CN104850752A - Parameter search range determining method based on adaptive random resonance - Google Patents

Abstract

The invention discloses a parameter search range determining method based on adaptive random resonance. The parameter search range determining method based on adaptive random resonance comprises the following steps of outputting an upper limit (amax) and a lower limit (amin) of a stable first parameter (a), which needs to be determined, of a random resonance system according to outputting of the random resonance system; determining a lower limit (bmin) of a second parameter (b) of the random resonance system according to the lower limit (amin) of the first parameter (a); transmitting a mixing signal to the random resonance system; calculating power spectra outputted by the random resonance system; continuously increasing the second parameter (b) of the random resonance system by using a fixed step length as a change interval until the power spectra are diverged; and using the second parameter (b), which corresponds to a previous operation of diverging of the power spectra, of the random resonance system as an upper limit (bmax) of the determined parameter (b) of the random resonance system. By the parameter search range determining method based on adaptive random resonance, the complexity of an algorithm is reduced effectively, and the success rate of induced random resonance is improved.

Description

A kind of parameter search range determining method based on self-adapting random resonant

Technical field

The present invention relates to Technique of Weak Signal Detection field, be specifically related to a kind of parameter search range determining method based on self-adapting random resonant.

Background technology

Since the research ancient meteorological glacier problems such as BenZi in 1981 propose accidental resonance concept, Stochastic Resonance Phenomenon receives to be paid close attention to widely.Stochastic Resonance Phenomenon is a kind of non-linear phenomena, it under certain condition, by partial noise energy trasfer on signal, the weak signal be submerged in noise can be made while reduction noise to obtain resonance strengthen, greatly improve the signal to noise ratio (S/N ratio) of output, thus realize the object detecting feeble signal from strong noise environment.

Self-adapting random resonant algorithm can according to different signals to be detected, and varitrol parameter with induced resonance, thus realizes the detection of feeble signal, brings certain convenience to practical implementation.But when self-adaptation obtains regulating system parameter, all need the hunting zone and the step-size in search that preset systematic parameter according to prior imformation or experience, then self-adaptation obtains optimum resonator system parameter in the hunting zone determined.The hunting zone of systematic parameter arranged conference and causes adaptive algorithm counting yield low, hunting zone is too small, likely misses optimum solution, causing cannot successful induced resonance, and only rule of thumb or prior imformation, is difficult to arrange a suitable parameter search scope.

Summary of the invention

Technical matters to be solved by this invention is the problem that traditional self-adapting random resonant can only rely on that experience comes parameter preset hunting zone, a kind of parameter search range determining method based on self-adapting random resonant is provided, it not only effectively reduces algorithm complex, and improves the success ratio of induction accidental resonance.

For solving the problem, the present invention is achieved by the following technical solutions:

Based on a parameter search range determining method for self-adapting random resonant, comprise the steps:

Step 1: the upper limit a needing the first parameter a determining stochastic resonance system according to stochastic resonance system stable output _max; Namely

a _max＝1/h

In formula, h is numerical evaluation step-length;

Step 2: the lower limit a needing the first parameter a determining stochastic resonance system according to stochastic resonance system resonance _min; Namely

$a_{\min }=2\sqrt{2}\mathrm{\π}{f}_0$

In formula, f ₀for carrier frequency;

Step 3: according to the lower limit a of step 2 determined stochastic resonance system first parameter a _min, determine the lower limit b of the second parameter b of stochastic resonance system _min; Namely

b _min＝ea _min ²/2D

In formula, D=σ ²h/2 is noise intensity, σ ²for noise variance, h is numerical evaluation step-length; E is natural constant; a _minbe the lower limit a of the first parameter a _min;

Step 4: mixed signal is sent in stochastic resonance system, calculates the power spectrum that this stochastic resonance system exports, and is that change interval constantly increases stochastic resonance system second parameter b until power spectrum is dispersed with fixed step size; Stochastic resonance system second parameter b corresponding to the back now dispersed by power spectrum is the upper limit b of the parameter b of the stochastic resonance system determined _max;

Step 5: by the scope [a of the first parameter a of determined for step 1-2 stochastic resonance system _min, a _max] and the scope [b of the second parameter b of the determined stochastic resonance system of step 3-4 _min, b _max] send in follow-up self-adapting random resonant inspection frequency process as hunting zone.

