This article is devoted to the study of a method for choosing the shape of orthogonal frequency division multiplexing signals in selective fading conditions. The result obtained is achieved by applying the concatenated coding method. The adequacy of the obtained results was verified by mathematical modeling of the orthogonal frequency division multiplexing channel with the proposed signals based on the components of the turbo codes. As a result of the study, it was found that the performance of the concatenated coding scheme in the orthogonal frequency division multiplexing channel depends on the number of decoding iterations, as well as the signal structure. The obtained results will be useful in choosing the coding method for orthogonal frequency division multiplexing communication channels.
At the current stage of the introduction of modern infocommunication technologies, the use of
channel coding (CC) plays a key role in improving the performance of channels in which Orthogonal
Frequency Division Multiplexing (OFDM) is used [
A review of relevant literature sources allows you to find out the following. OFDM modulation
method, the use of which allows you to effectively counteract the effects of interference and noise on
the quality of information transmission [
An important task of effective implementation of OFDM is to determine the change in the
probability of occurrence of bit errors in such systems due to distortions associated with the impact on
maintaining the orthogonality of individual subcarriers [
The above studies of well-known researchers are supplemented in the presented work. The
specificity of the presented studies is based on the fact that, despite the significant advantages of the
OFDM concept, a case is possible when deep fading accompanying the transmission of such signals
has a selective effect on some subcarriers by suppressing them, and on others, on the contrary,
amplifying them. This pattern ultimately leads to an increase in the average signal-to-noise ratio (SNR).
Selective fading effects occur in the OFDM channel. The article presents mechanisms to counteract
such effects in OFDM channels, which are based on the use of coding methods [
This section provides a description of the coding scheme. The features of the circuit implementation of encoders and decoders are given. Specific aspects when designing a decoder circuit are considered. 2.1.
In accordance with the formulated task, we analyzed systems with OFDM in order to evaluate CC and improve the noise immunity of systems with digital modulation. In parallel, the work compares the efficiency of signal-code construction (SCC) based on turbo codes (TC) as part of a concatenated coding (CnC) in the context of their application in OFDM. Separately, the efficiency of an OFDM system in the case of a multipath channel with fading was studied. The studies were carried out in channels using digital phase shift keying (PSK) of various orders.
We carry out the description, in Fig. 1 shows a block diagram of a turbo encoder (TE) containing
two convolutional recursive encoders (CRE) of length l=5 implemented on feedback shift registers.
This design has a feature compared to the classical one for convolutional codes (CoC) [
Puncturing was used at a coding rate of 1/2 (1/3). For frame lengths of 1632 to 16384 bits, different
interleaver were taken, including pseudo-random and regular, operating according to the permutation
rule with records of results in a look-up table. The diagram is shown in Fig. 1, synchronization elements
are provided. In particular, the synchronization marker serves to control the state of the input buffer
(control of the contents of the buffer); with respect to the original buffer (multiplexer), it fixes the frame
synchronization pointer; in component encoders, fixes the moment of expiration of the code block. It
should be noted that the encoding process is necessarily accompanied by signal time delays (timings)
[
Next, a construction of a CC based on the component TC and the Reed-Solomon (RS) code is proposed. Therefore, when synthesizing the structure of the TC encoder, we will place the following accents. The implementation of the TE has its own peculiarities. In particular, the information block must be buffered and read in a certain order during the encoding process. Such buffering is absent in the CoC, and the buffer size is comparable to the interleaver size for the RS code. Thus, the TE generally replaces the concatenated design based on the CoC and the RS code, and is not an alternative to the convolutional encoder (CoE) in the above concatenated design. However, a combination of such codes will be proposed to improve the efficiency of concatenated structures.
The design of the decoder is formed by two decoders (DC), organized according to the principle of "Soft in - Soft out" (SISO), one for each stream formed by a single CRE block. 2.2.
They process the systematic part of the code (denoted as Xk in Fig. 2) and the redundancy bits associated with the Yk1 (and/or Yk2) streams and external information coming from another encoder. The implementation of the decoder determines the process of its functioning, which can be parallel or sequential, but the recursiveness is preserved.
On Fig. 3 practical scheme of the TC decoder (TD). There are a number of algorithms for the
functioning of TC decoders. Minimum decoder scheme error probability (SISO) provides a maximum
a posteriori (MAP) decoding algorithm [
TE uses an iterative decoding algorithm. Each component decoder provides a likelihood estimate,
with the received uncoded information symbols available to both decoders to perform these estimates.
