Energy Efficiency for Faster-than-Nyquist Data Transmission Using Processing Algorithms with Decision Feedback

Abstract

One of the ways to increase the volume of transmitted information is to increase the bit rate above the Nyquist barrier. However, an increase in bit rate in the case of FTN (Faster-Than-Nyquist) signals leads to an increase in energy costs for receiving information on channels with limited bandwidth, for example, in Digital Video Broadcasting satellite systems like DVB-S2/S2X. It is possible to minimize energy losses by using the processing algorithm “maximum likelihood sequence estimation”. However, the computational complexity of this algorithm is extremely high, which limits its use, especially in terrestrial mobile satellite terminals. We propose a new bit-by-bit decision feedback algorithm with maximum likelihood ratio estimation of subsequent symbols in the observation interval. This algorithm provides minimal energy costs comparable to the method “maximum likelihood sequence estimation” at speeds 2–3 times higher than the Nyquist barrier. At the same time, the complexity is two orders of magnitude less. It is shown by simulation for a channel with additive noise that energy losses in relation to the potential bit error rate (BER) are less than 4.5 dB. In the presence of Rayleigh fading, the application of the proposed algorithm makes it possible to provide the processing of FTN signals for double bit rates in urban areas with energy costs equal to 12 dB when using an equalizer. We give numerical estimations of the increase in computational complexity for the proposed processing algorithm. It is shown that an increase in the bit rate by 1.5 times leads to an increase in the computational complexity by more than an order of magnitude. The same conclusion can be reformulated in another form: for the proposed algorithm, each decibel of energy gain is achieved by increasing the number of computational operations by $1.5\times {10}^5$. It is experimentally shown that additional energy losses due to non-ideal phase and timing synchronization are no more than 1 dB when the proposed algorithm is applied in a fading channel. The energy costs in fading channels relative to a stationary channel for twice the Nyquist rate are equal to 13.8 dB when using an equalizer.

1. Introduction

Satellite broadcasting systems DVB-S2/S2X, reverse channels of interactive broadcasting data transmission systems, are characterized by a limited frequency band. Increasing the bit rate in such channels can be achieved by the application of FTN (Faster-Than-Nyquist) signals [1,2,3,4]. The energy spectra of these signals have a close to rectangular shape. The duration of FTN signals is longer than the duration of the message bit. The correlation properties of signals are determined by the level of intersymbol interference, which is caused by the superposition of neighboring signals on each other [4,5,6]. Methods for constructing FTN signals are described in detail in [2,4,6,7,8] and can be divided into two groups: filtering methods and optimization methods of the signal envelope shape [9,10,11,12,13,14]. Their common feature is that the duration of the signals is significantly (8–16 times) greater than the duration of message bits for the binary channel alphabet. The energy efficiency of using such signals is determined by their correlation properties and is characterized by energy costs for various processing algorithms.

Modern trends in increasing the data transmission rate in cases of limited frequency bands are implemented by increasing the bit rate and by increasing the volume of the channel alphabet [4,6,7]. In this case, it is necessary to separate the concepts of the absolute data transmission rate and the transmission rate normalized to the frequency band. Restrictions on the computational complexity of the FTN signal demodulation algorithms are imposed for real-time data transmission systems. At the same time, when the data rate normalized to the band increases, there is no such limitation. An indicator of the effectiveness of the application of FTN signals in such conditions is the approach to the Shannon limit for continuous channels with additive Gaussian noise.

However, even in this case, the issue of computational complexity of FTN signal demodulation algorithms is debated. This is because additional intersymbol interference occurs as the bit rate of FTN signals increases. This interference leads to an increase in energy costs, the specific value of which is determined by the reception algorithms. Moreover, in this regard, most of the attention is paid to developing and obtaining the most effective algorithms, from the point of view of obtaining maximum reception reliability and ensuring minimal computational complexity in their implementation.

It is advisable to compare the reception efficiency under conditions of limited computational complexity of the proposed algorithms. Indeed, it is possible to obtain algorithms that provide low energy losses (for example, Viterbi-type algorithms [15,16]) but have high computational complexity. High complexity does not allow data processing in real time, which is often important, for example, when using portable receiving terminals of ground stations in DVB-S2/S2X satellite broadcasting systems. On the other hand, bit-by-bit algorithms are easy to implement, have low computational complexity and have significant energy losses. Moreover, energy losses increase with the increasing bit rate. In this regard, it is advisable to set the task of finding such processing algorithms that, with an increase in the data rate of FTN signals, allow an increase in energy losses of no more than specified for a fixed increase in computational complexity (for example, by more than 30%). Computational complexity can be estimated through the function $O\left(x\right)$, which actually shows the dependence of the amount of calculations performed on the size of the input data/parameters of the algorithm. For example, in the case where the number of addition and multiplication operations (and, as a consequence, running time) depends linearly on the size n of the input data, the algorithmic complexity can be defined as $O\left(n\right)$. When using an algorithm that is optimal according to the maximum likelihood criterion for receiving “in general” sequences of length N and signals with a volume of channel alphabet M, in the general case it is necessary to make $M^N$ comparisons of the observed implementation with all possible reference combinations of signals [17]. Then, the computational complexity of such an algorithm can be estimated as exponential: $O\left(M^N\right)$. In this way, it is possible to quantitatively compare different processing algorithms in terms of energy costs and computational complexity. In addition, it is possible to find the relationship between the increase in the number of addition and multiplication operations and the increase in the bit rate.

