An optimal design method for a modulator system

Abstract

This invention provides an optimization design method for a modulator system, comprising: designing an SDM system architecture, wherein the SDM system architecture includes a MIMO loop filter, a quantizer, and a MISO reconstruction filter; constructing a non-convex infinite constraint optimization model for the MIMO loop filter based on the time-domain discrete input signal and the feedback signal of the quantizer; solving the non-convex constraint optimization model for the MIMO loop filter to obtain an approximate global optimal solution; constructing a non-convex infinite constraint optimization model for the MISO reconstruction filter based on the quantized signal received by the MISO reconstruction filter and the filtering process of the received signal; and obtaining the approximate global optimal solution of the non-convex infinite constraint optimization model for the MISO reconstruction filter. The designed SDM system exhibits high signal-to-noise ratio performance.

Description

Technical Field

[0001] This invention relates to the field of modulator systems, and more specifically to an optimization design method for modulator systems. Background Technology

[0002] With the advent of the system-on-a-chip (SoC) era in integrated circuit design, Sigma Delta modulators (SDMs) are widely used in oversampling A / D and D / A conversions, becoming the dominant technology and development direction for data conversion chips. Their high performance has made them a research hotspot in recent years. Professor Paul R. Gary of the University of California, Berkeley, a world-renowned scholar, pointed out that SDM is the mainstream development direction of data conversion technology today. In today's big data era, with the development of wireless communication, high-fidelity digital video and audio, medical implantable electronic devices, and portable wearable devices towards ultra-high processing speed, high precision, and low voltage and low power consumption, data conversion technologies represented by SDM have been widely applied in many fields such as digital communication, audio, and biomedicine. Companies engaged in analog-to-digital converters have sprung up everywhere. In recent years, SDM research has been the subject of special discussions at annual international conferences on integrated circuits and signal processing. Professor Shanthi Pavan, an IEEE Fellow from the Indian Institute of Technology, gave a special report at the 2019 IEEE International Conference on Integrated Circuit Technology and Applications, highlighting the broad application prospects of SDM. The 2019 IEEE International Conference on Signal, Information and Data Processing also included a special discussion on SDM.

[0003] While SDM (Synchronous Modulator-Demodulator) is widely used in practical engineering, its theoretical research has also received attention from scholars. A typical SDM system architecture consists of a loop filter, a quantizer, etc. SDM uses a sampling frequency much higher than the Nyquist frequency for sampling, followed by quantization to obtain a feedback-type nonlinear modulator. This architecture is simple, applicable, and its high performance can be achieved by increasing the oversampling rate, increasing the modulator order, and increasing the quantizer bit depth. Increasing the oversampling rate reduces noise power within the signal band, but limits SDM applications in the high-frequency domain and is difficult to implement in terms of manufacturing processes. Increasing the modulator order can bring a high signal-to-noise ratio, but it also introduces stability issues. Although a cascaded structure of several low-order SDMs has been proposed for stability, this structure is highly sensitive to the parameters of analog devices in the circuit, and parameter deviations will degrade the modulator's performance. Increasing the quantizer bit depth can improve the signal-to-noise ratio and stability, but it introduces additional nonlinear errors and increases circuit complexity and system power consumption in practical applications.

[0004] Furthermore, in typical SDM system architectures, the loop filter is a single-input single-output (SISO) type. To ensure good frequency selectivity for both the signal transfer function (STF) and noise transfer function (NTF) of the SDM, the gain of the loop filter in the passband needs to be very high. However, in this case, the loop filter may not have stable boundary input and boundary output, making it difficult to guarantee the overall stability of the SDM. Even if the SDM system is locally stable, the dynamic range of the input signal and the allowable set of the state vector will be very small, thus limiting the use of SDM in many engineering applications.

[0005] With the rapid development of technology, the application of SDM (Self-Delta Analyzer) has been demonstrated in various engineering practices. For implantable biomedical devices requiring low voltage, low power consumption, and miniaturization of audio Sigma Delta ADCs, a corresponding SDM design flow has been proposed, providing compliant circuit design specifications. Low-power, high-resolution bandpass Sigma-delta ADCs have been realized, and SDM has been applied to micromechanical accelerometers. An offline calibration procedure has been proposed to correct nonlinearities caused by component mismatch in Sigma Delta DACs and applied to factory calibration. A multi-bit discrete-time SDM with good BER (Breakpoint) performance for OFDM-UWB systems has been designed. SDM has been applied to digital transmitters, improving the processing speed of digital signal systems. A fractional frequency divider based on phase-switching and negative feedback SDM has been proposed and applied to multi-mode, multi-standard communication systems. From an application perspective, in recent years, SDM has been widely used in biomedicine, integrated circuits, and communication systems, demonstrating its application value.

[0006] Early commonly used filter design methods included Butterworth design, Chebyshev design, and comb filter design. Subsequently, SDM (Self-Diffusing Filter) designs, such as those based on the Laguerre filter, gained attention. However, problems still exist in constructing high-performance SDMs, such as: ensuring the stability of the SDM during filter design is difficult because the quantizer must be selected to guarantee the quantization signal-to-noise ratio, but the precise selection of the quantizer is challenging; and there are also difficulties in the initial construction of the SDM and how to verify and evaluate the constructed SDM to determine its optimal performance. Summary of the Invention

[0007] This invention provides an optimized design method for a modulator system, and the designed SDM system has high signal-to-noise ratio performance.

[0008] To achieve the above objectives, embodiments of the present invention provide an optimization design method for a modulator system, comprising:

[0009] Design a modulator SDM system architecture, which includes a MIMO loop filter, a quantizer, and a MISO reconstruction filter. The MIMO loop filter receives a time-domain discrete signal and a feedback signal from the quantizer at its input, and outputs a noise-shaped digital signal. The quantizer quantizes the noise-shaped digital signal and feeds the quantized signal directly back to the MIMO loop filter. The MISO reconstruction filter receives multiple quantized digital signals at its input, and outputs a noise-reduced signal.

[0010] The spatial state equation of the MIMO loop filter is constructed based on the discrete-time signal of the input time domain of the MIMO loop filter and the digital signal fed back by the quantizer. The signal transfer function and noise transfer function of the MIMO loop filter are obtained by transforming the spatial state equation of the MIMO loop filter. With the goal of designing a high signal-to-noise ratio performance of the SDM system, a non-convex infinite constraint optimization model of the MIMO loop filter is constructed using the signal transfer function and noise transfer function of the MIMO loop filter.

[0011] The non-convex constraint optimization model of the MIMO loop filter is solved to obtain an approximate global optimal solution. Based on the approximate global optimal solution, the optimal MIMO loop filter is designed. The optimal MIMO loop filter is the one that makes the signal-to-noise ratio of the modulator meet the preset requirements in the case of multi-input multi-output linear time-invariant.

