The failure rate function is considered as one of the most significant reliability indicators, and this is due to its notable properties [1]. The failure rate has a good visualization and it affords to define reliability by means of a few values. Besides, the necessary failure rate parameters may be generally derived from the empirical data. Conventionally, the failure rate function is represented by the bathtub-shaped characteristic with three segments as shown in the Fig. 1 below [2].

Fig. 1 – The bathtub-shaped failure rate function

The legend of the Fig. 1 is following:

Segment I relates to the development period of the facility and generally has a higher failure rate due to faults at the start of production.

Segment II with a nearly constant failure rate corresponds to the main operating period of the facility.

Segment III is considered as the period of aging and therefore increasing of the failures associated with long-term using of the facility.

To date the modeling of the bathtub-shaped failure rate function mainly employs approach based on the Weibull failure rate distribution with specific parameters for each individual segment i.e. segmented approach [2, p. 8]. However, this approach has a number of disadvantages. The purpose of this article is to propose a functional presentation that would more effectively suit to the bathtub-shaped failure rate function using continuously a two-three parameters for all the segments. We define this as non-segmented modeling as opposed to the conventional segmented approach [3].

Segmented approach

The realization of the segmented approach is generally connected with the Weibull distribution and presented with the expression [3]:

                            (1)

where:

T – life cycle time

α  – the Poisson parameter

β₁ < 1 (in view of Segment I)

β₂ = 1 (in view of Segment II)

β₃ > 1 (in view of Segment III).

As will be shown below, such segmented modeling, which has a satisfactory performance on the Segment I, suits insufficient to the bathtub-shaped characteristic on two other segments.

Non-segmented approach

To improve the disadvantage mentioned above, consider the function using two components P(t) and W(t), which are Poisson and Weibull distribution function accordingly:

S(t) = β^-1P(t) – (T ln ln (1– W(t))^-1)^-1         ,     0 ≤ t < T                            (2)

Expression (2) may be written in a more practical manner as

S(t) = β^-1e^-δt – (β·T ln t/T)^-1                          ,     0 ≤ t < T                            (3)

where δ is the Poisson parameter.

Concerning the properties of this functional presentation, the function (3) is positive and regular. Besides, it has a special zero point:

S(0) = β^-1                                                                                                                                       (4)

which e.g. can be comfortably used for the parameter β evaluation.

According to expression (3), the Poisson component dominates on the Segment I, while the Weibull component has there a negligible effect. On the other hand, the Weibull component dominates on the Segment III, while the Poisson component has there a negligible effect. Finally, on the Segment II, the components are complementing each other, and that leads to a rather steady characteristic of the Segment II. As required, the function (3) uses an additional matching item γ which may be evaluated as the failure rate average value on the Segment II.

Numerical comparison

Below, to compare the efficiency of functional presentations (1) and (3) it will be used empirical data with a specific life cycle T = 100 e.g. months [4, p. 54]. The results of numerical comparison are shown on the diagram below.

Fig. 2 – Comparison of the failure rate presentations

Series1 – Empirical data,

Series2 – Non-segmented approach

Series3 – Segmented approach

Table 1– Numerical data to the Fig. 2

The characteristic of segmented approach in the Fig. 2 uses α = 0.01. Besides, it is supposed that β₁ = 0.5, β₂ = 1.0 and β₃ = 4. In turn, the characteristic of non-segmented approach in the Fig. 2 uses δ = 0.27, γ = 0.0161 and β = 5.9.

Another interesting aspect is the failure rate characteristic’s dependency on β which is a key parameter of the Weibull distribution [3]. The comparison results of modeling for the bathtub-shaped failure rate function using both functional presentations are shown in the Fig. 3 (the function (1)) and in the Fig. 4 (the function (3)).

Fig. 3 – Segmented failure rate functional presentation

Series1 – with β = 2.5

Series2 – with β = 7.5

Series3 – with β = 12.5

Fig. 4 – Non-segmented failure rate functional presentation

Series1 – with β = 2.5

Series2 – with β = 4.5

Series3 – with β = 6.5

Thus, as opposed to segmented approach, the non-segmented modeling outlined above distinctly preserves the bathtub-shape configuration by all values β.

Conclusions

The numerical comparison presented above makes it clear that the non-segmented modeling of the bathtub-shaped failure rate function is not only more effective from the point of view matching the bathtub-shape characteristic, but besides significantly better suits to the empirical data. At that as components the Poisson and Weibull distribution functions are used with their parameters which may be easily derived from the specific points of failure rate characteristic or evaluated by means of statistical estimation [5]. The non-segmented modeling proposed should improve the performance of reliability analyzes and its applications.

Generally signals and systems used to be treated without taking into account effects of special relativity [1]. However, their consideration could be useful in view of the current and future space developments. According to special relativity the reference system, travelling in the homogeneous and isotropic space with a steady velocity, is defined as moving inertial reference system (IRS). At the same time the IRS of the observer is defined as stationary IRS (in relation to the moving IRS). One of the consequences of special relativity is that the clocks in the moving IRS are measured as slower than the similar clocks in the stationary IRS. This relativistic time dilation has been empirically certified by the experiment with a pair of atomic clocks while one of them was sent on the space mission [2]. The same phenomenon has been registered also through measurements of μ-meson cosmic radiation in the atmosphere [3]. It is logical to assume that the relativistic time dilation should affect all processes associated with clocking.

Next for the sake of clarity it should be agreed on appropriate symbols which will be used further. So symbols with acute accent (´) belong to the moving IRS. On the other hand, symbols without acute accent belong to the stationary IRS.

Presentation of relativistic time dilation in frequency area

By definition, the relativistic time dilation is expressed by means of the Lorentz factor [1]

γ = 1 /(1+(v/c)²)^1/2 (1)

where

v – relative velocity of the IRS
c – speed of light

According to special relativity the speed of light which is the physical constant cannot be surpassed, and therefore γ > 1. So the times in both IRS are related as

t = t´ / γ (2)

From here

t < t´ (3)

that is the formal indication of the relativistic time dilation. Consider the signal in form of a simple harmonic oscillation[4]

g(t) = A cos (2π f t + φ) (4)

where

A – oscillation amplitude

f – oscillation frequency 

φ - oscillation phase

Then by substituting (2) in (4) we obtain

g(t´) = A cos (2π f´t´+ φ) (5)

where

f´ = f / γ (6)

This equality describes the presentation of the time dilation in frequency area. Hence, the frequency of harmonic oscillation in the moving IRS is decreased in relation to the stationary IRS. In view of that, for instance, the GPS system has required an appropriate adjustment of on-board synthesizer to provide correct frequency for terrestrial applications [5].