Described step 4 comprises the upper limit b of the parameter b tentatively determining stochastic resonance system _maxprocess and accurately determine the upper limit b of parameter b of stochastic resonance system _maxprocess; Namely

Step 4-1: tentatively determine b _max, namely mixed signal is sent in stochastic resonance system, calculates the initial adjustment power spectrum that this stochastic resonance system exports, and with the initial value b of the second parameter b of setting ₀based on, the initial adjustment fixed step size Δ of setting is change interval, constantly increases stochastic resonance system second parameter b until initial adjustment power spectrum is dispersed; Stochastic resonance system second parameter b corresponding to the back now dispersed by initial adjustment power spectrum is the upper limit b of the parameter b of the stochastic resonance system tentatively determined _max;

Step 4-2: accurately determine b _max, namely mixed signal is sent in stochastic resonance system, calculates the accurate adjustment power spectrum that this stochastic resonance system exports, and the upper limit b of the second parameter b tentatively determined with step 4-1 _maxbased on, the accurate adjustment fixed step size ε of setting is change interval, constantly increases stochastic resonance system second parameter b until accurate adjustment power spectrum is dispersed; Stochastic resonance system second parameter b corresponding to the back now dispersed by accurate adjustment power spectrum is the upper limit b of the parameter b of the stochastic resonance system finally determined _max;

Above-mentioned ε=Δ/n, n be greater than 1 positive integer.

In described step 4, the first parameter that the stochastic resonance system that mixed signal is sent into is all the time a is the initial value b of the second parameter b ₀be 1.

In described step 1 and 3, the numerical evaluation step-length h=1/f of setting _s, wherein f _sfor sample frequency.

Compared with prior art, the present invention has following features:

1, resonance parameter optimum solution is necessarily contained, the parametric optimal solution namely searched in determined scope in the determined parameter area of the present invention, necessarily can induced resonance.

2, do not need artificially to rely on experience to come parameter preset hunting zone, avoid parameter area and arrange excessive, cause algorithm convergence cross slow or parameter area too small causing is set can not the problem of successful induced resonance.

3, the present invention is according to the signal dynamics parameters hunting zone of input, adapts to fast changing environment therefore, it is possible to better.

Accompanying drawing explanation

Fig. 1 is a kind of process flow diagram of the parameter search range determining method based on self-adapting random resonant.

Fig. 2 (a) and (b) add the time domain after making an uproar and frequency-region signal waveform for small parameter signal.

Fig. 3 is the convergence curve figure of quanta particle swarm optimization in small parameter situation.

Fig. 4 (a) and (b) are time domain after small parameter stochastic resonance system and frequency-region signal waveform.

Fig. 5 (a) and (b) add the time domain after making an uproar and frequency-region signal waveform for large parameter signal.

Fig. 6 is the convergence curve figure of quanta particle swarm optimization in large parameter situation.

Fig. 7 (a) and (b) are time domain after excessive Parameter Signal Stochastic Resonance system and frequency-region signal waveform.

Embodiment

Based on a parameter search range determining method for self-adapting random resonant, as shown in Figure 1, comprise the steps:

Step 1: determine a _max.The upper limit a of the first parameter a determining stochastic resonance system is needed according to stochastic resonance system stable output _max; Namely

a _max＝1/h

In formula, h is numerical evaluation step-length.The numerical evaluation step-length of setting can set as required, in a preferred embodiment of the invention, and h=1/f _s, wherein f _sfor sample frequency.

Step 2: determine a _min.The lower limit a of the first parameter a determining stochastic resonance system is needed according to stochastic resonance system resonance _min; Namely

$a_{\min }=2\sqrt{2}\mathrm{\π}{f}_0$

In formula, f ₀for carrier frequency.

Step 3: determine b _min.According to the lower limit a of step 2 determined stochastic resonance system first parameter a _min, determine the lower limit b of the second parameter b of stochastic resonance system _min; Namely

b _min＝ea _min ²/2D

In formula, D=σ ²h/2 is noise intensity, σ ²for noise variance, h is numerical evaluation step-length; E is natural constant.The numerical evaluation step-length of setting can set as required, in a preferred embodiment of the invention, and h=1/f _s, wherein f _sfor sample frequency.The white Gaussian noise being zero by cycle feeble signal and average due to the mixed signal of stochastic resonance system input forms, so noise variance σ ²directly get the variance of mixed signal.

