CN1138369C - An Interference Suppression Method for Code Division Multiple Access System - Google Patents

Abstract

The present invention discloses an interference suppression method of a code division multiple access system. The statistics of estimating signal errors is added into the method on the basis of parallel interference suppression; weighted values of regenerative signals of all levels are controlled by checking corresponding data tables through the statistic of first orders and second orders of the error signals; the estimation of the variation of the weighted values of signals can be quickly and effectively controlled; the time delay of the signal process of a multi-user detector is short, the speed of the cancellation process of interference is high, and the detection performance can be quickly stabilized. Simultaneously, a multi-stage detector of the present invention is applied to realize simple structure, the complexity of equipment is low, and the hardware realization is convenient.

Description

A kind of disturbance restraining method of code division multiple access system

The present invention relates to the interference mitigation technology of CDMA (code division multiple access) mobile communication system.

In CDMA mobile communication system, be in the different mobile subscribers under the same base stations control because the difference of transmission channel can't guarantee its orthogonality between each subscriber signal, therefore, each subscriber signal that receives at base station receiver can exist multiple access to disturb.Multiple access disturbs (MAI) directly restricting the cdma system performance.Disturb in order to suppress multiple access, people have proposed many disturbance restraining methods, document " adaptive multistage parallel interference inhibition method in the cdma system " (Guaqiang Xue for example, Jianfeng Weng, Tho Le-Ngoc, Sofiene Tahar " Adaptive Multistage ParallelInterference Cancellation for CDMA " IEEE Journal on Selected Areas inCommun.Vol.17, No.10, Oct.1999) the adaptive weighted multistage parallel disturbance restraining method that proposes in the literary composition, adopted estimate after received signal and each the user's weighting regenerated signal and between the criterion of mean square error minimum determine weights, promptly by all just minimum (LMS) algorithm iteration realize the weights in each each path of user.Yet this method needs repeatedly iterative processing in realization, therefore needs certain convergence time, and convergence rate is slow, and systematic function prolongs stabilization time, has reduced the efficient of system.

At the problems referred to above, the purpose of this invention is to provide a kind of disturbance restraining method of code division multiple access system, use this method can make the systematic function fast and stable, and then improve the receptivity of system.

For achieving the above object, the technical solution used in the present invention is: a kind of disturbance restraining method of code division multiple access system, this method comprises following process: receiver at first is transformed into baseband signal with high-frequency signal, then, mask data information and Synchronization Control parameter information, from the Synchronization Control parameter information, extract three channel parameters, one is the call parameter that provides for each subscriber signal demodulation despreading, one is the parameter that provides for the regeneration interference signal, and another is the adjustment parameter that provides for the error statistics tables of data; After obtaining above-mentioned necessary channel parameter information, each subscriber signal is carried out despreading and demodulation, after carrying out decision process and weighted, bear each user's modulated spread spectrum signal again; Simultaneously, deduct interference user modulation signal after the regeneration with the received signal after postponing, carrying out the one-level multiaccess interference cancellation handles, the error signal that utilization offsets after the processing is carried out the error statistics processing, calculate its single order statistic and second-order statistic, search corresponding tables of data 1 and tables of data 2 according to above-mentioned single order and second-order statistic, the data that obtain with tabling look-up obtain the controlled quentity controlled variable that this grade weights are adjusted size, utilize the weighted operation of the size adjustment subscriber signal of above-mentioned controlled quentity controlled variable control weights, last, judge whether interference cancellation progression is zero, if it is non-vanishing, then enter the circulation of next stage interference cancellation, if zero, just detect desired user data symbol.

