Volume 18 Issue 1
May  2020
Article Contents

Sergey B. Makarov, Ilya I. Lavrenyuk, Anna S. Ovsyannikova, Sergey V. Zavjalov. BER Performance of Finite in Time Optimal FTN Signals for the Viterbi Algorithm[J]. Journal of Electronic Science and Technology, 2020, 18(1): 42-51. doi: 10.1016/j.jnlest.2020.100022
Citation: Sergey B. Makarov, Ilya I. Lavrenyuk, Anna S. Ovsyannikova, Sergey V. Zavjalov. BER Performance of Finite in Time Optimal FTN Signals for the Viterbi Algorithm[J]. Journal of Electronic Science and Technology, 2020, 18(1): 42-51. doi: 10.1016/j.jnlest.2020.100022

BER Performance of Finite in Time Optimal FTN Signals for the Viterbi Algorithm

doi: 10.1016/j.jnlest.2020.100022
Funds:  This work was supported by the Grant of the President of the Russian Federation for state support of young Russian scientists (agreement МК-1571.2019.8 No. 075-15-2019-1155)
More Information
  • Author Bio:

    Sergey B. Makarov was born in Leningrad in 1948. He received the Ph.D. degree and the doctor of science degree in technical science from Leningrad Polytechnic Institute, St. Petersburg in 1977 and 1991, respectively. He is currently a professor with the Higher School of Applied Physics and Space Technologies, Peter the Great St. Petersburg Polytechnic University (SPbPU), St. Petersburg. He is the author of more than 300 publications in the field of wireless communications. His current research interests include spectrally efficient signaling, optimal signals for wireless communications, meteor-burst and ultra wide band (UWB) communications networks, and industrial Internet of things

    Ilya I. Lavrenyuk was born in Kazakhstan in 1995. He received the B.S. and M.S. degrees in communications systems engineering from SPbPU in 2016 and 2018, respectively. He is currently pursuing the Ph.D. degree with the Higher School of Applied Physics and Space Technologies, SPbPU. His research interests include digital communications and spectrally efficient signaling

    Anna S. Ovsyannikova was born in Sochi in 1997. She received the B.S. degree in infocommunications technologies and communications systems from SPbPU in 2018. She is currently pursuing the M.S. degree with the Higher School of Applied Physics and Space Technologies, SPbPU. Her research interests include 5G, optimal signals, spectral efficiency, and signal processing

    Sergey V. Zavjalov was born in Leningrad in 1988. He received the B.S., M.S., and Ph.D. degrees from SPbPU in 2009, 2011, and 2015, respectively, all in communications systems engineering. He is currently an associate professor with the Higher School of Applied Physics and Space Technologies, SPbPU. His research interests include digital communications, spectrally efficient signaling, UWB signals, and WiFi

  • Corresponding author: S. B. Makarov, I. I. Lavrenyuk, A. S. Ovsyannikova, and S. V. Zavjalov are with the Higher School of Applied Physics and Space Technologies, Peter the Great St. Petersburg Polytechnic University, St. Petersburg 195251 (e-mail: makarov@cee.spbstu.ru; knaiser@mail.ru; anny-ov97@mail.ru; zavyalov_sv@spbstu.ru).
  • Received Date: 2019-08-14
  • Rev Recd Date: 2019-09-07
  • Available Online: 2020-05-06
  • Publish Date: 2020-03-01

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BER Performance of Finite in Time Optimal FTN Signals for the Viterbi Algorithm

doi: 10.1016/j.jnlest.2020.100022
Funds:  This work was supported by the Grant of the President of the Russian Federation for state support of young Russian scientists (agreement МК-1571.2019.8 No. 075-15-2019-1155)
  • Author Bio:

  • Corresponding author: S. B. Makarov, I. I. Lavrenyuk, A. S. Ovsyannikova, and S. V. Zavjalov are with the Higher School of Applied Physics and Space Technologies, Peter the Great St. Petersburg Polytechnic University, St. Petersburg 195251 (e-mail: makarov@cee.spbstu.ru; knaiser@mail.ru; anny-ov97@mail.ru; zavyalov_sv@spbstu.ru).

