INCREASING THE SPECIFIC RATE OF CONTINUOUS PHASE MODULATION SIGNALS

A method has been developed for increasing the specific rate of frequency modulation signals with a continuous phase based on the use of the properties of composite Walsh-Barker sequences. The proposed method is applicable for multi-position signals. Simultaneously, the tasks of synchronization and registration of processing results are solved. The increase in the efficiency of digital signal transmission systems is theoretically justified.


Introduction
The modern world lives in an era of rapid development of broadband wireless technologies. Transmission rate and noise immunity of digital information are the indicators largely determined by the signal modulation/demodulation methods used in the channel.
Discrete signals of continuous phase modulation (CPM) are characterized by a constant envelope, the absence of phase jumps during modulation, high rates of energy and frequency efficiency, as well as resistance to phase distortions introduced by a fading channel [1].
The traditional method of increasing the specific rate is the use of multi-position signals. In LTE, OFDM modulation is a widely used method. Each OFDM signal subcarrier is modulated by 4-, 16-, and 64-position quadrature amplitudephase modulation (QPSK, 16-QAM, or 64-QAM). Accordingly, one symbol on one subcarrier carries 2, 4, or 6 bits. In addition, for phase separation it is necessary to use coherent quadrature reference oscillations, information about which in the implemented LTE systems is transmitted as part of special sync packages.
Our preliminary studies [2] have shown that the ideas of the method for constructing composite signals can be effectively used to increase the specific transmission rate of digital information.
The task of this work is the development and experimental study of a method for increasing the specific transmission rate of digital information through channels with the CPM signals. Experimental results were obtained by modelling in the Hewlett Packard VEE object-oriented graphical programming environment.

Multi-position CPM
Discrete CPM signal has the form [1] where the current phase in the n-th interval [ Here 0 S is the amplitude, s  is the signal frequency, T is the duration of the elementary symbol, h is the modulation index, and k u are modulating symbols. From the expression (1) it follows that the CPM signal is a phase modulation signal of a harmonic carrier with a phase varying in time according to the law (2) with the amplitude 2 kk hu   of the phase pulse: where () gt  is the shape of the phase smoothing pulse. As with phase modulation, the limits of variation of the amplitude of the phase pulse k  are determined by the interval (0...2 )  .
In this case, the required number of positions M of the multiposition CPM signal can be obtained by splitting this interval with a step (phase index of the CPM signal) 2 1 If the information symbols i u are selected from the alphabet of volume M: , then the values of the phase amplitudes (3) of the CPM signal (1) will be () (5) Expression (5) together with (3) is the rule of the multi-position CPM modulation.
As usual, with multi-position modulation, the specific rate increases to That is, in this case, each character of the signal carries 2 log M bits of information. Consider, following the ideas of [2], a different (alternative) method of increasing the specific rate. This will be performed with simple example of the transmission of two different characters of multi-position messages. In the following, this example will be generalized to a large number of independent messages.

Forming a composite signal with high specific rate
As shown in [3], the composite signal is composed of elementary symbols (or sequences of symbols) according to the rules of the generating sequence, which coincides with the Barker code. We use Walsh orthogonal sequences as elementary sequences. (7) The sequences in the ensemble (7) are mutually orthogonal, their total number is M W = m.
(8) To send two independent messages U 0 and U 1 , we choose two sequences from the set (7), written below as row-matrices We use the string-matrix of the Barker code of length M B = 7 as the generating sequence.
Or, in a formal record   The matrix form of writing sequences allows a compact representation of the formation method. The composite sequence is determined by the product of matrices of selected Walsh elementary sequences (9) or (10) by the matrix of the generating Barker sequence (12): where i = (0..1) is the number of the selected elementary sequence. Since the Barker sequence (12) contains a set of constant numbers, the sequence (13) can be represented as     We assume that the sum of the composite modulating signals is transmitted over the channel with the chosen modulation method (for example CPM).
From the form of the modulating signal (16) the following statements follow: 1) In this case, each modulating signal contains information about two independent information symbols U 0 and U 1 . When using the sum of composite signals (17) for transmission over the channel, it is possible to double the specific rate, since each modulating signal carries information about two independent information symbols, 2) Removing the transmitted information symbols from the evaluation ˆc han S of the transmitted channel signal involves solving the following tasks: a) The task of separating composite signals contained in the sum (17), b) The task of optimal processing of each of the separated composite signals in order to extract the estimates of the transmitted information symbols U 0 and U 1 . To solve these problems, we use matched filtering of the transmitted sum (17).