[0012] Based on the multiple quantized digital signals input to the MISO reconstruction filter and the filtering process of the MISO reconstruction filter on these multiple quantized digital signals, a state-space equation for the MISO reconstruction filter is constructed. The signal transfer function of the MISO reconstruction filter is obtained by transforming the state-space equation. Taking the high signal-to-noise ratio (SNR) performance of the SDM system as the objective, a non-convex infinite constraint optimization model of the MISO reconstruction filter is constructed together with the signal transfer function of the MISO reconstruction filter. An approximate global optimal solution of the non-convex infinite constraint optimization model of the MISO reconstruction filter is obtained, and an optimal MISO reconstruction filter is designed based on the approximate global optimal solution. The optimal MISO reconstruction filter refers to a filter whose SNR meets a preset requirement.

[0013] The above technical solution has the following beneficial effects: the designed SDM system has high signal-to-noise ratio performance. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0015] Figure 1 This is a flowchart of an optimization design method for a modulator system according to an embodiment of the present invention;

[0016] Figure 2 This is a block diagram of a Sigma Delta modulator system according to an embodiment of the present invention;

[0017] Figure 3 This is a breakdown of the technical solutions of the embodiments of the present invention;

[0018] Figure 4 This is the optimal design scheme for the MIMO loop filter and MISO reconstruction filter in the embodiments of the present invention;

[0019] Figure 5 This is the optimal selection strategy for the quantizer in this embodiment of the invention. Detailed Implementation

[0020] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0021] like Figure 1 As shown, in conjunction with embodiments of the present invention, an optimization design method for a modulator system is provided, comprising:

[0022] S101: Design a modulator SDM system architecture, which includes a MIMO loop filter, a quantizer, and a MISO reconstruction filter. The input of the MIMO loop filter is a time-domain discrete signal and the feedback signal of the quantizer. The output of the MIMO loop filter is a noise-shaped digital signal. The quantizer quantizes the input noise-shaped digital signal and feeds the quantized digital signal directly back to the MIMO loop filter. The input of the MISO reconstruction filter is multiple quantized digital signals. The output of the MISO reconstruction filter is a noise-reduced signal, i.e., a high-resolution signal.

[0023] S102: Construct the spatial state equation of the MIMO loop filter based on the discrete-time signal of the input to the MIMO loop filter and the digital signal fed back by the quantizer. Transform the spatial state equation of the MIMO loop filter to obtain its signal transfer function and noise transfer function. With the goal of designing a high signal-to-noise ratio (SNR) performance for the SDM system, construct a non-convex infinite constraint optimization model of the MIMO loop filter using its signal transfer function and noise transfer function. Solve the non-convex constraint optimization model of the MIMO loop filter to obtain an approximate global optimal solution. Design an optimal MIMO loop filter based on this approximate global optimal solution. The optimal MIMO loop filter refers to the one that, under multi-input multi-output linear time-invariant condition, ensures that the signal-to-noise ratio of the modulator meets a preset requirement.

[0024] S103: Construct the state-space equation for the MISO reconstruction filter based on the multiple quantized digital signals input to the MISO reconstruction filter and the filtering process of the MISO reconstruction filter on the multiple quantized digital signals; obtain the signal transfer function of the MISO reconstruction filter by transforming the state-space equation of the MISO reconstruction filter; take the high signal-to-noise ratio (SNR) performance of the SDM system as the objective, and construct the non-convex infinite constraint optimization model of the MISO reconstruction filter together with the signal transfer function of the MISO reconstruction filter; find the approximate global optimal solution of the non-convex infinite constraint optimization model of the MISO reconstruction filter, and design the optimal MISO reconstruction filter based on the approximate global optimal solution; wherein, the optimal MISO reconstruction filter refers to the SNR meeting the preset requirements. The resolution is improved after the signal noise is reduced. High SNR is a common industry term; the ideal goal is a signal without impurities and noise.

[0025] Preferably, the MIMO loop filter has M+1 input terminals and M output terminals;

[0026] In step 102, the construction of the spatial state equation of the MIMO loop filter based on the time-domain discrete signal input to the MIMO loop filter and the digital signal fed back by the quantizer, and the obtaining of the signal transfer function and noise transfer function of the MIMO loop filter by transforming the spatial state equation of the MIMO loop filter, specifically includes:

[0027] The parameters for designing the MIMO loop filter are set as follows: Let N be the number of states of the MIMO loop filter, let x(k) be the state vector of the MIMO loop filter, u(k) be the signal input vector of the SDM system, y(k) and Q(y(k)) represent the signal input vector and signal output vector of the quantizer, respectively, and s(k) be the quantization noise vector, where k represents a discrete variable; A, B, and C are different state space matrices of the MIMO loop filter, and the specific formulas for A, B, and C are as follows:

[0028] A∈R N×N , B∈R N×(M+1) , C∈R M×N B = [B1 B2] ∈ R N×(M+1) ,

[0029] Where M represents the number of signal output channels of the MIMO loop filter, and B1 and B2 are two sub-matrices of B, respectively. The formulas for B1 and B2 are expressed as follows:

[0030]

[0031] The state-space equation of the MIMO loop filter is established using the parameters of the MIMO loop filter, and expressed as follows:

[0032]

[0033] y(k)=Cx(k) (2)

[0034] Where y(k) = [y0(k), ..., y M-1 (k)] T , Q(y(k))=[Q0(y0(k)),…,Q M-1 (y M-1 (k))] T ;

[0035] By performing Z-transform on vectors y(k), s(k), and Q(y(k)), the signal transfer function (STF) and noise transfer function (NTF) of the MIMO loop filter are derived; the formulas for STF and NTF are as follows:

[0036] STF(z)=C(zI N -(A+B2C)) -1 B1 = [STF0(z), ..., STF] M-1 (z)] T (6)

[0037]

[0038] Here, z represents the variable in the z-domain after the Z-transformation.

[0039] Preferably, the high signal-to-noise ratio performance of the SDM system design refers to the signal-to-noise ratio meeting a preset value;

[0040] In step 102, the step of designing a high signal-to-noise ratio (SNR) performance for the SDM system, and constructing a non-convex infinite constraint optimization model for the MIMO loop filter using the signal transfer function and noise transfer function of the MIMO loop filter, specifically includes:

[0041] The objective function of the MIMO loop filter that ensures a high signal-to-noise ratio for the SDM system is minimized when the sum of the inner products between corresponding elements of the STF and NTF is minimized is expressed as:

[0042]

[0043] For the objective function of the MIMO loop filter, by constraining the maximum mode difference between the amplitude response of the signal transfer function (STF) and the ideal STF amplitude response of the MIMO loop filter to be less than a preset threshold, and by constraining the maximum mode difference between the amplitude response of the noise transfer function (NTF) and the ideal NTF amplitude response of the MIMO loop filter to be less than a preset threshold, the signal transmitted by the STF and the noise transmitted by the NTF of the SDM system are respectively within a preset frequency band, thereby constructing the objective function of the MIMO loop filter into a non-convex constraint optimization model.

