Method for calculating error rate of pam-6 modulation system based on markov chain

Abstract

The application relates to the technical field of communication, and discloses a PAM-6 modulation system error rate calculation method based on a Markov chain, which is characterized in that, for 9 types of offset judgment results generated by joint judgment of two continuous unit intervals in PAM-6 modulation, Markov states are defined according to combinations of n groups of continuous judgment results; according to tap coefficients of a judgment feedback equalizer and a judgment feedback offset sequence of a current Markov state, basic probability values of single judgment results in each state are calculated, state transition probabilities are calculated by merging various judgment results, a state transition matrix is constructed, and a steady-state vector of each Markov state is solved; according to the number of error bits of each judgment point in a PAM-6 constellation diagram and position weights in the merging of judgment results, average error code numbers of various error judgments are calculated, and the system error rate is obtained. The application provides a reliable and efficient theoretical basis for performance evaluation and equalizer design of future ultrahigh-speed serial communication receivers.

Description

Technical Field

[0001] This invention relates to the field of communication technology, and specifically to a method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain. Background Technology

[0002] In communications, information is typically transmitted in unit intervals (UI), with each unit interval transmitting a signal called a symbol. In the era of low data rates, high-speed serial communication primarily used NRZ (non-return-to-zero) modulation, which uses two non-zero voltage levels of opposite polarities to represent bits 0 and 1. However, after data rates reached 50 Gbps, PAM4 (pulse-amplitude modulation with 4 levels) became the mainstream modulation method. Analysis indicates that in the commonly used 29.8-inch backplane channels of data centers, when the data rate is less than 50 Gbps, PAM4's additional signal-to-noise ratio (SNR) requirement exceeds the additional insertion loss of NRZ, meaning NRZ is superior to PAM4. Conversely, when the data rate is greater than 50 Gbps, PAM4's additional SNR requirement is lower than the additional insertion loss of NRZ, making PAM4 modulation the superior modulator. This is why mainstream communication standards such as Ethernet and OIF-CEI have chosen PAM4 as the modulation scheme for serial communication with data rates of 100 Gbps and above. Due to the ever-increasing demands for data rates, researchers are even considering using modulation schemes such as PAM6 and PAM8, which carry more bits per symbol and have lower baud rates, in future communication standards. 224G and even 448G serializers and deserializers (SerDes) are currently the main research directions, and PAM-6 encoding may be used in 448G SerDes; therefore, modeling the PAM-6 bit error rate is of practical significance.

[0003] Currently, modeling of the bit error rate (BER) for PAM2 / 4 / 8 encoding is relatively mature, but modeling of the BER for PAM-6 is lacking. Modeling of the BER for PAM2 / 4 / 8 encoding mainly utilizes Markov models. However, because PAM6 encoding involves two consecutive decision units (UIs), its Markov model is relatively complex. Therefore, current BER modeling methods for PAM2 / 4 / 8 encoding cannot be directly applied to modeling the BER for PAM6. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: A method for calculating the bit error rate (BER) of a PAM-6 modulation system based on a Markov chain is provided for use in a serial communication receiver. The receiver employs a decision feedback equalizer to process the PAM-6 signal. This method establishes a Markov model to address the BER propagation effect introduced by the decision feedback equalizer, including: For the nine types of offset decision results generated by the joint decision of two consecutive unit intervals in PAM-6 modulation, the Markov state is defined by the combination of n consecutive decision results, where n is determined by the number of taps N of the decision feedback equalizer. In an additive white Gaussian noise channel, based on the tap coefficients of the decision feedback equalizer and the decision feedback offset sequence of the current Markov state, the basic probability value of a single decision result in each state is calculated, the decision results of various types are merged to calculate the state transition probability, a state transition matrix is ​​constructed, and the steady-state vector of each Markov state is solved based on the state transition matrix. Based on the number of erroneous bits at each decision point in the PAM-6 constellation diagram and the position weight when merging decision results, the average number of erroneous decisions for each type is calculated. The average number of erroneous bits is then multiplied by the corresponding component of the steady-state vector and summed to obtain the system bit error rate under the influence of the decision feedback equalizer.

