Noisy channel

C. E. Shannon, Certain Results in Coding Theory for Noisy Channels, Inf. 4b Coni., 1 (1957). 

P. Elias, Coding for Noisy Channels, IRE Convention Record, Part 4, pp. 37-46, 1955<... 

A remarkable properties of very noisy channel with Gaussian noise distribution is that the channel capacity can be increased by discarding samples in... 

Figure 4. Channel capacity of noisy channel with discarded samples (SjN = 0.01)...

Figure 4.19 Transfer infidelity 1 - F T) for a modulated boundary-controlled coupling a tt) = a sin C y) as a function of (a) the transfer time T. (inset) the maximum value of the boundary coupling and (b) the perturbation strength Cj of the noisy channel, averaged over 10 noise realizations for = 0.6 and = 0.7. In static noisy channels, the infidelity obtained under static control p = 0 (empty circles) is shown to be strongly reduced under dynamical p = 2 control (empty squares). A fluctuating noisy channel is less damaging in the Markovian Unfit where the correlation time of the noise fluctuations 0 (p = 0, green solid circles), the infidelity converges to its unperturbed value.

This simple example is very instructive and shows the basic key features of classical error-correction. First, one has to assume a particular and physically motivated error model, one cannot fight a completely unknown enemy Then, one applies the following generic scheme. First, one encodes information on well-chosen states of an extended and redundant system of bits. For instance, in the repetition code, the original bit of information is encoded on two particular states of a three bit system. The idea is clear redundancy prevents information from serious damage due to the errors and assures very likely recovery (let us emphasize that one uses the same kind of trick in everyday life when asking someone to repeat a sentence or a question to make sure of every word). Finally, after the transmission through the noisy channel, one decodes... 

Let us consider the simple example, known as the bit flip code. The problem is the following we want to send one qubit of information through a noisy channel which flips the qubit with a probability p in other words, if the initial state of our qubit is a 0) + b 11), we get a 1) + b 0) with probability p, and a 0) + b 11) with probability (1 — p). The situation is much alike the classical case we have considered above, so we could be tempted to apply the same method. But if we try to do so, we rapidly run into major quantum trouble First, the no-cloning theorem forbids us to clone an arbitrary state. Moreover, even if cloning was possible, measurement of the qubits would completely destroy the information stored in the system. So, we have to find another way. 

The appropriate technique is the following. First, one encodes the original information a 10) + 6 1) on the two logical states 0/,) = 000) and 1 l) = 111) of a three qubit system. This is simply achieved by adding two physical qubits, initially prepared in the state 0), and by applying some well-chosen unitary transformation C to the compound system of three qubits this operation yields the state a 0/,) I 6 1/,) which is then submitted to the action of the noisy channel. Each of the three physical qubits of the system is likely to independently undergo a bit flip (with probability p). At the end of the channel, one performs the measurement associated with the four projectors... 

At the end of this brief introduction, quantum codes seem to be much alike their classical counterparts. Indeed, they are based on the same idea of redundancy, resulting from the addition of extra physical qubits. Moreover, quantum error-correcting schemes have the same frame as classical ones after encoding the information on well chosen codewords, one sends the system through a noisy channel then one measures the syndrome, which tells us exactly which error occurred and thus allows us to recover the original information. 

Let us now briefly show how error-correction works when conditions (7) are met. After transmission through the quantum noisy channel modeled by the errors e j, one diagnoses which error occurred by simply measuring in which... 

For PLC implementations on electrical Latin-American environments, most of the noise control regulations are difficult to complain or simply regulations do not exist, that s why this chapter tries to analyze the effects of noisy electrical wired channels over the network throughput in common scenarios, as well as define the residential noise control requirements for PLC implementation. The network model on noisy channels is described in section V. 

It is not possible to analyze power line channels as a traditional noisy channel with additive white Gaussian noise -AWGN-. Zimmermann [5, 6] classifies PLC noise into five classes according to Figure 2. 

In order to probe the throughput variations caused by noisy channels in electrical Latin-American environments, five scenarios were created with a PLC network, in a common electrical distribution home network, with 120 AC volts and 60Hz. Figure 3 shows the basic implementation. The electric noise source is connected in the outlet over the same electrical circuit using common wall sockets depending upon the scenario. In a first stage, the noise source is connected at the Tx host side, the measurements are taken and the noise source is connected at the receptor side for finally measurements. 

Shannon showed that to every communication channel we can associate a channel capacity C, and that there exist error correcting codes such that information can be transmitted across a noisy channel at rates less than C, with arbitrarily low bit error rate. In fact, an important implication of Shannon s theory is that it is wasteful to build a channel that is too good it is more economical to use an appropriate code. Until Shannon s discovery, it was believed that the only way to overcome channel noise was to use more powerful transmitters or build larger antennas. 

Hidden Markov Process A Markov chain observed through a noisy channel. 

Practical implementations of twirl operations - This paper, reported in 2005 by Anwar and co-workers [20], deals with a practical implementation of a proposal made by Bennett and co-workers in 1996 for the purification of entanglement from a mixed state. The typical situation would be that in which the qubits of an initially pure entangled state rjr ) = ( 01) - 110 V2 are sent through a noisy channel. The twirl operation is a step for the purification. This operation converts an arbitrary mixed state of two qubits into a pseudo-entangled state ... 

It is possible for an element to develop noise spontaneously, and in this case the conventional design has the advantage a Galperin design would mix the noisy channel into its vertical output (where it is most easily detected), whereas the conventional XYZ design would only manifest the noise on the output associated with one degraded axis. 

Claude E. Shannon is generally recognized as the founding father of information theory as we understand it today a mathematical theory or framework to quantitatively describe the communication of data. Irrespective of their nature or type, data need to be transmitted over channels, and a focal point of Shannon s pioneering work has been that channels available for communicating data are generally noisy. Shannon demonstrated that data can be communicated over noisy channels with a small probability of error if it is possible to encode (and subsequently) decode the data in a way that communicates data at a rate below but close to channel capacity. 

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