CN102697496B - Filtering method for functional electrical stimulation surface electromyogram signal - Google Patents

Abstract

The invention discloses a filtering method for a functional electrical stimulation (FES) surface electromyogram signal, and belongs to the technical field of instruments for helping rehabilitation of the disabled by using electric pulse stimulation. The filtering method comprises the following steps of: constructing an FES template signal r (n) of the filter; optimizing the coefficient of adaptive matched filter through a genetic algorithm; finally outputting key parameters of the filter; subsequently, intelligently optimizing the parameters while realizing adaptive filtering, and performing electromyogram noise filtering treatment by using the shaped filter to obtain relatively pure electromyogram information, thereby providing certain guide and control for the state of the muscle under the functional electrical stimulation of paralyzed patients such as paraplegia patients. According to the filtering method, the electromyogram nose filtering effect under FES can be effectively improved, certain guide and control are provided for the state of the muscle under the functional electrical stimulation of the paralyzed patients such as paraplegia patients, and considerable social benefit and economical benefit are achieved.

Description

A kind of filtering method of functional electric stimulation surface electromyogram signal

Technical field

The invention belongs to rehabilitation engineering and field of intelligent control, particularly a kind of filtering method of functional electric stimulation surface electromyogram signal.

Background technology

In recent years, the cerebrovascular such as spinal cord injury and apoplexy cause the sickness rate of paralysis to be remarkable ascendant trend, all bring larger burden not only to individual and family, also become day by day heavy social problem.The Executive Meeting of the State Council that China in 2011 holds is pointed out, strives 2015, China people with disability is lived and totally attain the well-off standard, and participates in and state of development is significantly improved, and tentatively realizes people with disability's " everybody enjoys rehabilitation service " target.

It is emphasis and the difficult point of concern when paralytic patient is carried out to rehabilitation that limb function is rebuild, and this is related to the raising problem of daily life active ability and quality of life.Aspect the recovery of paralytic patient locomotor activity, FES (Functional Electrical Stimulation, functional electric stimulation) is generally considered the more effective clinical tool of one at present.FES utilizes the excitable muscle of certain specific electric current (or voltage) signal stimulus, tissue or organ, to improve its muscle performance, recovery or to rebuild the technology of the limb activity function of being lost by nerve injury.The sixties in 20th century, Liberson successfully utilizes electricity irritation peroneal nerve to correct the gait of hemiplegic patient's drop foot first, has started functional electric stimulation for the new way of moving and Sensory rehabilitation is treated.

In FES, utilize neurocyte to transmit additional Artificial Control signal to the response of electricity irritation, by the effect of extrinsic current, neurocyte can produce a neural impulse similar to naturally exciting the action potential that causes, make the meat fiber of its domination produce contraction, thereby obtain the effect of motion.Although along with deepening continuously of understanding, FES has been applied to many fields of rehabilitation, but compared with the application prospect wide with it, a lot of new FES technology are also confined to laboratory stage, the FES stimulus modelity of clinical practice and the effect reaching are all very limited, and one of its reason is exactly to be also short of to some extent for the research of the mechanism aspect of FES effect.The explanation of cellular level can only illustrate that the contraction of FES stimulated muscle produces the reason of motion, but can not illustrate functional activity and the state of target muscle under FES stimulates in macroscopic view.And the contraction of skeletal muscle and diastole activity and state are the bases of the various motions of human body, limb motion is to be mainly that power completes by the mechanicalness reaction of skeletal muscle generation diastole and contraction in length, and the structure of skeletal muscle is the primary determiner of functional activity.So, want to utilize FES to reach skeletal muscle function and rebuild the object that specific action even moves, and it is rapidly developed and extensive use, just must need the lower structure of skeletal muscle of FES effect and the mechanism of functional activity and state thereof to carry out more deep probing into.

SEMG (surface electromyography, surface electromyogram signal) be the electric-physiology parameter of reaction musculation, it is the current most popular method that is used for evaluating musculation information, be widely used in the on-off control of artificial limb, the advantage such as sEMG has simply, harmless and real-time is good.