[0006] In one embodiment, the nine types of offset decision results specifically include: A set of two consecutive judgment results is divided into 9 categories from 0 to 8, and the corresponding offset sequence is as follows: Class 0: (0,0); Class 1: (2, -2); Class 2: (2,0); Class 3: (2,2); Class 4: (0,2); Class 5: (-2,2); Class 6: (-2, 0); Class 7: (-2, -2); 8 categories: (0, -2); Where 0 represents a correct judgment, and 1 to 9 represent 8 types of incorrect judgments.

[0007] In one embodiment, the nine types of offset decision results generated by joint decision of two consecutive unit intervals in PAM-6 modulation are defined as Markov states by the combination of n consecutive sets of decision results, where n is determined by the number of taps N of the decision feedback equalizer, specifically including: If N is even, then n = N / 2; If N is odd, then n = (N+1) / 2; Using the combination of n consecutive decision results as Markov states, there are a total of For each state, the n sets of decision results are arranged from left to right according to their numbers and order to form a base-9 number, which is then converted to a decimal number to obtain the state ordinal number.

[0008] In one embodiment, the step of calculating the basic probability value of a single decision result in each state under an additive white Gaussian noise channel, based on the tap coefficients of the decision feedback equalizer and the decision feedback offset sequence of the current Markov state, specifically includes: set up The basic level unit, The standard deviation of additive white Gaussian noise. Here are the tap values, and the current state's offset sequence is: , The offset of the result of a single decision at time k-1 is denoted as... for The probability of the first judgment; (Note: The original text contains several grammatical errors and inconsistencies. A more accurate for , for When the first judgment is At that time, record the second judgment result. for , for When the first judgment is At that time, record the second judgment result. for , for When the first judgment is At that time, record the second judgment result. for , for ; ; ; ; ; ; ; ; ; Where N is the number of taps in the decision feedback equalizer. This is the residual error function.

[0009] In one embodiment, the step of merging various decision results to calculate the state transition probability and constructing the state transition matrix specifically includes: The transition probability is non-zero only when the last n-1 sets of decisions of the previous state are the same as the first n-1 sets of decisions of the next state, and each state transitions to only 9 states. Calculate the transition probability of the current group's decision result being any of the classes using the following formula: ; ; ; ; ; ; ; ; ; This indicates that the current group's judgment result belongs to the group numbered... The transition probability of that type; Construct the state transition matrix, matrix elements Let $\mathbf{i}$ represent the probability of transitioning from state $i$ to state $j$. The calculated transition probabilities are then filled into the corresponding positions in the state transition matrix to obtain the state transition matrix. .

[0010] In one embodiment, the step of solving for the steady-state vector of each Markov state based on the state transition matrix specifically includes: The steady-state vector is obtained by solving the following system of equations. : ; in, Represents the steady-state vector The i-th component, This represents the state transition matrix.

[0011] In one embodiment, the step of calculating the average number of bit errors for each decision point in the PAM-6 constellation diagram and the position weights when merging decision results specifically includes: set up to The average number of bit errors for error decisions of types 1 to 8 are calculated using the following formula: ; ; ; ; ; ; ; .

[0012] In one embodiment, the average bit error rate is multiplied by the corresponding component of the steady-state vector and then summed to obtain the system bit error rate under the influence of the decision feedback equalizer, specifically including: System bit error rate for: .

[0013] Compared with the prior art, the beneficial technical effects of the present invention are: This invention utilizes a Markov model to model the bit error rate of PAM-6. Under low tap count conditions, the theoretical results can predict the system bit error rate well. Under high tap count conditions, at the same bit error rate, the difference between the ratio of energy per bit to noise power spectral density in the simulation results and the theoretical results is about 0.3 dB. Attached Figure Description

[0014] Figure 1 This is a schematic diagram illustrating the specific decision-making process for PAM6.

[0015] Figure 2 This is a schematic diagram showing the specific distribution of the nine judgment results of the present invention.

[0016] Figure 3 This is a schematic diagram showing the distribution of the eight types of error decision results and the number of error bits in this invention.

