Multi-channel sampling circuit hybrid error modeling and cyclic calibration method and system

Abstract

The invention discloses a multi-channel sampling circuit hybrid error modeling and cyclic calibration method, and belongs to the technical field of signal sampling. The method comprises the following steps: constructing a system error model, and introducing polynomial series to realize model simplification; generating a reference signal participating in error cycle calibration according to the simplified hybrid error model; initializing a frequency response function of each sampling channel, and calculating an initial tap coefficient of an error correction module; a cascade multi-order error correction structure is used for completing reference signal error iteration correction, so that the reference signal error continuously approaches to an input signal; correcting the signal to calculate a loss function, and completing estimation and updating of a tap coefficient; and judging whether the loss function reaches a convergence condition or not until hybrid error cyclic calibration is realized. On the basis of the reference signal constructed by the hybrid error model, low-complexity online blind cycle calibration is realized, the precision and completeness of the system error model are improved, the quantity of parameters to be estimated and the number of sample points for calibration are remarkably reduced, and the calibration efficiency is improved.

Description

Technical Field

[0001] The invention belongs to the technical field of signal sampling, and relates to a method and system for mixed error modeling and cyclic calibration of a multi-channel sampling circuit. Background Art

[0002] Multi-channel sampling uses several low-cost analog-to-digital converters (ADCs) to share the pressure of the system's performance requirements such as sampling rate and analog bandwidth in the time domain or frequency domain. Among them, time interleaving technology meets the high sampling rate requirements by alternately sampling in the time domain with multiple low-speed ADCs; frequency interleaving realizes the large bandwidth sampling function by splicing several narrow-band ADCs in the frequency domain. Considering the non-ideality of analog components caused by factors such as manufacturing process and environmental stress, circuit defects such as frequency response mismatch, harmonic folding, spectrum leakage, undersampling aliasing and I / Q mirroring between sampling channels are prone to cause system performance loss. Non-blind calibration technology relies on additional training signals, interrupts the normal operation of the system from time to time, and performs offline measurement or estimation of circuit error parameters. In order not to affect the normal sampling process of the system, blind calibration methods are usually more practical. The accuracy of signal reconstruction after blind calibration depends on the accuracy and completeness of the system mixed error model, and the calibration efficiency is also closely related to the implementation complexity of the algorithm. Summary of the invention

[0003] The purpose of the present invention is to provide a method and system for modeling and cyclic calibration of mixed errors of multi-channel sampling circuits, which can significantly reduce the amount of parameters to be estimated and the number of sample points used for calibration while improving the accuracy and completeness of the mixed error model of the multi-channel sampling circuit, thereby improving the calibration efficiency of the mixed errors of the multi-channel sampling circuit.

[0004] The objective of the present invention is achieved through the following technical solutions:

[0005] The present invention discloses a method for mixed error modeling and cyclic calibration of a multi-channel sampling circuit, comprising the following steps:

[0006] S1. Introduce frequency response mismatch, harmonic folding, spectrum leakage, undersampling aliasing and I / Q image error terms between sampling channels, establish a mixed error model for multi-channel sampling circuits, and improve the accuracy and completeness of the mixed error model for multi-channel sampling circuits. Use high-order polynomial series to replace the channel function in the mixed error model for multi-channel sampling circuits, and obtain a mixed error model for multi-channel sampling circuits based on polynomial series, eliminate the coupling relationship between the model error terms, and reduce the complexity of parameter estimation.

[0007] S1.1. Introduce frequency response mismatch, harmonic folding, spectrum leakage, undersampling aliasing and I / Q mirror error terms between sampling channels, establish a mixed error model of multi-channel sampling circuits, and improve the accuracy and completeness of the mixed error model of multi-channel sampling circuits.