Energy costs are the most important in cases of increasing the bit rate for various processing algorithms. The energy loss minimization for FTN signals under such conditions has been the topic of numerous research works [18,19,20,21]. In these papers, the authors use coherent optimal and suboptimal processing algorithms: maximum likelihood sequence estimation (MLSE), the Viterbi algorithm, and so on. It has been shown that binary data can be transmitted at bit rates approximately 25% above the Nyquist limit without energy losses. The MLSE-based processing algorithm is implemented by enumerating all possible implementations of the FTN signal sequence. Solutions are obtained by finding the minimum Euclidean distance between the resulting implementation of the FTN signal sequence and all possible enumeration implementations. The computational complexity of the MLSE-based processing algorithm limits practical implementation because we need to use processing of the entire transmitted sequence of symbols, which is only possible when scale of the operation of the transmission system is not in real time. Nevertheless, it is possible to take the BER performance in this case as a reference for comparing other bit-by-bit processing algorithms with an assessment of energy losses relative to this algorithm.

The struggle to reduce computational complexity for FTN signal processing has led to the emergence of a large class of bit-by-bit processing algorithms that have been discussed in scientific papers over the past 15 to 20 years. This emphasizes the relevance of solving the problem of demodulating FTN signals using simple methods under conditions of significant intersymbol interference. Among such algorithms, iterative processing algorithms with decision feedback can be distinguished [18,19,22,23]. There are several modifications of these algorithms, both with a fixed observation bit interval and with an increased observation interval. In this case, the value of the received symbol is re-evaluated depending on the values of subsequent received symbols falling within the observation interval. In other words, the evaluations of previous and subsequent symbols in the message are used to make a decision about a given symbol received.

In [18], the authors propose two methods for symbol sequence estimation in a received sequence: namely SSSSE (Sequential Character Sequence Estimation) and SSSgbKSE (Sequential Character Sequence Estimation with go-back-K Sequence Estimation). Simulation results show that the proposed methods are suitable for low intersymbol interference scenarios, allowing up to 11% increases in the data rate. However, the authors do not consider the possibility of further increasing the symbol rate beyond this value. The SSSSE and SSSgbKSE algorithms show a promising direction for low-computational-complexity FTN signal processing.

An interesting algorithm for processing FTN signals with a high level of intersymbol interference was proposed in [24,25]. This algorithm implements the rule of a decision-directed successive interference cancellation (DDSIC) scheme with Minimum Mean Square Error (MMSE) equalization for FTN signaling. FTN signals after equalization and decision making are regenerated and then subtracted from the original signals at the receiver input to estimate the level of intersymbol interference. This value is used for the next iteration of interference suppression. Based on the simulation results, the BER performance for the DDSIC scheme is better than those described in [18]. The advantage of this algorithm is its low computational complexity.

In [25], the authors propose a practical methodology for reducing intersymbol interference in an FTN signal sequence within linear equalization. The framework is based on a low-complexity linear equalizer and serial interference cancellation techniques in the frequency domain. This approach requires additional computational operations in the frequency domain and is characterized by high computational complexity.

Let us consider the energy costs for FTN signals with a binary alphabet as a quantitative illustration of the energy costs in case of increasing the data rate to 42% above the Nyquist barrier.

Table 1. The signal-to-noise ratios in dB at the input of the receiving device for various processing algorithms.

Algorithm	$\mathit{BER}\mathbf{=}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$	$\mathit{BER}\mathbf{=}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$
MLSE [18,19,20,21]	7.35	9.01
SSSSE [18]	11.53	-
SSSgbKSE [18]	9.92	-
DDSIC [24]	8.00	10.85

shows the signal-to-noise ratios in dB at the input of the receiving device for various processing algorithms. From this table, we can see that the energy costs are in the range of signal-to-noise ratios of 7–12 dB for $BER={10}^{-3}$ for various algorithms even with a relatively small increase in the bit rate. These values will increase with increases in the data rate. The indicated signal-to-noise ratio values can be used as quantitative guidelines when obtaining new algorithms for processing FTN signals.

Thus, we can identify general trends in improving methods and algorithms for signal processing with an increase in data rates above the Nyquist barrier. The greatest attention is focused on algorithms for bit-by-bit iterative processing with decision feedback, which provide sufficiently high BER performance for receiving FTN signals along with significantly lower computational complexity than with MLSE. Taking these trends into account, we formulated research objectives:

• Obtaining an iterative algorithm with decision feedback, using the generalized maximum likelihood criterion, that performs additional functions of estimating the values of subsequent symbols falling within the observation interval, providing a minimum number of computational procedures when determining the choice of maximum numbers in the sequence.
• Determining the BER performance of FTN signals by simulation modeling at data rates higher than the Nyquist barrier by 2–3 times, with an assessment of energy costs for transmission channels with additive noise and channels with signal fading.
• Determining the computational complexity of the algorithm to find the relationship between the increase in the number of addition and multiplication operations and the increase in the data transfer speed.
• Experimentally assessing the level of additional energy losses during packet transmission of sequences of FTN signals due to inaccuracies in the phase and clock synchronization system.

