Cyclic optimization method based on mismatch filtering model under uncertain information

Abstract

The application discloses a cyclic optimization method based on a mismatch filtering model under uncertain information, and comprises the following steps: step 1, determining a fuzzy function template based on an uncertain information mode; step 2, constructing a zero-sum game model of a radar side and a jamming side; step 3, improving based on the template model, and constructing and designing a model independent of the template; and step 4, deducing the model, and iterating an optimized waveform. The application can efficiently obtain an optimal transmitting waveform and a matching waveform in view of the problem that a traditional game solving algorithm is extremely difficult to solve in a large-scale network, thereby improving the anti-interference capability of a radar system, and effectively solving the calculation efficiency problem of attack and defense strategy selection in a large-scale network.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of radar waveform design, and particularly relates to a cyclic optimization method based on a mismatch filtering model under uncertain information. BACKGROUND

[0002] In modern radar systems, waveform design is one of the key technologies related to its performance and efficiency. Radar systems detect and identify targets by transmitting specific electromagnetic waveforms, while facing various complex electromagnetic environments and hostile jamming. Traditional waveform design methods often rely on fixed waveform patterns or empirical design, which, although meeting basic detection needs to some extent, are significantly limited in performance and effectiveness when facing variable electromagnetic environments and strong electronic warfare jamming.

[0003] The development of modern radar technology drives the continuous evolution and optimization of waveform design methods. Based on advanced signal processing technology and mathematical modeling means, the new generation of radar waveform design aims to optimize the energy distribution, spectral characteristics, and space-time characteristics of waveforms to improve the performance of radar systems in target detection, tracking, and anti-jamming capabilities.

[0004] The background technology of waveform design covers multiple aspects: first, a comprehensive understanding and analysis of the radar operating environment and use scenarios. Different task requirements may pose different challenges to the waveform design of radar systems, such as identification of high-resolution targets, implementation of low-probability interception, and effective signal transmission in complex electromagnetic environments. Second, waveform design also needs to consider the actual characteristics and limitations of radar hardware, such as the frequency response of the transmitter and receiver, power output capability, signal processing speed, etc.

[0005] The steps of waveform design include: 1. Requirement analysis: clarify the performance requirements of the radar system, such as resolution, detection probability, anti-jamming capability, etc. 2. Select waveform: select appropriate signal types and modulation methods according to requirements. 3. Parameter optimization: optimize signal parameters such as frequency, phase, amplitude, etc. through mathematical modeling and simulation. 4. Performance evaluation: evaluate the performance of the designed waveform through simulation and experiment to ensure that it meets system requirements. 5. System integration: integrate the optimized waveform into the radar system for actual testing and verification.

[0006] In traditional waveform design, researchers usually rely on mathematical models and simulation tools to evaluate the performance of different waveform schemes. These models involve the propagation characteristics of radar signals, target feature extraction methods, and interference suppression techniques. With the improvement of computing power and algorithm optimization, modern waveform design increasingly adopts methods based on statistical learning and optimization theory, maximizing signal processing efficiency in dynamic scenarios through big data analysis and adaptive algorithms. SUMMARY

[0007] Therefore, the application discloses a cyclic optimization method based on a mismatch filtering model under uncertain information.

[0008] The application aims to realize the cyclic optimization method based on the mismatch filtering model under uncertain information, and the method comprises the following steps:

[0009] Step 1, determining a fuzzy function template based on an uncertain information mode;

[0010] Step 2, constructing a zero-sum game model of a radar side and an interference side;

[0011] Step 3, improving based on the template model, and constructing and designing a model independent of the template;

[0012] Step 4, deducing an optimized waveform through the model.

[0013] Further, the uncertain information is random distribution based on an ideal mode: the radar does not know the accurate time delay t of the target target but has a certain probability distribution cognition of the target time delay; the radar does not know the accurate Doppler frequency shift f of the target d,target but has a certain probability distribution cognition of the target Doppler; the radar knows the interference mode of the jammer; the radar does not know the accurate time delay modulation t of the jammer based on the target time delay jammer but knows the probability distribution thereof; the radar does not know the accurate Doppler frequency shift modulation f of the jammer based on the target Doppler frequency shift d,jammer but knows the probability distribution thereof.

[0014] The uncertain information mode refers to that the radar does not have accurate cognition of parameters in the target and interference signals, but cognizes in the form of probability distribution.