[0008] The local oscillator clock signal is expressed as in The harmonic number corresponding to the mth channel. ξ Im,k =(4 / π)A Im,k sin(kπ / 2) / kξ Qm,k =(4 / π)A Qm,k / k,A I(Q)m,k and φ I(Q)m,k Respectively represent the deterministic amplitude and phase modulation terms of the kth harmonic in the mth complex local oscillator. Ω m =2πf m represents the reference frequency of the mth pair of orthogonal local oscillator clocks, Δ c is the local oscillator clock jitter,

[0009] The mixed error model of the M-channel multi-channel sampling circuit is expressed as

[0010] in,

[0011]

[0012] Where p = 0, 1, ..., M-1 corresponds to the aliasing error caused by undersampling, Indicates the frequency response of the I / Q branch, G 0,m,1 (e jω ) corresponds to the frequency response mismatch between channels, G 1,m,1 (e jω ) represents the channel frequency response including I / Q image errors, where Δ s is the sampling clock jitter. G r,m,1 (e jω ),k=3,5,...,2K m -1 indicates the kth harmonic folding function. Corresponding to the spectrum leakage error caused by non-ideal filtering.

[0013] S1.2. Introduce the I-order polynomial series and establish a mixed error model of multi-channel sampling circuit based on the polynomial series.

[0014] The I-order polynomial series is introduced into the mixed error model of multi-channel sampling circuit, so that

[0015]

[0016] The mixed error model of the multi-channel sampling circuit in S1 is converted into

[0017]

[0018] Where X i (e jω )=X(e jω )(jω) i represents X(e jω), that is, x i [n]=x[n]*d i [n],d i [n] represents the impulse response of the i-th order differentiator, and d0[n]=δ[n] is the Dirac function.

[0019] S2. Construct a reference signal to replace the sampling system output signal for error calibration, reduce the number of sample points required for the calibration process, and improve the error calibration efficiency.

[0020] The reference signal construction in step S2 is implemented by reference signal construction method 1 or reference signal construction method 2. Reference signal construction method 1: add the output y[n] and the signal after π / M frequency shift to obtain z0[n], z0[n]=y[n]+y[n]e j(π / M)n ; Reference signal construction method 2: by comparing y[n] and y[n]e j(π / M)n The difference is delayed by M samples, that is, z1[n]=y[nM]-y[nM]e j(π / M)(n-M) .

[0021] The frequency domain characteristics of the two constructed reference signals are described as follows:

[0022]

[0023] in

[0024] S3, randomly setting the parameters in the multi-channel sampling circuit mixed error model based on the polynomial series established in S1 and estimating the initial value of the system to achieve initialization setting of the calibration system.

[0025] S3.1. Convert the reference signal in S2 into a reference signal in vector form as shown in formula (6):

[0026]

[0027] in: Represented in matrix form

[0028]

[0029] in

[0030] z r′ =[...,z r′ [-1],z r′ [0],z r′ [1],...] T

[0031] x=[...,x[-1],x[0],x[1],...] T

[0032]

[0033] H d,r′ =[D r′,0 ,...,D r′,i ,...,D r′,I-1 ]

[0034]

[0035] Where O is a (2M-1)×I zero matrix. w and I x Respectively represent and vector z r’ The identity matrix, D, has the same dimensions as x 0,0 =I x .

[0036] S3.2, let k' = 0, randomly select the channel parameter design value to initialize the frequency response function

[0037] S3.3. By minimizing the objective function Make an initial prediction and get in Represents the k'th cycle estimation result.

[0038] S4, cascade I'-order successive over-relaxation iterative structure to correct the approximate value of the input signal x[n].

[0039] Let k'=k'+1. Keep The approximate value of the input signal x[n] is corrected by cascading the I'th order successive super-relaxation iterative structure. The i-th correction formula in the k'th cycle estimation is:

[0040]

[0041] Wherein i'=1,2,…,I'=1, η∈(1,2) is the relaxation factor, and L represents the number of transverse filter taps of the successive super-relaxation iterative structure. represents the output of the i-th order correction stage, and

[0042] S5. Construct the loss function update as follows with respect to x and w r’ The loss function gradient is calculated based on the joint optimization problem, and the calibration system parameters are updated.

[0043] S5.1. Construct the loss function update as follows with respect to x and w r’ The joint optimization problem

[0044]

[0045] The loss function is defined as follows:

[0046]

[0047] S5.2, Reset Update the inverse Hessian approximation

[0048]

[0049] Depend on calculate

[0050]

[0051] The gradient of the loss function is

[0052]

[0053] S5.3, let k'=k'+1. Keep Unchanged, using the error signal

[0054]

[0055] Update the following equation

[0056]

[0057] in is the step size determined by backtracking the straight line search.

