Optimization of signal parameters and discrete-continuous channel with memory to achieve its bandwidth
Authors:   Moscow Technical University of Communications and Informatics (MTUCI), Sukhorukov Alexander S. 
Published: September 2022

The article is devoted to the definition of the bandwidth of the channel with memory. A discrete-continuous deterministic channel with fixed-band memory DCDCP was chosen as a model. The encoder forms code combinations of discrete in time and continuous in level symbols. At the output of the channel with memory, these combinations turn into implementations of continuous processes. The channel is deterministic, i.e. the parameters of the channel, the set of transmitted and, uniquely corresponding to it, the set of received combinations, are known both on the transmission and on the reception. Only permitted combinations are used for transmission. The energy of each of them and the energy of the difference of combinations are constant for the whole set. The transmitted combinations are separated by a protective time interval. The duration of the code combination is much longer than the protective interval, which is much larger than the channel memory. The energy of combinations at the output of the channel is maximum if the frequency response of the channel is also constant in the specified frequency band. As a result, the requirements for the form of information impulses are determined, which should have the form of reference functions. These requirements determine the optimal structural scheme of the transmitter: the encoder forms discrete in time and continuous in level pulses, the transmitter filter limits their spectrum to the channel frequency band, the power amplifier provides the necessary energy of each of them and the energy of the difference of the allowed combinations. Pulses in the form of reference functions can be formed if the encoder supplies a sequence of delta pulses to the input of the forming filter. When passing through the channel with the same THF, the shape of the combinations is not distorted. Their parameters at the input of the decoder correspond to the parameters laid down in the combination when they are formed in the transmission. Since the allowed combinations have a given energy and the energy of the combination difference, the optimal reception is realized using a set of correlators. There are different estimates of the Shannon limit, exceeding which allows you to get an arbitrarily small probability of error. To do this, the energy of the combination must grow in proportion to the logarithmus of the number of allowed combinations. The article calculated the exact value of this limit, coinciding with the smaller of the estimates. With a comprehensive optimization of the transmission rate and noise immunity of reception, the bandwidth of the memory channel is increased due to the fact that the excess ratio of bit energy to noise energy compared to the Shannon limit is converted into an information transfer rate. At the same time, the aspiration of the probability of error to zero with an arbitrarily large number of allowed combinations remains, although slower. On a discrete-continuous deterministic channel with memory with a given bandwidth, at the output of which a signal with a certain average bit power and additive white Gaussian noise with a given spectral energy density can be transmitted at a speed arbitrarily close to the ratio of the average power of the bit to the spectral energy density of the noise. The analysis of the singular case of transmission of information pulses of a given shape allows us to conclude that in the unrealizable case of receiving information impulses differentiated an infinite number of times, the probability of error can be made arbitrarily small at an arbitrarily high transmission speed.