[0017] Figure 4 This is a schematic diagram illustrating the merging of Type 1 erroneous judgment results according to the present invention.

[0018] Figure 5 This is a flowchart of the method of the present invention. Detailed Implementation

[0019] A preferred embodiment of the present invention will now be described in detail with reference to the accompanying drawings.

[0020] like Figure 5 As shown, this invention discloses a method for calculating the bit error rate (BER) of a PAM-6 modulation system based on a Markov chain, used in a serial communication receiver. The receiver employs a decision feedback equalizer to process the PAM-6 signal. This method establishes a Markov model to address the BER propagation effect introduced by the decision feedback equalizer, including: S1, for the nine types of offset decision results generated by the joint decision of two consecutive unit intervals in PAM-6 modulation, the Markov state is defined by the combination of n consecutive decision results, where n is determined by the number of taps N of the decision feedback equalizer. S2, Under the additive white Gaussian noise channel, based on the tap coefficients of the decision feedback equalizer and the decision feedback offset sequence of the current Markov state, calculate the basic probability value of the single decision result in each state, merge the decision results of each type to calculate the state transition probability, construct the state transition matrix, and solve the steady-state vector of each Markov state based on the state transition matrix. S3. Based on the number of erroneous bits at each decision point in the PAM-6 constellation diagram and the position weight when merging decision results, calculate the average number of erroneous decisions for each type of decision. Multiply the average number of erroneous bits by the corresponding component of the steady-state vector and sum them up to obtain the system bit error rate under the influence of the decision feedback equalizer.

[0021] Using the method of this invention, the system bit error rate can be accurately predicted when the number of main taps is less than 0.5.

[0022] The specific modulation method of PAM6 is as follows: Figure 1 As shown, each UI has 6 levels, and two consecutive UIs are used as the vertical and horizontal axes respectively, together modulating 5 bits of information.

[0023] See Figure 2 A set of two consecutive judgments is divided into 9 cases, and each case is numbered as follows: 0 represents a correct judgment, and 1 to 9 represent 8 types of incorrect judgments.

[0024] The offset series for each type of judgment result is as follows: Class 0: (0,0); Class 1: (2, -2); Class 2: (2,0); Class 3: (2,2); Class 4: (0,2); Class 5: (-2,2); Class 6: (-2, 0); Class 7: (-2, -2); Category 8: (0, -2).

[0025] The order n is determined by the number of taps N: if N is even, n = N / 2; if N is odd, n = (N+1) / 2.

[0026] Using the combination of n consecutive decision results as Markov states, we know that there are a total of A state.

[0027] For all possible Markov states, arrange the n sets of decision results from left to right according to their numbers and order, with the earlier decision result on the left and the later decision result on the right, and combine them into a base-9 number. Convert the base-9 number into a decimal number to obtain the state ordinal number.

[0028] The transition probability between two Markov states is non-zero only when the last n-1 decisions of the current Markov state are the same as the first n-1 decisions of the next Markov state. Each state can only transition to one of the nine possible states.

[0029] To calculate the non-zero transition probabilities, we first determine the offset sequence of n decision outcomes from the previous Markov state. Combining these offset sequences from the previous Markov state yields a sequence of length 2n, which is the offset sequence itself, denoted as [equation missing]. , This represents the offset of the judgment result in the current single judgment. This represents the offset from the previous judgment, and so on.

[0030] remember for The probability of the first judgment. For a set of judgments, denote the probability of the first judgment. for , for When the first judgment result is At that time, record the second judgment result. for , for When the first judgment is At that time, record the second judgment result. for , for When the first judgment is At that time, record the second judgment result. for , for .

[0031] remember The basic level unit, The standard deviation of AWGN (Additive White Gaussian Noise) noise. The tap value represents the PAM level received by the receiver, which is equivalent to adding a value similar to the nearest tap to the ideal PAM level. Offset related to the outcome of the second judgment Add noise. Calculate the base probability value: ; ; ; ; ; ; ; .

[0032] The complementary error function is defined as follows: ; Where t is the integration variable.

