A low-complexity LDPC-hadamard code encoding and decoding method under OTFS modulation

Abstract

The embodiment of the present disclosure relates to a low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation, which comprises the following steps: the sending end adds check information to an information sequence through cyclic redundancy check, encodes the information sequence through LDPC-Hadamard code, and performs inverse symplectic finite Fourier transform and Heisenberg transform on complex value symbols after constellation mapping to output time domain signals; the time domain signals generate sending signals after time delay and Doppler shift through a multipath time-varying channel; the receiving end performs Wigner transform and symplectic Fourier transform on the received sending signals to output a symbol matrix; the received bit stream is processed through LDPC-Hadamard decoding, check information is removed through cyclic redundancy check, and the original bit stream is output. Through LDPC-Hadamard encoding and decoding and OTFS modulation and demodulation, the double-selectivity fading channel in a high-speed mobile scene can be better coped with; through the combination of LDPC-Hadamard code and OTFS, the error correction performance of the system under low signal-to-noise ratio is improved, the reliability of the system in a high-speed mobile environment is enhanced, and the error code performance of the OTFS system under low signal-to-noise ratio is improved.

Description

TECHNICAL FIELD

[0001] Embodiments of the present application relate to the field of communication technology, and particularly relate to a low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation. BACKGROUND

[0002] With the rapid development of emerging application technologies such as mobile Internet, Internet of Things, and 5G / 6G, modern communication systems must face changing channel conditions and complex interference environments. In order to cope with these challenges, there is an increasing demand for communication systems in high-noise and high-dynamic environments. Information theory shows that channel coding can overcome the interference of channel noise and improve communication capability. With the development of channel coding, according to the difference of decoding algorithm, channel coding can be divided into two categories: hard decision decoding and soft decision decoding. Channel coding using hard decision decoding mainly includes Hamming code, RM code, BCH code, and RS code, etc. The above-mentioned codes appeared early, and although they have the advantages of fixed algebraic structure, simple encoding method, easy hardware implementation, and low decoding complexity, the coding gain is low and cannot meet the requirements of modern communication systems. Channel coding using soft decision decoding mainly includes Turbo code and Polar code, etc. These codes are based on soft decision decoding algorithm of belief propagation. Because of the use of a variety of mathematical analysis tools, these codes have good mathematical structure and can approach or even reach the Shannon capacity limit. However, the above-mentioned codes face the problem of high encoding and decoding complexity, which is not conducive to hardware implementation.

[0003] Prior art solutions:

[0004] 1. OTFS combined with LDPC code: The original data is first encoded by an LDPC encoder. The LDPC code encodes the data and increases redundancy in order to correct errors during transmission. The encoded bit stream is converted into symbols by a mapper (such as a QAM modulator). Each symbol represents a specific complex number value, and these symbols will be placed into a time-frequency grid. The mapped symbols are arranged on the time-frequency grid and mapped to the delay-Doppler domain through SFFT transformation. The delay-Doppler domain can embed the symbols into the time-varying characteristics of the channel, ensuring the stability of the signal in high-speed scenarios. The modulated signal is transmitted through the wireless channel. At the receiving end, the signal is transformed back to the time-frequency domain through inverse transformation and channel equalization. Finally, the equalized symbols are corrected for errors that occur during transmission using the belief propagation algorithm through the LDPC decoder.

[0005] 2. OTFS combined with Polar code: The original data is first passed through a Polar encoder, which encodes the data and adds redundancy to correct errors during transmission. The encoded bit stream is converted into symbols by a mapper (e.g., QAM modulator). Each symbol represents a specific complex number value, and these symbols will be placed into a time-frequency grid. The mapped symbols are arranged on the time-frequency grid and mapped to the delay-Doppler domain through an SFFT transform. The delay-Doppler domain is able to embed the symbols into the time-varying characteristics of the channel, ensuring the stability of the signal in high-speed scenarios. The modulated signal is transmitted through the wireless channel. At the receiving end, the signal is transformed back to the time-frequency domain through an inverse transform and channel equalization. Finally, the equalized symbols are passed through a Polar decoder, which uses the belief propagation algorithm to correct errors that occurred during transmission.

[0006] 3. OTFS combined with Turbo code: The original data is first passed through a Turbo encoder, which encodes the data and adds redundancy to correct errors during transmission. The encoded bit stream is converted into symbols by a mapper (e.g., QAM modulator). Each symbol represents a specific complex number value, and these symbols will be placed into a time-frequency grid. The mapped symbols are arranged on the time-frequency grid and mapped to the delay-Doppler domain through an SFFT transform. The delay-Doppler domain is able to embed the symbols into the time-varying characteristics of the channel, ensuring the stability of the signal in high-speed scenarios. The modulated signal is transmitted through the wireless channel. At the receiving end, the signal is transformed back to the time-frequency domain through an inverse transform and channel equalization. Finally, the equalized symbols are passed through a Turbo decoder, which uses the belief propagation algorithm to correct errors that occurred during transmission.