Further any complex periodical signal G(t) can be presented as the sum of simple harmonic components [4] :

G(t) = ∑ A_n cos (2πn t f_n + φ_n) (n= 0,∞) (7)

Next in the moving IRS, by analogy to (6) for each component (7) should be

f_n´ = f_n / γ (8)

Thus, the frequencies of components in (7) are decreased in the moving IRS in relation to the stationary IRS. In other words, in the consequence of relativistic time dilation the signal frequency spectrum gets the red offset.

One comment on the red offset by optical signals

A class of harmonic oscillation includes also monochromatic optical signals [6]. Thus, as shown above, they are subject to decreasing of his frequency and accordingly increasing of his wavelength in the consequence of the relativistic time dilation. In this way it can be concluded that while the velocity of propagation of optical signals (say speed of light) is the physical constant and does not depend on the choice of IRS, the frequency or wavelength of the optical signal is variable and is determined in dependence on relative velocity of the moving IRS in which this signal is produced. That means that any optical signal, which is produced in the moving IRS, will be perceived in the stationary IRS with the red offset (not confuse with the Doppler’s red shift [1]).

Relativistic generalization of Nyquist–Shannon theorem

According to Nyquist–Shannon theorem [4] the sampled signal with the spectrum limited to the Nyquist frequency f_N can be correctly restored according to the sequence of its reference values, which are selected with the sampling rate interval

h < 1 / 2f_N (9)

Then in the moving IRS the sampling rate interval should be taken as

h´< 1 / 2f´_N , (10)

and combining (6) and (10) we obtain

h´ < γ / 2f_N (11)

Thus (11) presents relativistic generalization of the Nyquist–Shannon theorem. In case of γ = 1 the inequality (11) takes form of (9) which correspond to the stationary IRS. However, in the moving IRS the sampling rate interval should be selected in accordance with (11).

Relativistic generalization of Shannon-Hartley theorem

According to Shannon-Hartley theorem [4] the channel capacity is determined as

C = f_c log (1+S/N) (12)

where

f_c – cut frequency (channel bandwidth)S/N – signal-to-noise ratio

It is clear that on interval (-∞,∞) the signal-to-noise ratio

SNR =ʃ |S(f)|²df / ʃ |N(f)|²df (13)

is time-invariant. Thus, the channel capacity only depends on the bandwidth.

Then in the moving IRS

C´ = f´_c log (1+S/N) (14)

Since subject to (6)

f´_c = f_c / γ (15)

Shannon-Hartley theorem is generalized with

C´= f_c /γ log (1+S/N) , (16)

and in view of (12)

C´= C/ γ (17)

Thus, in the moving IRS occurs decreasing of the channel capacity. In the case of γ = 1, equality (15) corresponds to the stationary IRS. Note this effect should be taken into account also on evaluating of the achievable maximum of the channel capacity [7].

Relativistic time dilation impact on the impulse response of a discrete system

Consider the impulse response of a discrete system as a sampled signal [8]. Then in accordance with the generalized Nyquist-Shannon theorem the appropriate clock rate interval of the discrete system with a given cut frequency f_c should satisfy the inequality

η < γ / 2f_c (18)

Note that the case γ = 1 in (18) corresponds to the stationary IRS. On the other hand the time dilation impact on discrete system entails an increasing of the system time response. Indeed, any part of time response of the discrete system is the sum of the clock rate intervals. But according to (18) η is increasing exactly with the factor γ. Since the whole time response can be presented as the sum of such intervals, the system time response also increases.

Relativistic time dilation impact on the system service life period

One of the ways to represent the system service life period is an appropriate modelling of the so-called bathtub-shape failure rate function [9]. For the convenience of analysis, we make of use the model presented in [10], which includes the variable T (service life period )

S(t) = β^-1e^-δt – (β·T ln t/T)^-1 , (19)

where β and δ are specified parameters which are determined statistically.

In view of (2) in the moving IRS should be

S(t´) = β^-1e^-δt´/γ – γ(β·T´ ln t´/T´) (20)

where

T´= γ·T (21)

Thus in the moving IRS as opposed to the stationary IRS the system service life period increases exactly with the factor γ. This result being postulated by special relativity is confirmed here by means of the mathematical-statistical modelling.

In the Fig.1, the failure rate curves and hereby the service life period (in terms of time span of the curve) are provided for some values of γ. Note the case γ = 1 corresponds to the stationary IRS.

Relativistic time dilation impact on the mean time between failures

The mean time between failures (MTBF) is an important reliability characteristic which is often presented along with the failure rate function [11]. MTBF is useful in the phase of service life period where the failure rate practically may be assumed constant (the flat part of the curve in the Fig.1). By definition, MTBF is a reverse quantity to the parameter of Poisson distribution [9]

R(t) = e^-t/M (22)

where M is MTBF.

Then in the moving IRS subject to (2)

R(t´) = e^-t´/M´ (23)

where

M´= γ·M (24)

Thus, in the moving IRS the mean time between failures increases exactly with the factor γ. In the case of γ = 1, equality (24) corresponds to the stationary IRS.

Discussion

The severity of exposure of relativistic time dilation depends on the value of the Lorentz factor. In turn the last one is determined by the relative velocity of the moving IRS. In the case v ≪ c the impact of relativistic time dilation is not big enough to produce simply measurable effect for application. However, the relative velocity, which approaches to 30,000 km/s (ten per cent speed of light) and more, makes the relativistic time dilation rather tangible. In the Table 1 below are shown a number of the Lorentz factor values with the according relative velocities in the percent of speed of light.

Table 1. The Lorentz factor versus relative velocity

At present (perhaps also in the immediate future) the relative velocities shown above remain not technically feasible. However, some highly fine applications may require these effects to be taken into account also at the current stage of technological development.

Conclusions

A number of aspects of the relativistic time dilation impact on signals and systems were considered. This consideration led to generalization of the Nyquist-Shannon and Shannon-Hartley theorems. Thereupon it was shown that the time dilation impact is presented as follows:

• decrease of the oscillation frequency
• compress of the signal spectrum (red offset)
• decrease of the channel capacity
• increase of the system time response
• increase of the system service life period
• increase of the mean time between failures.

In principle, this requires an additional tuning of software or hardware, or appropriate taking these effects into account on designing. Anyway, on the actual stage of study this list should not to be considered as comprehensive. Besides, some of these effects perhaps are not confined with the framework of technology, and may occur in other systems. From this point of view a definite similarity between technical and biological systems may be also taken into account [12].

The relativistic time dilation is one of the fundamental issues of Einstein’s special relativity, which has to do with inertial reference systems (IRS) travelling in the homogeneous and isotropic space with a steady velocity [1]. At the same time, as opposed to the stationary IRS which is associated with observer all other inertial systems are defined as moving IRS. In the following, all the notations with acute accent (´) will be related to the moving IRS. The key quotient of the relativistic time dilation (named Lorentz factor [1]) is given by

γ = 1 / (1)

where

v – relative velocity of the moving IRS

c – velocity of light in free space.