[0033] Figure 1 Not all points in the algorithm have all nine possible decision outcomes. Except for decision 0 (correct decision), which can occur at any point, the other eight incorrect decisions are not possible for all points. Therefore, to calculate the state transition probability, we must first distinguish the possible decision outcomes for each point. Figure 3 Showing Figure 1 The distribution of various erroneous decision results for 32 points is shown. The blue part indicates that such a decision result exists at this point, and the green part indicates that such a decision result does not exist. The numbers in the figure represent the number of error bits generated when such an erroneous decision is made at this point. Figure 3 (a) to (h) in the table correspond to Class 1 to Class 8 judgments, respectively.

[0034] The decision results at certain special locations are merged, and it is precisely these merging of states that cause some decision results to be missing at these points. For example... Figure 4 As shown, Figure 3 In (a), the blue part of the Class 1 decision results is taken out, and the points marked in yellow have their Class 1, 2 and 3 decisions merged into Class 1 decisions, while the rest of the results do not have Class 1 decisions merged.

[0035] For the remaining seven types of erroneous decisions, the results are all merged. The merging of results from different decisions and at different locations varies, but the principle remains the same. Considering the distribution of different decision results and the merging of results, the transition probability calculation formula for the current set of decisions in the next Markov state having nine possible outcomes is as follows: (This indicates that the current set of judgments is of type *). ; ; ; ; ; ; ; ; .

[0036] Construct the state transition matrix Matrix elements Indicates the Markov state Transition to state The probability, here , Let be the state ordinal number. One round of calculation yields 9 probability values ​​corresponding to a Markov state, totaling [number missing]. For a given Markov state, such calculations require... The state transition matrix is ​​obtained by filling the calculated transition probabilities into the corresponding positions based on the correspondence between states. .

[0037] Calculate the steady-state vector : ; ; Represents the steady-state vector The i-th component represents the probability that the system is in Markov state i under steady state.

[0038] Figure 3 The graph displays the number of bit errors for each type of decision. However, when calculating the average number of bit errors for each type of decision, the number of bit errors in the graph cannot be directly averaged because it is known that the probability of a certain type of erroneous decision occurring varies depending on the location of the point. Figure 4 As shown, the points in the yellow section are more likely to produce a Class 1 decision than the points in the blue section. This is because of the merging of decision results; therefore, the points in the yellow section can be considered to have a higher weight when calculating the bit error rate. The Class 1, 2, and 3 decisions of the points in the yellow section are merged into a Class 1 decision, while the points in the blue section are not merged. Therefore, the weight ratio of the points in the blue section to those in the yellow section when calculating the bit error rate can be obtained as follows: .

[0039] Considering the number of bit errors at each point and their weight ratios, if we denote the average number of bit errors in error decisions from types 1 to 8 as... The average number of bit errors for each type of erroneous decision is expressed as follows: ; ; ; ; ; ; ; .

[0040] Calculate the bit error rate : .

[0041] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the invention. The terms “comprising,” “including,” etc., as used herein indicate the presence of the stated features, steps, operations, and / or components, but do not exclude the presence or addition of one or more other features, steps, operations, or components.

[0042] It should be understood that although the steps in the flowcharts of the accompanying drawings are shown sequentially as indicated by the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some of the steps in the flowcharts of the accompanying drawings may include multiple steps or stages, which are not necessarily completed at the same time, but may be executed at different times, and the execution order of these steps or stages is not necessarily sequential, but may be performed alternately or in turn with other steps or at least some of the steps or stages of other steps.

[0043] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0044] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention, and no reference numerals in the claims should be construed as limiting the scope of the claims.

[0045] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain, used in a serial communication receiver, characterized in that, The receiving end uses a decision feedback equalizer to process the PAM-6 signal, and the method establishes a Markov model to address the bit error propagation effect introduced by the decision feedback equalizer, including: For the nine types of offset decision results generated by the joint decision of two consecutive unit intervals in PAM-6 modulation, the Markov state is defined by the combination of n consecutive decision results, where n is determined by the number of taps N of the decision feedback equalizer. In an additive white Gaussian noise channel, based on the tap coefficients of the decision feedback equalizer and the decision feedback offset sequence of the current Markov state, the basic probability value of a single decision result in each state is calculated, the decision results of various types are merged to calculate the state transition probability, a state transition matrix is ​​constructed, and the steady-state vector of each Markov state is solved based on the state transition matrix. Based on the number of erroneous bits at each decision point in the PAM-6 constellation diagram and the position weight when merging decision results, the average number of erroneous decisions for each type is calculated. The average number of erroneous bits is then multiplied by the corresponding component of the steady-state vector and summed to obtain the system bit error rate under the influence of the decision feedback equalizer.