Step 4: determine b _max.

Step 4-1: tentatively determine b _max, namely mixed signal is sent in stochastic resonance system, calculates the initial adjustment power spectrum that this stochastic resonance system exports, and with the initial value b of the second parameter b of setting ₀based on, the initial adjustment fixed step size Δ of setting is change interval, constantly increases stochastic resonance system second parameter b until initial adjustment power spectrum is dispersed; Stochastic resonance system second parameter b (b=b corresponding to the back now initial adjustment power spectrum dispersed ₀+ (L-1) * Δ, wherein L is the step number corresponding when dispersing of preliminary power spectrum) be the upper limit b of the parameter b of the stochastic resonance system tentatively determined _max.

Step 4-2: accurately determine b _max, namely mixed signal is sent in stochastic resonance system, calculates the accurate adjustment power spectrum that this stochastic resonance system exports, and the upper limit b of the second parameter b tentatively determined with step 4-1 _maxbased on, the accurate adjustment fixed step size ε of setting is change interval, constantly increases stochastic resonance system second parameter b until accurate adjustment power spectrum is dispersed; Stochastic resonance system second parameter b (b=b corresponding to the back now accurate adjustment power spectrum dispersed _max+ (N-1) * ε, wherein N is the step number corresponding when dispersing of accurate power spectrum) be the upper limit b of the parameter b of the stochastic resonance system finally determined _max;

Above-mentioned ε=Δ/n, n be greater than 1 positive integer.First parameter a of the stochastic resonance system that mixed signal is sent into is constant all the time, and in a preferred embodiment of the invention, this value is chosen to be the initial value b of the second parameter b ₀be set as 1.

The present invention only adopts step 4-1 namely can determine the upper limit b of the parameter b of stochastic resonance system _max, but in order to obtain better Search Results, the present invention can also add step 4-2 further, or even repeatedly repeat the identical process of step 4-2, obtain the upper limit b of the parameter b of more accurate stochastic resonance system _max.

Step 5: by the scope [a of the first parameter a of determined for step 1-2 stochastic resonance system _min, a _max] and the scope [b of the second parameter b of the determined stochastic resonance system of step 3-4 _min, b _max] send in follow-up self-adapting random resonant inspection frequency process as hunting zone.

The present invention below, for bistable-state random resonance model and quantum particle swarm self-adapting random resonant algorithm, clearly determines the concrete grammar of parameter search scope.

1. bi-stable stochastic resonance theory model

According to Stochastic Resonance Theory, periodic signal and the coefficient bistable system model of noise are:

d _x/dt＝ax-bx ³+[s(t)+n(t)] (1)

In formula, a and b is the structural parameters of bistable system; S (t) is weak periodic signal; N (t) is white noise.

Above formula is a kind of Nonlinear Stochastic Differential Equation, carries out numerical solution by four step Runge-Kutta.Specific algorithm is as follows:

$\left\{\begin{array}{c}{k}_1=h({\mathrm{ax}}_n-{\mathrm{bx}}_n^3+s_n)\\ k_2=h[a(x_n+\frac{k_1}{2})-b{(x_n+\frac{k_1}{2})}^3+s_n]\\ k_3=h[a(x_n+\frac{k_2}{2})-b{(x_n+\frac{k_2}{2})}^3+s_{n+1}]\\ k_4=h[a(x_n+k_3)-b{(x_n+k_3)}^3+s_{n+1}]\\ x_{n+1}=x_n+\frac{1}{6}(k_1+{2k}_2+{2k}_3+k_4)\end{array}\right.n=\mathrm{1,2},...,\mathrm{Length}\left(s_n\right)-1---\left(2\right)$

In formula, s _nand x _nbistable system input S (t)=(s (t)+n (t)) and the n-th sampled value exporting X (t) respectively, h=1/f _s(f _sfor sample frequency) be numerical evaluation step-length.According to the different value of input signal and noise, regulating parameter a and b just can realize accidental resonance, transfers on signal s (t) by the energetic portions of noise n (t), improves output signal-to-noise ratio.