Above described utilization offset error signal after the processing and carry out error statistics and handle, be error signal to be carried out statistical property handle, promptly obtain the average and the variance of error signal, wherein, the average of error signal is: $E\left(e_{k,m}^{\left(s\right)}\left(t\right)\right)=E[r\left(t\right)-\underset{j=1}{\overset{K}{\mathrm{\Σ}}}{w}_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}\left(t\right)]=E\left[\underset{J=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}(t-{\widehat{\mathrm{\τ}}}_j)\right];$ Variance is: $\mathrm{var}\left(e_{k,m}^{\left(s\right)}\left(t\right)\right)=\mathrm{var}\left[n\left(t\right)\right]+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}(t-{\mathrm{\τ}}_j))\right].$ ${=\mathrm{\σ}}^2+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-\mathrm{\τ})-w_{j,m}^{\left(s\right)}{\hat{s}}_j(t-{\widehat{\mathrm{\τ}}}_j))\right]$

Above described tables of data 1 have the expectation information of the first-order statistics average of error signal, tables of data 2 has the expectation information of the second-order statistics variance of error signal.The data of storage obtain through training in described tables of data 1 and the tables of data 2.

Above the described usefulness controlled quentity controlled variable that the data that obtain obtain this grade weights adjustment size of tabling look-up obtain according to following method: weights are adjusted the controlled quentity controlled variable of size $w_{k,m}^{\left(s\right)}=w_{k,m-1}^{\left(s\right)}+f_1\left(E\right[e_{k,m}^{\left(s\right)}\left(t\right)\left]\right)+f_2\left(\mathrm{var}\right[e_{k,m}^{\left(s\right)}\left(t\right)\left]\right).$

The technical scheme that is adopted by the invention described above the invention has the advantages that as can be seen:

(1) because disturbing in the inhibition at multistage parallel, the present invention adopted the error second-order statistics to handle the weighted value of controlling regenerated signal, the Euclidean distance that signal is estimated is little, the weighted value that obtains is accurate, the multiple access that can effectively eliminate between each user disturbs, the systematic function capacity is improved, and simultaneously multistage detector progression is required few.

(2) because the present invention can make that the multi-user detector processing delay time is shorter by the method for fast zoom table control weighted value.Effectively guaranteed the real-time of message transmission, it is fast to carry out interference-cancellation process speed, and the weighted value that can control estimated signal fast and effectively changes, and makes to detect performance and can stablize rapidly.Experimental result shows, employing is not more than the system that 3 grades interference suppresses detector, and system detects performance and improves a lot, and therefore multistage detector implementation structure is simple relatively, and the equipment complexity is lower.

(3) because the data in the tables of data that the present invention adopts obtain through training, can carry out necessary adjustment and modification, make system can adapt to complex environment according to system user ruuning situation.

The present invention is described in further detail below in conjunction with drawings and Examples.

Fig. 1 is that statistical error control weighing parallel disturbs inhibition detector entire block diagram;

Fig. 2 is the handling process schematic diagram of the inventive method.

At first the background of the present invention and foundation thereof is carried out brief description in conjunction with DS-CDMA system (direct sequence spread spectrum codes division multiple access communication system).

In the DS-CDMA communication system, each user signal source data obtains data flow b after modulating through binary phase-shift keying (BPSK) _kAre multiplied each other and addition in they and frequency expansion sequence c (n) separately again, then, baseband signal is transformed to high frequency, and signal is launched with antenna.At receiving terminal, at first, will detect with reception antenna through the high-frequency signal that wireless mobile channel is propagated, again these high-frequency signals are transformed into baseband signal, then, to subscriber signal despreading and demodulation that expectation receives, obtain needed user profile with band receiver of base at last.Because the multiple access that traditional band receiver of base can't overcome between each user disturbs, therefore parallel interference inhibition technology is proposed.Though general parallel interference suppresses the detection performance that detection technique can improve signal, because the estimation of regenerated signals at different levels is that inclined to one side estimation is arranged, the existence of error makes signal detection performance stable inadequately.Though each regenerated signal weighted can reduce estimated bias, the choosing and become a big problem of best weight value.The present invention proposes to adopt the weighted value of the multistage regenerated signal that statistics regenerated signal and received signal error control, and improves the detection performance of system.Principle of the present invention is: at first, multiple user signals detects simultaneously, each user's modulated spread spectrum signal of regenerating, carry out single order and second-order statistics processing with the error of estimated signal to received signal, look into corresponding tables of data according to statistical value again, data sum value is used to control each user's weighted value in two tables, behind interference cancellation, detects the desired user signal.Experiment shows that the detection performance of the multistage parallel noise killer of principle is better than the detection performance of traditional receiver matched filter greatly on the foundation, compares with other parallel interference inhibitor, have simple in structure, progression is few relatively, and processing speed is fast, steady performance.