Inventor is realizing in process of the present invention, finds at least to exist in prior art following shortcoming and defect:

When the position of stimulation muscle and inducing myoelectric potential generation simultaneously and stimulating electrode and recording electrode is close, sEMG is mixed with to stimulate and disturbs, and has affected the collection of pure electromyographic signal; The amplitude of FES is mV scope, than the large manyfold of the amplitude of sEMG; The output of fes signal can be infected responsive myoelectricity acquisition system, causes the stimulation interference problem of closed loop nerve prosthesis control.

Summary of the invention

The invention provides a kind of filtering method of functional electric stimulation surface electromyogram signal, this method has realized and when the Electrophysiology that gathers muscle is movable, has weakened or suppress FES and disturb, and obtains purer surface electromyogram signal, described below:

A filtering method for functional electric stimulation surface electromyogram signal, said method comprising the steps of:

(1) by wireless myoelectricity system acquisition, stretch in knee joint process the first surface electromyographic signal of target muscle under functional electric stimulation, gather one group of second surface electromyographic signal S (n) that independently stretches target muscle in knee joint process simultaneously;

(2) by described first surface electromyographic signal, build FES template signal r (n);

(3), by described second surface electromyographic signal S (n), described FES template signal r (n) and filter weight obtain the estimated value of noise signal v (n)

(4) by the estimated value of described noise signal v (n) the estimation difference e (n) that obtains wave filter, upgrades filter weight;

(5) judge whether described estimation difference e (n) meets preset standard, if so, stops iteration, execution step (7); If not, execution step (6);

(6) n → n+1, repeating step (4) and step (5), until reach the iterations of setting, execution step (7);

(7) convergence parameter μ, the exponent number M to wave filter and weighting initial value w _m(0) carry out chromosome coding, obtain respectively the rear convergence of coding parameter μ, exponent number M and weighting initial value w _m(0) initial value;

(8), according to described estimation difference e (n), the rate of change of estimated bias and described second surface electromyographic signal S (n) build optimum index J, by described optimum index J, obtain appropriate function F;

(9) by genetic algorithm, restrain parameter μ, exponent number M and weighting initial value w after to described coding _m(0) initial value carries out optimizing, exports multiple appropriate function F, by described multiple appropriate function F, draws appropriate function curve;

(10) judge that whether described appropriate function curve is steady, if so, execution step (11); If not, execution step (12);

(11) output convergence parameter μ, exponent number M and weighting initial value w _m(0) end value, flow process finishes;

(12) re-execute step (7), until reach iterations, flow process finishes.

Described estimated value be specially:

$\hat{v}\left(n\right)=\underset{m=0}{\overset{M}{\mathrm{\Σ}}}{W}_m\left(n\right)r(n-m)=w_0\left(n\right)r\left(n\right)+w_1\left(n\right)r(n-1)+...+w_M\left(n\right)r(n-M)$

Initial value is w _m(0), M represents the rank of wave filter; W _m(n) weight of expression wave filter; R (n-m) is by r (n) delay acquisition.

Described convergence parameter μ, exponent number M and weighting initial value w to wave filter _m(0) carrying out chromosome coding is specially:

By convergence parameter μ, the exponent number M of wave filter and weighting initial value w _m(0) initial value is used respectively the binary code representation of any 8.

Describedly by genetic algorithm, restrain parameter μ, exponent number M and weighting initial value w after to described coding _m(0) initial value carries out optimizing, exports multiple appropriate function F, draws appropriate function curve be specially by described multiple appropriate function F:

To described convergence parameter μ, exponent number M and weighting initial value w _m(0) initial value carries out chromosome coding, gene Selection intersection, mutation operation and chromosome decoding and obtains convergence parameter μ, exponent number M and the weighting initial value w after renewal _m(0), by convergence parameter μ, exponent number M after each described renewal and weighting initial value w _m(0) be input in wave filter, after Filtering Processing by wave filter, obtain described estimation difference e (n), by described estimation difference e (n), the rate of change of described estimated bias and described second surface electromyographic signal S (n) build described optimum index J, by described optimum index J, obtain multiple appropriate function F, by described multiple appropriate function F, draw described appropriate function curve.

The beneficial effect of technical scheme provided by the invention is: the filtering method that the invention provides a kind of functional electric stimulation surface electromyogram signal, the present invention constructs the FES template signal r (n) of wave filter, by the coefficient of genetic algorithm optimization adaptive matched filter, final output filter key parameter, when then realizing adaptive-filtering, can intelligent optimizing parameter, wave filter after use is adjusted, carry out myoelectricity and filter the processing of making an uproar, obtained purer surface electromyogram signal, thereby for the state research of muscle under the functional electric stimulation of the paralysed patients such as paraplegia provides certain guidance and control, this invention can effectively improve myoelectricity under FES and filter the effect of making an uproar, and obtains considerable Social benefit and economic benefit.

