Application of homomorphic filtering for multiplicatively interacting signals and data sampling windows during periodic evaluation

Serdyukov Yurii Pavlovich Doctor of Technical Science Professor, Department of Medical Informatics and Physics, North-Western State Medical University named after I. I. Mechnikov 191015, Russia, g. Saint Petersburg, ul. Kirochnaya, 41 Yurii.Serdyukov@szgmu.ru Other publications by this author

Gelman Viktor Doctor of Technical Science Professor, Department of Medical Informatics and Physics, North-Western State Medical University named after I. I. Mechnikov 191015, Russia, Saint Petersburg, Kirochnaya str., 41 vyagelman@hotmail.com Other publications by this author

Introduction

Modern information transmission systems can be classified based on certain criteria or characteristics ^[1]. For example, by the type of description of their mathematical models, i.e. dividing them into classes of linear and nonlinear systems. The description of the first is based on a well-known and developed mathematical apparatus. Currently, there is no general mathematical theory for nonlinear systems that allows formalizing their design from a unified perspective ^{[2, 3]}. One of the many possible partial solution methods is to reduce such systems to a linear type using the so-called generalized superposition principle ^[4]. The use of his methodology makes it possible to form a procedure that significantly reduces the effect of intersymbol distortion (or intersymbol interference - MSI). This source of errors, as a rule, is dominant at information transfer rates close to the bandwidth of the communication channel ^[5-8]. It is this model of the information transmission system that has been adopted as the basis for research. In this case, the known processing methods do not provide an acceptable quality of information on the receiving side ^[9].  Therefore, research on reducing the influence of intersymbol distortions is important and relevant, and in recent years a significant number of domestic and foreign works have been devoted to them ^[10-13].

The sequence of operations described in this article is a homomorphic transformation and inherently implements a homomorphic filter ^[14]. The research is based on a model simulating the transmission of a sequence of non-modulated pulses of the meander type ^[15]. The accepted assumption, firstly, makes it possible to simplify the understanding of the essence of the described proposed method. And, secondly, even despite such a somewhat schematic approach of description, it allows us to obtain fairly simple and at the same time informative estimates of the sources of errors of the method itself and assess their impact. The procedure itself, which reduces the effects of intersymbol interference on the receiving side, is the implementation of an optimal filter based on a homomorphic transformation. That is why it is referred to further in the text as the method of concentrating integral transformations ^{[16, 17]}.

Description of the research model

We believe that the processing of the incoming pulse stream is performed in real time based on the procedures for periodic evaluation of each element. It consists of the following elements:

? sampling of the current signal of finite duration using a rectangular data window -;

? models of the communication channel as an ideal linear low-pass filter with a characteristic in the range. 

The channel model is dispersive, in the sense that the propagation speed of the frequency components of the signal depends on their value. In the time domain, this corresponds to the Cauchy-Duhamel formula of the form, where both the input and output signals, respectively, and is the impulse response of the communication channel.

The arbitrariness of the frequency response of this model assumes that only a non-empty class of functions is known. As that of a certain set of continuous functions from the class defined on the interval and , accordingly , also a non - empty set of impulse characteristics . They are interconnected by means of an integral transformation, for example, the Fourier transform.

It should be noted that a symmetric communication channel model is more convenient when conducting theoretical research. Its frequency response is symmetrical with respect to the zero of the frequency axis. It will be used in the future, which does not reduce the generality of the results obtained ^[18].

The sequence of information pulses in the communication channel is characterized by the period , duration of the signal and amplitude The information carrier is widely used rectangular pulses. They have a high degree of localization in the time domain and their parameters: amplitude, duration or time position (phase) can be modulated. There is no modulation of information carriers in the presented preliminary material. However, it should be borne in mind that this class of media has a significant drawback: the unlimited spectrum and the low rate of decrease of its components – 6dB / octave.

The main theoretical results

The general expression of the incoming signal sequence model is represented as follows

The following entry responds to the highlighted sequence signal on the receiving side

                             ,                                                   (1)

where is its envelope with amplitude. The k index indicates its position in the time sequence and characterizes its additive-temporal structure. The data window for the pulse is indicated through.

.                                         (2)

During periodic evaluation, each sequence pulse received at the receiver input is multiplied by a certain time window of data. With this in mind, we write expression (2) and relation (1) as follows

    .                                    (3)

In formula (3), the value is determined by the condition

,

where is the upper bound of the error resulting from the execution of the current operation. An illustration of this is Fig. 