Below the principle and the enforcement of institute of the present invention foundation are set forth.

In the cdma system of reality, the estimation that multiple access disturbs is not right-on, and it is a kind of inclined to one side estimation that has.For example for the DS-CDMA mobile communication system, the signal of receiver inlet is:

Here T is a symbol period, and i represents i symbol launching.ω wherein _k, _k, τ _kThe carrier angular frequencies of representing k user respectively, phase place and delay; c _k(t) be its spread-spectrum signal.ψ _T(t) be baseband signal waveform.N (t) is that average is zero, and variance is σ ²White Gaussian process.Adopt matched filter, can after demodulation expands, obtain each user's sufficient statistic, and can distinguish estimate symbol and signal power.Some parameter relies on and estimates to obtain in practice, but can't accurately learn.Adopt the multistage parallel interference mitigation technology of statistical error control, reduced evaluated error in the decision statistic amount process.Disturb the inhibition detector for multistage parallel, the detection performance of s level depends on the detection performance of s-1 level.And along with the increase of progression, the performance of detector can improve constantly.For estimated signal be through k user of s＞0 grade:

 wherein _kWith

It is the estimated value of k subscriber signal data bit and power.Just can effectively suppress the multiple access interference if obtain best weights.Adopt estimate after received signal and each the user's weighting regenerated signal and between minimum Europe apart from square method determine the weights method.Promptly seek the mean error that makes between received signal and the regenerated signal estimation $\mathrm{\ϵ}={\∫}_0^{T_b}|r\left(t\right)-\underset{k=1}{\overset{K}{\mathrm{\Σ}}}\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{w}_{k,l}{\mathrm{\α}}_{k,l}\sqrt{P/2}{\hat{b}}_k{\hat{s}}_k\left(t\right){|}^2\mathrm{dt}$ Be weighted value hour.By to received signal and the error signal between the estimated signal carry out statistical disposition and determine weights.Suppress (PIC) algorithm according to parallel interference, received signal can be expressed as with the instantaneous error of regeneration estimated signal: $e_{k,m}^{\left(s\right)}\left(t\right)=r\left(t\right)-\underset{j=1}{\overset{K}{\mathrm{\Σ}}}{w}_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}\left(t\right)$ $=n\left(t\right)+\underset{j=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}(t-{\mathrm{\τ}}_j))$ Error signal is carried out statistical property handle, its single order statistic average and second-order statistic variance can be expressed as respectively: $E\left(e_{k,m}^{\left(s\right)}\left(t\right)\right)=E[r\left(t\right)-\underset{j=1}{\overset{K}{\mathrm{\Σ}}}{w}_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}\left(t\right)]=E\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}(t-{\widehat{\mathrm{\τ}}}_j)\right]$ $\mathrm{var}\left(e_{k,m}^{\left(s\right)}\left(t\right)\right)=\mathrm{var}\left[n\left(t\right)\right]+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}\right(s_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}(t-{\mathrm{\τ}}_j)\left)\right]$ $={\mathrm{\σ}}^2+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}\right(s_j(t-\mathrm{\τ})-w_{j,m}^{\left(s\right)}{\hat{s}}_j(t-{\widehat{\mathrm{\τ}}}_j)\left)\right]$ The average of error signal has reflected the scope that estimated bias is bigger in a symbol, and the variance of error signal has reflected the little situation of change of evaluated error.Simultaneously,, improve the approximation ratio of estimated signal and received signal, adopt single order and second order database look-up method to determine weighted value in order to reduce operand.Work out two weight increment control data tables at first in advance, it can be according to signal to noise ratio, and community user number, the error range that offsets sum of series desired signal and estimated signal are determined and revised.Data owner will reflect the first-order statistics expectation information of error in the table 1.Data owner will reflect the second-order deviation information of error in the table 2.Data are that a large amount of training obtain in each table, and it can carry out necessary adjustment and modification according to system user ruuning situation and each user's Signal to Interference plus Noise Ratio.Then, combine by the data in table 1 and the table 2 and just can determine the increment of the weighting of wanting. $w_{k,m}^{\left(s\right)}=w_{k,m-1}^{\left(s\right)}+f_1\left(E\right[e_{k,m}^{\left(s\right)}\left(t\right)\left]\right)+f_2\left(\mathrm{var}\right[e_{k,m}^{\left(s\right)}\left(t\right)\left]\right)$ By from received signal, deducting each user after regeneration interference signal sum of regenerating, obtain the new received signal of desired user.In realization, for the simplification of structure, deduct whole user's estimated values with received signal, add that desired user signal estimated value finishes this computing.For k user, the received signal of this moment is ${\hat{r}}_k^{\left(s\right)}\left(t\right)=r\left(t\right)-\underset{j\≠k}{\overset{K}{\mathrm{\Σ}}}{w}_j{\hat{s}}_j^{\left(s\right)}\left(t\right)$