Accompanying drawing explanation

Fig. 1 is a kind of adaptive matched filter theory diagram provided by the invention;

Fig. 2 is the flow chart of the filtering method of a kind of functional electric stimulation surface electromyogram signal provided by the invention;

The schematic diagram of mixed signal when Fig. 3 a is 5000 provided by the invention;

The schematic diagram of original electromyographic signal when Fig. 3 b is 5000 provided by the invention;

The schematic diagram of filtered signal of telecommunication when Fig. 3 c is 5000 provided by the invention;

The schematic diagram of the signal of process low-pass filtering when Fig. 3 d is 5000 provided by the invention;

The schematic diagram of mixed signal when Fig. 4 a is 1500 provided by the invention;

The schematic diagram of original electromyographic signal when Fig. 4 b is 1500 provided by the invention;

The schematic diagram of filtered signal of telecommunication when Fig. 4 c is 1500 provided by the invention;

The schematic diagram of the signal of process low-pass filtering when Fig. 4 d is 1500 provided by the invention;

Signal to noise ratio when Fig. 5 is 1500 provided by the invention after Filtering Processing.

The specific embodiment

For making the object, technical solutions and advantages of the present invention clearer, below in conjunction with accompanying drawing, embodiment of the present invention is described further in detail.

In order to realize, when the Electrophysiology that gathers muscle is movable, to weaken or suppress FES and disturb, obtain purer surface electromyogram signal, referring to Fig. 1 and Fig. 2, the embodiment of the present invention provides a kind of filtering method of functional electric stimulation surface electromyogram signal, described below:

101: by wireless myoelectricity system acquisition, stretch in knee joint process the first surface electromyographic signal of target muscle under functional electric stimulation, gather one group of second surface electromyographic signal S (n) that independently stretches target muscle in knee joint process simultaneously;

Wherein, utilize wireless myoelectricity system to stimulate the relevant muscle group of lower limb, require experimenter healthy, without lower limb muscles, skeleton illness, impassivity illness and severe cardiac pulmonary disease.The Parastep functional electric stimulation system that experimental provision adopts U.S. SIGMEDICS company to produce, this system comprises microprocessor and boost pulse circuit for generating, containing six stimulation channels, battery powered.During experiment, experimenter sits on testboard, stimulating electrode is fixed on to the end positions of quadriceps femoris.While not applying electricity irritation, shank loosens, keeps vertical vacant state.Electric stimulation pulse sequence adopts classical Lilly waveform, and pulse frequency is 25Hz, pulsewidth 150, and pulse current is adjustable within the scope of 0～120m.In experiment, can adjust stimulus intensity to change the knee joint angle producing by stimulating by changing pulse current size.Under FES effect, since the 1st grade of effect, increase step by step, every grade of continuous action 3s, until the progression that shank stretches is designated as initial highest number.Press again same time decreasing series successively, until the 1st grade, be 1 cycle.This periodic process carries out continuously, and each highest number keeps initial highest number constant during this time.Meanwhile, gather and under major state, stretch the myoelectricity of knee joint process, for after Filtering Simulation prepare.

Wherein, adaptive-filtering is exactly the filtering result of utilizing a front iteration to obtain, automatically regulates the filter parameter of current iteration, with adaptation signal and noise the unknown or time dependent statistical property, thereby realizes optimal filter.Sef-adapting filter is exactly in fact that a kind of self transmission characteristic that can regulate is to reach optimum Wiener filter.Sef-adapting filter does not need the priori of input signal, and amount of calculation is little, is specially adapted to real-time processing.The characteristic variations of sef-adapting filter is realized by adjusting filter coefficient by adaptive algorithm.Generally speaking, sef-adapting filter is comprised of two parts, and the one, filter construction, the 2nd, the adaptive algorithm of adjustment filter coefficient.Adaptive noise cancellation based on Werner's theory needs unlimited weighting filter, with minimization output error.In order to realize Wiener filtering scheme, must use limited weighting filter.In other words, sef-adapting filter must suppose that Wiener filter is a finite impulse response (FIR) wave filter.And adaptive matched filter, under the prerequisite of same Signal Matching, meet the optimum filter of root-mean-square error minimum criteria, it can reach optimized object by the weight coefficient of adjusting self, is the adaptive matched filter functional-block diagram based on Wiener filter as shown in Figure 1.