To achieve this goal, it is first necessary to obtain algorithms for receiving FTN signals using an analytical method. In this case, it is necessary to use a criterion that provides the maximum likelihood and a minimum number of addition and multiplication operations as a criterion for the synthesis of the algorithm. As such, a generalized maximum likelihood ratio criterion is chosen, which is used in conjunction with an algorithm with decision feedback (reception of this and previous symbols) to process subsequent symbols falling within the observation interval.

This is discussed in Section 2, Section 3 and Section 4 of this article and in Appendix A. Section 5 presents the results of simulation modeling of the BER performance of FTN signals for various channels. The results of experimental studies are presented in Section 6. In the Conclusion, the main results of the research are presented.

2. Description of the FTN Signaling System

Signal $s_r\left(t\right)$ with the arbitrary form of amplitude pulse $a\left(t\right)$ with maximum value $A_0$ and carrier frequency $f_0$ is described by the following expression:

$$s_r\left(t\right)=A_{0}a\left(t\right)d_r^{\left(0\right)}cos\left(2\pi f_{0}t\right),$$

(1)

where $d_r^{\left(0\right)}$ is the value of the modulation symbol at time interval $[0,T]$, which is equal to a transferring time of one information bit.

A random sequence of N signals (1) for an arbitrary value of symbol rate $R=1/\xi T$ can be written as

$$y\left(t\right)=A_0\sum _{n=0}^{N-1}a(t-\xi nT)d_r^{\left(n\right)}cos\left(2\pi f_{0}t\right).$$

(2)

Random sequence (2) provides the transfer of binary data at rate $R=1/\xi T,0<\xi <1$. The amplitude pulse $a\left(t\right)$ determines the main spectral and correlation properties of random sequence (2). For signal duration $T_s=LT,L>1$, intersymbol interference appears even at a transmission rate below the Nyquist barrier. This interference increases significantly when transmitting at a rate that exceeds the Nyquist barrier. The appearance of significant intersymbol interference leads to a degradation in correlation properties and BER performance.

Let us consider a generalized block diagram of the data transmission system based on FTN signals (Figure 1). A sequence of binary symbols arrives at the transmitter input from the message source at rate $R=1/\xi T,0<\xi <1$. These binary symbols are converted into modulation symbols of the message in the “Mapper” block. For example, for binary phase modulation, $d_r^{\left(0\right)}=\pm 1,r=1,2$. A sequence of FTN signals on the basis of the amplitude pulse $a\left(t\right)$ is generated in the “FTN modulator”. The structure of this device is described in detail in [8]. The symbol rate for forming FTN signals is determined by the “Symbol rate R” block. Sequence of FTN signals (2) goes to block “Channel” from the transmitter output. Two types of channels are used as a transmission channel for simulation and for experimental studies: a channel with constant parameters and additive Gaussian noise (AWGN) $n\left(t\right)$ with an average power spectral density $N_0/2$ and a channel with multipath propagation. The random process $x\left(t\right)=y\left(t\right)+n\left(t\right)$ arrives at the input of the receiver from the output of the transmission channel. The realization of $x\left(t\right)$ is processed in accordance with one or another processing algorithm in the block “FTN demodulator”.

The choice of the reception algorithm is determined by the acceptable energy costs and the computational complexity of the implementation. We propose new bit-by-bit iterative processing algorithms with decision feedback, providing sufficiently high BER performance for FTN signals with limited computational complexity. A possible algorithm for processing the evaluation of the current symbol can be divided into two parts:

• The first part is the reception of a given symbol, taking into account decisions about previous accepted symbols (decision feedback), which is implemented with minimal computational complexity;
• The second part is the optimal processing of subsequent symbols falling within the observation interval.

Note that we cannot separate these parts of the algorithm. In this case, we make an analytical generation of the algorithm using the generalized maximum likelihood criterion. This general algorithm performs decision feedback processing and estimation of the values of subsequent symbols that fall within the observation interval. Also, this algorithm must provide a minimum number of computational procedures when determining the choice of maximum numbers in the sequence. The choice of this optimality criterion is due to the fact that the minimum possible number of addition and multiplication operations is required in the function of selecting the maximum numbers in the sequence to implement the proposed algorithm.

3. Proposed Algorithm with Decision Feedback with Estimation of Subsequent Symbols in Observation Interval by Maximum Likelihood Ratio

Let us present the process at the input of the receiver in the following form:

$$x\left(t\right)=\mu s_r^{\left(n\right)}\left(t\right)+\mu y_{-}(t,i)+\mu y_{+}(t,q)+n\left(t\right),$$

(3)

where $\mu$ is a channel transfer coefficient; $y_{-}(t,i)$ and $y_{+}(t,q)$ are the sequences of previous and subsequent signals. Note that $y_{-}(t,i)$ and $y_{+}(t,q)$ depend on $R=1/\xi T$. The indexes i and q denote the numbers of specific combinations of previous I and subsequent Q signals. For $M=2$ and $R=1/T$, $i=1,2,\dots ,2^I$ and $q=1,2,\dots ,2^Q$.