[0015] Further, Q target and Q jammer respectively represent a characteristic set of the target and the interference source, and according to the probability distribution on Q target , Q jammer :

[0016]

[0017] α k,p is the probability that the kth element in the target characteristic set is selected under the pth condition, and β k,pis the probability that the kth element in the set of target characteristics under the g function template is selected under the pth condition; k is the identifier of different targets or jammers, and p is the condition related to each target or jammer;

[0018] to determine the ambiguity function template g k,p is the weight corresponding to (k, p) ∈ T x F

[0019]

[0020] α k′,p′ is the probability that the kth element in the set of target characteristics under the g function template is selected under the pth condition; β k′,p′ is the probability that the kth element in the set of jammer characteristics under the g function template is selected under the pth condition, k' is the identifier of targets or jammers under the g function template, and p' is the condition related to each target or jammer under the g function template, Q zero is a set of characteristics, in which the weights of the elements contained are uniformly set to 1.

[0021] Further, according to the uncertain information mode + mismatch filtering + zero-sum game model of mutual ambiguity function:

[0022] Radar side:

[0023]

[0024] Jammer side:

[0025]

[0026] where χ cross,S,W (τ, f) is the mutual ambiguity function, which is a function of time delay τ and frequency offset f, τ represents the time delay parameter, and f represents the frequency offset parameter, and respectively represent the local signal-to-noise ratio and the jammer-to-noise ratio, the dynamic changes of which are determined by the state inside the system and the external control input u radar (t) and v jammer (t), and are functions for calculating the change rates of the local signal-to-noise ratio and the jammer-to-noise ratio, which describe the dynamic evolution;

[0027] A zero-sum game model is established:

[0028]

[0029] where min X,Y represents the minimum operation on variables X and Y, represents the summation on the index set T x F of time delay and frequency offset, gkp discrete template representing the cross ambiguity function, describing the characteristics of a signal at time delay k and frequency offset p, modulus of the expectation value of the cross ambiguity function template, X H denotes the conjugate transpose of X, J kp denotes the complex representation of the cross ambiguity function template, Y denotes the receive signal processing vector in the optimization model;

[0030] is equivalent to

[0031]

[0032] wherein, denotes the minimization operation on the variables X, Y and phase angle φ k,p denotes the complex exponential function, introducing a phase offset of the phase angle φ kp k,p denotes the phase angle, used to describe the phase offset of a signal;

[0033] its standard model form is

[0034]

[0035] wherein min denotes the minimization objective function, denotes the summation over the index set TxF of time delay and frequency offset, |X k denotes the modulus of the k-th element in X, denotes the constraint over all eligible indices k, s.t. denotes the constraint condition, denotes the square root of MN, φ kp ∈ [0, 2π) denotes the value range constraint of the phase angle φ kp

[0036] Further, a simplified model is established as follows:

[0037] determining the symbol vector of the first pulse

[0038] the M-1 symbol vectors of the subsequent pulses are derived from the symbol vector of the first pulse:

[0039]

[0040] At this time

[0041]

[0042] a simplified model is obtained:

[0043] ​​​

[0044] where min denotes that the goal is to minimize the subsequent expression, g kp denotes a discrete template of the cross ambiguity function, describing the characteristics of a signal at a time delay k and a frequency offset p, denotes a complex exponential function, introducing a phase angle φ kp denotes a phase offset, φ kp denotes a phase angle, used to describe a phase offset of a signal, X H denotes a conjugate transpose of X, J kp denotes a complex representation of the cross ambiguity function template, Y denotes a receive signal processing vector in the optimization model, s.t. denotes a constraint condition, |X 1,n | = 1 denotes a constraint condition, ensuring that the modulus of the elements X 1,n of the first part in X is 1 for all n, |ρ m | = 1 denotes a constraint condition, ensuring that the modulus of ρ m is 1 for all m, X m,n = ρ m X 1,n denotes a constraint condition, describing the way the subsequent part in X is constructed, where X m,n is obtained by multiplying ρ m with X 1,n , X m,N+l = 0 denotes a constraint condition, ensuring that the elements in the part of X are zero, denotes a constraint condition, ensuring that the modulus of Y is φ kp ∈ [0, 2π) denotes a constraint condition, limiting the range of values for the phase angle φ kp .