102: by first surface electromyographic signal, build FES template signal r (n);

Wherein, get the first surface electromyographic signal under FES stimulation, because the periodicity of first surface electromyographic signal is 0.08s, first surface electromyographic signal has randomness, so can utilize the method for superposed average, get the first surface electromyographic signal of same grade under FES, the signal plus of every 0.08s does on average again, by approximate the average signal obtaining, regard a FES template as, then this FES template is arranged in to FES template signal r (n) in turn.

103: by second surface electromyographic signal S (n), FES template signal r (n) and filter weight obtain the estimated value of noise signal v (n)

Referring to Fig. 1, mixed signal x (n) consists of second surface electromyographic signal S (n) and noise signal v (n), and wherein S (n) is uncorrelated with v (n).Because FES template signal r (n) is very large with respect to second surface electromyographic signal S (n) and noise signal v (n) amplitude, so can being similar to, noise signal v (n) thinks take FES template signal r (n) as main, because FES template signal r (n) is relevant with noise signal v (n) it is the best estimate of noise signal v (n).Wave filter will be removed its dependency at outfan, and concrete grammar is to deduct estimated value from mixed signal x (n)

the output of wave filter is exactly the estimation of second surface electromyographic signal S (n) so

$\hat{v}\left(n\right)=\underset{m=0}{\overset{M}{\mathrm{\Σ}}}{W}_m\left(n\right)r(n-m)=w_0\left(n\right)r\left(n\right)+w_1\left(n\right)r(n-1)+...+w_M\left(n\right)r(n-M)0\≤n\≤M---\left(1\right)$

Wherein, initial value is w _m(0), M represents the rank of wave filter; w _m(n) weight of expression wave filter; R (n-m) is by r (n) delay acquisition.

104: by the estimated value of noise signal v (n)

the estimation difference e (n) that obtains wave filter, upgrades filter weight;

$e\left(n\right)=x\left(n\right)-\hat{v}\left(n\right)\≈\hat{s}\left(n\right)---\left(2\right)$

w _m(n+1)=w _m(n)+2μe(n)r(n-m)0≤m≤M (3)

Wherein, μ is convergence parameter.

105: judge whether estimation difference e (n) meets preset standard, if so, stops iteration, execution step 107; If not, execution step 106;

Preset standard is specially: the amplitude of estimation difference e (n) is within the amplitude range of normal surface electromyographic signal, and this scope is 50 μ v-20mv.

106:n → n+1, repeating step 104-105, until reach the iterations of setting, execution step 107;

Wherein, iterations is set according to the needs in practical application, and on time, the embodiment of the present invention does not limit this specific implementation.

107: convergence parameter μ, the exponent number M to wave filter and weighting initial value w _m(0) carry out chromosome coding, obtain respectively the rear convergence of coding parameter μ, exponent number M and weighting initial value w _m(0) initial value;

The input of adaptive matched filter is mixed signal, in Filtering Processing process, according to the filter state of certain iteration n, revises the filter factor of n+1 time (restrain parameter μ, exponent number M and weighting initial value w by error _m(0)), improve the denoising effect of wave filter, 3 parameters that optimizing parameter is adaptive matched filter.

Wherein, owing to being Parametric optimization problem, and solution is real number value, therefore adopt multiparameter mapping binary-coding convergence parameter μ, the exponent number M to wave filter and weighting initial value w respectively _m(0) carry out chromosome coding, that is: by convergence parameter μ, the exponent number M of wave filter and weighting initial value w _m(0) initial value is used respectively the binary code representation of any 8.

For example: after 01000100|11110010|01101001| presentation code, restrain parameter μ, exponent number M and weighting initial value w _m(0) initial value.

108: according to estimation difference e (n), the rate of change of estimated bias and second surface electromyographic signal S (n) build optimum index J, by optimum index J, obtain appropriate function F;

$J={\∫}_0^n{w}_1{e}^2\left(n\right)+w_2{u}^2\left(n\right)+w_3{\left(\frac{e\left(n\right)}{n}\right)}^2\mathrm{dn}---\left(4\right)$

Wherein, w ₁, w ₂and w ₃be weights, generally all get w ₁=100, w ₂=10, w ₃=1.