Abstract: In this article, we consider the faster than Nyquist (FTN) technology in aspects of the application of the Viterbi algorithm (VA). Finite in time optimal FTN signals are used to provide a symbol rate higher than the “Nyquist barrier” without any encoding. These signals are obtained as the solutions of the corresponding optimization problem. Optimal signals are characterized by intersymbol interference (ISI). This fact leads to significant bit error rate (BER) performance degradation for “classical” forms of signals. However, ISI can be controlled by the restriction of the optimization problem. So we can use optimal signals in conditions of increased duration and an increased symbol rate without significant energy losses. The additional symbol rate increase leads to the increase of the reception algorithm complexity. We consider the application of VA for optimal FTN signals reception. The application of VA for receiving optimal FTN signals with increased duration provides close to the potential performance of BER, while the symbol rate is twice above the Nyquist limit.

Sergey B. Makarov, Ilya I. Lavrenyuk, Anna S. Ovsyannikova, Sergey V. Zavjalov. BER Performance of Finite in Time Optimal FTN Signals for the Viterbi Algorithm[J]. Journal of Electronic Science and Technology, 2020, 18(1): 42-51. doi: 10.1016/j.jnlest.2020.100022
Citation: Sergey B. Makarov, Ilya I. Lavrenyuk, Anna S. Ovsyannikova, Sergey V. Zavjalov. BER Performance of Finite in Time Optimal FTN Signals for the Viterbi Algorithm[J]. Journal of Electronic Science and Technology, 2020, 18(1): 42-51. doi: 10.1016/j.jnlest.2020.100022
    • Faster than Nyquist (FTN) random binary signal sequences s(t) with the signal piece of duration larger than one symbol interval T could provide a symbol rate higher than the “Nyquist barrier” without any encoding, as in [1] to [6]. The high bandwidth efficiency is achieved by transmitting information using signals with the duration Ts=LT ($L = 2,{\rm{ }}3,{\rm{ }} \cdots $) and energy Es, which is focused mainly on a relatively small (less than 20% of the duration Ts) time interval. Such signals are generated using a digital filter, which provides a narrow band of occupied frequencies ΔF and a specific information transfer rate. Message transmission occurs under intersymbol interference (ISI) conditions, caused by the overlapping of adjacent signals, which leads to the significant reduction of the bit error rate (BER) performance of reception at high channel message rates.

      To increase the BER performance of message reception, algorithms for coherent “reception in the whole” packets of sequences of FTN signals are used. It is shown in [3] and [4] that when using FTN signals built on the basis of root raised cosine (RRC) pulses with a value of the roll-off factor of the frequency response, the energy losses are about 12 dB with a probability of error per bit p=10–3. These costs exceed the potential signal-to-noise ratio for given BER of 5 dB.

      It is reasonable to formulate the problem of reducing energy costs due to the transition from the method of generating signals using bandpass filtering to the application of the method of constructing signals based on the solution of the optimization problem in the form of s(t), as in [6] to [10], with the introduction of a constraint on the cross-correlation coefficient or Euclidean distance, as in [7] and [11] to [14].

      The criterion for optimizing the shape of such FTN signals is based on the principle of choosing the maximum density of the energy spectrum G(f ), determined by the rate of out-of-band emissions of G(f ) outside the band ΔF. The occupied frequency band ΔF is determined by the level of the decline in the energy spectrum (for example, ΔF–50 dB). As shown in [13] to [15], the shape of the optimal FTN signals of the duration Ts=LT changes as the message transmission rate R increases. There is a decrease in the time interval, where the main part of the energy is concentrated; thus, this leads to an expansion of the occupied frequency band of the energy spectrum of the transmitted random sequence of signals.