[0058] Update the calibration system parameters according to equations (11) to (17).

[0059] S6. Establish an iterative termination mechanism based on convergence evaluation, perform convergence check on the loss function updated in step S5, terminate the loop calibration if the loss function is less than the set threshold, and obtain the updated calibration system parameters, otherwise return to S4. Improve the efficiency and stability of the calibration system according to the updated calibration system parameters.

[0060] Check the convergence of the loss function. If the loss function is less than the set threshold, the calibration cycle is terminated to obtain the updated calibration system parameters. At this time, k'=K'-1. Otherwise, return to S4.

[0061] The invention discloses a multi-channel sampling circuit mixed error modeling and cyclic calibration system, which is used to implement the multi-channel sampling circuit mixed error cyclic calibration method as described above. The multi-channel sampling circuit mixed error modeling and cyclic calibration system includes a reference signal generation module, a calibration initialization module, an error iteration correction module, a parameter estimation update module, and a convergence check module.

[0062] A reference signal generation module, used for generating a reference signal for error cycle calibration according to a mixed error model;

[0063] A calibration initialization module is used to initialize the frequency response function of each sampling channel and calculate the initial tap coefficients of the transversal filter of the error iteration correction module;

[0064] The error iterative correction module is used to iteratively correct the reference signal error through a cascaded multi-order successive super-relaxation structure, so that it continuously approaches the input signal;

[0065] A parameter estimation and updating module is used to update the loss function according to the output signal of the error iteration correction module, estimate the parameters of the error iteration correction module, and use the estimation result to update the tap coefficients of its transverse filter;

[0066] The convergence check module determines whether the cyclic calibration system is terminated by evaluating the convergence of the loss function.

[0067] Beneficial effects:

[0068] 1. The present invention discloses a mixed error modeling method for a multi-channel sampling circuit, which introduces frequency response mismatch, harmonic folding, spectrum leakage, undersampling aliasing and I / Q mirror error terms and high-order polynomial series between sampling channels. Compared with the classic multi-channel sampling modeling method, it can improve the model fitting accuracy and completeness while eliminating the coupling relationship between model error terms, reduce the amount of mismatch parameters to be estimated by half, reduce the design complexity of the error correction structure, and thus improve the accuracy and efficiency of the mixed error cyclic calibration of the multi-channel sampling circuit.

[0069] 2. The existing calibration method uses the sampled output signal of the sampling system for error calibration, which has a large amount of data and high calculation cost. The present invention discloses an efficient reference signal generation module, which constructs a reference signal for error calibration through basic operations such as frequency shift, addition, and delay, reduces the number of sample points required for the entire calibration process by half, and improves the calibration efficiency of mixed errors of multi-channel sampling circuits;

[0070] 3. The present invention discloses a parameter estimation and updating method based on a loss function, which constructs the loss function update as a joint optimization problem, and updates the parameters of the calibration system by calculating the gradient of the loss function. At the same time, a convergence check mechanism is introduced to improve the working efficiency and stability of the calibration system.

[0071] 4. The present invention discloses a mixed error cyclic calibration method for a multi-channel sampling circuit. It does not require a training signal and uses the real-time correction results of an online data stream to update the error compensation structure parameters, thereby realizing a mixed error cyclic blind calibration of a multi-channel sampling circuit with "foreground data correction + background parameter update". BRIEF DESCRIPTION OF THE DRAWINGS

[0072] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0073] Figure 1 Mixed error cyclic blind calibration workflow for multi-channel sampling circuits;

[0074] Figure 2 The convergence of the signal reconstruction residual and the parameter estimation residual with the number of cyclic calibration in the embodiment;

[0075] Figure 3 The effect comparison of the signal before and after the cyclic calibration in the embodiment is shown in FIG. DETAILED DESCRIPTION

[0076] The present invention is further described and described in detail below with reference to the accompanying drawings and embodiments.

[0077] like Figure 1 As shown, the present embodiment discloses a method for mixed error modeling and cyclic calibration of a multi-channel sampling circuit, and the specific implementation steps are as follows:

[0078] S1. Introduce sampling system error terms and high-order polynomial series, and establish a mixed error model of multi-channel sampling circuits based on polynomial series.