Each decoder sends its likelihood estimates to the other decoder and uses the corresponding estimates
from the other decoder to determine new likelihood estimates by extracting redundant information
contained in the other decoder's estimates based on parity symbols only available to it. In general, the
following decoding algorithms can be used in TE: MAP, logarithmic MAP (max-log-MAP, APP), Soft
Viterbi (SOVA) decoding algorithm [
Mathematically, for a systematic code, the soft decoding process Λ( ̂ ) can be represented as follows: Λ( ̂ ) = Λ ( ) + Λ( ) + Λ ( ̂ ) (1) where Λ( ) is the a priori logarithmic ratio of the likelihood function (LLF); Λ ( ) is the metric obtained during channel estimation; Λ ( ̂ ) is the metric is obtained on the basis of external information from the decoding process and is generally independent of the decoder input Хk.
Thus, the metrics Λ ( ), Λ ( ̂ ), and Λ( ̂ ) are used by the second decoder as an observation of in order to organize the iterative process.
On Fig. 4 shows a scheme for investigating a transmission system with OFDM and TC-based channel coding.
As follows from the diagram, the input frames of the TC are grouped, each block of packets forms
an OFDM frame, which begins with a synchronization symbol, followed by a reference symbol [
Synchronization of TC blocks is achieved through the use of a synchronization marker (Fig. 5). In this case, the length of the TC block and the synchronization marker is inversely proportional to the nominal coding rate R.
Therefore, the synchronization efficiency depends on the encoding rate. On the receiving side of the
telecommunication channel, the synchronization marker is identified by the features of its structure. It
should be noted that each of the decoders of the composite decoder (see Fig. 3) cannot always correctly
identify the effective size of the synchronization marker, which is very important. Indeed, when
decoding a long TC, the decoders of the constituent decoder must clearly identify the boundaries of the
code block, including taking into account the delimited and deinterleaved data in the decoder structure.
Therefore, the synchronization marker usually includes a channel symbol (domain) that defines the
framing method.
min1( ( ) − )2] ≡ | ( )|2 ,
and ,1 = −| [ ]| is the parameter for using QPSK; 16QAM (64QAM):
where H(i) is the channel response;
0 and
1 subsets described on the Gray code mapping [
2( [ ] + 1), [ ] < 2 ,2 = −| [ ]| + 2
Let us consider the issue of choosing an interleaver in the channel turbo coding scheme, having
analyzed the results shown in Fig. 6 dependencies of the initial positions (Output) of bits in the coded
blocks on the input (Input) for different types of interleavers [
In general, both periodic and pseudo-random data interleaving schemes can be implemented. Periodic interleavers are divided into block and convolutional interleavers. The advantage of convolutional interleavers is that the latency and storage capacity is half that of block interleavers. For TC, the choice of the interleaver is quite important, since its structure has a direct impact on the minimum distance and the number of low weights codewords, and these, in turn, according to formula (5), have a direct impact on the efficiency of the TC. In the case of a pseudo-random interleaver, the bits of the initial block in the interleaved block are located pseudo-randomly, and this ultimately has a rather positive effect on the overall efficiency of the TC.
Bit error probability for TC takes the form:
≤ ∑ = (5) where N is the length of the code block; l is the block size that can be interleaved; ci is the total information weight of all code words of weight i; pi is the probability of choosing an incorrect code word that differs from the correct one in i positions.
On Fig. 7 shows a scheme for studying a transmission system with OFDM and channel coding based
on a CnC (an external encoder based on a Reed-Solomon (RS) code, an internal encoder based on a
TC) [
To implement OFDM in the scheme of Fig. 7, we have carried out the formation of subsystems of
blocks of the OFDM modulator and demodulator. The blocks convert a sequence of complex symbols
into a multicarrier complex envelope OFDM waveform (see Fig. 8). In the diagram in Fig. 8, the bit
sequence enters the bit mapper (MAP) and then to the subcarrier signal generator (Subcarrier MAP)
[
On Fig. 9 we present part of the signal processing subsystem with OFDM. The scheme includes the SC DMAP block - a block that provides the allocation of data subcarriers from an OFDM symbol.