Appropriateness function F is

F=C/J （5）

Wherein, C=10 ^k, k is integer.

109: by genetic algorithm to restraining parameter μ, exponent number M and weighting initial value w after encoding _m(0) initial value carries out optimizing, exports multiple appropriate function F, by multiple appropriate function F, draws appropriate function curve;

Wherein, this step is specially: to convergence parameter μ, exponent number M and weighting initial value w _m(0) initial value carries out chromosome coding, gene Selection intersection, mutation operation and chromosome decoding and obtains convergence parameter μ, exponent number M and the weighting initial value w after renewal _m(0), by convergence parameter μ, exponent number M and weighting initial value w after upgrading at every turn _m(0) be input in wave filter, after Filtering Processing by wave filter, obtain estimation difference e (n), by estimation difference e (n), the rate of change of estimated bias and second surface electromyographic signal S (n) build optimum index J, by optimum index J, obtain multiple appropriate function F, by multiple appropriate function F, draw appropriate function curve.

Wherein, the processing procedure of chromosome coding, gene Selection intersection, mutation operation and chromosome decoding is known content in genetic algorithm, and the embodiment of the present invention does not repeat at this.

110: judge that whether appropriate function curve is steady, if so, execution step 111; If not, execution step 112;

Wherein, for example, when appropriate function curve levels off to a horizontal linear: the value of F remain unchanged or one very among a small circle in float, think that appropriate function curve is steady, otherwise, think that appropriate function curve is not steady.

111: output convergence parameter μ, exponent number M and weighting initial value w _m(0) end value, flow process finishes;

112: re-execute step 107, until reach iterations, flow process finishes.

To restrain parameter μ, exponent number M and weighting initial value w _m(0) end value is applied in wave filter, makes the wave filter after adjusting improve filter effect, and filtered signal noise is less.

With a feasibility of simply verifying the filtering method of a kind of functional electric stimulation surface electromyogram signal that the embodiment of the present invention provides, described below below:

Referring to Fig. 3 a, Fig. 3 b, Fig. 3 c, Fig. 3 d, Fig. 4 a, Fig. 4 b, Fig. 4 c, Fig. 4 d and Fig. 5, by this method, electromyographic signal under FES is carried out to Filtering Processing, basic filtering FES in electromyographic signal disturb.By the mode of superposed average, obtain FES template, again FES template is arranged in turn to FES template signal r (n), sample rate is 1500, get at one section 5000 and independently stretch knee joint myoelectricity as original myoelectricity, get the second surface electromyographic signal of same length, stack, as mixed signal, is carried out simulation filter experiment.By time-domain diagram, look like to find out, filtered surface electromyogram signal is comparatively pure.Intercept front 1500 points, and every 20 points are done to signal to noise ratio, can find out that signal to noise ratio obviously increases, show that the wave filter after adjusting by this method has obvious filtering and noise reduction effect, verified the feasibility of this method.

In sum, the embodiment of the present invention provides a kind of filtering method of functional electric stimulation surface electromyogram signal, the FES template signal r (n) of embodiment of the present invention structure wave filter, by the coefficient of genetic algorithm optimization adaptive matched filter, final output filter key parameter, when then realizing adaptive-filtering, can intelligent optimizing parameter, wave filter after use is adjusted, carry out myoelectricity and filter the processing of making an uproar, obtained purer surface electromyogram signal, thereby for the state research of muscle under the functional electric stimulation of the paralysed patients such as paraplegia provides certain guidance and control, the embodiment of the present invention can effectively improve myoelectricity under FES and filter the effect of making an uproar, and obtains considerable Social benefit and economic benefit.

It will be appreciated by those skilled in the art that accompanying drawing is the schematic diagram of a preferred embodiment, the invention described above embodiment sequence number, just to describing, does not represent the quality of embodiment.The foregoing is only preferred embodiment of the present invention, in order to limit the present invention, within the spirit and principles in the present invention not all, any modification of doing, be equal to replacement, improvement etc., within all should being included in protection scope of the present invention.