[0045] Further, the symbol vectors of the first and second pulses are optimized, and the symbol vectors of the other pulses are derived based on the symbol vectors of the first and second pulses through overall initial phase modulation, to establish multiple sub-models.

[0046] Further, a computationally feasible special model is constructed, that is, when N + L = 1, the symbol degenerates into a pulse, and a pulse repetition period is a symbol, and the phase encoding degenerates into initial phase modulation of the pulse X = (X1, X2, …, X m , …, X M ) T , which is the initial phase modulation of M pulses, and at this time the model becomes:

[0047]

[0048] At this time, the index set is T = [-M, …, 0, …, M],

[0049] Further, the template-independent model is:

[0050]

[0051] The index set here is a proper subset of the original index set

[0052] The model

[0053]

[0054] In the standard model form, we have

[0055]

[0056] Further, a simplified model is established for the template-independent model:

[0057] The symbol vector of the first pulse is determined

[0058] The symbol vectors of the following M-1 pulses are derived from the symbol vector of the first pulse

[0059]

[0060] At this time

[0061]

[0062] The simplified model is obtained

[0063]

[0064] Further, a computationally feasible special model is constructed, that is, when N+L=1, the symbol degenerates into a pulse, and a pulse repetition period is a symbol, and the phase encoding degenerates into the initial phase modulation of the pulse X=(X1,X2,…,X m ,…,X M ) T is the initial phase modulation of the M pulses, and at this time the model becomes

[0065]

[0066] At this time, the index set is

[0067]

[0068] Compared with the prior method, the method has the advantages that the network game is a research hotspot in recent years, however, the traditional game solving algorithm is extremely difficult to solve in a large-scale network, the optimal transmitting waveform and the matching waveform can be efficiently obtained, and the anti-interference capability of the radar system is improved, and the calculation efficiency problem of the attack and defense strategy selection in the large-scale network is effectively solved. BRIEF DESCRIPTION OF DRAWINGS

[0069] Figure 1 A flowchart of an embodiment of the application is shown;

[0070] Figure 2 A range-Doppler plot of a target echo and jamming before optimization of an embodiment of the application is shown;

[0071] Figure 3 A 2D cross ambiguity function plot of a transmitting waveform before optimization of an embodiment of the application is shown;

[0072] Figure 4 A range-Doppler plot of a target echo and jamming after optimization of an embodiment of the application is shown. DETAILED DESCRIPTION

[0073] The application will be further described below with reference to the drawings, but the application is not limited in any way by the drawings, and any transformation or replacement based on the teaching of the application is within the protection scope of the application.

[0074] In the embodiment, a linear frequency modulation is adopted, noise and jamming are added to the waveform, and the effectiveness of the method is verified by comparing the signal-to-jamming-and-noise ratio before and after optimization.

[0075] As shown in Figure 1 the cyclic optimization method based on the mismatch filtering model under uncertain information, the method comprises the following steps:

[0076] Step 1, determining the target of the radar and the jamming according to the prior information;

[0077] Step 2, designing a self-ambiguity function template under the prior information;

[0078] Step 3, constructing a zero-sum game model of the radar side and the jamming side;

[0079] Step 4, generating the optimized waveform according to the model derivation iteration method.

[0080] Specifically, the uncertain information is:

[0081] The uncertain information mode is a random distribution based on the ideal mode: the radar does not know the accurate time delay t target of the target, but has a certain probability distribution cognition of the target time delay; the radar does not know the accurate Doppler frequency shift fd,target but with some probability distribution knowledge of target Doppler; radar knows the jammer's jamming pattern (assumed to be repeater deception jamming); radar does not know the jammer's precise time delay modulation t jammer but knows its approximate probability distribution; radar does not know the jammer's precise Doppler frequency modulation f d,jammer but knows its approximate probability distribution.

[0082] The so-called uncertain information mode means that the radar has no precise knowledge of the parameters in the target and jamming signals, but knows them in the form of probability distribution. For example

[0083]

[0084] wherein are discrete probability distributions. The uncertain mode is set differently according to different problems, and is uncertain priori.