Then, basically, relativistic time dilation is expressed by the equality

t´ = γ t , γ > 1 (2)

where

t´ – time interval measured in the moving IRS

t – corresponding time interval measured in the stationary IRS

Until now, the relativistic time dilation has been tested and has received confirmation through physical experiments [2]. The human perceiving of the time dilation was studied in [3]. In technical context, the impact of the relativistic time dilation on signals and systems has been discussed in [4].

Dynamics of linear system in the moving IRS

The dynamics of the linear system configured in the stationary IRS is described by the ordinary differential equation with constant coefficients [5]

a_nx⁽ⁿ⁾(t) + a_n-1x^(n-1)(t) + … + a_оx(t) = f(t) (3)

where

a_n - coefficients associated with the system parameters in the stationary IRS

x(t) – system response to the input action f(t)

Representing equality (2) in differential form we obtain

dt´ = γ dt (4)

This clearly implies

dt = dt´/ γ (5)

Then substituting (5) in (3) one immediately gets the differential equation for the moving IRS in the form

γⁿa_nx⁽ⁿ⁾(t´) + γ^n-1a_n-1x^(n-1)(t´) + … + a_оx(t´) = f(t´) (6)

or otherwise

a_n´x⁽ⁿ⁾(t´) + a_n-1´x^(n-1)(t´) + … + a_о´x(t´) = f(t´) (7)

where

a_k´ = γ^ka_k , k = 0 … n (8)

Thus, the relativistic time dilation in the moving IRS leads to the differential equation, which coefficients are modified by the Lorentz factor γ. At the same time, these coefficients are associated in a certain way with the elements of the analyzed system. Therefore, along with the change in the characteristics of the system, one can also speak about a change of parameters of its elements due to the relativistic time dilation. This will also be shown in the Example 1 below.

Stability of a linear system placed in the moving IRS

Consider a stable linear system configured in the stationary IRS. The question is how the relativistic time dilation affects the stability of this system to be placed in a moving IRS. As well-known, for stability of a linear system it is necessary and sufficient that all roots of the characteristic equation of the system should have the negative real parts [5].

Represent the characteristic polynomial associated with the equation (3) as a product

P(s) = a_n (γs – s_o) (γs – s₁) … (γs – s_n-1) (9)

where s_o, …, s_n-1 are the roots (generally, complex) of the characteristic equation.

Next, the polynomial (9) can be written in the form

P(s) = a_n´ (s – s_o´) (s – s₁´) … (s – s_n-1´) (10)

where a_n´ = γⁿa_n and s_k´ = s_k / γ.

Then for all roots of the polynomial (10)

Re(s_k`) = Re(s_k) / γ , k=0, …, n-1 (11)

Since by definition γ is positive the sign in (11) is retained. Therefore, a stable linear system retains its stability despite the relativistic time dilation impact.

On the other hand, since γ > 1 the equality (11) implies

Re(s_k`) < Re(s_k) , k=0, …, n-1 (12)

That means that the stability margin of the analyzed system decreases. In other words, if the stability margin of the linear system configured in the stationary IRS is given through the quotient η₀, then in the moving IRS it will be equal η₀/γ.

Example 1

Consider the second-order system with its differential equation:

x⁽²⁾(t) + a₁x⁽¹⁾(t) + a_ox(t) = γ² , γ > 1 (13)

where a₁=2δ and a_o=ω² are attenuation coefficient and resonance frequency accordingly.

Then in view of (7) and (8) we obtain

x⁽²⁾(t´) + 2δ´x⁽¹⁾(t´) + ω´²x(t´) = 1 (14)

where in view of (6) δ´ = δ/γ and ω´ = ω/γ.

In particular, in case of series RLC circuits [6]: δ = R/L and ω = 1/.

In the moving IRS these parameters are given by

δ´ = R / γL (15-a)

ω´ = 1 / γ (15-b)

Here, δ´ plays also the role of the stability margin. Thus according to equality (15-a) stability of the circuit decreases.

On the other hand 1/δ is considered as response time Δ of RLC circuit [6], and the latter in regard to (15-a) increases with the factor γ. Moreover, the same effect follows from (15-b) for oscillation cycle. Hence, in the moving IRS response time and oscillation cycle are given by

Δ´ = γ·Δ (16-a)

T´ = γ·2π (16-b)

At the same time

T = 2π (17)

Then in view of (16-b) and (17) we obtain

T´ = γ·T (18)

Hence, the oscillation period in the moving IRS increases with factor γ. This is in full accordance with the equality (2) for the relativistic time dilation.

Moreover, the equality (15-a) can be written as

δ´ = R / L´ (19)

where

L´ = γ·L (20)

On the other hand in the moving IRS

T´ = 2π (21)Then under consideration of (19), comparing (16-b) and (21) we obtain

C´= γ·C (22)

Hence, formally it can be admitted that in an oscillation circuit to be placed in the moving IRS the values of C and L are increasing in accordance with (20) and (22). Note that the physical aspect of this statement is beyond the scope of this paper.

Example 2

Consider the characteristic equation of a stable three-order linear system

a₃ s³ + a₂ s² + a₁ s¹ + a_о = 0 (23)

According to Hurwitz’s criterion [5], the coefficients in (23) are positive and also

a₁ a₂ – a_o a₃ > 0 (24)Then in case of the system placed in the moving IRS the polynomial (23) takes the form

a₃´ s³ + a₂´ s² + a₁´ s¹ + a_о´ = 0 (25)

where (in accordance with (7) by γ >1) all coefficients stay positive.

Next, using (8) and (24) we obtain

a₁´ a₂´ – a_o´ a₃´ = γ³ (a₁ a₂ – a_o a₃) > 0 (26)

Therefore, the system retains its stability in the moving IRS despite the relativistic time dilation impact.

Discussion

As it was shown above, the relativistic time dilation leads to a change in parameters of the system without loss of its stability. The stationarity of the system is retained by definition since it is about inertial reference. In this case, the Lorentz factor and coefficients of the differential equation describing the system remain constant. Moreover, this effect may turn out to be more or less significant, depending on the value of the Lorentz factor. For example, if in the coming decades the speeds in the space field can reach about 0.01% of the velocity of light in free space, the values ​​of the Lorentz factor will remain less than 1.000000005, which in many cases entails that the time dilation impact remains within the limits of measurement error or manufacturing tolerance. Nevertheless, by the applications which require rather precise measurements such as GPS [7], the relativistic time dilation requires an appropriate fine-tuning of the equipment already at the current stage of technical development.