2. The method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain according to claim 1, characterized in that, The nine types of offset decision results specifically include: A set of two consecutive judgment results is divided into 9 categories from 0 to 8, and the corresponding offset sequence is as follows: Class 0: (0,0); Class 1: (2, -2); Class 2: (2,0); Class 3: (2,2); Class 4: (0,2); Class 5: (-2,2); Class 6: (-2, 0); Class 7: (-2, -2); 8 categories: (0, -2); Where 0 represents a correct judgment, and 1 to 9 represent 8 types of incorrect judgments.

3. The method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain according to claim 1, characterized in that, The nine types of offset decision results generated by the joint decision of two consecutive unit intervals in PAM-6 modulation are defined as Markov states based on the combination of n consecutive sets of decision results, where n is determined by the number of taps N of the decision feedback equalizer. Specifically, they include: If N is even, then n = N / 2; If N is odd, then n = (N+1) / 2; Using the combination of n consecutive decision results as Markov states, there are a total of For each state, the n sets of decision results are arranged from left to right according to their numbers and order to form a base-9 number, which is then converted to a decimal number to obtain the state ordinal number.

4. The method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain according to claim 3, characterized in that, In the additive white Gaussian noise channel, the basic probability value of a single decision result in each state is calculated based on the tap coefficients of the decision feedback equalizer and the decision feedback offset sequence of the current Markov state. Specifically, this includes: set up The basic level unit, The standard deviation of additive white Gaussian noise. Here are the tap values, and the current state's offset sequence is: , Let the offset of a single decision at time k-1 be denoted as . for The probability of the first judgment; (Note: The original text contains several grammatical errors and inconsistencies. A more accurate for , for When the first judgment is At that time, record the second judgment result. for , for When the first judgment is At that time, record the second judgment result. for , for When the first judgment is At that time, record the second judgment result. for , for ; ； ； ； ； ； ； ； ； Where N is the number of taps in the decision feedback equalizer. This is the residual error function.

5. The method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain according to claim 4, characterized in that, The process of merging various decision results to calculate state transition probabilities and construct a state transition matrix specifically includes: The transition probability is non-zero only when the last n-1 sets of decisions of the previous state are the same as the first n-1 sets of decisions of the next state, and each state transitions to only 9 states. Calculate the transition probability of the current group's decision result being any of the classes using the following formula: ； ； ； ； ； ； ； ； ； This indicates that the current group's judgment result belongs to the group numbered... The transition probability of that type; Construct the state transition matrix, matrix elements Let $\mathbf{i}$ represent the probability of transitioning from state $i$ to state $j$. The calculated transition probabilities are then filled into the corresponding positions in the state transition matrix to obtain the state transition matrix. .

6. The method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain according to claim 5, characterized in that, The process of solving for the steady-state vector of each Markov state based on the state transition matrix specifically includes: The steady-state vector is obtained by solving the following system of equations. : ； in, Represents the steady-state vector The i-th component, This represents the state transition matrix.

7. The method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain according to claim 6, characterized in that, The calculation of the average number of bit errors for each decision point in the PAM-6 constellation diagram and the position weights when merging decision results specifically includes: set up to The average number of bit errors for error decisions of types 1 to 8 is calculated using the following formula: ； ； ； ； ； ； ； 。 8. The method for calculating the bit error rate of a PAM-6 modulation system based on a Markov chain according to claim 1, characterized in that, The system bit error rate under the influence of the decision feedback equalizer is obtained by multiplying the average bit error rate by the corresponding components of the steady-state vector and summing the results. Specifically, this includes: System bit error rate for: 。

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