[0085] Further, the fuzzy function template is:

[0086] The so-called uncertain information mode means that the radar has no precise knowledge of Q target jammer is uncertain, so the discrete template g k,p designed for the mutual fuzzy function is not accurate enough, and needs to be determined according to Q target jammer The above probability distribution

[0087]

[0088] determines the fuzzy function template g k,p The weight corresponding to (k, p) ∈ T × F

[0089]

[0090] Further, the zero-sum game model is:

[0091] The probability distribution essentially needs to be converted into a weight distribution, so it is also embodied that the uncertainty can be directly given without giving the probability distribution. Then, according to the uncertain information mode + mismatch filtering + mutual fuzzy function zero-sum game model:

[0092] Radar side:

[0093]

[0094] Jammer side:

[0095] ​​

[0096] A model can be built

[0097]

[0098] is equivalent to

[0099]

[0100] Writing in the standard model form gives

[0101]

[0102] Many simplified models can be built, for example, we determine the symbol vector of the first pulse

[0103]

[0104] The symbol vectors of the following M-1 pulses are derived from the symbol vector of the first pulse

[0105]

[0106] Then at this time

[0107]

[0108] The simplified model can be obtained

[0109]

[0110] In one embodiment, the symbol vectors of the first and second pulses are further optimized, and the symbol vectors of the other pulses are derived from the symbol vectors of the first and second pulses by overall initial phase modulation. By this idea, a large number of sub-models are built.

[0111] In one embodiment, a special model that is computationally feasible is also constructed, that is, when

[0112] $N+L=1$, at this time the symbol degenerates into a pulse, and a pulse repetition period is a symbol, and at this time the phase encoding degenerates into initial phase modulation of the pulse

[0113] X=(X1,X2,…,X m ,…,X M ) T

[0114] is the initial phase modulation of the M pulses, and at this time the model becomes

[0115]

[0116] At this time, the index set is T=[-M,…,0,…,M],

[0117] The design of the template needs considerable accuracy to get good results, which needs the help of intelligent technology. We can improve the above model based on the template model, and construct and design the model which does not depend on the template.

[0118]

[0119] It must be noted that the index set here

[0120]

[0121] Generally speaking, it is a proper subset of the original index set. That is because, if the proper subset is not selected, when w kp = C, then according to the energy conservation, there is

[0122]

[0123] Then the optimization loses its meaning.

[0124] Model

[0125]

[0126] In the form of the standard model, we can get

[0127]

[0128] In one embodiment, a plurality of simplified models are further established. For example, the symbol vector of the first pulse is determined

[0129]

[0130] The symbol vectors of the following M-1 pulses are changed from the symbol vector of the first pulse

[0131]

[0132] Then at this time

[0133]

[0134] The simplified model can be obtained

[0135]

[0136] In one embodiment, the symbol vectors of the first and second pulses are further optimized, and the symbol vectors of the other pulses are based on the symbol vectors of the first and second pulses through the overall initial phase modulation. Through such an idea, a large number of sub-models can be established.

[0137] In one embodiment, a computationally feasible special model is also constructed, that is, when

[0138] When N+L=1, the symbol degenerates into a pulse, and a pulse repetition period is a symbol, and the phase encoding degenerates into initial phase modulation of the pulse:

[0139] X=(X1,X2,…,X m ,…,X M ) T

[0140] is the initial phase modulation of M pulses, and the model becomes

[0141]

[0142] At this time, the index set is

[0143] T=[-M,…,1,…,M],

[0144] According to the above, the distance Doppler diagram before optimization of the embodiment is as shown in Figure 2 , and the mutual ambiguity function diagram before optimization is as shown in Figure 3 . The waveform is optimized and designed, first, the initial waveform, filtering and other necessary parameters of the radar are given. The ambiguity function two-dimensional color diagram and three-dimensional function diagram of the initial waveform and filtering are calculated. According to the model of the target and the interference, the range Doppler image is formed, and the local signal-to-interference-and-noise ratio of the target unit is calculated. The initial waveform and filtering of the radar are modulated by using the model and algorithm based on the ambiguity function in the foregoing, and the waveform and filtering are output. The ambiguity function two-dimensional color diagram and three-dimensional function diagram of the modulated waveform and filtering are calculated. According to the signal model of the target and the interference again, the range Doppler image is formed, and the local signal-to-interference-and-noise ratio of the target unit is calculated. The shapes of the ambiguity functions before and after are compared, the clarity and the local signal-to-interference-and-noise ratio of the target unit before and after are compared, and the optimized range Doppler diagram is as shown in Figure 4 .