[0007] Drawbacks of existing technologies:

[0008] 1. OTFS combined with LDPC code: LDPC code can provide good error correction performance in noisy environments, but its error performance decreases in complex Doppler effect scenarios. The combination of Hadamard code can further reduce the bit error rate, and Hadamard transform can evenly distribute signal energy in multiple dimensions, reducing the impact of single-point interference on the overall signal.

[0009] 2. OTFS combined with Polar code: The core advantage of Polar code is its efficient transmission in ideal channels, but in high-speed motion scenarios such as vehicle communication or unmanned aerial vehicle communication, Polar code does not have sufficient error correction capability. The LDPC-Hadamard scheme can improve the stability of the signal in high dynamic environments through Hadamard transform.

[0010] 3. OTFS combined with Turbo code: Turbo code performs well at short code length, but at long code length, the performance is inferior to LDPC code, and cannot approach the Shannon limit sufficiently. LDPC code has good error correction performance at long code length, and Hadamard code can further improve the reliability and bit error rate performance of the overall system, forming better long-distance communication capability.

[0011] Therefore, it is necessary to improve one or more problems existing in the above-mentioned related technical solutions.

[0012] It should be noted that this section aims to provide background or context for the technical solutions of the present application stated in the claims. The description herein is not admitted to be prior art merely because it is included in this section. SUMMARY

[0013] The purpose of the present application is to provide a low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation, and to at least partially solve one or more problems caused by the limitations and defects of the related art.

[0014] The present application provides a low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation, comprising the following steps:

[0015] The sending end adds check information to the information sequence after cyclic redundancy check, and then performs LDPC-Hadamard code encoding to form encoded data;

[0016] Conducting constellation mapping on the encoded data, and conducting OTFS modulation on the complex-valued symbols after constellation mapping, wherein the OTFS modulation comprises inverse-sine finite Fourier transform and Heisenberg transform on the complex-valued symbols to output time-domain signals,

[0017] The time-domain signals are subjected to time delay and Doppler shift through a multipath time-varying channel to generate a transmission signal;

[0018] The receiving end conducts OTFS demodulation on the received transmission signal, wherein the OTFS demodulation comprises Wigner transform and sine Fourier transform on the transmission signal to output a symbol matrix,

[0019] Conducting constellation inverse mapping on the symbol matrix to recover the received bit stream;

[0020] Conducting LDPC-Hadamard decoding processing on the received bit stream, removing the check information through cyclic redundancy check on the decoded bit stream, and outputting the original bit stream.

[0021] In an example embodiment of the present disclosure, the LDPC-Hadamard code is a concatenated code combining LDPC code and Hadamard code, and includes:

[0022] The LDPC code is constructed by using a low-complexity quasi-cyclic method and encoding is performed by using a shift register.

[0023] The output of the LDPC code is subjected to Hadamard encoding.

[0024] In an example embodiment of the present disclosure, the LDPC code is constructed by using a low-complexity quasi-cyclic method and encoding is performed by using a shift register, and includes:

[0025] The check matrix is composed of a plurality of permutation matrices or shift matrices, and each sub-matrix in the check matrix is a cyclic shift matrix, wherein the cyclic shift matrix cyclically shifts elements of a vector.

[0026] The check matrix is converted into a systematic form by using a Gaussian elimination method based on the quasi-cyclic structure of the check matrix, and a generator matrix is obtained.

[0027] The generator matrix and an information bit vector are subjected to matrix multiplication for encoding.

[0028] In an example embodiment of the present disclosure, the check matrix and the cyclic shift matrix are represented by the following expressions, respectively:

[0029]

[0030] P i = I i

[0031] wherein H is a check matrix, P i,j is a cyclic shift matrix or a zero matrix with a size of z×z, referred to as a permutation matrix or a unit shift matrix, and the entire matrix is composed of m b ×n b sub-matrices, wherein each sub-matrix is a cyclic shift matrix or a zero matrix with a size of z×z, and I i represents that a column of a unit matrix is cyclically shifted to the right by i positions.

[0032] In an example embodiment of the present disclosure, the systematic form is represented by the following expression:

[0033] H = [P | I]

[0034] wherein P is a sub-matrix and I is a unit matrix.

[0035] In an example embodiment of the present disclosure, the expression for encoding the generator matrix and the information bit vector by using matrix multiplication is as follows:

[0036] c = uG

[0037] where c is the code, u is the information bit vector, and G is the generator matrix.