Conclusions

As shown above, owing to the relativistic time dilation impact on linear system the coefficients of its differential equation change their values in the moving IRS. This leads, first of all, to the change in system characteristics which are associated with these coefficients (such as, attenuation coefficient or response time). On the other hand, these changes can be formally treated equivalent to the modification of parameters of individual elements. And finally, the time the relativistic time dilation impact does not make the system unstable though reduces its stability margin.

The effect of time dilation relates to the Einstein’s special relativity [1]. According to the latter, the relativistic time dilation is a difference in the course of time recorded by two observers, which are respectively in the moving inertial reference system (IRS) and in the IRS, which is stationary with respect to the first, while in the moving IRS the clock turns out to be slower than the course of time in a stationary IRS This phenomenon was confirmed, for example, by the Hafele-Keating experiment. [2] The relativistic time dilation has a certain effect on the signals and on the linear system characteristics [3], [4]. The present article discusses the possible ways to compensate for this effect in the frequency and time domain. Subsequently, all the notations with acute accent (´) will be related to the moving IRS.

Formulation in the time domain

The linear system can be represented by an ordinary differential equation with constant coefficients [5]:

a_nx⁽ⁿ⁾(t) + a_n-1x^(n-1)(t) + … + a_оx(t) = f(t)     (1)

Placed in a moving IRS the system is affected by relativistic time delay, and the coefficients in (1) are expressed in the form [3]:

a_k´ = γ^ka_k , k = 0 … n      (2)

where γ - Lorentz factor, γ = 1 / (v – relative velocity of IRS, c – light speed in free space and γ > 1) [1].

Consider the impulse characteristics of a linear system in a stationary and moving IRS as G(t) and G´(t) respectively. In turn these characteristics depend implicitly on the parameters of the linear system. If we represent both characteristics by the corresponding set of samples, taking into account the Nyquist frequency and the attenuation [6], then the ordinary least-squares technique can be used for adjustment of the system parameters [7]. We define the objective function as a proximity measure of two sample sets:

Φ(p₁,p₂, …,p_m) =(g_k – g_k´)²     (3)

where
g_k´ - samples of the impulse response G´(t), depending on the variable parameters (p₁,p₂, …,p_m)
g_k - samples of the impulse response G(t)

The choice of parameters and their quantity in (3) depends on system features. Within this statement of the problem, the minimization of the function (3) is next step to be performed. Herewith, existing numerical methods of multidimensional optimization can be used which are varying in specific strategies for selecting of the increment vector for variable parameters (Δp₁,Δp₂, …,Δp_m) [7]. 
The block diagram for the adjustment in time domain by using ordinary least-squares is shown in Fig. 1.

Fig. 1 – Block diagram of the adjustment in the time domain

On this diagram, the input of the linear system (LS) is fed by delta function δ´(t). The impulse response G´(t) is transferred to the sampler (S), at the output of which appears a set of samples g_k´. The specific of this procedure is that the samples g_k and g_k´ are produced with a time step which is the subject to the Nyquist-Shannon theorem [6] and the Lorentz transformation [1].

Note that in the case of a pulse (discrete) linear system the sampler is the standard part of it [5]. However, for temporal matching, additional oversampling with subsequent digital filtering may be required to remove high-frequency components that do not satisfy the conditions of the Nyquist-Shannon theorem [6].

Next, the appliance realizing the ordinary least-squares (OLS) is used, which includes a multidimensional optimization algorithm generates a vector of parameter increments transmitting to the linear system. It is assumed that the system has a gear for adjustment of parameters by using these increments.

The parameter adjustment process must be stopped at the moment when the specified value (σ) of the objective function is reached, that is:

Φ(p₁,p₂, …,p_m) < σ      (4)

As another termination condition, the value of the time interval may be set, by which no further increment vector was found, leading to the further decrease of the objective function.

Formulation in the frequency domain

The frequency response of a linear system can be obtained on the basis of the differential equation (1). Next, assuming in (1) zero initial conditions f(t) = δ´(t) follows the operator representation with transfer function [5]:

R(s) = (a_nsⁿ + a_n-1s^n-1 + … + a_о)^-1      (5)

With the substitution s = jω the system frequency response is determined. On the other hand, the frequency response can be obtained from the impulse response by means of the Fourier transform [6].

Consider the amplitude-frequency characteristics of the linear system in a stationary and moving IRS as R(ω) and R´(ω) respectively. In turn these characteristics depend implicitly on the parameters of the linear system. Then, by analogy with the formulation of the problem in the time domain, the objective function may be defined as a proximity measure in the form:

Φ(p₁,p₂, …,p_m) =(r_k – r_k´)²      (6)

where
r_k´ – samples of the impulse response R´(t), depending on the variable parameters (p₁,p₂, …,p_m)
r_k – samples of the frequency response R(ω)

The sum on the right-hand side of (6) can be supplemented by the same components but containing the samples of the phase characteristics or one can confine with only these components. However, the final choice of the function (6) must be made by taking into account the features of the system and the actuality of specific characteristic.

The block diagram for adjustment in the frequency domain by using ordinary least-squares is shown in Fig. 2.

Fig. 2 – Block diagram of the adjustment in the frequency domain

This block diagram differs from the Fig. 1 only in the way that the discrete Fourier transform appliance (DFT) is additionally used [6] in order to obtain the samples of the amplitude-frequency characteristic obtained from the of generated impulse response. By analogy with the formulation of the problem in the time domain, the parameter adjustment process must be stopped when inequality (4) is fulfilled or when the run time is exceeded, during which no single incremental vector has been found, that would lead to a further decrease of the objective function.

Computer simulation

For the experimental verification of the compensation methods described above, computer simulation of a second-order linear system, represented by a series LCR circuit, is carried out. Moreover, the following parameters are used: γ equal 2.5, L equal 25mH, C equal 0.5nF and R equal 50kΩ.

When simulating in the time domain (in accordance with the block diagram in the Fig. 1) L´ was taken as variable parameter. The simulation results by two OLS iterations are shown in the Fig. 3, where, for clarity, the points of impulse characteristics are connected by lines.

When simulating in the frequency domain (in accordance with the block diagram in the Fig. 2) C´ was taken as variable parameter. The simulation results by two OLS iterations are shown in the Fig. 4, where, for clarity, the points of amplitude-frequency characteristics are connected by lines.

From these results it may be concluded with confidence that further iterations lead to complete compensation of the relativistic time dilation impact on the linear system under consideration. Thus, the principal possibility of the compensating on the bases of the represented block diagrams is confirmed.