      A random sequence consisting of binary modulated N time-limited single optimal FTN signals sopt(t) with the duration Ts=LT ($L = 2,{\rm{ }}3,{\rm{ }} \cdots $) and energy Eopt has the following form:

      where the values cn=±1 have equal probabilities of occurrence for each value n. The coefficient ξ (0<ξ<1) determines the symbol rate, which is equivalent to the bit rate in the case of the binary alphabet. A feature of such signals is the controlled level of ISI. The energy reception efficiency is estimated by calculating the Euclidean distance. The square of the Euclidean distance d 2(i,k) between different realization of two random sequences yi(t) and yk(t) in (1) determines BER and can be calculated by the following formula:

      The minimum normalized Euclidean distance is defined as

      As a transmission channel, we consider the channel with the additive white Gaussian noise (AWGN) n(t) with an average power spectral density of N0, which allows us to estimate the place of optimal FTN signals on the Shannon plane and to compare their efficiency with the Shannon boundary.

      This paper considers the possibility of using time-limited optimal binary FTN signals to increase the BER performance of message reception when approaching high specific transmission rates in a channel with AWGN.

    • For example, Fig. 1 shows the implementation forms of a random sequence as (1) of optimal FTN signals[10],[16]-[18] with the duration Ts=8T for ξ=1/2 (R=2 /T, Fig. 1 (a)) and ξ=1/2.5 (R=2.5/T, Fig. 1 (b)). In these figures, the sequence of cn has the form: +1, +1, –1, and +1. Thin lines indicate the forms of single signals corresponding to cn, and thick lines indicate the total sequences of signals. It can be seen that the ISI level is quite high in these sequences.

      Figure 1.  Sequence of optimal FTN signals with symbol rates: (a) R=2/T and (b) R=2.5/T.

      The shape of the optimal FTN signal was obtained for the transmission rate R=2/T (Fig. 1 (a)) with a restriction on the cross-correlation coefficient K0=0.01 for the rate of the out-of-band emission level 1/f 4, as shown in [10] and [13] to [15]. Fig. 1 (b) shows the sequence of these signals transmitted at a speed of R=2.5/T.

      As can be seen from a comparison of the forms of the sequences shown in Figs. 1 (a) and (b), as the transmission speed increases, the influence of neighboring signals falling into this analysis interval increases significantly.

      Let us consider the energy spectra of FTN signals based on the optimal pulses and, for comparison, based on RRC pulses with a roll-off factor β=0.3. The normalized energy spectra ${{{{\left| {S(f\,)} \right|}^{\,2}}}\! / {{{\left| {S(0)} \right|}^{\,2}}}}$ are represented in Fig. 2. Here $ S(f\,) $ means the spectrum of the single pulse and S(0) means the absolute value of the spectrum at zero frequency.

      Figure 2.  Comparison of the normalized energy spectra of optimal FTN signals.

      Fig. 2 demonstrates the energy spectra for FTN pulses obtained for the transmission rates R=1/T and R=2/T and the energy spectra for FTN signals constructed on RRC pulses[3]-[6] for the transmission rate R=1/T. It can be seen that, at R=1/T, the shapes of the energy spectra of optimal FTN signals and signals based on RRC pulses are very close, but for the double-speed, they differ significantly. This is due to the fact that with an increase in the transmission rate by a factor of 2, the duration of the main lobe sopt(t) decreases (see Fig. 1 (a)), which leads to the expansion of the spectrum.

    • The presence of ISI when transmitting FTN signals significantly worsens the reception conditions even in channels without fading. The use of coherent elementwise reception algorithms involves the determination of the phase of the carrier wave, clock frequency, and cyclic synchronization in packet message transmission modes. These tasks are solved by the methods of constructing the message preamble. However, even in these close to ideal reception conditions, we need to consider the influence of ISI. The most effective reception algorithms in these conditions are the maximum likelihood sequence estimation (MLSE) reception algorithms with the implementation of weighted enumeration of all possible combinations of the received signals. The hardware complexity of implementing such algorithms does not allow us to achieve high absolute transmission rates. Therefore, it is advisable to consider well-known approximations, for example, the Viterbi algorithm (VA)[19],[20].