[0038] In an example embodiment of the present disclosure, the OTFS modulation comprises performing inverse symplectic Fourier transform and Heisenberg transform on the complex-valued symbols to output time-domain signals, including:

[0039] performing inverse discrete symplectic Fourier transform on the complex-valued symbols to convert them from Delay-Doppler domain to time-frequency domain;

[0040] converting the time-frequency domain signals to time-domain signals by Heisenberg transform.

[0041] In an example embodiment of the present disclosure, the OTFS demodulation comprises performing Wigner transform and symplectic Fourier transform on the received signals to output a symbol matrix, including:

[0042] performing Wigner transform on the received signals to recover the representation of the signals in Delay-Doppler domain;

[0043] re-mapping the signals from time-frequency domain to Delay-Doppler domain by symplectic Fourier transform to output a symbol matrix.

[0044] In an example embodiment of the present disclosure, the constellation mapping employs QAM or PSK.

[0045] In an example embodiment of the present disclosure, the LDPC-Hadamard decoding comprises:

[0046] performing iterative decoding on the received signals using LDPC code by BP algorithm;

[0047] after completing the LDPC decoding, the output soft information is passed to the Hadamard decoding part, and fast Hadamard transform is used to decode the Hadamard code.

[0048] The OTFS modulation low-complexity LDPC-Hadamard code encoding and decoding method provided by the present disclosure can have the following beneficial effects: on the one hand, through OTFS preprocessing and OTFS post-processing, the double-selectivity fading channel in the high-speed mobile scenario can be better coped with; on the other hand, through the combination of LDPC-Hadamard code and OTFS, the error correction performance of the system under low signal-to-noise ratio is improved, more information can be transmitted under the same bandwidth condition, the spectrum efficiency and transmission rate are improved, and the reliability of the system in the high-speed mobile environment is enhanced; on the other hand, through LDPC-Hadamard encoding, the low-density characteristics of the LDPC code and the strong error correction performance of the Hadamard code are combined, and the error performance of the OTFS system under low signal-to-noise ratio is further improved. BRIEF DESCRIPTION OF DRAWINGS

[0049] The drawings incorporated into the specification and forming a part of the specification, illustrate embodiments consistent with the present disclosure and, together with the specification, serve to explain the principles of the present disclosure. It is apparent that the drawings described below are only some embodiments of the present disclosure, and other drawings can be obtained according to these drawings without creative labor for those skilled in the art.

[0050] Figure 1 A step schematic diagram of the OTFS modulation low-complexity LDPC-Hadamard code encoding and decoding method in the exemplary embodiment of the present application is shown.

[0051] Figure 2 A basic OTFS architecture schematic diagram in the exemplary embodiment of the present application is shown.

[0052] Figure 3 An LDPC-Hadamard encoding structure schematic diagram in the exemplary embodiment of the present application is shown.

[0053] Figure 4 An LDPC-Hadamard decoding structure schematic diagram in the exemplary embodiment of the present application is shown.

[0054] Figure 5 A fast Hadamard transform schematic diagram in the exemplary embodiment of the present application is shown.

[0055] Figure 6 An error performance comparison schematic diagram of the LDPC-Hadamard code and the LDPC code in the OTFS system in the exemplary simulation experiment of the present application is shown. DETAILED DESCRIPTION

[0056] Example implementations will now be described more fully with reference to the accompanying drawings. Example implementations can be implemented in any number of ways, and example implementations should not be construed as limited to only those described herein; rather, embodiments should be construed more broadly. It will be appreciated that structural and / or logical subdivisions of any component described herein can be combined or sub-divided in any manner.

[0057] In addition, the accompanying drawings are included to provide a further understanding of the present disclosure and are incorporated in and constitute a part of this specification. The drawings illustrate embodiments and, together with the description, serve to explain principles and operations. In the drawings:

[0058] The present example implementation provides a low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation, as shown in Figures 1-2 may include the following steps:

[0059] S101: The sending end appends check information to the information sequence after cyclic redundancy check, and then performs LDPC-Hadamard code encoding to form encoded data.

[0060] S102: The encoded data is subjected to constellation mapping, and the complex-valued symbol after constellation mapping is subjected to OTFS modulation, wherein the OTFS modulation includes inverse-sine finite Fourier transform and Heisenberg transform of the complex-valued symbol to output a time-domain signal.

[0061] S103: The time-domain signal is subjected to time delay and Doppler shift by a multipath time-varying channel to generate a sending signal.

[0062] S104: The receiving end performs OTFS demodulation on the received sending signal, wherein the OTFS demodulation includes Wigner transform and sine Fourier transform of the sending signal to output a symbol matrix.

[0063] S105: The symbol matrix is subjected to constellation inverse mapping to restore a received bit stream.

