for digital comm. systems (CS) are based on Asymptotic Equipartition Property (AEP), and are valid approximately, for long sequences of messages. Transmission of the shorter blocks inevitably causes deviation of measured characteristics from those established by the theory. This is one of the reasons why theoretical limits in IT can be achieved only asymptotically.

The second reason is more “prosaic” and concerns digital transmission of the analog signals. Achievement of theoretical limits is impossible due to quantization errors. A/D conversion also transforms the continuous set of the signal values to the finite set of digital numbers that “activates” AEP.

In the analog feedback communication systems (AFCS), independent transmission of the samples removes these problems. Assumption that each sample is transmitted in the same way permits to reduce the analysis of the system functioning to consideration of a single sample transmission. This makes the results of study valid for arbitrarily long signal transmission.

Answering to the question of Dr. Zekry, I should emphasize that in analog communications, basic criterion of performance is MSE of transmission errors (in full concordance with Shannon). He even determined, in the rate distortion theory, connection between the capacity of the channel and MSE of the signal recovering.

In AFCS: Capacity of the system is expressed by the formula: Cn = (B/n) log2(Sinp/MMSEn) where n is the number of cycles of the sample transmission; Sinp is the power of the signal at AFCS input, and MMSEn is the low boundary of MSE of transmission at the last cycle.

If the number n is not greater than threshold number n* [Comm.Letters , no. 5, 2012, pp. 573-576], then MMSEn = Sinp ۠(1+S/N)^-n, and capacity of AFCS is equal to the capacity of forward channel expressed by Shannon’s formula C= = B log2 (1+S/N). For n> n*, capacity of the system considered as a generalized communication channel, quickly diminishes. Upper boundary of power-bandwidth of transmission also lowers down for n> n*.

Additional difference between known and our results is that the power S of received signals depends on the saturation factor “alfa” determined by established by the designers “permissible” probability “mi” of transmitter saturation in each cycle of transmission. We also count dependence of S on the distance between transmitter and base station r , channel gain “gamma”, therefore S = (A0 gamma / r alfa)^2, where A0 is maximal value of the emitted signal (saturation level of the transmitter).

Unlike digital IT, its analog version allows derivation of optimal transmission – receiving algorithm which determines basic parameters of the system attaining the Shannon’s boundary. Expressions for these parameters (modulation index values and the gains of restoring the signal algorithm in receiver of based station, as well as the threshold number n*) are given in CL no 5, 2012. Their values depend on MMSE, S/N and variance of feedback transmission errors “sigma_ni”. These results contain basic information for designing of (simple, low energy, low size, low cost, top efficient) optimal AFCS. Fields of applications: short range communications (1m-1000m)., where AFCS will be significantly more efficient, in all senses, than digital systems designed for the same aims.

Concerning a question “Is your system continuous times or discrete time? Sending samples n times would increase the required bandwidth n times since it increases the symbol rate by n times. This would decrease the bandwidth efficiecny n times.”

- Sampling is sampling. The time is discrete, but the analog parameters of transmitter should be set to corresponding and constant during each cycle values.

To the second sentencenot defined term “symbol”. AFCS transmits not bits but analog values of the error of estimates formed in BS and delivered to modulator over the feedback channel. Really, n=B/F is the spectrum spread gain (F – signal baseband frequency). Dependently on decision, one may fix F or B and design a system with spread bandwidth (greater number of cycles/s) or decreased signal baseband (under fixed B, and longer sampling periods). Bandwidth is typically a given fixed resource predefined by standards. Therefore, we solve a task assuming that B is fixed.

Welcome and thank you Dr, Platonov for answering my question about how the AFCS can satisfy the Shanon limit of the maximum channel capacity.It is convincing. The analog forward communication channel maximum capacity can reach that of shanon with a specific methods for minimization of the mean square error between the real signals and the detected signals

What about the feed back communication channel.It can be considered as an additional bandwidth that must be attached to the AFCS. Therefore , it seems for me that the overall effective channel capacity is smaller than that given by Shanon. Would you please give an estimation for the band width required for the wireless feedback CS ?

Best regards

Welcome and answer.

Thank you for interesting questions. In the local CS and networks, feedback channels are always present. Attractiveness of our approach is that we do not determine the feedback transmission scenario. Only final errors (MSE) “sigma_ni^2” of feedback transmission are to be known. This is valid for every stationary, memoryless feedback channel with AWGN), and single constraint is its bandwidth is to be not smaller than bandwidth 2B of the forward channel.

In turn, in work "The zero-error capacity of a noisy channel." IRE Trans. on Inf. Theory, vol.2, no.3, 1956, pp. 8-19, Shannon shown that capacity of the forward channel systems with feedback does not depend on the characteristics of feedback channels with AWGN.

The said above means that one may choose arbitrary method of feedback transmission and parameters of units influencing the errors “sigma_ni^2”: power of feedback transmitter, directed or non-directed antennas, coding and modulation methods, even analog transmission can be used. Final choice depends on assumed application of AFCS, required quality and cost of feed/forward transmission (i.e. on the energy consumption, lifetime, emitted power, cost of the node transmitter and allocated bandwidth, as well as on allocated resources of base station including cost, power, bandwidth, and other factors. Our results give analytical support for a proper decision making.

Dear professor Platonov,

It seems me, that your communication system needs a bandwidth of 2B for the forward channel and 2B for the feedback channel. Therefore an overall bandwidth of 2B is required. This would decrease the effective channel capacity to half its value for the forward channel.

Is that right Professor or i misunderstood you? Thank you my dear colleague for your patience.

Dear Dr. Zekry,

Yes, bandwidth is 2B. We send the samples in n=T/t0 cycles each, T=1/2F is sampling period, t0 is duration of transmission cycle, and t0 = 1/2B where 2B is the channel bandwidth. Capacity C in bit/cycle is 1/2 log2 (1+S/N), being expressed in bit/s takes the form C[bit/s]=C/t0 = B log2 (1+S/N). All is OK.

Wish you new successes and questions.

Thank you Professor Plantonov. You contribution is very appreciable.

In his article, Shannon assumes a very high dimensional alphabet of transmitted messages.

This allows to convert stochastic properties to practical ones (because they are averaged out).

Since the band width is fixed, this high dimension is obtained by stretching the transmission in time and this also gives a hint for the implementation of a practical system, approaching the channel capacity.

Two ways come to mind to extend a signal in time:

1) assume symbols carrying few bits but a coding in between the symbols themselves. This is done in e.g. trellis coded modulation.

2) assume symbols carrying many bits. The most practical way to construct such symbols is to use OFDM, in which the spectrum is divided into many narrow bands. Because of the narrow bands, the extension in time is increased, which is what we want.

As a surplus, this approach also allows to use Shannon's water-filling algorithm. Probably this is the best one can do.

Thanks for a question.

Initial comments (i have rigorous time): 1) The high dimensional alphabet of transmitted messages means that origin analog signal should be passed through ADC of very high resolution. The encoders should be also of corresponding complexity. 2) Naturally, the longer block length, the nearer bit rate can be approached to the Shannon's limit. But asymptotically. Never exactly. 3) It is not easy to understand points 2,3 - term "symbol" is not determined (sample, bit, or bit-word?).

Independently, there appear a question about interchannel interference. Finally - why "we want" increase of extension in time, and which "time" have you in mind?

Dear Mr. Platonov. Thanks for you remarks. My "symbol" is equivalent to your "block". We want to increase the extension in time (=the block length) because, as you mention, this allows to approach the Shannon limit.