The normalized symbol rate $R_n$ is equal to $RT=1/\xi$. In this case, the number of previous signals in the observation interval $T_s$ is equal to $I=⌈L{R}_n-1⌉$. The number of subsequent signals is equal to $Q=⌈L{R}_n-1⌉$. Then, for $n=0$ in (1),

$$y_{-}(t,i)=\sum _{p=-1}^{-⌈L{R}_n-1⌉}{d}_{ri}^{\left(p\right)}a(t-p\xi T)cos\left(2\pi f_{0}t\right),$$

(4)

$$y_{+}(t,q)=\sum _{p=+1}^{⌈L{R}_n-1⌉}{d}_{rq}^{\left(p\right)}a(t-p\xi T)cos\left(2\pi f_{0}t\right),$$

(5)

where $d_{ri}^{\left(p\right)}$ and $d_{rq}^{\left(p\right)}$ are the channel symbols of previous (i) and subsequent (q) sequences of signals; p is the ordinal number of the symbol in the sequence of signals. For example, values of I and Q are equal to 7 for $R_n=1/\xi =2$ and have a symmetrical pulse shape $a\left(t\right)$ with duration $T_s=8T$. Note that ISI increases significantly at such values of R and affects the correlation properties.

Let us present $n\left(t\right)$, $s_r\left(t\right)$ and $x\left(t\right)$ in terms of in-phase $A_{nc},A_{rc}$ and $A_{xc}$ and quadrature $A_{ns},A_{rs}$ and $A_{xs}$ baseband components:

$$n\left(t\right)=A_{nc}\left(t\right)cos\left(2\pi f_{0}t\right)-A_{ns}\left(t\right)sin\left(2\pi f_{0}t\right),$$

(6)

$$s_r\left(t\right)=A_{rc}\left(t\right)cos\left(2\pi f_{0}t\right)-A_{rs}\left(t\right)sin\left(2\pi f_{0}t\right),$$

(7)

$$x\left(t\right)=A_{xc}\left(t\right)cos\left(2\pi f_{0}t\right)-A_{xs}\left(t\right)sin\left(2\pi f_{0}t\right).$$

(8)

Processing algorithms with decision feedback can be obtained provided that the functions $y_{-}(t,i)$ are known and the functions $y_{+}(t,q)$ are processed in accordance with the generalized maximum likelihood criterion. The derivation of this algorithm is given in Appendix A.

The analytical notation of the algorithm with decision feedback with the evaluation of subsequent symbols in accordance with the maximum likelihood ratio criterion has the following form for $M=2$:

$$\begin{array}{c}2{\int }_0^{LT}{A}_{xc}\left(t\right)a\left(t\right)dt\underset{s_2\left(t\right)}{\overset{s_1\left(t\right)}{\gtrless }}\delta (i,q),\\ \delta (i,q)={\displaystyle \underset{\left(q\right)}{max}}({\int }_0^{LT}{A}_{xc}\left(t\right){\displaystyle \sum _{(p=1)}^{⌈L{R}_n-1⌉}}{d}_{rq}^{\left(p\right)}a(t-p\xi T)dt\\ -\mu {\int }_0^{LT}a\left(t\right){\displaystyle \sum _{p=-⌈L{R}_n-1⌉}^{(-1)}}{d}_{ri}^{\ast \left(p\right)}a(t-p\xi T)dt\\ -\mu {\int }_0^{LT}a\left(t\right){\displaystyle \sum _{p=1}^{⌈L{R}_n-1⌉}}{d}_{rq}^{\left(p\right)}a(t-p\xi T)dt)\\ -{\displaystyle \underset{\left(q\right)}{max}}({\int }_0^{LT}{A}_{xc}\left(t\right){\displaystyle \sum _{(p=1)}^{⌈L{R}_n-1⌉}}{d}_{rq}^{\left(p\right)}a(t-p\xi T)dt\\ +\mu {\int }_0^{LT}a\left(t\right){\displaystyle \sum _{p=-⌈L{R}_n-1⌉}^{(-1)}}{d}_{ri}^{\ast \left(p\right)}a(t-p\xi T)dt\\ +\mu {\int }_0^{LT}a\left(t\right){\displaystyle \sum _{p=1}^{⌈L{R}_n-1⌉}}{d}_{rq}^{\left(p\right)}a(t-p\xi T)dt).\end{array}$$

(9)

where $d_{ri}^{\ast \left(p\right)}$ is the evaluation of the p-th received symbol.

This algorithm is characterized, as will be shown later, by limited computational complexity with energy costs close to the MLSE algorithm (see Table 1).

4. Realization of the Processing Algorithm

Analog processing of input process (8) with transfer to the zero frequency of the spectrum of the sequence of FTN signals is carried out when implementing the preliminary reception algorithm. The radio modem is built on the SDR NI USRP 2920 platform and the signal spectrum is transferred from the carrier frequency to the zero frequency in the analog (hardware) part of the platform. Phase Locked Loop (PLL) and timing are used to achieve coherent reception. PLL and timing are implemented in the platform in the controller and processor in PCHost.