[0044] Preferably, in step 102, solving the non-convex constraint optimization model of the MIMO loop filter to obtain an approximate global optimal solution, and designing the optimal MIMO loop filter based on the approximate global optimal solution, specifically includes:

[0045] For the non-convex infinite constraint optimization model of the MIMO loop filter, the initial values ​​of the solution are randomly generated by an evolutionary algorithm, and all constraint functions are evaluated based on the initial values.

[0046] If the initial value does not satisfy the constraint function, the initial guess value of the non-convex infinite constraint optimization model of the MIMO loop filter is regenerated, and all constraint functions are evaluated based on the regenerated initial guess value. The iterative process is repeated until the regenerated initial guess value satisfies all constraint functions.

[0047] Once the regenerated initial guesses satisfy all constraint functions, the objective function value of the MIMO loop filter is evaluated based on the initial guesses that satisfy all constraint functions. The objective function value between two iterations is calculated, and the initial guesses corresponding to the solutions with smaller objective function values ​​are retained. The iteration calculation is repeated until the objective function value is lower than a given threshold. A, B, and C are regenerated through permutation and crossover operations, and local optima are eliminated to obtain an approximate global optimum. The optimal MIMO loop filter is then designed based on the approximate global optimum.

[0048] Preferably, in step 103, the process of constructing the state-space equation for the MISO reconstruction filter based on the multiple quantized digital signals input to the MISO reconstruction filter and the filtering of the multiple quantized digital signals by the MISO reconstruction filter; and obtaining the signal transfer function of the MISO reconstruction filter by transforming the state-space equation of the MISO reconstruction filter, specifically includes:

[0049] Set the parameters for designing the MISO reconstruction filter: Let The number of states of the reconstruction filter is denoted as . Let u(k) be the state vector of the MISO reconstruction filter, u(k) be the signal input vector of the SDM system, y(k) and Q(y(k)) be the signal input vector and signal output vector of the quantizer, respectively, and s(k) be the quantization noise, where k represents the discrete variable.

[0050] For the MISO reconstruction filter, different state space matrices, The specific formula is expressed as follows:

[0051]

[0052] The state-space equation of the MISO reconstruction filter is established using the parameters of the MISO reconstruction filter. The state-space equation of the MISO reconstruction filter is expressed as follows:

[0053]

[0054]

[0055] The transfer function of the MISO reconstruction filter is obtained by performing a Z-transform on the vectors y(k), s(k), and Q(y(k)) and deriving the result; the transfer function of the MISO reconstruction filter is expressed as:

[0056]

[0057] Preferably, in step 103, the step of taking the high signal-to-noise ratio performance of the SDM system as the objective and constructing the non-convex infinite constraint optimization model of the MISO reconstruction filter together with the signal transfer function of the MISO reconstruction filter specifically includes:

[0058] The objective function of the MISO reconstruction filter that ensures a high signal-to-noise ratio for the SDM system is minimized is expressed as:

[0059]

[0060] The maximum modulus constraint of the difference between the amplitude response of the MISO reconstruction filter and the amplitude response of the desired signal transfer function is used as the constraint condition of the objective function of the MISO reconstruction filter. Based on the objective function model of the MISO reconstruction filter, a non-convex infinite-form constraint optimization model of the MISO reconstruction filter is constructed.

[0061] Preferably, in step 103, the step of finding the approximate global optimal solution of the non-convex infinite constraint optimization model of the MISO reconstruction filter, and designing the optimal MISO reconstruction filter based on the approximate global optimal solution, specifically includes:

[0062] For the non-convex infinite constraint optimization model of the MISO reconstruction filter, initial values ​​of the solution are randomly generated by an evolutionary algorithm, and all constraint functions are evaluated based on the initial values.

[0063] If the initial values ​​do not satisfy the constraint functions, the initial guesses of the non-convex infinite constraint optimization model of the MISO reconstruction filter are regenerated, and all constraint functions are evaluated based on the regenerated initial guesses. The iterative process is repeated until the regenerated initial guesses satisfy all constraint functions.

[0064] Once the regenerated initial guesses satisfy all constraint functions, the objective function value of the MISO reconstruction filter is evaluated based on the initial guesses that satisfy all constraint functions, and the objective function value between the two iterations is calculated.

[0065] Retain the initial guesses corresponding to the smaller objective function values, and repeat the iterative calculation until the objective function value is below a given threshold; regenerate the reconstruction filter coefficients through permutation and crossover operations. By eliminating local optima, an approximate global optima is obtained. Based on the approximate global optima, an MISO reconstruction filter is designed to obtain the optimal MISO reconstruction filter.

[0066] Preferably, the derivation of the Z-transform of vectors y(k), s(k), and Q(y(k)) to obtain the signal transfer function (STF) and noise transfer function (NTF) of the MIMO loop filter specifically refers to simplifying STF and NTF by using the matrix inversion lemma.

[0067] The technical solutions of the present invention will be described in detail below with reference to specific application examples. For technical details not described in the implementation process, please refer to the relevant descriptions above.

[0068] This invention provides an optimized design method for SDM systems. Considering the problems faced in typical SDM system architectures, this invention focuses on achieving high signal-to-noise ratio and stability in SDM systems, and addresses the following technical problems:

[0069] (1) How to construct non-convex infinite constraint optimization models for MIMO loop filters and MISO reconstruction filters, and design effective algorithms to find the optimal solution.

[0070] When the loop filter is in the MIMO linear time-invariant case, it offers greater freedom in noise shaping, while the MISO reconstruction filter can better separate signal and noise. Based on the high signal-to-noise ratio and stability requirements of SDM, a non-convex infinite constraint optimization model for the loop filter and reconstruction filter is constructed and solved. This invention addresses challenging issues such as complex modeling, high computational cost, and difficulty in finding the global optimum due to non-convexity.

[0071] (2) How to construct a non-smooth constraint optimization model of the quantizer based on the absolute stability criterion and perform stability analysis of SDM.

[0072] Stability is a crucial issue in SDM design. This invention, based on the absolute stability criterion formula and combined with quantization theory, identifies the key parameters in the formula that significantly impact system stability, constructs a non-smooth constraint optimization model for the quantizer, and solves it. The absolute stability criterion is also used in the evaluation and verification of the SDM, analyzing the key parameters in the formula and verifying the stability conditions.

[0073] Therefore, based on the high signal-to-noise ratio and stability of the SDM system, this invention establishes a novel SDM system framework including a MIMO loop filter, a MISO reconstruction filter, and a multi-bit quantizer. It provides in-depth analysis of issues such as the design of the MIMO loop filter and the MISO reconstruction filter, the selection of the quantizer, and the evaluation and verification of the SDM. Specific research objectives are:

[0074] 1) To address the optimal design problem of MIMO loop filters and MISO reconstruction filters, a non-convex infinite constraint optimization model is constructed, an effective algorithm is proposed to find the optimal solution, and a filter with high signal-to-noise ratio and stability is designed, providing guidance for the research of SDM filter theoretical models;

[0075] 2) To address the optimal selection strategy for quantizers, a non-smooth constraint optimization model is proposed based on the absolute stability criterion and quantization theory. The model is then solved and verified to select the optimal quantizer (purpose: to improve system stability).