2. quantum particle swarm adaptive algorithm

Quanta particle swarm optimization proposes from quantum-mechanical angle.In vector subspace, the speed of particle and position are can not be simultaneously exactly determined, wave function ψ (the x of particle can be obtained by solving schrodinger equation, t), wave function square be the probability density function of particle a certain appearance in space, after obtaining probability density function, the position equation being obtained particle by the mode of Monte-Carlo Simulation is:

y(t)＝P±L/2ln(1/μ) (3)

In formula, L is the parameter of DELTA Well strength, and P is the random point between the desired positions of individual particles and the optimal location of all particles, and μ is the random number of change in [0,1] scope, when μ >=0.5 formula (3) gets minus sign, otherwise, get plus sige.The position equation finally obtaining particle is:

y(t+1)＝P±β×m _best-y(t)×ln(1/μ) (4)

In formula, m _bestbe local optimum center, β generally changes according to the following formula:

β＝0.5+0.5(T _max-t)/T _max(5)

In formula, T _maxfor maximum iteration time, t is current iteration number of times.

Constraint condition is that output signal-to-noise ratio is maximum, and signal to noise ratio (S/N ratio) formula is:

SNR _out＝10lg(S(ω ₀)/N(ω ₀)) (6)

3. parameter search method of determining range

Be provided with a signals and associated noises, signal frequency is f, and sample frequency is f _squantum particle swarm self-adapting random resonant method is adopted to detect this signal, namely be cost function with output signal-to-noise ratio, adopt quanta particle swarm optimization self-adaptation to obtain the optimal value of bistable stochastic resonance system parameter a and b, to detect the feeble signal frequency in very noisy.

3.1 small parameter signals

The signal that in accidental resonance, the amplitude of signal and frequency are less than 1 is called small parameter signal, and the parameter search range determining method example of a small parameter signal accidental resonance is as follows, and wherein, signal is amplitude A ₀=0.2, carrier frequency f ₀the cycle Sine wave signal of=0.02Hz, sample frequency fs=1Hz, noise to be average be zero, noise intensity D=1 white Gaussian noise.Fig. 2 (a) and (b) add the time domain after making an uproar and frequency-region signal waveform for small parameter signal.

(1) parameter a is determined _max.According to stochastic resonance system stable output demand fulfillment ah≤1, obtain ah≤1.This example a _max=1.

(2) parameter a is determined _min.Stochastic resonance system resonance demand fulfillment r _k>=2f ₀, $a^2/4\mathrm{bD}\≤\ln (a/\left(2\sqrt{2}\mathrm{\π}{f}_0\right)),$ Obtain according to right side non-negative $a_{\min }=2\sqrt{2}\mathrm{\π}{f}_0.$ This example a _min=0.1777.

(3) parameter b is determined _min.Determining a _minin situation, have obtain b _minobtained by left side minimum value, namely exist time (e is natural constant), obtain b _min=ea _min ²/ 2D, noise intensity D can pass through D=σ ²h/2 tries to achieve, σ ²for noise variance, because input is trigonometric function signal, therefore, the variance of mixed signal directly can be got.This example b _min=0.0438.

(4) tentatively parameter b is determined _max.Mixed signal is sent into parameter b ₀=1, h=1/f _sbi-stable stochastic resonance theory system, the power spectrum that computing system exports and is that step-length constantly increases parameter b with Δ ₀until power spectrum is dispersed.This example gets Δ=0.1.

(5) accurately parameter b is determined _max.Note b=b ₀/ Δ, is that step-length constantly increases parameter b until power spectrum is dispersed with ε, remembers b respectively _max=b/ (1+ ε), b _min=ea _min ²/ 2D (e is natural logarithm).This example b _max=0.2633, ε=0.001.

(6) quantum particle swarm adaptive algorithm initialization.The maximum iteration time T of this algorithm is set _max=50, population quantity M=50 and dimension D _dim=2 (parameter a and b), Search Range is respectively [b _min, b _max], this example Search Range is [0.1777,1.0000] and [0.0438,0.2633], then position vector of initialization particle within the scope of this.

(7) initialization optimal-adaptive angle value.Calculate fitness value corresponding to each particle according to formula (6), and using the self-adaptation angle value of first generation particle self as single particle local optimum fitness value, be designated as P _best(i) (i=1,2 ..., M), by P _besti the maximal value in (), as global optimum's fitness value, is designated as g _best.