[0064] S106: The received bit stream is subjected to LDPC-Hadamard decoding processing, and the decoded bit stream is subjected to cyclic redundancy check to remove check information and output an original bit stream.

[0065] The OTFS modulation low-complexity LDPC-Hadamard code encoding and decoding method provided by the disclosure can have the following beneficial effects: on the one hand, through OTFS preprocessing and OTFS postprocessing, the double-selectivity fading channel in a high-speed mobile scenario can be better coped with; on the other hand, through the combination of the LDPC-Hadamard code and the OTFS, the error correction performance of the system under a low signal-to-noise ratio is improved, more information can be transmitted under the same bandwidth condition, the spectrum efficiency and the transmission rate are improved, and the reliability of the system in a high-speed mobile environment is enhanced; on the other hand, through the LDPC-Hadamard encoding, the low-density characteristics of the LDPC code and the strong error correction performance of the Hadamard code are combined, and the error performance of the OTFS system under a low signal-to-noise ratio is further improved.

[0066] Next, the various steps of the above method in the present example embodiment will be described in more detail.

[0067] Step S101: After the transmitting end appends the check information to the information sequence through the cyclic redundancy check, the LDPC-Hadamard code encoding is performed to form the encoded data.

[0068] Specifically, the input bit stream u is first processed through the cyclic redundancy check (CRC) to generate a bit stream c' with check code.

[0069] The LDPC-Hadamard code is encoded in the channel encoding link.

[0070] First, the LDPC encoding is performed. In the traditional Tanner graph of the LDPC code, the information bits are allocated to the variable nodes, and the variable nodes are connected to the check nodes to form the structure of the graph. The check node represents the parity check constraint, and each check node is connected to multiple variable nodes to ensure that the weighted sum of these variable nodes satisfies a certain parity check rule.

[0071] Secondly, the Hadamard encoding is performed. The LDPC-Hadamard code introduces the Hadamard constraint on the basis of the standard LDPC code, and its bipartite graph is similar to that of the standard LDPC code, but the difference is that the check node becomes a super check node, and the super check node is connected to multiple variable nodes, but their check relationship is not a simple parity check, but is based on the Hadamard code. These nodes are connected through the introduction of the Hadamard transform, so that the encoding process is more complex and has stronger error correction capability. The output of the Hadamard encoding is x, and its formula is represented as:

[0072] x = H·c' (1)

[0073] H is the Hadamard transform matrix.

[0074] S102: performing constellation mapping on the coded data, and performing OTFS modulation on the complex-valued symbol after the constellation mapping, wherein the OTFS modulation comprises performing inverse symplectic finite Fourier transform and Heisenberg transform on the complex-valued symbol to output a time-domain signal.

[0075] Specifically, the data x after Hadamard coding is subjected to constellation mapping, and is converted into a complex-valued symbol for transmission. A constellation mapping scheme such as QAM (Quadrature Amplitude Modulation) or PSK (Phase Shift Keying) is adopted. Using QAM, the mapped symbol is x[k, l], and k and l represent the Doppler index and delay index in the Delay-Doppler domain, respectively. The expression is:

[0076] x[k, l] = Modulate(x) (2)

[0077] The symbol x[k, l] is subjected to inverse discrete symplectic Fourier transform (ISFFT) to convert it from the Delay-Doppler domain to the time-frequency domain. The expression of the symbol after ISFT processing is:

[0078] x[n, m] = ISFFT(x[k, l]) (3)

[0079] where n and m are the indices of the time-frequency domain. Then, the time-frequency domain signal is converted into a time-domain signal x(t) by Heisenberg transform, which is used to prepare the transmission signal:

[0080]

[0081] where g tx (t) is a transmission window function, T is the symbol duration, and Δf is the subcarrier spacing.

[0082] S103: generating a transmission signal by subjecting the time-domain signal to time delay and Doppler shift through a multipath time-varying channel.

[0083] Specifically, after Heisenberg transform, the signal x(t) is transmitted through a multipath time-varying channel h(τ, v). The channel causes time delay and Doppler shift to the signal, and the received signal r(t) can be expressed by convolution as:

[0084] r(t) = ∫∫h(τ, v)x(t-τ)e j2πν(t-τ) dτdν+w(t) (5)

[0085] where w(t) is noise in the channel, and h(τ, v) is the channel response.

[0086] S104: The receiving end performs OTFS demodulation on the received transmission signal, wherein the OTFS demodulation includes performing Wigner transform and symplectic Fourier transform output symbol matrix on the transmission signal.