[0076] 3) The signal-to-noise ratio (SNR) and stability conditions of the SDM system are evaluated, an evaluation model is proposed, and the model and algorithm are further optimized based on feedback results. Combining computer simulation and mathematical theoretical analysis, theoretical guidance is provided for the application of SDM in fields such as digital audio.

[0077] This invention establishes a novel SDM system architecture: a new multi-bit SDM system architecture based on a Multiple-Input Multiple-Output (MIMO) loop filter. This SDM architecture mainly includes a MIMO loop filter, multiple quantizers, and a Multiple-Input Single-Output (MISO) reconstruction filter. This invention aims to optimally design the MIMO loop filter and MISO reconstruction filter, optimally select the quantizer, and evaluate and verify the SDM system. The approach of this invention is as follows: First, a non-convex infinite constraint optimization model is established for the MIMO loop filter and MISO reconstruction filter, and an algorithm is designed to find the optimal solution; second, a non-smooth constraint optimization model is established, and the optimal quantizer is determined through mathematical theory and algorithms; finally, an evaluation model for SDM is determined, and a highly accurate evaluation and verification scheme is proposed, providing theoretical guidance for the application of SDM in digital audio and other fields. Further explanation: Combining signal processing and optimization theory, and based on the requirements of high signal-to-noise ratio and stability of the SDM system, a non-convex constraint optimization model is constructed to optimize the filter design in the proposed new SDM architecture; based on the absolute stability criterion, a non-smooth constraint optimization model is established to determine the optimal selection strategy for the quantizer; considering the signal-to-noise ratio index and stability conditions, the evaluation and verification methods for the SDM system are determined.

[0078] This invention establishes a new SDM system architecture, such as Figure 2As shown, the SDM system mainly includes a MIMO loop filter, multiple quantizers, and a MISO reconstruction filter. The main purpose is to provide greater freedom in noise shaping when the loop filter operates under MIMO linearity and time invariance (noise shaping reduces noise within the signal band and shapes it outside the signal band, effectively improving the overall performance of the modulator. Greater freedom can be understood as lower circuit complexity and higher expansion freedom, thereby improving the modulator's high precision and high energy efficiency). When multiple quantizers are used and their outputs are fed back to the loop filter input instead of being subtracted from the input signal, the input signal and the quantizer outputs will not mix. The MISO reconstruction filter can better separate the signal from noise (this reconstruction filter is used to reduce the sampling frequency, filter out quantization noise outside the signal band, and reduce noise energy within the signal band, outputting a high-resolution signal). This SDM system architecture is proposed to improve the shortcomings of general systems in terms of slightly poor local or overall stability and to enhance signal-to-noise ratio performance.

[0079] Figure 3 The invention presents three aspects of its technical solution: optimal design of MIMO loop filter and MISO reconstruction filter, optimal selection strategy for quantizer, and determination of evaluation and verification method for SDM.

[0080] I. The three aspects of the technical solution of this invention are briefly introduced as follows:

[0081] (1) Optimal design of MIMO loop filter and MISO reconstruction filter

[0082] In SDM systems, rational causal IIR filters are commonly used. Their optimal design problem usually involves writing the filter coefficients in the numerator and denominator into vector form based on the filter's transfer function and frequency response expression. For design requirements such as high signal-to-noise ratio, an optimization model with constraints is established and an algorithm is designed to solve for the coefficient vector.

[0083] Considering factors such as the noise shaping characteristics of filters, optimizing the noise transfer function can reduce quantization noise in the baseband and improve the signal-to-noise ratio. However, since the frequency response of the noise transfer function is defined in the frequency domain, and the frequency domain is a continuous set, each element in the frequency domain corresponds to a constraint condition. This results in significant computational complexity, and the stability of the SDM cannot be guaranteed solely by the frequency response of the noise transfer function. Therefore, in... Figure 2 In the SDM system shown, if the MIMO loop filter has M+1 inputs and M outputs, and the MIMO single-output reconstruction filter has M inputs and single outputs, give the state space matrices of the loop filter (which filters some noise) and the reconstruction filter, respectively. The techniques used to address the optimal design problem of the MIMO loop filter and the MISO reconstruction filter are as follows:

[0084] 1) Establish the state-space equations for the MIMO loop filter and the MISO reconstruction filter;

[0085] 2) Give the expressions for the signal transfer function (STF), noise transfer function (NTF), and reconstruction filter transfer function;

[0086] 3) Based on the design requirements and stability requirements of SDM with high signal-to-noise ratio, the objective functions for the design of the loop filter and reconstruction filter are constructed, and the optimal design problem model is built according to the constraints. Among them, the constraints consider the following conditions: in order to ensure that the STF and NTF of the SDM system are within the design frequency band, the maximum mode constraint of the difference between the designed STF amplitude response and the ideal STF amplitude response, and the maximum mode constraint of the difference between the NTF amplitude response and the ideal NTF amplitude response should be less than a given threshold. Stability conditions are also considered.

[0087] 4) Obtain the optimal solution based on the non-convex constraint optimization model.

[0088] (2) Determining the optimal selection strategy for the quantizer

[0089] Single-bit quantization is a commonly used choice for quantizers due to its simple structure. To achieve higher resolution with single-bit quantization, the modulator needs to have a higher order or a higher oversampling rate. However, modulators with too many orders have poor stability, and excessively high oversampling rates increase the system power consumption of the modulator. With the same signal-to-noise ratio, multi-bit quantization can reduce the oversampling rate of the SDM, thereby reducing the modulator power consumption. Furthermore, with the same sampling frequency, multi-bit quantization is beneficial for designing SDMs with larger bandwidths. In addition, multi-bit quantization helps maintain the loop stability of the modulator and can also appropriately increase the maximum out-of-band gain of the noise transfer function; however, its nonlinearity increases the complexity and uncertainty of circuit implementation.

[0090] Quantizers are mainly classified into two types: flat quantizers and rising quantizers. For all input values ​​within the zero neighborhood, the output of a flat quantizer is zero, while the output-input transfer function of a rising quantizer has a rising edge at zero input values. Furthermore, quantizers can be classified into uniform quantization and non-uniform quantization based on the quantization level division. Uniform quantization, also known as linear quantization, has equal quantization steps and is suitable for signals with uniform amplitude distribution. Non-uniform quantization, proposed in response to uniform quantization, is also known as nonlinear quantization, with unequal quantization steps. It is suitable for situations where signal amplitude is not uniformly distributed, such as speech signals, i.e., using small quantization steps for small amplitude signals to ensure a large quantization signal-to-noise ratio. μ-law and Lloyd-Max are commonly used non-uniform quantizers. The optimal selection strategy for quantizers in SDM is essentially a non-smooth constraint optimization problem. The challenge is to construct an optimal selection strategy model for quantizers based on the design criteria and scheme of SDM, considering the characteristics of different types of quantizers, and using an efficient algorithm to select the best quantizer. Comparing uniform and non-uniform quantization in the SDM system, this invention requires the SDM output to be bounded and stable for any initial conditions and input values, necessitating a high stability margin. Based on the absolute stability criterion, let Q be the transfer function of the uniform quantizer, and K be the maximum output-to-input ratio of the quantizer, then:

[0091] If Q(0) = 0, Make Make Then H(z) in SDM is controllable, and satisfy:

[0092]

[0093] Here, H(z) refers to the system function of the loop filter.