      Let us consider the maximum likelihood reception method. The task of receiving according to this method of MLSE is to find a sequence of symbols $\left\{ {c_n^*} \right\}_{k = 0}^{N - 1}$ that would minimize the Euclidean distance between the corresponding sequence of optimal FTN signals y(t) in (1) and the received noisy signal x(t)=y(t)+n(t). It is necessary to determine the number $p \!=\! 1,{\rm{ 2, }} \cdots ,{\rm{ }}{{\rm{2}}^N}$ of the sequence $\left\{ {c_n^*} \right\}_{k = 0}^{N - 1}$, in the total number of variants of such sequences:

      where yp(t) is the pth sequence of optimal FTN signals in (1), providing the minimum Euclidean distance; ${\left\| \cdot \right\|_2}$ is the operator of the Euclidean norm (the Euclidean distance in the functional space of signals), and Y is the set of all possible sequences of optimal FTN signals in (1). The complexity of this reception algorithm depends on the sequence length N. For example, for the binary alphabet and the duration of the packet of signals equal to N=10, it is necessary to analyze 1024 possible sequences of optimal FTN signals with ISI and calculate (4) for each of these sequences.

      The search for such a sequence can be implemented using VA, the computational complexity of which depends on the depth of ISI (duration Ts), the size of a constellation, and the message transfer rate.

      Let us consider the application of VA for the example of receiving binary optimal FTN signals with the duration of Ts=2T for a transmission rate of R=1/T. Suppose that during the formation of a message package, a known initial group of symbols is added, which allows starting the process of message demodulation using VA with known initial states.

      At each clock interval T, the waveform is determined by a binary combination of 2 symbols (states). With each new clock interval, a combination change occurs. This process could be described conveniently by a trellis transition diagram, which is shown in Fig. 3. The vertical axis shows the states that determine the possible combinations of signals in the current time interval. For the duration Ts=2T there are 4 possible states (00, 01, 10, and 11). In the time interval T, the values of current and previous symbols influence the shape of the total signal. The movement along the lattice diagram horizontal axis corresponds to the transition along the edges from the state in the interval (k–1)T to the state in the clock interval kT. From each state, a transition to certain two states is possible.

      Figure 3.  Search by the trellis transition diagram with a known initial state.

      In the general case, when the duration of the signal sopt(t) is Ts=LT, the size of the binary state will be L, and at each step of the algorithm, M L Euclidean distances between the received signal will be calculated at a given time interval (the number of states in the array will be M L).

      The solid lines in Fig. 3 indicate the appearance of the symbol “–1”, and the dotted lines indicate the appearance of “1”; the black states are the states corresponding to the transmitted sequence of symbols $c_n^*$ in (4) equal to [1, 1, –1, 1, –1, –1, –1] with the initial state [00] (Fig. 3). In this example, the initial and final states are considered to be known. Their numbers depend on the duration Ts of the signal and the constellation size. During demodulation, a part of sopt(t) is received at each time interval. After that, the Euclidean distances between the received signal x(t) and the four possible forms of the expected reference signals y(t)=sopt(t)+sopt(tT) are calculated on the considered time interval T. When the signal length sopt(t) is equal to Ts=2T and the transmission rate R=1/T, the shape of the expected signal at a given time interval is affected by the given transmitted signal and one previous signal. Each iteration VA is used to calculate the Euclidean distance and add the obtained values to the distance values (metrics) of all paths calculated in the previous iterations on the trellis diagram (Fig. 3). After updating the metrics, a search for surviving paths is made. The metrics of all paths leading to the same state are compared, and the path with the smallest metric is selected. For a known initial and final state, the only sequence of transitions from the initial state [00] to the final state [00] is determined from the diagram in Fig. 3. This sequence is optimal, as (4), in the considered sense of estimating the sequence of the transmitted symbols $\left\{ {c_n^*} \right\}_{k = 0}^{N - 1}$.

      The condition of a known initial and final state is not necessary, since the sequence of transitions starting with erroneously selected values of the first state is most likely to be interrupted at some stage of the calculation. An example of a grid search with an unknown initial state is shown in Fig. 4.

      Figure 4.  Search by the trellis diagram of transitions with an unknown initial state.