(8) optimal-adaptive angle value is upgraded.Upgrade particle position by formula (4), recalculate the fitness value of each particle, if the single particle local optimum fitness value P obtained _best(i) or global optimum fitness value g _bestbe better than previous generation particle, then upgrade corresponding P _best(i) or g _best.

(9) optimal value of the parameter is determined.Particle position corresponding to the final global optimum's fitness value exported obtains final parameter optimization result, optimal value of the parameter is substituted into bistable resonator system, exports Detection of Weak Signals result.Quanta particle swarm optimization convergence map is as Fig. 3.Optimized parameter is substituted into bi-stable stochastic resonance theory system, the time domain shown in Fig. 4 (a) He (b) and frequency-domain waveform thereof is obtained respectively after accidental resonance process is carried out to original signal, its periodic component is fairly obvious, and very outstanding at the spectrum peak at 0.02Hz place, so the method that the present invention carries can determine the hunting zone of parameter under small parameter condition fast.

3.2 large parameter signals

The signal that in accidental resonance, the amplitude of signal and frequency are less than 1 is called small parameter signal, and the parameter search range determining method example of a small parameter signal accidental resonance is as follows, and wherein, signal is amplitude A ₀=1, carrier frequency f ₀the cycle Sine wave signal of=10KHz, sample frequency fs=1MHz.Noise to be average be zero, noise intensity D=25 white Gaussian noise.Fig. 5 (a) and (b) add the time domain after making an uproar and frequency-region signal waveform for large parameter signal.

(1) parameter a is determined _max.According to stochastic resonance system stable output demand fulfillment ah≤1, obtain ah≤1.This example a _max=1e+06.

(2) parameter a is determined _min.Stochastic resonance system resonance demand fulfillment r _k>=2f ₀, $a^2/4\mathrm{bD}\≤\ln (a/\left(2\sqrt{2}\mathrm{\π}{f}_0\right)),$ Obtain according to right side non-negative $a_{\min }=2\sqrt{2}\mathrm{\π}{f}_0.$ This example a _min=8.8858e+004.

(3), parameter b is determined _min.Determining a _minin situation, have obtain b _minobtained by left side minimum value, namely exist time (e is natural constant), obtain b _min=ea _min ²/ 2D, noise intensity D can pass through D=σ ²h/2 tries to achieve, σ ²for noise variance, because input is trigonometric function signal, therefore, the variance of mixed signal directly can be got.This example b _min=4.4520e+014.

(4) tentatively parameter b is determined _max.Mixed signal is sent into parameter b ₀=1, h=1/f _sbi-stable stochastic resonance theory system, the power spectrum that computing system exports and is that step-length constantly increases parameter b with Δ ₀until power spectrum is dispersed.This example gets Δ=0.1.

(5) accurately parameter b is determined _max.Note b=b ₀/ Δ, is that step-length constantly increases parameter b until power spectrum is dispersed with ε, remembers b respectively _max=b/ (1+ ε), b _min=ea _min ²/ 2D (e is natural logarithm).This example, b _max=1.0822e+16, ε=0.001.

(6) quantum particle swarm adaptive algorithm initialization.The maximum iteration time T of this algorithm is set _max=50, population quantity M=50 and dimension D _dim=2 (parameter a and b), Search Range is respectively [b _min, b _max], this example Search Range is [8.8858e+004,1e+6] and [4.4520e+014,1.0822e+16], then position vector of initialization particle within the scope of this.

(7) initialization optimal-adaptive angle value.Calculate fitness value corresponding to each particle according to formula (6), and using the self-adaptation angle value of first generation particle self as single particle local optimum fitness value, be designated as P _best(i) (i=1,2 ..., M), by P _besti the maximal value in (), as global optimum's fitness value, is designated as g _best.

(8) optimal-adaptive angle value is upgraded.Upgrade particle position by formula (4), recalculate the fitness value of each particle, if the single particle local optimum fitness value P obtained _best(i) or global optimum fitness value g _bestbe better than previous generation particle, then upgrade corresponding P _best(i) or g _best.