Direct compensating

In some cases it is possible to obtain analytical relationships that establish a specific connection between parameters of the linear system and Lorentz factor [4]. As such an example, consider series LC circuit without losses. As well-known, the period of damped oscillations in this circuit is determined by the Thomson formula [8]:

T = 2π      (7)

Hence the oscillation frequency will be equal:

ω = 1 /     (8)

Then for this system in a moving IRS:

ω ´ = 1 /     (9)

In this case, as was shown in [4]:

L´ = γ·L     (10)
C´ = γ·C            

Thus, in order to provide the equality ω´ = ω (for compensation in the frequency domain) it is sufficient to change the values of L’ and C ‘, that is

L´ => L´ / γ     (11)
C´ => C` / γ           

Note that this adjustment of the parameters leads to complete compensation of the time dilation impact on the frequency response. Another example of the direct compensating is described in [9], where parameter adjustment was completed for the on-board equipment of the global navigation system.

Conclusions

The methods for compensating the relativistic time dilation impact on the characteristics of linear systems are proposed. In connection with this, two formulations of the problem for the parameter adjustment of linear systems – in the time domain and frequency domain – is considered. In both cases, the compensation for the time dilation effect is formulated as OLS and minimizing of the objective function as proximity measure of the two characteristics. The computational complexity of this minimization procedure (generally, multidimensional) mostly depends on the number of variable parameters and may require rather powerful computing appliances. Hereby, it seems appropriate to use specialized processors of the kind FFT [10]. On the other hand, by known analytical relationships, reflecting the relativistic time dilation impact on the characteristics of a linear system, direct adjustment of the parameters is possible, up to the complete compensation.

In [1] it was shown that the relativistic time dilation impact on linear dynamical systems does not lead to the loss of stability, but at the same time reduces the stability margin of the system. At the same, linear systems consist of elements that are in itself an idealization of real nonlinear components. On the other hand, there are intrinsically nonlinear systems. So it is also important to consider the relativistic time dilation impact on the stability of nonlinear dynamical systems.

Recall, in accordance with special relativity the time dilation is a difference between the clock times, measured by two observers, one of which is in the moving inertial reference frame (IRF), and the other in the IRF, which is stationary relative to the first [2]. This time dilation is expressed by the Lorentz transformation:

Δt^~ = γ·Δt             (1)

where 
Δt^~ - time interval measured in the moving IRF
Δt – time interval measured in the stationary IRF
γ - Lorentz factor, γ = 1 / 
v – relative velocity of the moving IRF
c – light velocity in free space.

Thus, by definition the value of the Lorentz factor is equal to one for stationary IFR and greater than one for moving IRF.

Next, the notations with the upper tilde (^~) will refer to the moving IRF.

Nonlinear system representation in the moving IRF

A nonlinear dynamical system may be formally described by means of nonlinear differential equation, where at least one the derivative of the calculated function (including the zero-order derivative, which actually is the calculated function itself) enters into the equation as a nonlinear factor [3]. Thus, the nonlinear differential equation has the following form:

a_n y⁽ⁿ⁾(t) + a_n-1 y^(n-1)(t) + … ā_ky^(k)(t) … +a_оy(t) = 0            (2)

where в_k is one of the function f(y) or f(y^(k)), that are relating to two types of the equation.

Under consideration of (1), i.e. in the moving IRF, equation (2) takes the form [1]:

a_n^~y⁽ⁿ⁾(t) + a_n-1^~y^(n-1)(t) + … ā_k^~y^(k)(t) … +a_о^~y(t) = 0            (3)

where
a_n^~ = гⁿ a_n           (4)

As for the element в_k^~, its representation is similar to (4) as

ā_k^~ = г^k f(y)          (5)

for the first type

or

ā_k^~ = г^k f(г^ky^(k))           (6)

for the second type, and г is Lorentz factor.

Note, that by г=1 in (3) it follows a_n^~ = a_n.

Generally, for the nonlinear differential equations there are no useable analytical solution methods with exception to some particular cases [3]. Therefore, to study stability of a nonlinear dynamical system we will use numerical approach. Specifically Van der Pol’s and Rayleigh’s non harmonic oscillators [4] are considered, which are described by nonlinear second-order differential equations of the first and second type, respectively. Moreover, the estimation of stability is carried out by using the phase portrait techniques [5].

Van der Pol’s oscillator in the moving IRF

The Van der Pol’s oscillator is formally represented with the nonlinear differential equation of the second order related to the first of the types mentioned above:

y^‘‘(t) + f(y) y^’(t) + ?² y(t) = 0          (7)

where ? is the oscillation frequency, and the nonlinear factor is expressed by the equality

f(y) = α (1 – y²)         (8)

where α as a fixed parameter.

Further, in accordance with the quotients (4) and (5), and under consideration (8), the equation of the oscillator specifically in the moving IRF takes the form:

г² y^’’(t^~) + α (1 – y(t^~)²) γ y^’(t^~) + ?² y(t^~) = 0          (9)

or after reducing:

y^’’(t^~) + α_о (1 – y(t^~)²) y^’(t^~) + β_о y(t^~) = 0             (10)

where

α_о= α/γ

β_о= ?²/γ²

and γ is Lorentz factor.

Let us call (9) as Van der Pol’s equation in the generalized form, which takes into account the Lorentz factor.

By numerically study of the relativistic time dilation impact on the stability of Van der Pol’s oscillator, the following values of the fixed parameters were used: α=0.025 and ?²=1.

The results of computer simulation for different values of Lorentz-factor and under initial conditions y(0)=0 and y’(0)=1 are presented in the Table below.

Table 1. The stability of Van der Pol’s oscillator vs Lorentz factor

Lorentz factor	Oscillator phase portrait	Type of bifurcation
γ = 1.0		Steady focus (stationary IRF)
γ = 2.0		Stable limit cycle (moving IRF)
γ = 2.5		Unsteady focus (moving IRF)

The Table above illustrates the feature of a nonlinear system that can be formulated as follows: upon transition to the mobile IRF, the nonlinear system, which is stable in the stationary IRF (γ=1.0), for other values of the Lorentz factor (γ>1) may go to the mode of stable limit cycle or become unstable. Moreover, this feature also depends on the initial conditions.

Rayleigh’s oscillator in the moving IRF

Similar to Van der Pol’s oscillator considered above, Rayleigh’s oscillator is represented by nonlinear differential equation of the second order related to the second the type with the nonlinear factor which contains the derivative of the calculated function:

f(y^‘) = α (1 – y^‘2)            (11)

where α as a fixed parameter.

Thus, after reducing, the Rayleigh’s equation takes the form:

y^’’(t^~) + α_о(1 – γ² y^‘2(t^~)) y^’(t^~) + β_оy(t^~) = 0        (12)

where 

α_о= α/γ 

β_о=ꙍ²/γ²          

(γ is Lorentz factor and ꙍ is the oscillation frequency)               .                                                                            

Let us call (12) as Rayleigh’s equation in the generalized form, which takes into account the Lorentz factor.