[0087] Specifically, the receiving end first performs Wigner transform on the received signal r(t) to restore the representation of the signal in the Delay-Doppler domain. Then, through SFFT (Symplectic Fourier Transform), the signal is remapped from the time-frequency domain to the Delay-Doppler domain, and the symbol matrix is output:

[0088] y[n,m] = SFFT(r(t)) (6)

[0089] S105: The symbol matrix is recovered to the received bit stream through constellation inverse mapping.

[0090] Specifically, the received symbol matrix y[n,m] is recovered to the received bit stream y[k,l] through constellation inverse mapping:

[0091] y[k,l] = Demodulate(y[n,m]) (7)

[0092] S106: The received bit stream is processed by LDPC-Hadamard decoding, and the decoded bit stream is removed by cyclic redundancy check to output the original bit stream.

[0093] Specifically, error correction is performed through LDPC-Hadamard decoding. First, the received bit stream y[k,l] is decoded by Hadamard, and its expression is:

[0094] c" = H -1 ·y[k,l] (8)

[0095] Then, the final bit stream c' is recovered by LDPC decoding to correct errors:

[0096] c' = LDPC-Decode(c") (9)

[0097] The decoding process of LDPC-Hadamard code first performs preliminary error correction through iterative message passing decoding of LDPC code. In this stage, the belief information is exchanged between the variable nodes and the check nodes, and the bit errors are gradually repaired. Then, in the Hadamard check node, the received bit sequence is processed by Hadamard transform, and the most likely codeword is selected by maximum likelihood decoding. The strong error correction ability of Hadamard helps to solve the decoding in low signal-to-noise ratio environment.

[0098] The decoded bit stream c' is checked by CRC to remove redundant bits and check errors, and the original bit stream is recovered

[0099]

[0100] The entire transmission process is completed.

[0101] In the embodiments of the present disclosure, the LDPC-Hadamard encoding structure, for a standard LDPC code, can be represented by a Tanner graph, as shown in the left part of the figure. Figure 3 The variable node with degree j is a (j, 1) repetition code, where the edges connecting the left and right nodes need to satisfy the constraints of the repetition code at the variable node and the constraints of the SPC at the check node at the same time. It is defined by replacing the SPC code in the LDPC code with any other block code. Considering the use of Hadamard code in the right check node, i.e., LDPC-Hadamard, as shown in the right part of the figure, the LDPC-Hadamard code has good performance under very noisy channel conditions. Figure 3 The present application only considers the case where the check node degrees of all Hadamard codes are the same.

[0102] First, the construction of QC-LDPC codes, a low-complexity quasi-cyclic construction method is used to construct LDPC codes, i.e., quasi-cyclic LDPC codes (QC-LDPC for short). QC-LDPC codes are a class of structured LDPC codes, and their check matrices have quasi-cyclic structures, which can greatly reduce the complexity in hardware implementation, especially in storage and decoding. The construction process generally combines the quasi-cyclic properties of the generator matrix or the check matrix, and is generated by cyclic shift. The construction process of QC-LDPC codes is as follows:

[0103] 1) Construction of check matrix

[0104] The check matrix H of the QC-LDPC code is usually composed of several permutation matrices or shift matrices, where each submatrix is a cyclic shift matrix. Therefore, the check matrix of the QC-LDPC code can be expressed as the splicing form of multiple block matrices:

[0105]

[0106] where P i,j is a cyclic shift matrix or a zero matrix with a size of z×z, called a permutation matrix or a unit shift matrix. Therefore, the entire matrix is composed of m b ×n b submatrices, where each submatrix is a z×z cyclic shift matrix or a zero matrix.

[0107] 2) Circulant shift matrix

[0108] Circulant shift matrix P i is a special matrix that cyclically shifts the elements of a vector. Assuming that I

[0109] P i = I i (12)

[0110] where I i denotes a right cyclic shift of the columns of the identity matrix by i positions. For example, a simple 4x4 circulant shift matrix P1 can be represented as:

[0111]

[0112] 3) Generator matrix of QC-LDPC codes

[0113] The generator matrix G of a QC-LDPC code can be constructed by exploiting the quasi-cyclic property of the parity check matrix H. The construction method of the generator matrix G is usually based on the following steps: first, the parity check matrix H is converted into a systematic form, i.e., H = [P | I], where P is a submatrix and I is an identity matrix. Then, by exploiting the quasi-cyclic structure of the parity check matrix, the parity check matrix can be converted into a systematic form by Gaussian elimination, thereby obtaining the generator matrix G.

[0114] 4) Encoding of QC-LDPC codes

[0115] The encoding process of QC-LDPC codes is relatively simple. Due to its quasi-cyclic structure, encoding can be performed using shift registers, thereby greatly reducing hardware complexity. Specifically, the encoding of QC-LDPC codes can be achieved through matrix multiplication. Given the generator matrix G and the information bit vector u, the code word c can be generated by matrix multiplication:

[0116] c = uG (14)

[0117] Since the generator matrix G has a quasi-cyclic structure, the encoding result can be quickly calculated through shift operations.