      For an unknown initial state, the movement along the diagram in Fig. 4 occurs simultaneously from all initial states and at the zero step, the metrics of all states (4 metrics) are calculated. The paths are determined from all states (00, 01, 10, and 11) to all possible (8 transitions) in the next step. As a result, after the first iteration (step 0 to step 1 in Fig. 4), 2 transitions lead to each state. The metrics of these transitions are added to the metrics of the initial states. There are 2 paths to each state in step 1. The path that has a smaller metric is selected, and the other is discarded. Thus, there are four surviving paths remained. Then in the second iteration (step 1 to step 2), 8 transitions are calculated again. Their metrics are added to the previous metrics of the surviving paths (path metrics are updated). According to this algorithm, the movement along the diagram in Fig. 4 occurs.

      When moving along the trellis diagram and removing non-surviving paths, there is a high probability that at the kth step in the first column of the diagram all paths merge into one and only one edge comes out from only one node. In this case, a decision about the first group of the received symbols can be done and then free up the memory resources of the demodulator. Thus, when passing through the trellis with a fixed interval (depth of reverse lookup), we can regularly make decisions about the next group of symbols.

    • Let us evaluate the possibility of using time-limited optimal binary FTN signals to increase the BER performance of receiving messages using a simulation model. Fig. 5 shows the schematic representation of the operation of VA, discussed in the previous section.

      Figure 5.  Schematic representation of the VA operation.

      In accordance with the description of the work on the trellis transition diagram (Figs. 3 and 4), the received signal x(t) is demodulated at time intervals [kT, (k+1)T ]. The Euclidean distances between x(t) and the reference signals are determined. Then, the path metrics are updated and the surviving paths are searched. The depth of the backtrace in the simulation model may be equal to the duration of the entire message packet. In this case, the sequence of the estimates of the transmitted symbols will be generated immediately after processing all the received signals. The depth of the backtrace also may be less than the duration of the packet.

      For example, if the packet size is N=100 symbols, then the depth of the reverse lookup is 20 symbols. Then the evaluation of the first symbol will be made after processing 20 times of interavls. The evaluation of the second symbol will be made after processing the 21st interval, etc. With this demodulation mode, there will be a fixed delay in receiving a message equal to the depth of the reverse lookup.

      Fig. 6 shows a simulation model for transmitting and receiving optimal FTN signals, where S corresponds to an array of reference signals, Ld indicates the duration of the reference signal in the demodulator, and Tbacktrace indicates the depth of the reverse lookup.

      Figure 6.  Simulation model of message transmission using optimal FTN signals and VA demodulator.

      The simulation model is implemented in the MATLAB environment. It implements three procedures: The formation of a sequence of optimal FTN signals in (1) with increased duration and a given symbol rate and the transmission over the channel with AWGN and reception according to the Viterbi algorithm. The generation of optimal FTN signals is implemented using the formation of digital delta pulses and further linear convolution with the impulse response of the forming digital filter, equivalent to a sequence of signal samples. The duration of the transmitted useful signal is Ts=LT, and the duration of one clock interval is equal to n discrete samples. The Baud rate is controlled by the parameter ξ. There are ξn samples per time interval.

    • To estimate the losses associated with the operation of VA with known and unknown initial states of the trellis, we consider the results of the simulation for different lengths of the message packet. Fig. 7 shows the error probability values from the signal-to-noise ratio (Eb/N0, where Eb=Eopt) when using the VA reception of optimal FTN signals with the duration of Ts=8T (Fig. 1) and a message transmission rate R=2/T. The packet lengths are 10 bits (Fig. 7 (a)), 50 bits (Fig. 7 (b)), 100 bits (Fig. 7 (c)), and 1000 bits (Fig. 7 (d)), respectively.

      Figure 7.  BER vs. Eb/N0 for different packet lengths: (a) 10 bits, (b) 50 bits, (c) 100 bits, and (d) 1000 bits.

      As can be seen from a comparison of the curves in these figures, the following conclusions can be done. Firstly, with the short packet duration N=10 bits (Fig. 7 (a)) and the absence of information about the initial and final states of the message packet, the BER performance of reception is significantly reduced compared with the reception mode when these states are known.