[0145] Compared with the traditional waveform design method, the cyclic optimization method based on the mismatch filtering model under uncertain information proposed in the application can significantly improve the performance of the radar system in a complex electromagnetic environment. Through optimization of the waveform design, not only the detection and identification ability of the target can be improved, but also various forms of interference signals can be effectively suppressed, so that the anti-interference ability and the survival ability of the radar system are improved.

[0146] As used herein, the word "preferably" is used interchangeably with the word "preferably," to mean that an example, implementation or embodiment is preferred, but not necessarily advantageous over other examples, implementations or embodiments. As used herein, the word "preferably" is used to mean that an aspect, design, or implementation so described is preferred, but not necessarily advantageous over other aspects, designs, or implementations. The use of the terms "preferably," "more preferably," "most preferably," and the like are intended to further clarify the order of preference of various aspects, designs, or implementations. As used herein, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or." That is, unless specified otherwise, or clear from context, "X employs A or B" is intended to mean any of the natural inclusive permutations. That is, if X employs A; X employs B; or X employs both A and B, then "X employs A or B" is satisfied under any of the foregoing instances. In addition, the articles "a," "an," and "the" as used in the context of this document are to be construed to be open-ended terms (i.e., to mean "one or more"). Follow-on Related Application

[0147] Moreover, although the present disclosure has been illustrated and described with respect to one or more implementations, equivalent alterations and modifications will occur to others skilled in the art based on the foregoing description and accompanying drawings. The present disclosure includes all such modifications and alterations and is limited only by the scope of the following claims. In particular, with reference to the various functions performed by the above described components (e.g., elements, etc.) to achieve the results, the terms used to describe certain claim components should not be construed to be limited to the specific aspects as set forth above but rather should be interpreted as broadly as is reasonable. For example, the term "comprises" is used in the specification and claims to mean that the disclosed implementations include, but are not limited to, the features, elements, and / or components described in the specification and / or claims. In addition, the term "comprises" is used in the specification and claims to mean that the disclosed implementations include, but are not limited to, the features, elements, and / or components described in the specification and / or claims. Further, the term "comprises" is used in the specification and claims to mean that the disclosed implementations include, but are not limited to, the features, elements, and / or components described in the specification and / or claims.

[0148] The various functional units in the embodiments of the present application can be integrated in one processing module, or each unit can exist physically, or a plurality of or more units can be integrated in one module. The integrated module can be realized in the form of hardware, or in the form of a software functional module. If the integrated module is realized in the form of a software functional module and sold or used as an independent product, it can also be stored in a computer readable storage medium. The storage medium mentioned above can be a read-only memory, a magnetic disk or an optical disk, etc. The above-mentioned devices or systems can execute the storage method in the corresponding method embodiments.

[0149] In summary, the above-mentioned embodiments are one embodiment of the present application, but the embodiments of the present application are not limited by the above-mentioned embodiments, and any changes, modifications, substitutions, combinations and simplifications made without departing from the spirit and principle of the present application are equivalent replacement modes and are included in the protection scope of the present application.

Claims

1. A cyclic optimization method based on a mismatch filtering model under uncertain information, characterized in that, Comprise the following steps: Step 1, determine the fuzzy function template based on uncertain information mode, which means that the radar has no accurate cognition on the parameters in the target and jamming signal, but cognizes in the form of probability distribution; Step 2, build the zero-sum game model of radar side and jamming side; Step 3, improve based on template model, build and design the model independent of template; Step 4, deduce the optimized waveform through model iteration; Wherein, the standard model form of zero-sum game model is wherein, denotes a minimization objective function, denotes a set of indices for time delays and frequency offsets a summation is performed, denotes the modulus of the element in denotes a phase angle for describing a phase offset of a signal, denotes a constraint for all eligible indices , denotes a constraint condition, denotes a square root of MN denotes a range constraint for a phase angle . The simplified model is established as follows: determining a symbol vector for the first pulse ; The last M-1 pulse symbol vectors are changed from the first pulse symbol vector: At this time The simplified model is obtained wherein denotes that the goal is to minimize the subsequent expression, denotes a discrete template of the cross ambiguity function, describing the behavior of the signal at time delays and frequency offsets , denotes a complex exponential function, introducing a phase offset of the phase angle , denotes a conjugate transpose of , denotes a complex representation of the cross ambiguity function template, denotes a receive signal processing vector in the optimization model, denotes a constraint condition, denotes a constraint condition, ensuring that the modulus of the elements of the first part in is 1 for all , denotes a constraint condition, ensuring that the modulus of is 1 for all , denotes a constraint condition, describing the way of construction of the subsequent part in , wherein is obtained by multiplying , denotes a constraint condition, ensuring that the partial elements in are zero, denotes a constraint condition, ensuring that the modulus of is , denotes a constraint condition, limiting the value range of the phase angle ; A special case of the model is computationally feasible, namely when N + L = 1, in which case the symbols degenerate to pulses and a pulse repetition period is a symbol. In this case the phase encoding degenerates to pulse initial phase modulation which is the initial phase modulation of M pulses. In this case the model becomes: At this time the indicator set is ; The model independent of template is The set of indicators here is a proper subset of the original indicators.