[0118] Secondly, Hadamard encoding is performed, and a Hadamard matrix is a square matrix defined over a binary alphabet. An n-order (n = 2 r ) Hadamard matrix over {1, -1} can be constructed as follows:

[0119]

[0120] Let the columns of the Hadamard matrix ±H n be denoted as {±h j:j=0,1,...,2 r -1}. It can be proven that ±H n The columns of form a linear space with dimension r+1. Therefore, ±H n Replacing {+1,-1} in each column with {0,1} will yield a parameter of (2 r The linear block code of (r+1), also known as the Hadamard code.

[0121] The generator matrix G of the systematic form of the Hadamard code H,r+1 It can be obtained through recursion:

[0122]

[0123] Among them, J r Is with G H,r Matrices of the same order have the following form:

[0124]

[0125] The recursion termination condition is:

[0126]

[0127] The codeword c of the systematic form of Hadamard code where r is even has the following properties:

[0128]

[0129] That is, the bits at corresponding positions in the codeword satisfy the parity check relationship. This property plays an important role in LDPC-Hadamard.

[0130] The Hadamard code encoding process maps information bits to rows of a Hadamard matrix, generating codewords through this mapping. Assuming there are k information bits, the Hadamard code encoding process maps these bits to rows of a Hadamard matrix. For m bits of information, it can encode a codeword of length n=2. m The code words.

[0131] Let the input information vector be u = [u0, u1, ..., u m-1 There are m information bits in total. According to the encoding rules of Hadamard codes, each information vector corresponds to a row in the Hadamard matrix. The information bit u is mapped to a row of the Hadamard matrix. Define the Hadamard matrix H. n The i-th line is h i Then the encoded codeword c is a row in the matrix, that is:

[0132] c = h u (20)

[0133] where h u denotes a certain row in the Hadamard matrix, the position of which is determined by the binary value of the information bit u. We will illustrate the above encoding process by an example.

[0134] Let m = 2, then n = 2 2 = 4, the corresponding Hadamard matrix H4 is:

[0135]

[0136] If the input information bit u = [1, 0] with binary value 2, the encoded codeword is the 3rd row of H4:

[0137] c = [1, 1, -1, -1] (22)

[0138] As can be seen from the above process, the LDPC-Hadamard encoding method is a concatenated encoding scheme combining LDPC code and Hadamard code. The system first encodes the input data with LDPC code to generate a codeword with strong sparsity and easy hardware implementation. The sparsity of LDPC encoding makes it have lower complexity in large data transmission. Next, the output of LDPC encoding is Hadamard encoded to further improve the anti-noise performance. As the maximum distance separable code, Hadamard encoding can effectively correct errors in a very low signal-to-noise ratio environment. Through this concatenated way, LDPC-Hadamard encoding not only has the efficiency of LDPC code, but also can enhance the robustness of the system in the noise and interference environment by using Hadamard code, which is suitable for communication systems with high reliability requirements.

[0139] In the embodiments of the present disclosure, the LDPC-Hadamard decoding structure is as shown in Figure 4 , the information is iteratively exchanged between the variable nodes and the Hadamard check nodes, and after a certain number of iterations, the decoder outputs the decoding result.

[0140] The BP decoding algorithm of LDPC code is an iterative process that approximates the maximum likelihood estimate through message passing. The LDPC code can be represented as a sparse check matrix H, where each row corresponds to a check node and each column corresponds to a variable node. The BP decoding algorithm of LDPC is performed on this factor graph.

[0141] 1) Initially, the received signal y = [y1, y2,..., y n ] is transmitted through the channel. For each variable node i, initialize the message (i.e. the prior probability of each bit):

[0142]

[0143] where P(c i = 0|y i ) and P(c i = 1|y i ) are the probabilities that the code bit c i is 0 or 1, respectively, given the received signal y i . For the AWGN channel, these probabilities can be computed from the Gaussian distribution.

[0144] 2) Message passing: BP decoding of LDPC codes is mainly based on message passing on the factor graph. In each iteration, messages are passed from variable nodes to check nodes, processed at the check nodes, and then passed back to the variable nodes.

[0145] Variable nodes send messages to the connected check nodes, which are updated based on the signal received at the variable node and the information passed from other check nodes. For each variable node i and its connected check node j, the passed message m i→j is given by:

[0146]

[0147] where L(q i ) is the log-likelihood ratio of the received signal, r k→i is the information passed from other check nodes k to variable node i, and N(i)\j denotes all check nodes connected to variable node i, except j.