Fig. The model of the envelope k of the signal on the receiving side, highlighted using a window of duration T

The figure shows a rectangular sampling window and the envelope of the signal at the output of the communication channel, taking into account its filtering frequency properties. The maximum length of the sampling window is equal to the period of the transmitted signals. The middle of the data window is located at the maximum point of the envelope. Note that the time length or aperture of the data window determines the spectral resolution, i.e. the ability to distinguish closely spaced signals in the spectrum. Taking into account the fact that the transmission rate is close to the bandwidth of the communication channel, periodic evaluation involves the use of fairly short time window apertures. Their length is limited only by the pulse repetition period, which is also an additional source of error, leading to errors in the estimation of their spectrum. To some extent, this component of the error can be reduced by an appropriate choice of optimal time windows and their parameters, but this is not the subject of our study ^{[9, 19]}. The periodical literature describes various approaches and methods that provide, to one extent or another, a solution to the problem of reducing the influence of intersymbol distortions. According to some authors, these methods can be conditionally divided into "soft" and "hard" ^[20]. At the same time, according to the authors of the presented article, an approach based on improving mathematical methods for processing typical signals within the framework of a single design and technological base of information transmission systems and minimizing the influence of intersymbol distortions is more constructive. Of course, the design of information transmission systems aimed at optimizing the shape of the information carrier ^{[21, 22]}, as well as modern approaches based on the use of software and hardware complexes ^{[23, 24]} do not contradict the approach used by the authors, but soon complement it.

The signal processing procedure described below results in the maximum concentration of its energy in the shortest possible time interval. It represents a certain sequence of integral transformations leading to the achievement of the formulated goal. We use a rectangular type as the selection window. Figure 1 serves as an illustration. Note that rectangular-type time windows, like signals of a similar type, have the narrowest main lobe of the spectral characteristic in the frequency domain and a set of lateral decreasing ones with a velocity.  The low rate of their decrease leads to a significant manifestation of the "seepage" effect. As a result, the amplitudes of the spectral components of the signal itself are distorted, which can lead to masking of weak signals ^[25].

Let's consider expression (3) in more detail. It is easy to see that it consists of two groups of components interacting with each other additively. The first group includes a description of the signal itself transmitted over the communication channel, the impulse response of the communication channel itself and the data sampling window. These components interact with each other through the operations of convolution and multiplication. The second group characterizes the upper bound of the linear filtering error arising from the operation of isolating the envelope of an information signal. The data window affects both groups through a multiplication operation.

The transformations below are essentially a detail of the sequence of transformations of expression (3). In the first step, we find the inverse Fourier transform and as a result we have

                    (4)

or in the expanded form

The latter ratio can also be represented as follows

,      (5)

where is the inverse Fourier transform of the impulse function and is the frequency response of the communication channel.

Let's rewrite expression (5) in the following form

,                                                       (6)

where is indicated

,                                                       (7)

,                                  (8)

.                                                   (9)

The implemented sequence of integral transformations eventually led to the desired result (8). It is a description of the information part of the received signal. As for the component of the form (9), it is an error function and its source is actually the imperfection of the implementation of the filtering operation.

Remark. It should be borne in mind that the case of the absence of any type of modulation in the transmitted signal sequence is considered. Generalization of the obtained results for the case of transmission of modulated signals using the described method is quite possible and this is the subject of further research.

It should be noted that in the obtained ratios (6) - (9), other parameters corresponding to communication channels with other characteristics can be used.

Concentrating integral transformations

for an idealized communication channel model

We concretize the results obtained above for the accepted model of a communication channel with an amplitude-frequency response of the type , bandwidth and pulse response

.

This allows you to write the ratio (8) as follows

.              (10)

So, expression (10) gives a description of the amplitude-time parameters of the information signal.

Consider the relations (9). It follows from the definition of convolution that

.

Thus, the component is equal to

.                                               (11)

Taking into account the transformations performed, the record (10) is presented in the following form

.                      (12)

An integral in square brackets is a convolution with finite integration limits. Further, due to the fact that the time parameters of the sequence and data windows, as well as the frequency range of the bandwidth of the communication channel, are known to us a priori, this allows us to find their convolution in the form of some function depending on the variable of the inverse Fourier transform. We also take the notation for the integral in expression (12), which is a normalizing factor and has the form

,                                 (13)

Let's turn to the relations (11) and (12). Renormalize them. Divide the right and left sides of these ratios by the normalizing factor (13), excluding pre-determined points of uncertainty.                                 