[0079] S1.1. Establish a mixed error model for a 2-channel sampling circuit:

[0080]

[0081] in,

[0082]

[0083] Among them, the channel response H I / Qm (e jω ) Using the second-order low-pass elliptical characteristic model, the peak-to-peak ripple R peak =1dB, minimum stopband attenuation R stop =40dB. Deterministic gain mismatch A I(Q)m,kand phase parameter φ I(Q)m,k Select any value from the interval [0.995, 1.005] and [-0.005, 0.005] respectively; the local oscillator clock jitter and sampling clock jitter are set to Δ c =Δ s =0.01.

[0084] S1.2. Introduce a 16th-order polynomial series, i.e., I=16, and establish a mixed error model of a multi-channel sampling circuit based on the polynomial series.

[0085] Under the condition of I = 16, we have:

[0086]

[0087] The model in S1 is converted to

[0088]

[0089] Where X i (e jω ) corresponds to x i [n]=x[n]*d i [n], the order L of the first-order differentiator d1[n] d Take 40 (the i-th order differentiator is obtained by cascading i d1[n]).

[0090] S2. Construct a reference signal.

[0091] The two reference signals designed are as follows:

[0092]

[0093] The corresponding frequency domain feature is

[0094]

[0095] in

[0096] S3, calibration system initialization

[0097] S3.1. Convert the reference signal into vector form:

[0098]

[0099] in

[0100] w r′ =[w r′,0 ,...,w r′,i ,...,w r′,15 ] T

[0101] X d[n]=[x0[n],...,x i [n],...,x 15 [n]] T

[0102] Represented in matrix form

[0103]

[0104] z r′ =[...,z r′ [-1],z r′ [0],z r′ [1],...] T

[0105] x=[...,x[-1],x[0],x[1],...] T

[0106]

[0107] H d,r′ =[D r′,0 ,...,D r′,i ,...,D r′,15 ]

[0108]

[0109] S3.2, let k'=0, randomly initialize λ r,i λ0=[0.388,2.213,-4.173,5.726,-5.362,4.508,-2.659,1.620,-0.647,0.309,-0.083,0.032,-0.005,0.001,-1.44×10 -4 ,4.13×10 -5 ], λ1=[-0.136,0.416,-0.722,0.889,-0.839,0.645,-0.403,0.220,-0.097,0.041,-0.012,0.004,-8.38×10 -4 ,2.16×10 -4 ,-2.24×10 -5 ,4.58×10 -6 ] to initialize the frequency response function

[0110] S3.3. Minimize the objective function get Initial prediction =[1.156,0.504,-0.944,1.291,-1.169,0.976,-0.548,0.334,-0.124,0.061,-0.015,-0.006,9.21×10 -4 ,-3.37×10 -4 ,-2.28×10 -5 ,7.62×10 -6 ].

[0111] S4, iterative error correction: set k'=k'+1. Keep The approximation value of the input signal x[n] is corrected by cascading an I'=2nd order successive super-relaxation iterative structure, the relaxation factor η is 1.135, and the number of transversal filter taps L is 9.

[0112] S5. Calculate the loss function gradient and update the calibration system parameters:

[0113] S5.1. Loss function update:

[0114]

[0115] S5.2, Reset Update the inverse Hessian approximation

[0116]

[0117] Depend on calculate

[0118]

[0119] The gradient of the loss function is

[0120]

[0121] S5.3, let k'=k'+1. Keep Unchanged, using the error signal

[0122]

[0123] Update the following equation

[0124]

[0125] S6. Convergence check: If the loss function is less than the set threshold, the calibration process is terminated, and k'=K'-1 is recorded at this time. Otherwise, return to S4. In the embodiment, the convergence of the signal reconstruction residual and the parameter estimation residual with the number of cyclic calibration times is as follows: Figure 2 shown.

[0126] The embodiment discloses a multi-channel sampling circuit mixed error modeling and cyclic calibration system, which is used to implement the multi-channel sampling circuit mixed error cyclic calibration method as described above. The effect comparison of the signal in the sampling circuit of the embodiment before and after the cyclic calibration is as follows: Figure 3 The multi-channel sampling circuit mixed error modeling and cyclic calibration system comprises a reference signal generation module, a calibration initialization module, an error iteration correction module, a parameter estimation update module, and a convergence check module.