(9) optimal value of the parameter is determined.Particle position corresponding to the final global optimum's fitness value exported obtains final parameter optimization result, optimal value of the parameter is substituted into bistable resonator system, exports Detection of Weak Signals result.Quanta particle swarm optimization convergence map is as Fig. 6.Optimized parameter is substituted into bi-stable stochastic resonance theory system, after carrying out accidental resonance process to original signal, result is as shown in Fig. 7 (a) He (b), and time domain and Frequency domain noise all obviously reduce, and very outstanding at the spectrum peak at 10KHz place.Its time domain and frequency domain output waveform amplitude are far smaller than input waveform, this is because when regulating parameter b carrys out equivalent adjustment mixed signal enlargement factor, certain multiple can be reduced to the amplitude of output signal, but can't affect form or the figure of solution, so the method that the present invention carries can determine the hunting zone of parameter under the faint 2FSK signal conditioning of large parameter fast.

The present invention is not limited only in the category of the present embodiment, and namely the present invention can be applied in other nonlinear accidental resonance models, equally also can be applied to other adaptive algorithms.

Claims (4)

1., based on a parameter search range determining method for self-adapting random resonant, it is characterized in that, comprise the steps:

Step 1: the upper limit a needing the first parameter a determining stochastic resonance system according to stochastic resonance system stable output _max; Namely

a _max＝1/h

In formula, h is numerical evaluation step-length;

Step 2: the lower limit a needing the first parameter a determining stochastic resonance system according to stochastic resonance system resonance _min; Namely

$a_{\min }=2\sqrt{2}\mathrm{\π}{f}_0$

In formula, f ₀for carrier frequency;

Step 3: according to the lower limit a of step 2 determined stochastic resonance system first parameter a _min, determine the lower limit b of the second parameter b of stochastic resonance system _min; Namely

b _min＝ea _min ²/2D

In formula, D=σ ²h/2 is noise intensity, σ ²for noise variance, h is numerical evaluation step-length; E is natural constant; a _minbe the lower limit a of the first parameter a _min;

Step 4: mixed signal is sent in stochastic resonance system, calculates the power spectrum that this stochastic resonance system exports, and is that change interval constantly increases stochastic resonance system second parameter b until power spectrum is dispersed with fixed step size; Stochastic resonance system second parameter b corresponding to the back now dispersed by power spectrum is the upper limit b of the parameter b of the stochastic resonance system determined _max;

Step 5: by the scope [a of the first parameter a of determined for step 1-2 stochastic resonance system _min, a _max] and the scope [b of the second parameter b of the determined stochastic resonance system of step 3-4 _min, b _max] send in follow-up self-adapting random resonant inspection frequency process as hunting zone.

2. a kind of parameter search range determining method based on self-adapting random resonant according to claim 1, it is characterized in that, described step 4 comprises the upper limit b of the parameter b tentatively determining stochastic resonance system _maxprocess and accurately determine the upper limit b of parameter b of stochastic resonance system _maxprocess; Namely

Step 4-1: tentatively determine b _max, namely mixed signal is sent in stochastic resonance system, calculates the initial adjustment power spectrum that this stochastic resonance system exports, and with the initial value b of the second parameter b of setting ₀based on, the initial adjustment fixed step size Δ of setting is change interval, constantly increases stochastic resonance system second parameter b until initial adjustment power spectrum is dispersed; Stochastic resonance system second parameter b corresponding to the back now dispersed by initial adjustment power spectrum is the upper limit b of the parameter b of the stochastic resonance system tentatively determined _max;

Step 4-2: accurately determine b _max, namely mixed signal is sent in stochastic resonance system, calculates the accurate adjustment power spectrum that this stochastic resonance system exports, and the upper limit b of the second parameter b tentatively determined with step 4-1 _maxbased on, the accurate adjustment fixed step size ε of setting is change interval, constantly increases stochastic resonance system second parameter b until accurate adjustment power spectrum is dispersed; Stochastic resonance system second parameter b corresponding to the back now dispersed by accurate adjustment power spectrum is the upper limit b of the parameter b of the stochastic resonance system finally determined _max;

Above-mentioned ε=Δ/n, n be greater than 1 positive integer.

3. a kind of parameter search range determining method based on self-adapting random resonant according to claim 1 and 2, is characterized in that, in step 4, the first parameter that the stochastic resonance system that mixed signal is sent into is all the time a is the initial value b of the second parameter b ₀be 1.

4. a kind of parameter search range determining method based on self-adapting random resonant according to claim 1, is characterized in that, in step 1 and 3, and the numerical evaluation step-length h=1/f of setting _s, wherein f _sfor sample frequency.