[0148] Check nodes update their information based on the messages from the variable nodes connected to them. For each check node j and its connected variable nodes i, the passed message r j→i is given by:

[0149]

[0150] where tanh -1 is the hyperbolic tangent function, and N(j)\i denotes all variable nodes connected to check node j, except i.

[0151] 3) Decision and hard decision

[0152] At the end of each iteration, each variable node updates its estimate based on the initial received information and the messages received from the check nodes. For each variable node j, we compute its posterior log-likelihood ratio:

[0153]

[0154] Then, for each qi Make hard decision:

[0155]

[0156] After each iteration, check if all check conditions are met, i.e. If yes, decoding succeeds; if no, proceed to next iteration.

[0157] Fast APP decoding for Hadamard codes. In the iterative step of the APP decoding algorithm for Hadamard codes, the probability information is updated using the product of the Hadamard matrix and the current vector at each step. However, the complexity of directly calculating these products is high. The introduction of the fast Hadamard transform (FHT) can significantly reduce this computational complexity. The FHT uses the structural properties of the Hadamard matrix to speed up the operation by recursive decomposition, making the iterative decoding of Hadamard codes more efficient.

[0158] Hadamard matrix of order n, the general method of calculating y = H n x requires n(n-1) additions, i.e. 2 r (2 r -1) times, but using the properties of the Hadamard matrix can significantly reduce the complexity. Let

[0159]

[0160] In this way, it can be transformed into two sub-problems of H n / 2 x′ and H n / 2 x″ to solve. This process is recursive until it is reduced to H1. It can be proved that this method only requires r2 r additions, and the computational complexity is greatly reduced. The above method of calculating y = H n x is called FHT.

[0161] The fast Hadamard transform is very similar to the fast Fourier transform, as shown in the attached Figure 5 The steps of the fast APP decoding algorithm for Hadamard codes are as follows:

[0162] Let x = (X0, X1,..., X n-1 ) be the transmitted sequence (x i ∈{+1,-1}), and y = (y1, y2,..., y n-1 ) be the received sequence, then the log-likelihood ratio of the posterior probability is:

[0163] ​

[0164] where, To compute γ(±h j ) one needs to compute the inner product which is exactly the element in Note that can be computed using the fast Hadamard transform, so γ(±h j ) can be obtained conveniently.

[0165] In the decoding process of LDPC-Hadamard code, first, the LLR-BP (Log Likelihood Ratio Belief Propagation) algorithm is used to perform iterative decoding of the received signal for the LDPC code. This algorithm updates the LLR values of the variable nodes and check nodes by message passing on the bipartite graph corresponding to the check matrix of the LDPC code, and gradually approaches the optimal decoding result. After completing the LDPC decoding, the output soft information is passed to the Hadamard decoding part. For the decoding of the Hadamard code, the fast Hadamard transform (FHT) is used for processing. The FHT algorithm can quickly and efficiently perform Hadamard transform, significantly reducing the computational complexity, thereby recovering the original information data. The entire LDPC-Hadamard decoding process realizes efficient and low-complexity decoding performance by combining the two algorithms, and is suitable for communication systems in high-noise environments.

[0166] In order to verify the method of the present application, the following simulation experiment is performed. Comparison of error performance of LDPC-Hadamard code and LDPC code under OTFS system.

[0167] In the OTFS system, the signal is severely affected by time delay spread and Doppler shift after passing through the multipath time-varying channel, while the LDPC-Hadamard code has better error correction performance in complex environments than the LDPC code alone due to the anti-interference characteristics of the Hadamard encoding.

[0168] As shown in Figure 6 , the experimental parameters are set as the number of information bits K = 1024, the code rate R = 1 / 9, and the simulation is performed under different signal-to-noise ratio conditions (Eb / N0). The bit error rate (BER) comparison curve shows the error performance of the two encoding methods under the same conditions. The experimental results show that as the signal-to-noise ratio increases, the bit error rate of the LDPC-Hadamard code is significantly lower than that of the traditional LDPC code, especially in the low signal-to-noise ratio region (such as Eb / N0 below 0 dB), the performance advantage is particularly obvious. The bit error rate of the LDPC-Hadamard code remains at a low level in the signal-to-noise ratio range of -0.8 to 0.6 dB, showing higher stability and anti-interference ability.

[0169] The simulation results show that the LDPC-Hadamard code significantly improves the error performance of the system in the low SNR condition by introducing the anti-noise characteristics of Hadamard code, and makes it more suitable for communication requirements in high-speed mobile environment.

[0170] In the description of the present specification, the description referring to the terms "one embodiment", "some embodiments", "an example", "a specific example" or "some examples" and the like means that the specific features, structures, materials or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present application. In the present specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Also, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art can combine and combine different embodiments or examples described in the specification.