When s=0, ${\hat{s}}_k^{\left(0\right)}\left(t\right)=0,k=\mathrm{1,2},...,K.$ K user in the sufficient statistic (behind the Interference Cancellation through the s-1 level) of m position s level is:

Wherein be

Estimated signal regeneration.

Being applied to spreading factor with the present invention below is that the DS-CDMA system is that example is discussed enforcement of the present invention.This system spread-spectrum sequence adopts the orthogonal variable frequency expansion sequence, and adds scrambler.Tables of data 1 and tables of data 2 be the average and the mean-square value of corresponding statistical error respectively.They can obtain by a large amount of training, and make amendment and adjust according to system user number and signal to noise ratio.

With reference to figure 1.Fig. 1 uses parallel interference of the present invention to suppress the detector entire block diagram.Among the figure, receiver adopts matched filter to carry out the demodulation despreading at first to received signal and handles 1; Then, judgement data bit information 2, and to after detection signal weighting 3 and each subscriber signal 4 of regenerating, each user is offseted processing 5, at its output, one road signal carries out error signal statistical disposition 6 with estimated signal to received signal, first-order statistics signal for error, the data 7 in 1 of can tabling look-up obtain the mapping value of signal average, second-order statistics signal for error, the data 8 in 2 of can tabling look-up obtain the mapping value of signal variance, the weighted value 9 of both sums control regenerated signals.Another road and each signal to be detected are done further processing.Error signal is added subscriber signal to be detected 10, this signal is being detected 11, just finish the one-level multiaccess interference cancellation and handle.Statistical error control weights PIC model comprises input, sign estimation, weighted, and signal regeneration is handled, and interference cancellation, error statistics are tabled look-up and processing such as weights control.Through second level PIC algoritic module be 12 and 13 3 grades of interference cancellations of third level PIC algoritic module handle, just detect desired user data symbol.