The software part of algorithm (9) and its block diagram are shown Figure 2 and Figure 3. The software part determines the computational complexity of the receiving device. The input of the software part of algorithm (9) receives digital samples from the function. These samples are distributed among the correlator blocks. The upper correlator (the left side of algorithm (9)) performs processing on the observation interval $[0;LT]$, equal to the duration of the FTN signal. The remaining correlators are used for preliminary assessment of subsequent symbols falling within the observation interval (the right side of inequality (9)). The introduction of decision feedback when receiving previous symbols occurs in the block “Decision feedback with estimation of subsequent symbols in observation interval by maximum likelihood ratio” and the block “Correlation coefficient of previously received symbols” $R_{ri}^{\left(p\right)}$ in accordance with $\delta (i,q)$ in (9). The correlator block process is performed with digital “sliding” integration by using memory integrators and continuously performing the integration process. This is due to the fact that the duration of the FTN signal exceeds the duration T, so several consecutive signals will be included in integration processes during integration. Note that a digital matched filter can be used instead of a digital “sliding” integrator. The processing of subsequent signals is performed on the observation interval $[0;LT]$ in accordance with the maximum likelihood estimate in the digital block “Correlation coefficient of subsequent symbols” $K_{rq}^{\left(p\right)}$.

Evaluations of the received symbols $d_r^{\ast \left(p\right)}$ are formed at the output of the decision unit (DU). The threshold value of the DU is equal to $\delta (i,q)$. These values are formed by taking into account the correlation coefficients of the subsequent symbols $K_{r,q}^{\left(p\right)}={\int }_0^{LT}a\left(t\right){\sum }_{p=1}^{⌈L{R}_n-1⌉}{d}_{rq}^{\left(p\right)}a(t-p\xi T)dt$ and the correlation coefficients of the evaluations of the received symbols $R_{r,i}^{\left(p\right)}={\int }_0^{LT}a\left(t\right){\sum }_{p=-⌈L{R}_n-1⌉}^{(-1)}{d}_{ri}^{\ast \left(p\right)}a(t-p\xi T)dt$ with the amplitude pulse shape $a\left(t\right)$. Blocks “$d_{rq}^{\left(p\right)}$” are needed to calculate the values of the correlation integral ${\int }_0^{LT}{A}_{xc}\left(t\right){\sum }_{(p=1)}^{⌈L{R}_n-1⌉}{d}_{rq}^{\left(p\right)}a(t-p\xi T)dt$. Note that the value of $\delta (i,q)$ depends on the coefficient $\mu$, which is determined by the attenuation level of the channel.

The computational complexity of proposed algorithm (9) is determined by the number of operations for the calculation of the maximum. The upper limit for the number of additions is equal to $2\left(M^{⌈L{R}_n-1⌉}\right)$ and no multiplications, but there are many effective realizations. Note that the complexity of proposed algorithm (9) does not depend on the length of the message packet. The complexity gain for the comparison with the processing algorithm based on the MLSE is more than two orders in terms of the memory consumption for the case of the length of the received message packet equal to $N=100$.

The computational complexity of algorithm (9) is given in

Table 2. Computational complexity of (9) for $T_s=8T$ and $R=1.42/T$.

Algorithm	Number of Additions	Number of Multiplications	Memory Consumption
Proposed algorithm with decision feedback with evaluation of subsequent symbols in accordance with the maximum likelihood ratio criterion (9)	$M^{\left(\mathit{Q}\right)}(4ML{N}_s-2M+1)$	$M^{\left(\mathit{Q}\right)}4ML{N}_s$	$M^{(\mathit{Q}+\mathit{L})}$

. It is determined by the number of additions and multiplications and depends on the number Q of subsequent symbols which fall within the observation interval. Also, it is determined by the number of samples per bit $N_s$. Compared to the MLSE, algorithm (9) has a lower memory consumption since it does not depend on the packet length. For example, when packet length is $N=100$, the gain in memory consumption reaches more than two orders.

We can make a subconclusion that the proposed decision feedback with the estimation of subsequent symbols in the observation interval by the maximum likelihood ratio can be implemented on the basis of a universal digital platform using universal microprocessors (e.g., starting from family Intel 3000) as a PCHost. We obtained the values of the computational complexity of the proposed algorithm depending on signal parameters (signal duration, level of ISI). Using these values, it is possible to estimate the increase in the computational complexity under conditions of fixed energy losses and an increased transmission rate. In conditions of specific transmission rates up to 3 bps/Hz and signal duration (8–16)T, the computational complexity of the proposed algorithm is more than two orders lower compared to that of MLSE.

5. Simulation Results—BER Performance

Let us see the simulation results for proposed algorithm (9). An assessment of the BER performance of FTN signals at bit rates higher than the Nyquist barrier by a factor of 2–3 was carried out using the simulation method to calculate the energy costs for algorithm (9). Channels with additive noise and constant characteristics and channels with Rayleigh fading of signals were considered transmission channels.

FTN signals based on RRC pulses with a duration $T_s=8T$ and a roll-off factor $\beta =0.3$ of the frequency response of the shaping low-pass filter were used in the simulation of packet transmission. A high level of intersymbol interference in FTN signals, caused by the superposition of adjacent signals on each other, was observed under these conditions. The values of error probability of the FTN signals were obtained depending on the signal-to-noise ratio. The signal-to-noise ratio is understood as the ratio $E_b/N_0$, where $E_b$ is the energy of the received signal (for the binary alphabet). The sample size (number of transmitted bits) was equal to ${10}^6$ in BER measurements. The modulation type of FTN signals was selected as BPSK.