[0171] Other embodiments of the application will be apparent to those skilled in the art from consideration of the specification and practice of the application disclosed herein. It is intended that the specification and examples be considered as exemplary only, with the true scope and spirit of the application being indicated by the following claims.

Claims

1. A low complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation, characterized in that, The method comprises the following steps: The sending end adds check information to the information sequence through cyclic redundancy check, and then encodes the information sequence through LDPC-Hadamard code to form coded data; The coded data is subjected to constellation mapping, and the complex value symbol after the constellation mapping is subjected to OTFS modulation, wherein the OTFS modulation comprises inverse symplectic finite Fourier transform and Heisenberg transform to output a time domain signal; The time domain signal is subjected to time delay and Doppler shift through a multipath time-varying channel to generate a sending signal; The receiving end subjects the received sending signal to OTFS demodulation, wherein the OTFS demodulation comprises Wigner transform and symplectic Fourier transform to output a symbol matrix; The symbol matrix is subjected to constellation inverse mapping to recover a received bit stream; The received bit stream is subjected to LDPC-Hadamard decoding, and the decoded bit stream is subjected to cyclic redundancy check to remove the check information and output an original bit stream.

2. The method of claim 1, wherein the low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation is characterized by, The LDPC-Hadamard code is a concatenated code combining LDPC code and Hadamard code, comprising the following steps: An LDPC code is constructed by using a low-complexity quasi-cyclic method and a shift register is used for encoding; The output of the LDPC code is subjected to Hadamard encoding.

3. The method of claim 2, wherein the low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation is characterized by, The LDPC code is constructed by using a low-complexity quasi-cyclic method and a shift register is used for encoding, comprising the following steps: A check matrix is composed of a plurality of permutation matrices or shift matrices, and each sub-matrix in the check matrix is a cyclic shift matrix, wherein the cyclic shift matrix cyclically shifts elements of a vector; The check matrix is converted into a systematic form through Gaussian elimination by using the quasi-cyclic structure of the check matrix to obtain a generator matrix; The generator matrix and an information bit vector are subjected to matrix multiplication for encoding.

4. The method of claim 3, wherein the low-complexity LDPC-Hadamard code encoding and decoding method under OTFS modulation is characterized by, The check matrix and the cyclic shift matrix are represented by the following expressions: P i =I i where H is a check matrix, P i,j is a z x z circulant shift matrix or a zero matrix, called a permutation matrix or unit shift matrix, the entire matrix is composed of m b x n b sub-matrices, where each sub-matrix is a z x z circulant shift matrix or a zero matrix, I i denotes that the columns of the identity matrix are cyclically shifted to the right by i positions.

5. The method of claim 4, wherein the low-complexity LDPC-Hadamard code encoding and decoding method is based on OTFS modulation. The systematic form is represented by the following expression: H=[P|I] Wherein, P is a sub-matrix, and I is an identity matrix.

6. The method of claim 5, wherein the low-complexity LDPC-Hadamard code encoding and decoding method is based on OTFS modulation. The generator matrix and the information bit vector are subjected to matrix multiplication for encoding, and the expression is as follows: c=uG Wherein, c is encoding, u is an information bit vector, and G is a generator matrix.

7. The method of claim 1, wherein the low complexity LDPC-Hadamard code encoding and decoding method is based on OTFS modulation. The OTFS modulation comprises inverse symplectic finite Fourier transform and Heisenberg transform to output a time domain signal, comprising the following steps: The complex value symbol is converted from a Delay-Doppler domain to a time-frequency domain through inverse discrete symplectic Fourier transform; The time-frequency domain signal is converted into a time domain signal through Heisenberg transform.

8. The method of claim 1, wherein the low complexity LDPC-Hadamard code encoding and decoding method is based on OTFS modulation. The OTFS demodulation comprises Wigner transform and symplectic Fourier transform to output a symbol matrix, comprising the following steps: The sending signal is subjected to Wigner transform to recover the representation of the signal in a Delay-Doppler domain; The signal is remapped from a time-frequency domain to a Delay-Doppler domain through symplectic Fourier transform to output a symbol matrix.

9. The method of claim 1, wherein the low complexity LDPC-Hadamard code encoding and decoding method is based on OTFS modulation. The constellation mapping adopts QAM or PSK.

10. The method of claim 1, wherein the low complexity LDPC-Hadamard code encoding and decoding method is based on OTFS modulation. The LDPC-Hadamard decoding comprises the following steps: The BP algorithm is used to perform iterative decoding on the received signal through LDPC code; After the LDPC decoding is completed, the output soft information is passed to the Hadamard decoding section, which uses fast Hadamard transform to decode the Hadamard code.

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