The multistage parallel of statistical error control disturbs and suppresses detector operational processes flow process as shown in Figure 2.In Fig. 2, receiver at first is transformed into high-frequency signal baseband signal 1, then, mask data information 2 Synchronization Control parameter informations 3, from the Synchronization Control parameter information, extract three channel parameters 4, one is the call parameter that provides for each subscriber signal demodulation despreading, and one is the parameter that provides for the regeneration interference signal, and another is the adjustment parameter that provides for the error statistics tables of data; After obtaining above-mentioned necessary channel parameter information, each subscriber signal is carried out despreading and demodulation 5, after carrying out decision process and weighted 6, bear each user's spread-spectrum signal 7 again; Simultaneously, deduct interference user signal after the regeneration, carry out the one-level multiaccess interference cancellation and handle 8 with the received signal after postponing.It has two-way output, one the tunnel is output as the error signal after offseting, utilize statistical module that it is carried out error statistics calculation process 9, according to resulting first-order statistics amount and second-order statistics value, search data 10 corresponding in corresponding tables of data 1 and the tables of data 2, with tabling look-up to such an extent that show numerical value addition 11, obtain respectively the regenerate controlled quentity controlled variable of estimated signal weights size of this grade.Utilize the size of above-mentioned this grade of control weighted value, to adjust the weighted operation of subscriber signal.Another road signal that offsets module is desired user estimated signal and error signal sum.It is detected, just obtain the Temporal Data signal of this grade desired user.At last, judge whether interference cancellation progression is 0 12,, then enter the circulation of next stage interference cancellation if non-vanishing, if zero, desired signal 13 expanded with regard to demodulation; At last, detect desired user data symbol 14 with decision device.

The error signal that above-mentioned utilization offsets after the processing is carried out the error statistics processing, is error signal to be carried out statistical property handle, and promptly obtains the average and the variance of error signal, and wherein, the average of error signal is obtained by following formula: $E\left(e_{k,m}^{\left(s\right)}\left(t\right)\right)=E[r\left(t\right)-\underset{j=1}{\overset{K}{\mathrm{\Σ}}}{w}_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}\left(t\right)]=E[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}(t-{\widehat{\mathrm{\τ}}}_j)];$

The variance of error signal is obtained by following formula: $\mathrm{var}\left(e_{k,m}^{\left(s\right)}\left(t\right)\right)=\mathrm{var}\left[n\left(t\right)\right]+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(s\right)}{\hat{s}}_j^{\left(s\right)}(t-{\mathrm{\τ}}_j))\right]$ $={\mathrm{\σ}}^2+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}(s_j(t-\mathrm{\τ})-w_{j,m}^{\left(s\right)}{\hat{s}}_j(t-{\widehat{\mathrm{\τ}}}_j))\right]$

The above-mentioned usefulness controlled quentity controlled variable that the data that obtain obtain this grade weights adjustment size of tabling look-up is to obtain according to following formula: promptly weights are adjusted the controlled quentity controlled variable of size $w_{k,m}^{\left(s\right)}=w_{k,m-1}^{\left(s\right)}+f_1\left(E\right[e_{k,m}^{\left(s\right)}\left(t\right)\left]\right)+f_2\left(\mathrm{var}\right[e_{k,m}^{\left(s\right)}\left(t\right)\left]\right).$

Emulation can be verified the stability and the reliability of this detector by experiment.It is to adopt the chip-level interference cancellation that the multistage interference that the present invention adopts suppresses detector, and the control of symbol level weights can be finished a plurality of users and detect simultaneously, computation complexity only with the linear proportional relation of number of users.

Claims (6)