Claims (3)

1. a filtering method for functional electric stimulation surface electromyogram signal, is characterized in that, said method comprising the steps of:

(1) by wireless myoelectricity system acquisition, stretch in knee joint process the first surface electromyographic signal of target muscle under functional electric stimulation, gather one group of second surface electromyographic signal S (n) that independently stretches target muscle in knee joint process simultaneously;

(2) by described first surface electromyographic signal, build FES template signal r (n);

(3), by described second surface electromyographic signal S (n), described FES template signal r (n) and filter weight obtain the estimated value of noise signal v (n)

(4) by the estimated value of described noise signal v (n)

the estimation difference e (n) that obtains wave filter, upgrades filter weight;

(5) judge whether described estimation difference e (n) meets preset standard, if so, stops iteration, execution step (7); If not, execution step (6);

(6) n → n+1, repeating step (4) and step (5), until reach the iterations of setting, execution step (7);

(7) convergence parameter μ, the exponent number M to wave filter and weighting initial value w _m(0) carry out chromosome coding, obtain respectively the rear convergence of coding parameter μ, exponent number M and weighting initial value w _m(0) initial value;

(8), according to described estimation difference e (n), the rate of change of estimated bias and described second surface electromyographic signal S (n) build optimum index J, by described optimum index J, obtain appropriate function F;

(9) by genetic algorithm, restrain parameter μ, exponent number M and weighting initial value w after to described coding _m(0) initial value carries out optimizing, exports multiple appropriate function F, by described multiple appropriate function F, draws appropriate function curve;

(10) judge that whether described appropriate function curve is steady, if so, execution step (11); If not, execution step (12);

(11) output convergence parameter μ, exponent number M and weighting initial value w _m(0) end value, flow process finishes;

(12) re-execute step (7), until reach iterations, flow process finishes;

Described estimated value

be specially:

$\hat{v}\left(n\right)=\underset{m=0}{\overset{M}{\mathrm{\Σ}}}{W}_m\left(n\right)r(n-m)=w_0\left(n\right)r\left(n\right)+w_1\left(n\right)r(n-1)+...+w_M\left(n\right)r(n-M)$

Initial value is w _m(0), M represents the rank of wave filter; W _m(n) weight of expression wave filter; R (n-m) is by r (n) delay acquisition;

Wherein,

$e\left(n\right)=x\left(n\right)-\hat{v}\left(n\right)\≈\hat{s}\left(n\right)$

w _m(n+1)=w _m(n)+2μe(n)r(n-m) 0≤m≤M

μ is convergence parameter;

Wherein,

$J={\∫}_0^n{w}_1{e}^2\left(n\right)+w_2{u}^2\left(n\right)+w_3{\left(\frac{e\left(n\right)}{n}\right)}^2\mathrm{dn}$

Wherein, w ₁, w ₂and w ₃be weights, generally all get w ₁=100, w ₂=10, w ₃=1;

Appropriateness function F is

F=C/J

Wherein, C=10 ^k, k is integer.

2. the filtering method of a kind of functional electric stimulation surface electromyogram signal according to claim 1, is characterized in that, described convergence parameter μ, exponent number M and weighting initial value w to wave filter _m(0) carrying out chromosome coding is specially:

By convergence parameter μ, the exponent number M of described wave filter and weighting initial value w _m(0) initial value is used respectively the binary code representation of any 8.

3. the filtering method of a kind of functional electric stimulation surface electromyogram signal according to claim 2, is characterized in that, describedly by genetic algorithm, restrains parameter μ, exponent number M and weighting initial value w after to described coding _m(0) initial value carries out optimizing, exports multiple appropriate function F, draws appropriate function curve be specially by described multiple appropriate function F:

To described convergence parameter μ, exponent number M and weighting initial value w _m(0) initial value carries out chromosome coding, gene Selection intersection, mutation operation and chromosome decoding and obtains convergence parameter μ, exponent number M and the weighting initial value w after renewal _m(0), by convergence parameter μ, exponent number M after each described renewal and weighting initial value w _m(0) be input in wave filter, after Filtering Processing by wave filter, obtain described estimation difference e (n), by described estimation difference e (n), the rate of change of described estimated bias and described second surface electromyographic signal S (n) build described optimum index J, by described optimum index J, obtain multiple appropriate function F, by described multiple appropriate function F, draw described appropriate function curve.

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