[0127] A reference signal generation module, used for generating a reference signal for error cycle calibration according to a mixed error model;

[0128] A calibration initialization module is used to initialize the frequency response function of each sampling channel and calculate the initial tap coefficients of the transversal filter of the error iteration correction module;

[0129] The error iterative correction module is used to iteratively correct the reference signal error through a cascaded multi-order successive super-relaxation structure, so that it continuously approaches the input signal;

[0130] A parameter estimation and updating module is used to update the loss function according to the output signal of the error iteration correction module, estimate the parameters of the error iteration correction module, and use the estimation result to update the tap coefficients of its transverse filter;

[0131] The convergence check module determines whether the cyclic calibration system is terminated by evaluating the convergence of the loss function.

[0132] The specific description above further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above 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 in the scope of protection of the present invention.

Claims

1. A method and system for modeling mixed errors and cyclic calibration of multi-channel sampling circuits, characterized in that: The following steps are included: S1. Introduce frequency response mismatch, harmonic folding, spectrum leakage, undersampling aliasing and I / Q image error terms between sampling channels, establish a mixed error model of multi-channel sampling circuits, and improve the accuracy and completeness of the mixed error model of multi-channel sampling circuits; use high-order polynomial series to replace the channel function in the mixed error model of multi-channel sampling circuits, obtain a mixed error model of multi-channel sampling circuits based on polynomial series, and eliminate the coupling relationship between the error terms of the model; S2, construct a reference signal to replace the sampling system output signal for error calibration, reduce the number of sample points required for the calibration process, and improve the error calibration efficiency; S3, randomly setting the parameters in the multi-channel sampling circuit mixed error model based on the polynomial series established in S1 and estimating the initial value of the system to achieve the initialization setting of the calibration system; S4, cascade I'-order successive super-relaxation iterative structure to correct the approximate value of the input signal x[n]; S5. Construct the loss function update as follows with respect to x and w r’ A joint optimization problem is solved, and a loss function gradient is calculated based on the joint optimization problem to update the calibration system parameters; S6. Establish an iterative termination mechanism based on convergence evaluation, and perform a convergence check on the loss function updated in step S5. If the loss function is less than the set threshold, terminate the loop calibration and obtain the updated calibration system parameters. Otherwise, return to S4; improve the efficiency and stability of the calibration system according to the updated calibration system parameters.

2. A method for hybrid error modeling and cyclic calibration of a multi-channel sampling circuit as claimed in claim 1, characterized in that: The implementation method of step S1 is: S1.1, introduce frequency response mismatch, harmonic folding, spectrum leakage, undersampling aliasing and I / Q image error terms between sampling channels, and establish a mixed error model for multi-channel sampling circuits; The local oscillator clock signal is expressed as Where Ω m =2πf m represents the reference frequency of the mth pair of orthogonal local oscillator clocks, The harmonic number corresponding to the mth channel; Δ c is the local oscillator clock jitter, ξ Im,k =(4 / π)A Im,k sin(kπ / 2) / k,ξ Qm,k =(4 / π)A Qm,k / k,A I(Q)m,k and φ I(Q)m,k denote the deterministic amplitude and phase modulation terms of the k-th harmonic in the m-th complex local oscillator respectively; The mixed error model of the M-channel multi-channel sampling circuit is expressed as in, Where p = 0, 1, ..., M-1 corresponds to the aliasing error caused by undersampling, Indicates the frequency response of the I / Q branch, G 0,m,1 (e jω ) corresponds to the frequency response mismatch between channels, G 1,m,1 (e jω ) represents the channel frequency response including I / Q image errors, where Δ s is the sampling clock jitter; G r,m,1 (e jω ),k=3,5,...,2K m -1 indicates the kth harmonic folding function; Corresponding to the spectrum leakage error caused by non-ideal filtering; S1.2, introduce the I-order polynomial series and establish a mixed error model of multi-channel sampling circuit based on the polynomial series; The I-order polynomial series is introduced into the mixed error model of multi-channel sampling circuit, so that The mixed error model of the multi-channel sampling circuit in S1 is converted into Where X i (e jω )=X(e jω )(jω) i represents X(e jω ), that is, x i [n]=x[n]*d i [n],d i [n] represents the impulse response of the i-th order differentiator, and d0[n]=δ[n] is the Dirac function.