1, a kind of interference suppression method of code division multiple access system, it is characterized in that, this method comprises the following process: receiver first transforms high-frequency signal into baseband signal, then, separates data information and synchronous control parameter information, from synchronous control Three channel parameters are extracted from the parameter information, one is the necessary parameter provided for the demodulation and despreading of each user signal, the other is the parameter provided for the regenerated interference signal, and the other is the adjustment parameter provided for the error statistics table; After necessary channel parameter information, despread and demodulate each user signal, and regenerate the spread spectrum modulation signal of each user after judgment processing and weighting processing; then, subtract the regenerated interference user signal from the delayed received signal Spread spectrum modulated signal, perform first-level multiple access interference cancellation processing, use the error signal after cancellation processing to perform statistical processing, calculate its first-order statistics and second-order statistics, and search according to the above-mentioned first-order and second-order statistics Corresponding data table 1 and data table 2, use the data found in the two tables to obtain the control amount of weight adjustment at this level, use the size of the above-mentioned control weight to adjust the weighted processing operation of the user signal, and finally, judge Whether the level of interference cancellation is zero, if it is not zero, enter the next level of interference cancellation cycle, if it is zero, detect the desired user data symbol. 2, the interference suppression method of CDMA system according to claim 1, it is characterized in that: the error signal after described utilizing cancellation processing is carried out error statistical processing, is to error signal is carried out statistical characteristic processing, promptly obtains The mean and variance of the error signal, where the mean of the error signal is: $\mathrm{E.}\left(e_{k,m}^{\left(\mathrm{the\; s}\right)}\left(t\right)\right)=\mathrm{E.}[r\left(t\right)-\underset{j=1}{\overset{K}{\mathrm{\Σ}}}{w}_{j,m}^{\left(\mathrm{the\; s}\right)}{\widehat{\mathrm{the\; s}}}_j^{\left(\mathrm{the\; s}\right)}\left(t\right)]=\mathrm{E.}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}({\mathrm{the\; s}}_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(\mathrm{the\; s}\right)}{\widehat{\mathrm{the\; s}}}_j^{\left(\mathrm{the\; s}\right)}(t-{\widehat{\mathrm{\τ}}}_j)\right];$ The variance is: $\mathrm{var}\left(e_{k,m}^{\left(\mathrm{the\; s}\right)}\left(t\right)\right)=\mathrm{var}\left[\mathrm{no}\left(t\right)\right]+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}({\mathrm{the\; s}}_j(t-{\mathrm{\τ}}_j)-w_{j,m}^{\left(\mathrm{the\; s}\right)}{\widehat{\mathrm{the\; s}}}_j^{\left(\mathrm{the\; s}\right)}(t-{\mathrm{\τ}}_j))\right],$ $={\mathrm{\σ}}^2+\mathrm{var}\left[\underset{j=1}{\overset{K}{\mathrm{\Σ}}}({\mathrm{the\; s}}_j(t-\mathrm{\τ})-w_{j,m}^{\left(\mathrm{the\; s}\right)}{\widehat{\mathrm{the\; s}}}_j(t-{\widehat{\mathrm{\τ}}}_j))\right]$ Wherein, k represents the kth user, and k is an integer; m represents the mth symbol, and m is an integer; t represents time, and t>=0. 3. The interference suppression method of CDMA system according to claim 1, characterized in that: said data table 1 stores the expected information of the first-order statistical mean value of the error signal, and data table 2 stores the second order statistical mean value of the error signal. The expected information of the first-order statistical variance. 4. The interference suppression method for a code division multiple access system according to claim 3, characterized in that: the data stored in the data table 1 and the data table 2 are obtained through training. 5. The interference suppression method for a code division multiple access system according to claim 1, characterized in that: the weighting processing of the estimated signals of each user is adjusted in units of symbols. 6, the interference suppression method of CDMA system according to claim 1 is characterized in that: the data that described obtains with table look-up obtains the control quantity that this class weight adjusts the size and obtains according to the following method: weight Amount of control for value resizing $w_{k,m}^{\left(\mathrm{the\; s}\right)}=w_{k,m-1}^{\left(\mathrm{the\; s}\right)}+f_1\left(\mathrm{E.}\right[e_{k,m}^{\left(\mathrm{the\; s}\right)}\left(t\right)\left]\right)+f_2\left(\mathrm{var}\right[e_{k,m}^{\left(\mathrm{the\; s}\right)}\left(t\right)\left]\right),$ Wherein, k represents the kth user, and k is an integer; m represents the mth symbol, and m is an integer; t represents time, and t>=0.

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