[0094] Considering the different types of quantizers, and to improve system stability, the optimal quantizer selection strategy for this research is addressed through the following steps:

[0095] 1) Establish the input-output transfer function of the quantizer and determine the form of the input-output ratio of the quantizer;

[0096] 2) The problem of selecting the optimal quantizer strategy is transformed into a non-smooth constraint optimization problem;

[0097] 3) Determine the quantizer type based on the solution results.

[0098] (3) Determination of evaluation and verification methods for Sigma Delta modulators

[0099] In SDM design, signal-to-noise ratio (SNR) is a crucial performance evaluation metric, and stability is a fundamental requirement for SDM systems. During stability analysis, the invariant set method requires significant computation for real-time applications, while the non-overload method is too stringent, failing to meet the requirements for many bounded-input, bounded-output SDMs. Root locus methods are suitable for stable interpolated SDM loop filters, but may not guarantee high SNR. Strict stability criteria reduce the designable range of the noise transfer function, thereby shortening SDM design time and increasing stability margin, but these are critical for real-time applications. Therefore, it is essential to explore SDM evaluation models and strategies to assess and verify the optimality of the designed filters and selected quantizers within the SDM system.

[0100] In the design of the SDM system of this invention, the signal-to-noise ratio is an important performance evaluation indicator of the SDM system, and stability is a basic requirement of the system. This invention will adopt the following technical means in the evaluation and verification of SDM:

[0101] (1) Due to the discontinuous and nonlinear characteristics of the quantizer in the SDM system, the entire system is difficult to analyze. In the quantization noise analysis, a continuous function is selected to approximate the quantization operator based on the transfer function of the quantizer; the approximation function is evaluated to determine whether it meets the error requirements, laying the groundwork for the next step of deriving and solving the signal-to-noise ratio.

[0102] (2) Based on the absolute stability criterion, examine the factors affecting stability, establish an evaluation model, and verify whether the designed SDM system meets the stability conditions.

[0103] (3) A new evaluation scheme is proposed based on the information capacity of the noise-shaping channel.

[0104] II. The specific technical solution of this invention is as follows:

[0105] (1) Optimal design scheme for MIMO loop filter and MISO reconstruction filter

[0106] like Figure 4 As shown, in this invention, the loop filter of the SDM system is assumed to be M+1 input and M output, and the reconstruction filter is M input and single output. Let N and Let x(k) be the state vector of the loop filter and the state vector of the reconstruction filter, u(k) be the SDM input, y(k) and Q(y(k)) be the quantizer input and output, respectively, and s(k) be the quantization noise (quantization noise refers to the quantization error generated during the quantization process; this error is regenerated as noise and is called quantization noise). The different state space matrices of the loop filter are A, B, and C (A, B, C can also be called the coefficient matrices). The different state space matrices (or matrix coefficients) of the reconstruction filter are... A∈R N×N , B∈R N×(M+1) , C∈R M×N Here, B = [B1 B2] ∈ R N×(M+1) B1 and B2 are two submatrices of B, denoted as: For the reconstruction filter, Where k represents a discrete variable.

[0107] 1) Establish state-space equations for the MIMO loop filter and the MISO reconstruction filter respectively.

[0108] For a MIMO loop filter, the state-space equation can be expressed as:

[0109]

[0110] y(k)=Cx(k), (2)

[0111] Where y(k) = [y0(k), ..., y M-1 (k)] T , Q(y(k))=[Q0(y0(k)),…,Q M-1 (y M-1 (k))] T .

[0112] For the MISO reconstruction filter, the state-space equation can be expressed as:

[0113]

[0114]

[0115] 2) Perform Z-transform on vectors such as y(k) and s(k) (Z represents Z-transform), and through derivation, obtain the signal transfer function STF, noise transfer function NTF, and the transfer function of the reconstruction filter.

[0116] For the MIMO loop filter, the following can be calculated:

[0117] Q(Y(z))=C(zI N -(A+B2C)) -1 B1U(z)+(C(zI N -(A+B2C)) -1 B2+I M )s(z).(5)

[0118] From equation (5) above, we can derive STF(z) and NTF(z), that is:

[0119] STF(z)=C(zI N-(A+B2C)) -1 B1 = [STF0(z), ..., STF] M-1 (z)] T (6)

[0120]

[0121] Where z represents the variable in the z-domain after the Z-transformation.

[0122] For the MISO reconstruction filter, the following calculations were performed:

[0123]

[0124] Let the transfer function be:

[0125]

[0126] Among them, I N Represents an N-order identity matrix; the left-hand side of equation (8) Representing the expression (4) The z-transform of (4) means that (8) is obtained by performing the z-transform of (4).

[0127] It is worth noting that since the calculations of matrix inversion involved in equations (6), (7) and (9) are relatively large, the mathematical matrix inversion lemma is used to simplify, transform and calculate the corresponding formulas.

[0128] 3) Establish optimal design problem models for the MIMO loop filter and the MISO reconstruction filter respectively.

[0129] For a MIMO loop filter, to achieve high signal-to-noise ratio performance in an SDM system, the sum of the inner products between corresponding elements of the STF and NTF should be minimized. The objective function can be written as:

[0130]

[0131] To ensure that the STF and NTF of the SDM system are within the designed frequency band, the maximum mode constraint of the difference between the designed STF amplitude response and the ideal STF amplitude response, and the maximum mode constraint of the difference between the NTF amplitude response and the ideal NTF amplitude response should be less than a given threshold. Therefore, conditions such as absolute stability are required for limitation.

[0132] For the MISO reconstruction filter, consider establishing an objective that minimizes the sum of the absolute values ​​of the differences between the amplitude response of the designed reconstruction filter and the amplitude response of the signal transfer function, i.e.:

[0133]

[0134] Furthermore, a constrained optimization model is established using the maximum modulus constraint of the difference between the amplitude response of the designed reconstruction filter and the amplitude response of the desired signal transfer function as conditions.

[0135] The two models above are non-convex and have infinite constraints. By modeling the problem and establishing an algorithm to find the optimal solution, the technical problem of this invention can be solved.

[0136] 4) Analyze the optimization model and design an effective algorithm to find the optimal solution.

[0137] For optimization models (10) and (11), commonly used methods such as gradient descent can only find local optima and cannot guarantee finding the global optimum. This invention considers using evolutionary algorithms such as genetic algorithms to find approximate global optima. The basic idea of ​​this invention is to randomly generate initial values ​​for the solution and evaluate all constraint functions based on the initial values. Constraint functions refer to a series of constraints in the optimization model.