By numerical study of the relativistic time dilation impact on the stability of the Rayleigh’s oscillator, the following values of the fixed parameters were used: α=0.05 and ꙍ²=4.75.

The results of computer simulation for different values of Lorentz-factor and under initial conditions y(0)=0 and y’(0)=1 are presented in the Table below.

Table 2. The stability of Rayleigh’s oscillator vs Lorentz factor

Lorentz factor	Oscillator phase portrait	Type of bifurcation
γ = 1.0		Stable limit cycle (stationary IRF)
γ = 2.0		Unsteady focus (moving IRF)
γ = 2.5		Unsteady focus (moving IRF)

The Table above illustrates the feature of a nonlinear system, that can be formulated as follows: the nonlinear system, which stays in the mode of stable limit cycle in the stationary IRF (γ=1.0), may lose its stability by certain values of the Lorentz factor (γ>1) upon transition to the mobile IRF. Moreover, this feature also depends on the initial conditions.

Harmonic oscillator as a special case

As shown above, the stability of nonharmonic oscillators depends on Lorentz factor. From this point of view it is substantial to compare it to a harmonic oscillator.

Note that the nonlinear differential equation (7) may be easily reduced to the case of the harmonic oscillator while taking f(y) equal to a constant. So the numerical study here was produced with f(y)=0.05 and ꙍ²=4.75.

The results of the computer simulation under initial conditions y(0)=0 and y’(0)=1 are presented in the Table below. 

Table 3. Stability of harmonic oscillator vs Lorentz factor

Lorentz factor	Oscillator phase portrait	Type of bifurcation
γ = 1.0		Steady focus (stationary IRF)
γ =2.0		Steady focus (moving IRF)
γ = 2.5		Steady focus (moving IRF)

From the Table above follows that the linear system stays stable regardless of the value of Lorentz factor. Nevertheless, the stability margin is obviously diminishing as Lorentz factor increases (at the same time it is symptomatic, that the “density” of the oval pattern in the pictures above is increasing). That confirms our conclusion in [1], that stability of a linear dynamical system is invariant to the type of IRF, though the system stability margin (which is obviously related to the system response time) has a decreasing tendency.

Conclusions

It is shown above, that a nonlinear dynamical system as distinct from linear systems may become unstable in the moving inertial reference frame due to relativistic time dilation impact. On the other hand it is numerically confirmed that the linear dynamical systems stay stable regardless of the inertial reference frame, though the system stability margin is diminished. In addition, the formal representation of nonlinear dynamical systems and, in particular, Van der Pol’s and Rayleigh’s oscillators is generalized to the form which takes into account Einstein’s relativity.

Analog and digital filters belong to the class of linear dynamical systems, which are eventually influenced by relativistic time dilation (RTD) [1]. The latter is one of the fundamental consequences of the special relativity formulated for moving and stationary inertial frames through the Lorentz transformation [2, p. 316]:

t` = t  = t·γ      (1)

where 
t´ – time measured in the moving inertial frame
t – time measured in the stationary inertial frame
γ - Lorentz factor ( γ >1 )
v – relative velocity of the moving inertial frame
c – speed of light in free space

In the following, all the notations with acute accent (´) will be relating to the moving frame. Moreover, without loss of generality, only low-pass filters are discussing in detail below.

Synthesis of analog filters under consideration of RTD

The frequency response of analog filters is normally represented as

G(ω) = 1/ [1+K(ω/σ)²]      (2)

where 
K(ω/σ) – characteristic function
σ - cut-off frequency

Generally, specific choice of the characteristic function determines the type of filter. The basic representations of the characteristic function with the corresponding types of filter are given in the Table 1.

Table 1. The basic representations of the characteristic function.

Item	Characteristic function representation	Type of filter
1	K(x) = εTn(x), where Tn(x) is the n-th order Chebyshev polynomial and ε is unevenness index	Chebyshev filter of the first kind with pulsations in the passband [3]
2	K(x) = ε-1Tn(x)-1, where Tn(x) is the n-th order Chebyshev polynomial and ε is unevenness index	Chebyshev filter of the second kind with pulsations in the stopband [3]
3	K(x) = εRn(x), where Rn(x) is the rational n-th order elliptic function and εis unevenness index	Elliptical filter with pulsations, both in the passband and in the stopband [3]
4	K(x) = xn, where n – filter order	Butterworth filter with a smooth frequency response [3]
5	K(x) =  - n-th order eigen-polynomial, which coefficients are elements of the eigenvector associated with the minimum or maximum eigenvalue of a Gram matrix in the basis of orthogonal polynomials (Chebyshev, Legendre, etc.) [4]	Eigenfilter with pulsations in passband or stopband, depending on the formulation of extreme concentration criterion [5]

To take into account the RTD by synthesis procedure, one can use the Lorentz transform (1) for the frequency domain [1]:

ω`= ω / γ      (3)

Then, after substitution (3) into (2), we obtain:

G(ω´) = 1 / [1+K(ω`γ/σ)²]      (4)

or otherwise

G(ω´) = 1 / [1+K(ω`/σ`)²]      (5)

and
σ` = σ / γ      (6)

Thus, the bandwidth of the filter in the stationary inertial system decreases in the moving inertial system by factor of γ. In order to compensate this phenomenon is sufficient to adjust preliminary the cut-off frequency as

σ => γ·σ     (7)

Note that this ratio is also usable by synthesis of the recursive and non-recursive digital filters under consideration of RTD.

Synthesis of recursive filters under consideration of RTD

For the synthesis of recursive filters with regard to RTD, the following two approaches are well suited.

The first one uses the bilinear z-transform method [3]. According to this method, the synthesis is performing in the following way: based on the desired frequency response, one should select the characteristic function of the analog prototype, get its operator image in the s-plane, and then apply the bilinear z-transform:

s =      (8)

where T is quantization period (the inverse of the sampling frequency). As a result, the transmission gain of the non-recursive digital filter in z-plane is obtained.

In this approach, to consider the RTD, one should adjust the analog prototype in accordance with (7), and then apply the bilinear z-transform.

The second approach uses the representation of the characteristic function in the form [4]:

K(jω) =       (9)

where 
n – order of filter
T- quantization period (inverse to sampling frequency)
v_k - coefficients of characteristic function

Next, we have to minimize the criterion [6]:

Φ_min = min { ω / ω }     (10)

where |σT| < π.

Minimization in (10) leads to the homogeneous system:

A V = λ V      (11)

where 
V – set of coefficient
A – square (n x n) matrix

and

A = [ ]           (12)

Herewith, the minimum of the criterion (10) is provided by eigenvector of the matrix (12) associated with the minimal eigenvalue. Note, since the matrix (12) is real and symmetric, all eigenvalues and eigenvectors of the matrix are different and real [7]. To take here RTD into account, it suffices to adjust the integration limits in the criterion (10) in accordance with (7).