Its application allows you to rewrite expression (11) and (12) as follows

,                                                    (14)

,                                                (15)

where

, .

Taking into account the transformations carried out, we will write

.                       (16)

The ratio (16) contains an integral of the form (13). Let's evaluate it, taking into account the expression (4). The well-known impulse response of the communication channel for the model we have adopted allows us to record

or in the expanded form

.

The limitation of the function in time allows the latter relation to be represented as follows

.

After performing the integration operation, we get

.           (17)

Next, we will use the result obtained as follows To find the inverse Fourier transform of expression (17), given that the function is limited in time. Let's write it down

       (18)

Firstly, the integral sine in expression (18) has a time shift, which, with the inverse Fourier transform, is equivalent to multiplication by a multiplier. The sign of the exponent is determined by the direction of the time shift. Thus, the ratio (18) is represented as follows

.

Secondly, let's take into account that the function is odd and therefore true for it

,

where is the inverse sine-Fourier transform.

It should be noted that the function of the prototype can be continued for an interval in both even and odd ways. Therefore, considering a variable as well as a variable to be positive, you can use both the cosine and sine Fourier transforms. However, the results of these transformations may generally be different. Therefore, such a continuation into the domain of a negative argument seems more reasonable and logical, when the function retains the properties of continuity and odd. In addition, if we proceed from the symmetry property of the Fourier transform, then the result of the transformation should also remain an odd function.

Let's use the transformation tables ^[26] to express in curly brackets and find that

or in the form of

The latter ratio can be represented as follows

,                           (19)

Expressions (6) and (19) are inherently equivalent. Taking into account the notation in the form of ratios (7) - (9) and (13), we write

           (20)

The left part of formula (20) is, in essence, a convolution of the inverse Fourier transform of a pulse signal of a single amplitude with a duration and a time window of data with an aperture equal to . The limits of the convolution calculation are finite and determined by the interval.

Consider the behavior of the function on the right side of the expression (20). First, the multiplier is continuous on the entire axis and has a maximum of 1 at and oscillates along the axis with a period and amplitude decreasing as .

Secondly, when the function is identically zero. Thus, based on the result (19), it can be concluded that for a single pulse, as the limiting case of a pulse sequence, the level of intersymbol distortion is zero at a point. The error contained in the transmitted pulse amplitude is determined only by the characteristics of the linear filter that emits this pulse. Based on the fact that the duration of the transmitted pulse, we consider the level of intersymbol distortion within the peak of the pulse to be a duration, a constant value, and in this case equal to zero.

Thirdly, since the model of a consistent communication channel ^[1] is considered, for which the inequality is fulfilled, it is obvious that .

It should be noted that for values shifted by integral sines, the difference behaves like a function of the type having a maximum equal to at . Therefore, the function in square brackets has a minimum at zero. Thus, the function of the form (20) is continuous and regular on the entire axis.

Obtaining an analytical dependence for the case in question in the form of the ratio (20) is rather an exception. Using an ideal low-pass filter as a communication channel model made it possible to achieve this result. In the case of using more general communication channel models or non-rectangular data windows, the function will not be expressed in quadratures. The necessary information about the behavior and influence of the function for analysis or use in applications can be obtained only on the basis of numerical methods. This, of course, leads to the appearance of some methodological error such that

.                                                  (21)

On the other hand, even knowledge of the functional dependence for does not guarantee an objective reflection of the properties of the normalizing factor. This is due to the fact that the size of the aperture of the data window, its shape, the parameters of the information carrier – the pulse signal, actually contain errors in their practical implementation, technological dispersion and the influence of destabilizing factors. Therefore, the real dependence, taking into account these factors, is defined as follows

or taking into account inequality (21)

.                                          (22)

Formula (22) is a description of the total error of the method of concentrating integral transformations or a homomorphic filter.

Conclusion

The presented work describes the implementation of a homomorphic filter that provides separation of the multiplicatively interacting transmitted information signal and the sampling window in the communication channel. The information part of the signal itself is highlighted, as well as the error caused by the interaction of the signal itself and the sampling window. This interaction at speeds approaching the potential bandwidth of the communication channel is the source of intersymbol interference. The expressions are obtained in the most general form and can be detailed within the framework of the described information transmission model, which is the subject of further research.