5.1. Channel with Constant Parameters

BER performance for different bit rates R in the case of applying algorithm (9) are shown in Figure 4 and

Table 3. BER performance (values of $E_b/N_0$ for fixed BER). Channel with constant parameters. Processing algorithm (9) in case of high rates.

BER	${\mathbf{10}}^{\mathbf{-}\mathbf{2}}$	${\mathbf{10}}^{\mathbf{-}\mathbf{3}}$	${\mathbf{10}}^{\mathbf{-}\mathbf{4}}$
$R=1.11/T$	4.34 dB	6.79 dB	8.38 dB
$R=1.25/T$	4.63 dB	6.94 dB	8.44 dB
$R=1.42/T$	5.47 dB	7.98 dB	9.77 dB
$R=1.66/T$	6.83 dB	9.17 dB	10.74 dB
$R=2.00/T$	8.34 dB	10.73 dB	12.71 dB

. The same figure shows the potential BER without intersymbol interference.

First of all, let us compare the BER at a bit rate $R=1.25/T$ with the potential BER. It can be seen that the energy losses are no more than 0.1 dB at $BER={10}^{-4}$. When the bit rate is increased to values above the Nyquist barrier by a factor of two ($R=2/T$), energy losses are equal to 4.5 dB at $BER={10}^{-4}$.

Figure 5 and

Table 4. BER performance (values of $E_b/N_0$ for fixed BER). Channel with constant parameters. Processing algorithm (9) in case of high rates.

R	$\mathbf{2.00}\mathbf{/}\mathit{T}$	$\mathbf{2.10}\mathbf{/}\mathit{T}$	$\mathbf{2.22}\mathbf{/}\mathit{T}$	$\mathbf{2.50}\mathbf{/}\mathit{T}$
$E_b/N_0$ $(BER={10}^{-2})$	8.34 dB	9.10 dB	12.93 dB	17.61 dB

show the BER dependences for high bit rates. We can see that the energy costs for $R=2.5/T$ are equal to 20 dB for $BER=7\times {10}^{-3}$. The energy consumption is equal to 8 dB at the same BER value for $R=2/T$. It can be stated that when the data rate increases above the Nyquist barrier by 2.5 times, energy losses increase significantly and amount to more than 15 dB relative to $R=1/T$.

Let us evaluate algorithm (9) in comparison with known processing algorithms. In Figure 6 and

Table 5. $E_b/N_0$ at different levels of BER ($R=1.42/T$ and $\beta =0.2$).

Algorithm	$\mathit{BER}\mathbf{=}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$	$\mathit{BER}\mathbf{=}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$
Proposed algorithm (Simulation)	8.33 dB	10.23 dB
Bit-by-bit algorithm (Simulation)	-	-
MLSE (Simulation)	7.35 dB	9.01 dB
MLSE [18,19,20,21]	7.35 dB	9.01 dB
SSSSE [18]	11.53 dB	-
SSSgbKSE [18]	9.92 dB	-
DDSIC [24]	8.00 dB	10.85 dB

, the dependence of the BER on the ratio $E_b/N_0$ for the bit rate $R=1.42/T$ for various processing algorithms (Table 1) is shown.

Proposed algorithm (9) provides an energy gain compared to the SSSSE [18] and SSSgbKSE [18] algorithms. DDSIC [24] provides a gain of less than 0.4 dB in the area $E_b/N_0$ = (6–10) dB. However, at higher values of $E_b/N_0$, proposed algorithm (9) allows one to achieve better values of energy costs (up to 4 dB). Note that this case ($E_b/N_0>10\mathrm{dB}$) is common for 5G and 6G communication systems.

5.2. Rayleigh Channel

Let us consider the energy costs for algorithm (9) for FTN signal processing in Rayleigh fading channel. At the same time, we will take into account that it is advisable to use an equalizer when receiving FTN signals in the presence of fading. The MMSE (Minimum Mean Square Error) equalizer was used in this work. Figure 7 shows the dependences of the BER on the $E_b/N_0$ ratio for bit rates up to $R=2.5/T$.

Let us analyze the dependence in Figure 7. It can be concluded that in the presence of Rayleigh fading, the application of algorithm (9) makes it possible to ensure the reception of FTN signals at least up to bit rates of $R=2.5/T$. Energy costs are equal to 12 dB for a bit rate of $R=2/T$ at $BER={10}^{-2}$. When bit rates increase above the values of $R=2.22/T$, a well-known effect of stabilization of BER performance with increasing $E_b/N_0$ is observed in case of fading. This can be explained by the fact that at such values of bit rates, the level of intersymbol interference, which is caused by the formation of FTN and interference signals and passing through the transmission channel, is quite high. This effect is especially noticeable in the area of high signal-to-noise ratios.

6. Simulation Results—Comparison of Energy Efficiency and Computational Complexity of Decision Feedback Detection Algorithm with Estimation of Subsequent Symbols in Observation Interval by Maximum Likelihood Ratio

Let us estimate the increase in the computational complexity of decision feedback algorithm (9) caused by an increase in data transmission rate. We try to answer the following questions:

• To what extent must the computational complexity of the digital demodulator of spectrally efficient signals be increased under conditions of a fixed BER and increased symbol rate?
• What is the value of the energy loss caused by an increased symbol rate under conditions of a fixed BER and computational complexity?