Synthesis of non-recursive filters under consideration of RTD

The frequency response of a non-recursive filter has the form [3]:

G(jω) =       (13)

where 
n – filter order
T – quantization period (inverse to sampling frequency) 
v_k - samples of the impulse response

While synthesizing non-recursive filters with a frequency response (13) under consideration of RTD, two approaches might be used.

In the first one, in the range given through the frequency sampling applied, the approximation error relating to the ideal frequency response D(jω/σ) is minimized.

Herewith, in accordance with the approximation criterion, the problem of determining of the coefficients in (13) is formulated. For instance, this criterion can be chosen as a minimum of the maximum error (weighted Chebyshev approximation [8]):

Φ_T = min max {W(ω/σ) (|D(jω/σ)| – |G(jω/σ)|)²} (14)

or as a minimum of the squared error (weighted least squares approximation [9])

Φ_L = min { }      (15)

where ωT ∈ (- π,π), and W(ω/σ) is a positive weight function

The second approach uses the maximization criterion [10]:

Φ_max = max { ω / ω }      (16) 

where |σT| < π.

Hereby, the maximum in (16) is given by eigenvector of the matrix (12) associated with the maximum eigenvalue.

The main feature of the approaches considered above is that the adjustment can be done directly in the criteria.

Discussion

Generally, by synthesis of non-recursive filters, the phase response is non-linear. If it is necessary to obtain a filter with the linear phase, the criteria (14), (15) and (16) should be supplemented with a restriction concerning the symmetry type of the impulse response [11]. Hereby, the coefficients in (13) should satisfy for the symmetric type

v_k = v_n-k-1      (17)

and, otherwise, for the antisymmetric type

v_k = -v_n-k-1      (18)
where
k = 0 … n/2 , by even n
k = 0 … (n-1)/2 , by uneven n

As appropriate, the way for taking the RTD into account proposed above can be easily extended to the synthesis of bandpass filters (pass-through or rejection), for which the methods described above are well suited. In this case, all the specified cut-off frequencies can be adjusted in accordance with (7).

Conclusions

The approach for analog and digital filters synthesis under consideration of the relativistic time dilation is proposed. Generally, the adjustment of one or several cut-off frequencies should be done in the stage of approximation of the frequency response of a filter. Concerning analog and digital filters with the approximation based on extreme concentration functions, such an adjustment may be part of the appropriate criterion.

Pseudo-random sequence generators (PRSG) today are an integral part of fast any application software [1]. Among them, an important role is played by PRSGs with the uniform distribution law from which PRSGs with any other distribution law can be obtained through a transform. In this regard, it is important to be ensured what is about a good quality of the uniform distribution to be used. To do this, there are various methods for PRSG testing [2].

In this paper, we offer the simple to implement test, as well as a method based on Monte Carlo methods for optimizing PRSGs with the uniform distribution law [3].

Using Monte Carlo methods

In Monte Carlo methods, a PRSG is usually first selected, and on its basis the required estimation of the object under study is formed. We will proceed, on the contrary, from the reference value of the required evaluation, in order to estimate on its basis the real quality of PRSG. To do this, we consider a definite integral:

S = (1)

Note that since this integral is a well-known one (namely, S = π/2 [4]), it can be just used as the required reference value.

Calculation of the integral (1) by Monte Carlo methods has to do with the well-known mathematical model [3]. In this model, points within a unit square in which the unit circle is inscribed are randomly sampled, and then the ratio of the number of points M being sampled in the unit circle to the total number of selected points N is determined. In this case, it can be obtained:

π ≈ 4·M / N (2)

In the case of an ideal PRSG, the calculation accuracy through this model increases in proportion to the number of sampled points. However, real PRSGs (for example, due to the pseudo-random sequence generation method being used) often do not provide the desired quality of uniform distribution [5].

In this case, the quality of a PRSG estimated by means of how accurately the number π is determined. This procedure, which we call then “π-Test”, can be used to evaluate the quality of any PRSG with the uniform distribution law. Below we provide two modules the first of which implements the π-Test (specifically using the function Math.GetRandomNumber [6]), and the second provides a pseudo-random sequence based on the linear congruential method (LCM) [2].

Notes:

1. The program modules presented below are intellectual property, and their use requires a reference to this article.

2. The values of Par1, Par2, Par3 by the second module are part of the author’s know-how and can be provided upon request.

Table 1 - Module π-Test

Table 2 - Module LCM

The results of the π-Test with Math.GetRandomNumber are shown in the diagram below. For comparison, here are also the results based on the LCM. The reference points on the diagram correspond to the average estimation for π as the number of cycle’s increases.

Series1 – Based on Math.GetRandomNumber
Series2 – Based on the LCM
Series3 – Level of π

This diagram clearly shows that the quality of the standard PRSG generator leaves much to be desired. Here, indeed, compared to the LCM, the standard PRSG has two disadvantages.

First, it has a relatively large variance, which is indicated by the phase of “acceleration” (the first 10 points of the diagram).

Secondly, at the stationary section (points 11 to 23), the accuracy of approximation π does not exceed the 2nd decimal place, while in the case of the LCM it is provided by 4 decimal place.

This can be explained by the fact that in this model the generated points are distributed unevenly in the unit square. At the same time, their density in the unit circle occurs to be higher.

In addition, the results of the π-Test illustrate the fact that the quality of different PRSG can vary significantly [5].

Optimization of PRSG

To optimize a PRSG with the uniform distribution law, one can use the quadratic criterion [7]:

E => min (ӯ – a)² (3)
Ф(μ,y))

where
ӯ –sample mean 
а –reference value

Note, the function Ф(μ,y) has a similarity to the functions that are used to the pseudo-random sequence, for example, to a normal distribution (Box-Muller transform [2]). However, in this mathematical model, this function serves only to correct the distribution law in order to improve the quality of PRSG.

We consider the above formulation of the problem as applied to a correcting function of the form:

Ф(μ,y) =(y +μ·sin(yπ)) (4)

where y is a random variable in the {0,1} interval formed by the standard PRSG generator and μ is a small parameter (| μ | << 1), since, as a rule, only an insignificant correction of the distribution law of the PRSG is required for optimization. Note, with this definition, in addition, the following necessary relations are satisfied:

0 > Ф(μ, y) < 1 (5)
Ф(0, y) = y

Below are the configurations of the function (5) for the three main cases of the parameter μ.