Separation and optimal processing of composite signals
Separation of composite signals contained in the sum (17) is performed using matched filters (MF). A feature that can be used for separation is the presence of different orthogonal elementary Walsh sequences (w 0 and w 1 ) in the composition of the divided signals (15) and (16). The orthogonality properties of these sequences are realized by applying matched filters. In accordance with the signal processing theory, the impulse response of a filter matched with a discrete signal s(t) coincides with an accuracy of a constant factor (a) with a mirror image of the signal waveform: where t 0 is the MF output sample moment. In processing sequences, the concept of a filter matched with a sequence is also applicable. The impulse response of the filters matched with the elementary sequences (9) and (10) mentioned above are The orthogonality properties of the said elementary sequences are manifested in the values of the correlation functions at the reference time t 0 , and the values of the autocorrelation (ACF) (20) and the cross-correlation (CCF) (21) functions at the reference time t 0 are equal to: In accordance with (21) From the form of results (22), (24) that are consistent with the elementary sequences w 0 and w 1 used, it is possible to obtain estimates of the transmitted information symbols U 0 , U 1 . To solve this problem, the subsequent processing of the results (22) and (24) with a filter matched with the C7 generating sequence, whose impulse response is determined by the expression matched with (10), is necessary: Similarly to (16) we form the aggregate modulating composite signal: Thus, the cumulative modulating composite signal contains information on W M independent transmitted information symbols. This increases the specific rate. When using the M-positional modulation method (for example, M-CPM), the specific rate increases to ( ), 2 The separation of composite signals and the extraction of information symbols is made similarly to the above (25..27) using a set of filters matched with the "working" subset {} i w .

Self-synchronization property of composite signals
In addition to the above advantages of composite signals, their important property of self-synchronization should be noted. As noted in [3], the author R.H. Barker proposed a kind of sequence of characters (called in digital technology as the Barker code), which has a unique property (the so-called self-synchronization property) and is currently used to build time synchronization systems. In [2], it was shown that this property can be enhanced by applying the composition method based on the use of Barker sequences instead of binary symbols in the Barker code. In the case of the use of composite Walsh-Barker sequences should also be expected to enhance the selfsynchronization property. Consider this question in more detail. The result of processing the Barker C7 sequence with a matched filter is shown in Fig. 2. In accordance with the theory [4], the result of processing coincides with the aperiodic autocorrelation function (ACF) of such a sequence. When calculating the ACF, it is necessary to take into account that due to the processing of Walsh sequences in separating matched filters, the "weight" of each «1» in the received Barker sequence increases by an amount equal to the length m of the Walsh sequence.
With this, ACF has the form where R(k) = mM B when k = 0.  In the simulation, the composite signals (15) and (16) in the form of sum (17) were transmitted over a channel with the CMP. At the same time, a sequence of binary independent equiprobable symbols, which imitated digital interference, was transmitted over the same channel. Registration of processes within the test cycle of a given length N c was envisaged. In the middle of this cycle, the above-mentioned composite signals were transmitted. The separation of signals is implemented using matched filters as described above.
On the question of the optimal processing of composite signals, the following should be noted. If the energy of each symbol of the composite sequence is Es, then, in accordance with formula (30), when forming the ACF spike of composite signal with the length of the elementary sequence m and the length of the generating sequence M B , the amplitude of the spike increases to max () (32) This is due to the symbols energy accumulation effect of the composite sequence in the matched filters during the formation of the ACF. It provides an increase in the noise immunity of the ACF spikes registration. As in the construction of any time synchronization systems, in this case, one should evaluate the reliability of the synchronization signal extraction (and the reliability of the above-described method of the processing results recording) when there is possible interference in the channel. For registration of clock signals, we use a threshold detector. For registration of synchronization signal, we use a threshold detector.
The threshold sync detector consists of a filter matched with a clock signal. The filter output is fed to the detector with a fixed threshold L. When applying a composite signal to the MF input the response at its output is determined by the ACF form (Fig. 3,  Fig. 4). Exceeding the threshold L is registrated as a mark used for time synchronization. The quality of such systems is determined by the following indicators: 1) False detection probability, P fd . False detection of the sync signal occurs whenever a segment of digital interference at the input of the SF takes the form of a sync signal. 2) Probability of skipping the sync signal, P skip , on account of suppressing the sync signal with random noise coming from the channel. A sync skipping occurs whenever random noise coming from a channel takes the form inverse to the shape of the sync signal and, accordingly, reduces the response level at the output of the matched filter. These parameters (P fd and P skip ) depend on the magnitude of the threshold level L of the sync signal detector, the properties of the Walsh-Barker signal, digital interference, and random noise. As can be seen from the curves in Fig. 3 and Fig. 4, the threshold level L of the sync signal can be set from the minimum R min = 0 to the maximum R max = R(0) = mM B = 56.
Moreover, the probabilities of each of the erroneous events depend on the threshold value in such a way that the optimum can be between these extreme values. In order to determine the value of the optimal threshold, a simulation of the sync signals false detection registration of a composite signal was performed using HP VEE software. Fig. 5 illustrates the simulation results. 1) The ammount of false sync detections has the highest value when setting the threshold level L close to zero, 2) As the L value increases, the number of false detections decreases significantly. In particular, for L = 24, the number of false detections is 5 for the length of the measurement cycle N c = 1000 and the probability of sync signal false detection R fd = 5·10 -3 , 3) From the data in Fig. 5 it is clear that with a subsequent increase in the threshold level, the number of false detections sharply decreases.

Conclusion
The ways to increase the specific rate of the modulated signal are considered: 1) The use of M-position modulation, in which the entire dynamic range of the modulating signal is divided into M levels. The transition from binary information symbols to M-level symbols is carried out using the Gray code. The canonical example of using M-positional modulation is QAM. 2) A new method for increasing the specific rate is described above in this article (based on the use of composite sequences). 3) The method described is applicable to multi-position signals.

5)
In the case when the transmission system developer pursues the goal of increasing the frequency efficiency, it is necessary to find a compromise between the equipment complexity with the use of the above method and the gain from saving bandwidth.