By the energy loss, we mean the difference between the signal-to-noise ratios $E_b/N_0$ in the case of the ISI-free BER performance and in the case of using proposed algorithm (9) at $BER={10}^{-4}$.

Then, it is possible to construct dependences between the energy losses of algorithm (9) in relation to the potential BER depending on the number of addition and multiplication operations for different bit rates. Figure 8 shows the values of the energy loss for the bit rates from $R=1.11/T$ to $R=2/T$.

Figure 8 shows the energy loss values obtained as a result of simulation modeling for bit rates from $R=1.11/T$ to $R=2/T$ for FTN signals, the parameters of which are given in Section 5. In this figure, the X-axis shows the number of addition and multiplication operations (computational complexity). Moreover, each point along this axis corresponds to the value of parameter q in formula (9). This parameter determines the number of subsequent signals falling within the observation interval, which are taken into account when implementing the scheme in Figure 5. For example, for $R=1.11/T$ and $q=1$ (one previous symbol is used in the feedback), the energy loss is equal to 0.6 dB at the minimum values of the computational complexity. When $q=4$ (four previous values of symbols are used in the feedback ), the energy loss practically becomes zero, that is, the BER coincides with the potential one. By increasing the transmission rate to $R=1.25/T$, approaching the potential BER is achieved by increasing the computational complexity, which reaches a value of $0.4\times {10}^5$ addition and multiplication operations.

With a further increase in the bit rate up to $R=1.42/T$, the minimum energy loss will be equal to 1.2 dB with the number of operations $3\times {10}^5$. When the bit rate $R=2/T$ is reached, the number of operations increases up to $6.5\times {10}^5$ with a minimum energy loss of 3.5 dB.

Thus, the following conclusions can be made for proposed processing algorithm (9) with decision feedback. First, it is possible to estimate resulting energy losses from an increase in the bit rate at a fixed error probability ($BER={10}^{-4}$) using the developed methodology. Second, the energy loss in relation to potential BER increases by approximately 2 dB with an increase in the bit rate speed by 1.5 times (above the Nyquist barrier). Third, an increase in the bit rate by 1.5 times leads to an increase in the energy loss from 1.2 dB to 3.7 dB for a fixed $BER={10}^{-4}$ with a fixed computational complexity (which is often the main criterion for the practical implementation of a digital demodulator). Finally, an increase in the bit rate by 1.5 times leads to an increase in computational complexity from $0.8\times {10}^5$ to $3.3\times {10}^5$ numbers of operations for acceptable increases in energy loss equal to 2 dB. Summarizing this conclusion, we can estimate how much additional increases in the number of computational operations will cost a gain in energy efficiency, for example, 1 dB. For the proposed processing algorithm with decision feedback, each decibel is achieved by increasing the number of computational operations by $1.5\times {10}^5$.

7. Experimental Research

Proposed algorithm (9) was experimentally tested in real conditions of a channel with noise and fading and non-ideal phase and timing synchronization. The purpose of these studies was to evaluate the level of additional energy losses during packet transmission of FTN signals at high bit rates. We used the NI USRP 2920 (

Table 6. NI USRP 2920 specifications.

Parameter	Values
Frequency range	50 MHz to 2.2 GHz
Frequency step	<1 kHz
Frequency accuracy	2.5 ppm
Digital-to-analog converter (DAC)	2 channels, 16 bit
Analog-to-digital converter (ADC)	2 channels, 14 bit

) SDR platform as the experimental setup in our research. The parameters of the FTN signals corresponded to the parameters of the signals in the simulation (described in Section 5). Packets consisted of a preamble in the form of a 64-bit clock synchronization sequence and an information part of the packet (2048 bits). These values are typical and provide an estimation of BER performance in real conditions.

The experimental setup (Figure 9) consisted of a transmitting (Tx) and receiving part (Rx). The carrier frequency was equal to 429 MHz. The area of the room was equal to 40 square meters. A spectrum analyzer (Agilent Technologies N9342C, manufacturer—Keysight Technologies (Agilent Technologies), Santa Clara, CA, USA) was located between the transmitter and receiver to study the energy spectrum of FTN signals.

An example of the signal sequence at the output of the transmitter is shown in Figure 10a, $E_b/N_0$ = 10 dB. The spectral characteristics are shown in Figure 10b.

7.1. Energy Losses Due to Phase and Timing Synchronization

Additional energy losses occur during packet transmission due to inaccuracies in phase and timing synchronization. Figure 11 (red line) shows the experimental BER for the bit rate $R=1.42/T$. The same figure shows the results of calculating the BER using simulation (Section 5). As we can see, the difference between simulation and experiment at $BER={10}^{-4}$ is about 1 dB.

These additional energy losses are explained by the fact that the experimental SDR setup used a standard phase-locked frequency control system and a timing synchronization system designed to process classical BPSK signals with a rectangular amplitude pulse of duration T. The resulting additional energy losses of 1 dB can be reduced to 0.1–0.2 dB by using improved phase and clock synchronization systems working with FTN signals, which can provide less phase and timing errors.