[0138] If the constraint functions are not satisfied, the initial guess of the solution is regenerated and all constraint functions are re-evaluated. This iterative process is repeated until all constraint functions are satisfied. Then, based on the initial guess of the solution that satisfies all constraints, the objective function value is evaluated. The objective value between two iterations is calculated, and the initial guess value corresponding to the solution with the smaller objective value is retained. This iterative calculation is repeated until the objective function value is below a given threshold. Filter coefficients are regenerated through permutation and crossover operations during the computation. Locally optimal solutions are eliminated, and finally, an approximate global optimum is reached.

[0139] (2) Research scheme for the optimal selection strategy of quantizer

[0140] like Figure 5 The technical means adopted in this invention are shown below, and are described in detail below:

[0141] 1) Considering that the output of a uniform quantizer is zero for all input values ​​in the zero neighborhood, this invention considers an N-bit uniform quantizer type and assumes that the quantizer is uniform, with the transfer function Q(·) as follows:

[0142]

[0143] Where n represents the discrete-time variable, `sign(·)` is the sign function, `ceil(·)` is the floor function, and the quantization region is [-L, L]. Define the quantization boundary, the ratio of two consecutive quantization boundaries, and determine the form of the output-input ratio.

[0144] 2) Based on the research of the absolute stability criterion, since K is the maximum output-to-input ratio of the quantizer, and a smaller K value corresponds to a more stable system, the problem of minimizing K can be transformed into a Min-Max problem model. It is worth noting that simply minimizing K is insufficient, as it would cause all quantization boundaries to reach saturation levels. To avoid this, a constraint is imposed on the quantization region: the length of the quantization region corresponding to small input values ​​is less than or equal to the length of the quantization region corresponding to large input values.

[0145] Therefore, by writing the output-input ratio of the quantizer as a function of each quantization boundary, it is further transformed into a Min-Max optimization problem model. Considering the constraints such as the different lengths of each quantization region, the problem is finally transformed into solving a mathematical non-smooth constraint optimization problem.

[0146] 3) For the non-smooth constraint optimization model established in the previous step, use mathematical induction to verify whether the uniform quantizer is the best (high robustness). Otherwise, try μ-law and Lloyd-Max non-uniform types.

[0147] (3) Evaluation and verification scheme for Sigma Delta modulator

[0148] 1) The signal-to-noise ratio is an important indicator for SDM evaluation. In order to better analyze the system and perform quantization noise analysis, in combination with the quantizer transfer function (12), we consider using a polynomial function to approximate the quantization operator, that is, using a polynomial in y to approximate Q(y).

[0149] This step prepares for the following method of calculating the signal-to-noise ratio (SNR). Specifically, we first approximate the quantizer using a continuous function. If a suitable approximation function is found, the next step is to select an approximate quantizer, then perform a Fourier transform on the quantizer's inputs and outputs to define the SNR in the frequency domain and derive its value.

[0150] Let vector y = [yy] 3 … y 2M-1 ] T , p = [p1 … p M ] T , where p m (m = 1, 2, ..., M) represents the coefficients of the polynomial, and 2M-1 represents the order of the polynomial. An optimization problem model is established with the objective of minimizing the absolute squared difference between the actual quantizer and the approximate quantizer. It is easy to obtain p = -A -1 b, where Then, the analysis of Q(y)-y under different bit conditions is performed. TThe error value of p. If the error value is less than a given threshold, then the approximation function is reasonable. Otherwise, consider other continuous functions.

[0151] After selecting an approximate quantizer, Fourier transforms are performed on the quantizer's inputs and outputs to define the signal-to-noise ratio (SNR) in the frequency domain and derive its value. To comprehensively evaluate the performance of the SDM, two evaluation schemes are considered: one is the commonly used SNR, and the other is to perform Fourier transforms on the quantizer's inputs and outputs, define the SNR in the frequency domain, and derive its value. A suitable scheme will be selected for different scenarios.

[0152] 2) According to the absolute stability criterion, if Make:

[0153]

[0154] Among them, H r (ω) and H i If ω represents the real and imaginary parts of the loop filter's frequency response, then the absolute stability criterion is satisfied. Let...

[0155]

[0156] According to equation (13), a larger T(q) corresponds to a more stable system. The stability condition is verified by comparing the results of the designed filter in numerical experiments.

[0157] 3) The information capacity of the SDM design can only be achieved if and only if the NTF is at its minimum phase. However, when loop filters satisfying or not satisfying the minimum phase NTF condition were designed using semi-infinite programming theory and solved using the dual parameter method: designs satisfying the minimum phase NTF can achieve the ideal information capacity of the noise-shaping channel, but the signal-to-noise ratio is low; designs not satisfying the minimum phase NTF achieve a positive information capacity of the noise-shaping channel, but the signal-to-noise ratio is very high. Therefore, there should be a trade-off between the signal-to-noise ratio and the information capacity of the noise-shaping channel. Therefore, the evaluation and verification of the SDM in this invention further improves the performance evaluation strategy design of the SDM through theoretical verification and computational simulation to achieve a balance between the signal-to-noise ratio and the information capacity of the noise-shaping channel. Combining 1), 2), and 3), considering various evaluation and verification methods, numerical experiments were conducted using actual signals, and the results were applied to the evaluation and testing of actual SDM design and optimization.

[0158] In summary, considering the theoretical development and application requirements of SDM, and taking into account the shortcomings of general SDM system architectures such as slightly poor local or overall stability, this invention establishes a new SDM system framework based on the high signal-to-noise ratio performance of SDM. This framework mainly includes a MIMO loop filter, multiple quantizers, and a MISO reconstruction filter. The performance requirements of the SDM system are achieved through three parts: optimal design of the MIMO loop filter and the MISO reconstruction filter, optimal selection strategy of the quantizer, and evaluation and verification methods for the SDM. Specifically:

[0159] The first part concerns the optimal design schemes for MIMO loop filters and MISO reconstruction filters. In this section, the objective function considers factors such as stability and the maximum mode constraint of the difference between the passband and stopband amplitude responses, constructing a non-convex optimization model. Computationally, the matrix inversion lemma is cleverly applied to simplify the corresponding matrix calculation formulas. A genetic algorithm is used to design the solution path.

[0160] The second part discusses the research scheme for the optimal selection strategy of quantizers: based on the absolute stability criterion, a Min-Max optimization model is proposed, taking into account other non-uniform types of quantizers, and the modeling and solution are achieved through a non-smooth constraint optimization algorithm.

[0161] Part Three concerns the research scheme for the evaluation and verification of SDM: For the signal-to-noise ratio (SNR), a polynomial function is considered to approximate the quantization operator, which is simple to solve and easy to implement error analysis. Two technical solutions are proposed for SNR calculation: one is a common SNR calculation method, and the other is to select an appropriate scheme for different situations, making the SNR calculation more reasonable and feasible.

[0162] The beneficial effects achieved by this invention are as follows:

[0163] 1) A novel SDM system framework including MIMO loop filter and MISO reconstruction filter with multi-bit quantization is proposed. This improves the shortcomings of the general SDM architecture in the prior art, which has poor local stability or overall stability, and improves the signal-to-noise ratio to meet the high performance requirements of SDM in practical applications.