2. The method of claim 1, wherein, The uncertain information is random distribution based on ideal mode: the radar does not know the accurate time delay of the target but has certain probability distribution cognition on the time delay of the target The radar is unaware of the precise Doppler shift of the target but has some probabilistic knowledge of the target Doppler Radar aware jammer's jamming pattern; Radar unaware jammer's accurate time delay modulation based on target time delay But aware of its probability distribution; The radar is unaware of the jammer's precise Doppler frequency modulation based on the target Doppler frequency shift but is aware of its probability distribution.

3. The method of claim 2, wherein, and represent the set of characteristics of the target and of the interferer, respectively, according to a probability distribution over ​ It is the first in the target feature set The element in the first... The probability of being selected under certain conditions. It is the first in the set of interference source characteristics The element in the first... The probability of being selected under each condition; k is the identifier of the different target or interference source, and p is the condition associated with each target or interference source; T and F are index sets of time delays and frequency offsets, determining the ambiguity function template corresponding weights is the probability that the element number of the target characteristic set under the fuzzy function template is selected under the condition number , is the probability that the element number of the interference source characteristic set under the fuzzy function template is selected under the condition number , is the identifier of the target or interference source under the fuzzy function template, is the condition related to each target or interference source under the fuzzy function template, is a characteristic set, the weight of the elements contained in which is uniformly set to 1.

4. The method of claim 3, wherein, According to the uncertain information mode + mismatch filtering + mutual fuzzy function zero-sum game model: Radar side: Jamming side: wherein is a mutual blurring function which is a function of time delay and frequency offset f, denotes a time delay parameter, f denotes a frequency offset parameter, and denote the local signal-to-noise ratio and the interference-to-noise ratio, respectively, whose dynamics are determined by the states within the system and the external control inputs and , and are functions describing the dynamics of evolution, used to calculate the rate of change of the local signal-to-noise ratio and the interference-to-noise ratio; Establish zero-sum game model: wherein, denotes a minimization operation on the variables and denotes a summation over an index set of time delays and frequency offsets denotes a discrete template of the cross ambiguity function, describing the characteristics of the signals at a time delay and a frequency offset denotes the modulus of the expectation value of the cross ambiguity function template, denotes the conjugate transpose of denotes the complex representation of the cross ambiguity function template, denotes a receive signal processing vector in the optimization model;​​​​ Equivalent to wherein represents a minimization operation on the variable , and the phase angle , represents a complex exponential function introducing a phase offset of the phase angle , represents a phase angle describing a phase offset of the signal.

5. The method of claim 4, wherein, Optimize the first and second pulse symbol vectors, and the other pulse symbol vectors are based on the first and second pulse symbol vectors through the overall initial phase modulation to establish multiple sub-models.

6. The method of claim 5, wherein, Model Write into the standard model form, and get 。 7. The method of claim 6, wherein, The simplified model is established for the model independent of template: determining a symbol vector for the first pulse The last M-1 pulse symbol vectors are changed from the first pulse symbol vector At this time The simplified model is obtained 。 8. The method of claim 7, wherein, A special case of the model is computationally feasible, namely when N + L = 1, in which case the symbols degenerate to pulses and a pulse repetition period is a symbol. In this case the phase encoding degenerates to pulse initial phase modulation is the initial phase modulation of M pulses, in which case the model becomes At this time, the index set is 。

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