The performance of the system was evaluated by conducting a series of numerical simulation
experiments in various configurations of coding/decoding schemes in the AWGN channel. For research,
constructions of 100 frames in size were used, which contained 20 packets each. Programmatically, the
model had settings according to the known structure of information transmitted in the channel, in
addition, in the case of studying the effectiveness of coding, the synchronization mechanism was
organized as ideal. The research process included steps to determine the impact of the TC structure on
performance and evaluated the impact of the number of decoding iterations, encoding rate and SCC
shape. In the experiment, the characteristics of the TC as a composite encoder in the OFDM channel
were studied. The idea of using the concatenated design was to achieve the channel performance at the
level of the bit error probability BER=10E-9. It was proved in [
Thus, in the scheme, the internal decoder reduces the probability of obtaining a BER to the level of 10E-4÷10E-5 (Fig. 7), in order to obtain a concatenated code (RS +TC) of performance 10E-9 at the output. On Fig. 10 shows a typical distribution of errors at the output of the TC in terms of the probability of a bit error at the level of 10E-5, depending on the number of iteration cycles during decoding.
In the diagram in Fig. 4, it is proposed to use an outer interleaver in order to ensure the correct
distribution of errors formed by the grouped areas of damaged data that can enter the channel from the
TC constituent convolutional encoder (Fig. 2). The spectrum of the OFDM signal [
Fig. 12 a shows graphs of noise immunity (versus carrier-to-noise ratio (C/N)) of a telecommunication channel with OFDM and CnC. During the experiment, 3 iterations, coding rate 1/3, codes (7, 5), (15, 17), (37, 21), (117, 121) and QAM modulation were used. The best indicators have a code like (117, 121), the energy gain of coding (EGC) was almost 2 dB.
On Fig. 12 b shows the resulting graphs of the noise immunity of a telecommunication channel with OFDM signals and CnC (RS + TC) for the configuration of codes (37, 21) and (117, 121) 5 iterations and a coding rate of 1/3 for various modulation schemes. We used multiposition signals QPSK and 16QAM.
The simulation result indicates performance losses in the case of increasing modulation levels, and for the case of the same code structure (37, 21) when using QPSK and 16QAM, the performance drop was about 3.9 dB, while the transition to the code structure (117, 121) for a similar type of signals resulted in a loss of performance of 3.1 dB.
a b Figure 12: Graphs of noise immunity of OFDM signals with CnC (RS+TC): 1 is the (7, 5), 2 is the (15, 17), 3 is the (37, 21); 4 is the (117,121); QPSK - a and b is the graphs of noise immunity of OFDM signals with CnC (RS+TC): 1 is the QPSK (37, 21); 2 is the QPSK (117, 121); 3 is the 16QAM (37, 21); 4 is the 16QAM (117, 127)
On Fig. 13 shows the results of studying the performance of a channel with OFDM signals in the case of a change in the number of iterations when decoding an external TC decoder.
During the study, the number of iterations was changed from 1 to 5 (coding rate 1/3). As a result of the study, EGC was set when the number of iterations changed. So, in the case of an increase in the number of iterations from 1 to 3, the channel performance increased by 2 dB, and an increase in the number of iterations to 5 led to an increase in performance by 0.25 dB.
The results of the study (Fig. 13) suggest that the performance of the CnC scheme for an OFDM channel depends on the number of decoding iterations, as well as the SCC structure. Increasing the length of the code sequence improves the performance of the circuit, but has a direct impact on the complexity of decoding. Increasing the number of iterations increases the decoding performance, but affects the speed and, accordingly, the computational complexity of decoding. However, optimal in terms of speed and computational complexity, the selection of a circuit and a TC decoding algorithm makes it possible to find a compromise on the design of concatenated circuits in OFDM channels and ultimately obtain a satisfactory performance gain.
A technique for improving the noise immunity of OFDM systems with efficient coding is described. Recommendations on the use of FEC coding methods are given. The scientific problem of increasing the noise immunity of signal reception in an OFDM channel under conditions of selective fading and multipath propagation is solved by applying ECC methods. A coding scheme with FEC in an OFDM channel based on CnC SCC is proposed.
Based on the results of the study, it was determined that with an increase in the length of the code sequence and the number of decoding iterations, the productivity of the proposed SCC increases to almost 2 dB. However, performance decreases by 3.9 dB when changing the waveform and moving to higher order modulation in SCC. When deploying networks based on OFDM, it is necessary to optimize the selection of SCCs in terms of coding rate and computational complexity, which will ultimately make it possible to find a compromise in performance indicators.
Future research will focus on the design of SCC for OFDM based on the use of polar codes and LDPC codes and their implementation through Field Programmable Gate Array (FPGA) for 5G (6G) information systems.