[0164] 2) The optimal design problem of the MIMO loop filter and the MISO reconstruction filter is transformed into a non-convex optimization problem, and the optimal selection problem of the quantizer is transformed into a non-smooth optimization problem. Due to the non-convex and non-smooth characteristics, a new model and algorithm are proposed in combination with the genetic algorithm to solve the optimal design of the SDM.

[0165] It should be understood that the specific order or hierarchy of steps in the disclosed process is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process may be rearranged without departing from the scope of this disclosure. The appended method claims provide elements of various steps in an exemplary order and are not intended to limit the specific order or hierarchy described.

[0166] In the above detailed description, various features are combined together in a single embodiment to simplify this disclosure. This approach to disclosure should not be construed as reflecting an intention that embodiments of the claimed subject matter require more features than are explicitly stated in each claim. Rather, as reflected in the appended claims, the invention is presented with fewer features than all of the features of the single disclosed embodiment. Therefore, the appended claims are hereby explicitly incorporated into the detailed description, wherein each claim stands alone as a preferred embodiment of the invention.

[0167] The disclosed embodiments have been described above to enable any person skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein can be applied to other embodiments without departing from the spirit and scope of this disclosure. Therefore, this disclosure is not limited to the embodiments given herein, but is consistent with the broadest scope of the principles and novel features disclosed in this application.

[0168] The foregoing description includes examples of one or more embodiments. It is certainly impossible to describe all possible combinations of components or methods in order to describe the above embodiments, but those skilled in the art will recognize that further combinations and arrangements of the various embodiments are possible. Therefore, the embodiments described herein are intended to cover all such changes, modifications, and variations that fall within the scope of the appended claims. Furthermore, the term "comprising" as used in the specification or claims is interpreted in a manner similar to the term "including," as interpreted when used as a conjunction in the claims. Additionally, the use of any term "or" in the specification of the claims is intended to mean "non-exclusive or."

[0169] Those skilled in the art will also understand that the various illustrative logical blocks, units, and steps listed in the embodiments of the present invention can be implemented by electronic hardware, computer software, or a combination of both. To clearly demonstrate the interchangeability of hardware and software, the functions of the various illustrative components, units, and steps described above have been generally described. Whether such functionality is implemented through hardware or software depends on the specific application and the overall system design requirements. Those skilled in the art can implement the described functions using various methods for each specific application, but such implementation should not be construed as exceeding the scope of protection of the embodiments of the present invention.

[0170] The various illustrative logic blocks or units described in the embodiments of this invention can be implemented or operate the described functions using a general-purpose processor, digital signal processor, application-specific integrated circuit (ASIC), field-programmable gate array or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof. The general-purpose processor can be a microprocessor; alternatively, it can be any conventional processor, controller, microcontroller, or state machine. The processor can also be implemented using a combination of computing devices, such as a digital signal processor and a microprocessor, multiple microprocessors, one or more microprocessors combined with a digital signal processor core, or any other similar configuration.

[0171] The steps of the methods or algorithms described in the embodiments of this invention can be directly embedded in hardware, a software module executed by a processor, or a combination of both. The software module can be stored in RAM, flash memory, ROM, EPROM, EEPROM, registers, hard disk, removable disk, CD-ROM, or any other form of storage medium in the art. Exemplarily, the storage medium can be connected to the processor so that the processor can read information from and write information to the storage medium. Optionally, the storage medium can also be integrated into the processor. The processor and storage medium can be housed in an ASIC, which can be housed in a user terminal. Optionally, the processor and storage medium can also be housed in different components of the user terminal.

[0172] In one or more exemplary designs, the functions described in the embodiments of the present invention can be implemented in hardware, software, firmware, or any combination of these three. If implemented in software, these functions can be stored on a computer-readable medium or transmitted on a computer-readable medium in the form of one or more instructions or code. Computer-readable media include computer storage media and communication media that facilitate the transfer of computer programs from one place to another. Storage media can be any available media that can be accessed by a general-purpose or special-purpose computer. For example, such computer-readable media can include, but is not limited to, RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to carry or store program code in the form of instructions or data structures and other forms that can be read by a general-purpose or special-purpose computer, or a general-purpose or special-purpose processor. Furthermore, any connection can be suitably defined as a computer-readable medium, for example, if the software is transmitted from a website, server or other remote resource via a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL) or wirelessly, such as infrared, wireless and microwave, it is also included in the defined computer-readable medium. The disks and discs mentioned include compressed disks, laser discs, optical discs, DVDs, floppy disks, and Blu-ray discs. Disks typically copy data magnetically, while disks typically copy data optically using lasers. Combinations of the above can also be contained in computer-readable media.

[0173] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method of optimal design of a modulator system, characterized by, include: Design a modulator SDM system architecture, which includes a MIMO loop filter, a quantizer, and a MISO reconstruction filter; The input to the MIMO loop filter is a discrete-time signal and a feedback signal from the quantizer. The output of the MIMO loop filter is a noise-shaped digital signal. The quantizer quantizes the noise-shaped digital signal and feeds the quantized digital signal directly back to the MIMO loop filter. The input to the MISO reconstruction filter is multiple quantized digital signals. The output of the MISO reconstruction filter is a noise-reduced signal. The spatial state equation of the MIMO loop filter is constructed based on the discrete-time signal of the input time domain of the MIMO loop filter and the digital signal fed back by the quantizer. The signal transfer function and noise transfer function of the MIMO loop filter are obtained by transforming the spatial state equation of the MIMO loop filter. With the goal of designing a high signal-to-noise ratio performance of the SDM system, a non-convex infinite constraint optimization model of the MIMO loop filter is constructed using the signal transfer function and noise transfer function of the MIMO loop filter. The non-convex constraint optimization model of the MIMO loop filter is solved to obtain an approximate global optimal solution. Based on the approximate global optimal solution, the optimal MIMO loop filter is designed. The optimal MIMO loop filter is the one that makes the signal-to-noise ratio of the modulator meet the preset requirements in the case of multi-input multi-output linear time-invariant. Based on the multiple quantized digital signals input to the MISO reconstruction filter and the filtering process of the MISO reconstruction filter on these signals, a state-space equation for the MISO reconstruction filter is constructed. The signal transfer function of the MISO reconstruction filter is obtained by transforming its state-space equation. Taking the high signal-to-noise ratio (SNR) performance of the SDM system as the objective, a non-convex infinite constraint optimization model for the MISO reconstruction filter is constructed together with the signal transfer function of the MISO reconstruction filter. An approximate global optimal solution for the non-convex infinite constraint optimization model of the MISO reconstruction filter is obtained, and an optimal MISO reconstruction filter is designed based on this approximate global optimal solution. The optimal MISO reconstruction filter refers to a filter whose SNR meets a preset requirement. The MIMO loop filter has M+1 input terminals and M output terminals; The process of constructing the spatial state equation of the MIMO loop filter based on the time-domain discrete signal input to the MIMO loop filter and the digital signal fed back from the quantizer, and obtaining the signal transfer function and noise transfer function of the MIMO loop filter by transforming the spatial state equation of the MIMO loop filter, specifically includes: Configure the parameters of the MIMO loop filter: Let N be the number of states of the MIMO loop filter, and denote... Let be the state vector of the MIMO loop filter. Here is the signal input vector of the SDM system. and These represent the signal input vector and signal output vector of the quantizer, respectively. Let k be the quantization noise vector, where k represents the discrete variable; These are different state-space matrices for the MIMO loop filter. The specific formula is expressed as follows: ， ； Where M represents the number of signal output channels of the MIMO loop filter. , These are two submatrices of B. and The formulas are expressed as follows: ， ； The state-space equation of the MIMO loop filter is established using the parameters of the MIMO loop filter, and expressed as follows: （1） （2） in, , ; For vectors , , By performing a Z-transform, the signal transfer function (STF) and noise transfer function (NTF) of the MIMO loop filter are obtained; the formulas for STF and NTF are as follows: （6） （7） Here, z represents the variable in the z-domain after the Z-transformation.