3. A method for hybrid error modeling and cyclic calibration of a multi-channel sampling circuit as claimed in claim 2, characterized in that: The reference signal construction in step S2 is implemented by reference signal construction method 1 or reference signal construction method 2. Reference signal construction method 1: add the output y[n] and the signal after π / M frequency shift to obtain z0[n], z0[n]=y[n]+y[n]e j (π / M)n ; Reference signal construction method 2: by comparing y[n] and y[n]e j(π / M)n The difference is delayed by M samples, that is, z1[n] = y[nM] - y[nM]e j(π / M)(n-M) ; The frequency domain characteristics of the two constructed reference signals are described as follows: in 4. A method for hybrid error modeling and cyclic calibration of a multi-channel sampling circuit as claimed in claim 3, characterized in that: The implementation method of step S3 is: S3.

1. Convert the reference signal in S2 into a reference signal in vector form as shown in formula (6): in: in: Represented in matrix form With r′ =[...,z r′ [-1],from r′ [0],from r′ [1],...] T x=[...,x[-1],x[0],x[1],...] T H d,r′ =[D r′,0 ,...,D r′,i ,...,D r′,I-1 ] Where O is a (2M-1)×I zero matrix; I w and I x Respectively represent and vector z r’ The identity matrix, D, has the same dimensions as x 0,0 =I x ; S3.2, let k' = 0, randomly select the channel parameter design value to initialize the frequency response function S3.

3. By minimizing the objective function right Make an initial prediction and get in Represents the k'th cycle estimation result.

5. A method for hybrid error modeling and cyclic calibration of a multi-channel sampling circuit as claimed in claim 4, characterized in that: The implementation method of step S4 is: Let k'=k'+1; keep The approximate value of the input signal x[n] is corrected by cascading the I'th order successive super-relaxation iterative structure. The i-th correction formula in the k'th cycle estimation is: Where i'=1,2,…,I'=1, η∈(1,2) is the relaxation factor, and L represents the number of transverse filter taps of the successive super-relaxation iterative structure; represents the output of the i-th order correction stage, and 6. A method for hybrid error modeling and cyclic calibration of a multi-channel sampling circuit as claimed in claim 5, characterized in that: The implementation method of step S5 is: S5.

1. Construct the loss function update as follows with respect to x and w r’ The joint optimization problem The loss function is defined as follows: S5.2, Reset Update the inverse Hessian approximation Depend on calculate The gradient of the loss function S5.3, let k'=k'+1; keep Unchanged, using the error signal Update the following equation in is the step size determined by backtracking straight line search; Update the calibration system parameters according to equations (11) to (17).

7. A method for hybrid error modeling and cyclic calibration of a multi-channel sampling circuit as claimed in claim 6, characterized in that: In step S6, Check the convergence of the loss function. If the loss function is less than the set threshold, the calibration cycle is terminated to obtain the updated calibration system parameters. At this time, k'=K'-1. Otherwise, return to S4.

8. A system for mixed error modeling and cyclic calibration of a multi-channel sampling circuit, used to implement a mixed error modeling and cyclic calibration method of a multi-channel sampling circuit as claimed in claim 1 or 2, characterized in that: It includes a reference signal generation module, a calibration initialization module, an error iteration correction module, a parameter estimation update module, and a convergence check module; A reference signal generation module, used for generating a reference signal for error cycle calibration according to a mixed error model; A calibration initialization module is used to initialize the frequency response function of each sampling channel and calculate the initial tap coefficients of the transversal filter of the error iteration correction module; The error iterative correction module is used to iteratively correct the reference signal error through a cascaded multi-order successive super-relaxation structure, so that it continuously approaches the input signal; A parameter estimation and updating module is used to update the loss function according to the output signal of the error iteration correction module, estimate the parameters of the error iteration correction module, and use the estimation result to update the tap coefficients of its transverse filter; The convergence check module determines whether the cyclic calibration system is terminated by evaluating the convergence of the loss function.

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