      Energy losses are more than 7 dB with an error probability of р=10–2. With a packet length of 50 bits (Fig. 7 (b)), the energy loss is about 4 dB at р=2×10–3. An increase in the packet length to N=100 allows one to reduce energy losses to 3 dB at р=2×10–3 (Fig. 7 (c)). Secondly, starting from a packet length of 1000 bits (Fig. 7 (d)), the lack of information about the initial and final states of a message packet during demodulation using VA does not affect the BER performance of message reception.

      Consider the results of receiving optimal FTN signals of the duration Ts=8T (Fig. 1) with a reduced observation interval for tanalysis (see Fig. 1). In this case, during the demodulation procedure, only the central part closest to the main lobe of the signal sopt(t) will be considered. Such a statement of the demodulation problem is interesting from the point of view of reducing the computational cost to perform the search procedure by the trellis transition diagram (Figs. 3 and 4). Fig. 8 shows the dependence of the BER performance on Eb/N0 for R=2/T for various values of L with the duration of the transmitted optimal FTN signal Ts=8T. It is seen that a decrease in the observation interval of signals from Ld =8 to Ld =6 and Ld =4 leads to the appearance of energy losses in the range of error probabilities p=10–4 no more than 0.2 dB. With a further decrease in the observation interval to Ld =2, the energy losses increase significantly and reach 4 dB at p=10–2.

      Figure 8.  BER vs. Eb/N0 for various parameters Ld.

      Let us compare the BER performance of receiving optimal FTN signals and signals based on the RRC pulse. Fig. 9 shows the results of the simulations of transmission and reception of optimal FTN signals and signals based on RRC pulses of the duration of Ts=8T for message rates R=2/T and R=2.5/T with the packet duration of N=1000.

      Figure 9.  BER performance of optimal FTN signals and FTN RRC signals for different symbol rates.

      As can be seen from the comparison of the dependence in Fig. 9, for optimal FTN signals at a transmission rate of R=2/T, the error probability p=10–3 is achieved at Eb/N0=7.1 dB, and for signals based on RRC pulses that is achieved at Eb/N0=9.0 dB. With an increase in the transmission rate of messages to R=2.5/T, the probability of errors p=10–2 is achieved at Eb/N0=9.5 dB, and for signals based on RRC pulses the probability of errors p=8×10–2 is achieved at Eb/N0=10 dB. For comparison, Fig. 9 shows the potential BER curve for receiving binary signals with a rectangular envelope of the duration T and energy Es=Eb. As can be seen from a comparison of the curves in Fig. 9, the application of VA for receiving optimal FTN signals with the duration of 8T and a symbol rate 2 times higher than the Nyquist limit[2],[3],[15],[21] provides close to the potential performance of BER. Energy losses are less than 0.3 dB with an error probability of p=10–4.

    • The use of finite time optimal FTN signals in channels with AWGN makes it possible to obtain the high BER performance at message rates above the Nyquist barrier. It is shown that the application of VA for receiving optimal FTN signals with the duration of 8T when using a message symbol rate twice as high as the Nyquist limit provides the performance close to potential BER. Energy losses are not more than 0.3 dB with an error probability of p=10–4.

      The use of optimal FTN signals at a symbol rate greater than the Nyquist limit by a factor of 2 is achieved at lower energy costs than that when using signals based on RRC pulses for the same symbol rate. So for the probability of errors p=10–3, the energy gain is about 2 dB. With an increase in the transmission speed up to 2.5 times higher than the Nyquist limit, this gain reaches 4.5 dB with an error probability of p=8×10–2.

      Using VA to receive message packets consisting of optimal FTN signals is advisable with a packet length of at least 1000 bits when using the binary constellation. In this case, it is not required to use additional reception algorithms at the initial stage of demodulation or information about the initial and final states of the message packet.

    • The authors would like to express sincere appreciation to Peter the Great St. Petersburg Polytechnic University Supercomputing Center (http://www.scc.spbstu.ru) for the used computational resources in this paper.

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