7.2. Multipath Channel

The application of algorithm (9) was considered for a channel with multipath wave propagation. We conducted experimental studies on the reception of FTN signals in burst mode in an urban area. In this case, the experiment was carried out repeatedly to obtain the necessary statistics in the presence of traffic and people.

We can see BER performance in a case of fading channel in an urban area in Figure 12. The transmission rate of symbols of the channel binary alphabet is equal to $R=2/T$. There is potential BER performance of BPSK signals. We used an MMSE equalizer for work in fading channels. Figure 12 shows the BER performance without an MMSE equalizer for comparison with the obtained results. The simulation results (Section 5) in the channel with AWGN are also presented here.

The following conclusions can be drawn. The energy costs for transmission channels in an urban area are equal to 13.8 dB at $BER={10}^{-3}$ in a case of application of an MMSE equalizer. The energy losses in a multipath channel relative to a channel with AWGN are equal to 3 dB for $BER={10}^{-3}$ for the application of proposed algorithm (9). Energy losses in a multipath channel in relation to the potential BER are equal to 7 dB at $BER={10}^{-3}$ for a symbol transmission rate twice the Nyquist barrier.

7.3. Multipath Channel and Increased Bit Rate

Let us consider the results of experimental research for $R=2.22/T$. Intersymbol interference caused by the FTN signal generation and transmission conditions increases significantly under these conditions. Added to this is intersymbol interference caused by multipath propagation.

Figure 13 shows the BER performance for a channel with fading in urban areas at $R=2.22/T$ with the application of an MMSE equalizer. The same figure shows the results of simulation modeling (Section 5) for an AWGN channel for the same bit rate. As follows from the analysis of the curves in this figure, at $R=2.22/T$ with an MMSE equalizer, the additional energy costs for $BER=2\times {10}^{-2}$ are about 7 dB relative to the costs in a channel with constant parameters. This comes at the cost of fading and an increased bit rate of 2.22 times the Nyquist barrier. Note that classical signals under such signal transmission conditions cannot be used due to the fact that the error probability will be equal to $BER=0.5$ for any signal-to-noise ratio.

8. Conclusions

We proposed a new bit-by-bit algorithm, (9), for processing FTN signals designed for bit rates above the Nyquist barrier. The advantage of this algorithm is BER performance close to MLSE with minimal computational complexity. Algorithm (9) is based on an iterative algorithm with decision feedback. It is complemented by an algorithm for receiving subsequent signals falling within the observation interval, which is optimal according to the generalized maximum likelihood criterion, providing a minimum number of computational procedures when determining the choice of maximum numbers in the sequence. It is shown that the computational complexity of the proposed algorithm is more than two orders less than the complexity of the MLSE algorithm.

The energy costs for receiving FTN signals at bit rates above the Nyquist barrier for transmission channels with additive noise (AWGN) and fading channels for proposed algorithm (9) are given in

Table 7. $E_b/N_0$ for proposed algorithm (9).

Channel	$\mathit{R}\mathbf{=}\mathbf{1.42}\mathbf{/}\mathit{T}$	$\mathit{R}\mathbf{=}\mathbf{2.00}\mathbf{/}\mathit{T}$	$\mathit{R}\mathbf{=}\mathbf{2.10}\mathbf{/}\mathit{T}$	$\mathit{R}\mathbf{=}\mathbf{2.22}\mathbf{/}\mathit{T}$	$\mathit{R}\mathbf{=}\mathbf{2.50}\mathbf{/}\mathit{T}$
AWGN, simulation ($BER={10}^{-4}$)	9.77 dB	12.82 dB	15.68 dB	>20 dB	>20 dB
Rayleigh channel, simulation ($BER=6\times {10}^{-2}$)	-	8.21 dB	-	12.81 dB	24.02 dB
AWGN, experiment ($BER={10}^{-4}$)	11.11 dB	-	-	-	-
Multipath channel, experiment	-	13.82 dB ($BER={10}^{-3}$)	-	19.19 dB ($BER=2\times {10}^{-2}$)	-

. This table contains the results of simulation and the results of experimental research.

In this paper, we give a numerical estimation of the increase in computational complexity for proposed algorithm (9) at bit rates exceeding the Nyquist barrier. An increase in the bit rate by 1.5 times (from $R=1.42/T$ to $R=2/T$) leads to an increase in energy losses from 1.2 dB to 3.7 dB in the area of $BER={10}^{-4}$ and under conditions of fixed computational complexity. If we fix the possible energy losses, for example, at a value of 2 dB, then an increase in the bit rate by 1.5 times (from $R=1.11/T$ to $R=1.66/T$) leads to an increase in computational complexity by more than an order, from $0.1\times {10}^5$ to $3.3\times {10}^5$ operations. The same conclusion can be reformulated in another form: for proposed algorithm (9), each decibel of energy gain is achieved by increasing the number of computational operations by $1.5\times {10}^5$ operations.

The application of proposed algorithm (9) in the DVB standard of satellite broadcasting systems S2/S2X can provide increases in the bit rate without significant degradation in the BER with minimal computational complexity of the demodulator. This is especially important for mobile terminals of satellite broadcasting ground stations. Potential future directions for this research are related to the optimization of pulse shapes for different channels to achieve additional increases in spectral and energy efficiency without increasing computational complexity.