2. The optimization design method for the modulator system according to claim 1, characterized in that, The high signal-to-noise ratio performance of the SDM system design refers to the signal-to-noise ratio meeting a preset value. The goal is to design a high signal-to-noise ratio (SNR) system for the SDM system. A non-convex infinite constraint optimization model for the MIMO loop filter is constructed using the signal transfer function and noise transfer function of the MIMO loop filter. Specifically, this includes: The objective function of the MIMO loop filter that ensures a high signal-to-noise ratio for the SDM system is minimized when the sum of the inner products between corresponding elements of the STF and NTF is minimized is expressed as: （10） For the objective function of the MIMO loop filter, by constraining the maximum mode difference between the amplitude response of the signal transfer function (STF) and the ideal STF amplitude response of the MIMO loop filter to be less than a preset threshold, and by constraining the maximum mode difference between the amplitude response of the noise transfer function (NTF) and the ideal NTF amplitude response of the MIMO loop filter to be less than a preset threshold, the signal transmitted by the STF and the noise transmitted by the NTF of the SDM system are respectively within a preset frequency band, thereby constructing the objective function of the MIMO loop filter into a non-convex constraint optimization model.

3. The optimization design method for the modulator system according to claim 2, characterized in that, Solving the non-convex constraint optimization model of the MIMO loop filter yields an approximate global optimal solution. Based on this approximate global optimal solution, the optimal MIMO loop filter is designed, specifically including: For the non-convex infinite constraint optimization model of the MIMO loop filter, the initial values ​​of the solution are randomly generated by an evolutionary algorithm, and all constraint functions are evaluated based on the initial values. If the initial value does not satisfy the constraint function, the initial guess value of the non-convex infinite constraint optimization model of the MIMO loop filter is regenerated, and all constraint functions are evaluated based on the regenerated initial guess value. The iterative process is repeated until the regenerated initial guess value satisfies all constraint functions. Once the regenerated initial guesses satisfy all constraint functions, the objective function value of the MIMO loop filter is evaluated based on the initial guesses that satisfy all constraint functions. The objective function value between two iterations is calculated, and the initial guesses corresponding to the solutions with smaller objective function values ​​are retained. The iteration calculation is repeated until the objective function value is lower than a given threshold. A, B, and C are regenerated through permutation and crossover operations, and local optima are eliminated to obtain an approximate global optimum. The optimal MIMO loop filter is then designed based on the approximate global optimum.

4. The optimization design method for the modulator system according to claim 1, characterized in that, The process of constructing the state-space equation for the MISO reconstruction filter based on the multiple quantized digital signals input to the MISO reconstruction filter and the filtering of the multiple quantized digital signals by the MISO reconstruction filter; and obtaining the signal transfer function of the MISO reconstruction filter by transforming the state-space equation of the MISO reconstruction filter, specifically includes: Set the parameters for designing the MISO reconstruction filter: Let The number of states of the reconstruction filter is denoted as . Let be the state vector of the MISO reconstruction filter. Here is the signal input vector of the SDM system. and These represent the signal input vector and signal output vector of the quantizer, respectively. To quantize the noise, where k represents the discrete variable; For the MISO reconstruction filter, different state space matrices, The specific formula is expressed as follows: ； The state-space equation of the MISO reconstruction filter is established using the parameters of the MISO reconstruction filter. The state-space equation of the MISO reconstruction filter is expressed as follows: （3） （4） Through vector pairs , , By performing a Z-transform and deriving the result, the transfer function of the MISO reconstruction filter is obtained; the transfer function of the MISO reconstruction filter is expressed as: （9）。 5. The optimization design method for the modulator system according to claim 4, characterized in that, The step of designing a high signal-to-noise ratio (SNR) performance for the SDM system, along with the signal transfer function of the MISO reconstruction filter, to construct a non-convex infinite constraint optimization model for the MISO reconstruction filter, specifically includes: The objective function of the MISO reconstruction filter that ensures a high signal-to-noise ratio for the SDM system is minimized is expressed as: （11） The maximum modulus constraint of the difference between the amplitude response of the MISO reconstruction filter and the amplitude response of the desired signal transfer function is used as the constraint condition of the objective function of the MISO reconstruction filter. Based on the objective function model of the MISO reconstruction filter, a non-convex infinite-form constraint optimization model of the MISO reconstruction filter is constructed.

6. The optimization design method for the modulator system according to claim 4, characterized in that, Solving the non-convex constraint optimization model of the MIMO loop filter yields an approximate global optimal solution. Based on this approximate global optimal solution, an optimal MISO reconstruction filter is designed, specifically including: For the non-convex infinite constraint optimization model of the MISO reconstruction filter, initial values ​​of the solution are randomly generated by an evolutionary algorithm, and all constraint functions are evaluated based on the initial values. If the initial values ​​do not satisfy the constraint functions, the initial guesses of the non-convex infinite constraint optimization model of the MISO reconstruction filter are regenerated, and all constraint functions are evaluated based on the regenerated initial guesses. The iterative process is repeated until the regenerated initial guesses satisfy all constraint functions. Once the regenerated initial guesses satisfy all constraint functions, the objective function value of the MISO reconstruction filter is evaluated based on the initial guesses that satisfy all constraint functions, and the objective function value between the two iterations is calculated. Retain the initial guesses corresponding to the smaller objective function values, and repeat the iterative calculation until the objective function value is below a given threshold; regenerate the reconstructed solution through permutation and crossover operations. By eliminating local optima, an approximate global optima is obtained. Based on the approximate global optima, a MISO reconstruction filter is designed to obtain the optimal MISO reconstruction filter.

7. The optimization design method for the modulator system according to claim 1, characterized in that, The pair vectors , , The derivation is performed using the Z-transform, for vectors , , The signal transfer function (STF) and noise transfer function (NTF) of the MIMO loop filter are derived by performing a Z-transform. Specifically, the STF and NTF are simplified by using the matrix inversion lemma.

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