Series1 - μ < 0
Series2 - μ > 0
Series3 - μ = 0

Discussion

In computer simulation, as a result of optimization of the standard PRSG, the value of μ is obtained by the above-described scheme, which provides a better approximation of π and thus a higher quality of the implementation with the standard function Math.GetRandomNumber after its correction. We note that in this case the parameter μ turned out to be greater than zero, i.e. the configuration of the function (4) corresponds to the red curve in Fig. 2. The average estimations obtained in the Monte Carlo model with optimized PRSG are presented in the diagram below, together with the estimation for the standard PRSG from the diagram in Fig. 2.

Series1 – Before optimization
Series2 – After optimization
Series3 – Level of π

From this diagram, it can be seen that the Monte Carlo model with an optimized PRSG generator has a smaller dispersion and improves the approximation of the π (according to the results – up to the third decimal place).

Conclusion

Above, we considered the formulation of a problem in which the Monte Carlo methods are used to evaluate the quality PRSG with the uniform distribution law. In the framework of this problem formulation, a π-Test with a simple implementation considered above is proposed that can be used to verify the quality of any PRSG with the uniform distribution law. Therefore, this test should be a good supplement to existing testing methods. In addition, it has been shown that this π-Test can be successfully used to optimize PRSG with the uniform distribution law. Performance of the proposed approach is confirmed by the results of computer simulation.

To date, a pseudo-random number generator (PRNG) is considered indispensable for any modern software package [1]. However, the standard software functions in use generating pseudo-random numbers (PRN) differs among themselves in regard to the quality that may exercise unpredictable effects on the results of statistical calculations [2]. From this point of view, the quality of a PRNG plays an important role [3]. In this article, for the purpose of evaluation of PRNG quality, a new two-stage PRNG test procedure is presented, the core of which is the Inverse Box-Muller Transform (BMT) [4]. Below the Direct BMT is described and on the basis of this description the Inverse BMT is formulated.

Direct Box-Muller Transform

For the transition from the Gaussian type PRN to the PRN with the uniform distribution mainly use the well-known Direct BMT, that can be presented as follows [4]:

u =  (1)
v = 

where
x и y – two statistically independent random variable distributed uniformly on 
interval (0,1)
u и v – two statistically independent random variable distributed normally with

average and variance equal to 0 and 1 accordinglyThe standard use of the Direct BMT in the form (1) allows obtaining PRNG with a normal distribution on the basis of an eventually ideal PRNG with the uniform distribution.

As a matter of fact, the PRNGs used in software do not provide the perfect uniform distribution. That is due to the methods of PRN generating and limitations of the decimal places of the variable used by calculations in the course of statistical calculations.

Inverse Box-Muller Transform

Opposing to the formulation given above, an approach may be considered where conversion of the PRN with the normal distribution leads to the adequate PRN with the uniform distribution. As will be shown below, due to such inverse transform, a quality check of the Gaussian type PRNG can be reduced to the evaluation of the quality of the PRNG with the uniform distribution.

The Inverse BMT can be derived immediately from the expression (1). To do this, at first one should divide the second entity in (1) by the first one, as a result of which we obtain:

v/u =  (2)

On the other hand, after squaring and subsequent addition of the both entity in (1), follows:

u²+v² = - 2 ln y (3)

Finally, on the basis of equalities (2) and (3), the Inverse BMT can be formulated as follows:

x = / 2р (4)
y = e^-s
where 
p = v/u 
s = (u²+v²)/2
u и v – two statistically independent random variable distributed normally with average and variance equal to 0 and 1 accordinglyx и y – two statistically independent random variable distributed uniformly oninterval (0,1)Inverse BMT is unique, and its superposition with Direct BMT leads to an identity. The formal proving of this statement is beyond the scope of this article, but its validity was confirmed through computer simulation with the modules shown in the following tables. Below, there is also the original function GetNormRandNumber (not given here) that allows obtaining the Gaussian type PRN.

Table 1. Program “Superposition of Direct & Inverse BMT”

Table 2. Module “Direct BMT”

Table 3. Module “Inverse BMT”

Table 4. The screenshot “Superposition of Direct & Inverse Box-Muller”.

In the screenshot above, it is easy to see that the consistent use of the Direct and Inverse BMTs leads to the initial PRN. Thus, it shows that the superposition of the both transforms is unique and leads to identity.

Application of the Inverse Box-Muller Transform

In regard to the representation above, the procedure for the quality evaluation of PRNG can be implemented in the form of a two-stage procedure in which the PRN with the uniform distribution will be formed by means of Inverse BMT, and then the π-Test is applied [5].

Generally, to check the effectiveness of the described approach the Central Limit Theorem Probability can be used [6]. According to this theorem, a PRN which elements defined as the average of the original set of statistically independent random variable has the Gaussian distribution. The program generating the Gaussian type PRN on the basis of the N-set mentioned above is presented below.

Table 5. Module “Gaussian type PRNG”

Note, a special function RndLCM is applied above, namely one on the basis of Linear Congruential Method (LCM), which provides a smart PRN with statistically independent variable [5]. The distribution density curves calculated in such way for some values N are represented below.

Fig. 1 Distribution density diagrams for sampled values N on the basis of the Central Limit Theorem Probability.

The diagram below shows results of computer simulation of the proposed two-stage test procedure to the quality evaluation of the Gaussian type PRNG by N = 6 vs. results for two other (reference) PRNGs with uniform distribution.

Fig. 2 PRNGs quality diagram.

The curves in the Fig. 2 represent the dynamics of the 23 sequential test sessions.

The legend of the Fig. 2 is as follows:

GRN – immediate π-Test of the PRNG with the use of function GetRandomNumber

TRF –      two-stage procedure with the use of Central Limit Theorem Probability on the

first step and Inverse BMT on the second

LCM – immediate π-Test of the PRNG based on the Linear Congruential Method

PI –     the value of π ( 3.14159…)

Discussion

The blue curve
in the Fig. 2 corresponds to the π-Test by the standard function GetRandomNumber with uniform distribution [7], while the second diagram (red curve) was calculated by means of the two-stage procedure described above. It should be emphasized that hereby a comparatively “simple” generator on the basis of the Central Limit Theorem Probability was applied. At the same time, it is easy to see that the use of GetRandomNumber leads to the worst indices.
It is all the more remarkable that in this case the two-stage testing procedure using Inverse BMT has a fairly smart match with the “LCM”
curve that is meanwhile the best of all and practically provide the approximation of the number π with the same accuracy. However, this of course does not preclude the use of the standard statistical testing methods for quality check of PRNGs and can be a good complement to them.

Conclusion

The Inverse Box-Muller transform was considered, which allows the transition from Gaussian type PRN to the adequate PRN with uniform distribution. A procedure for testing a Gaussian type PRNG is proposed by which after applying of the described Inverse BMT the quality of the PRNG with the uniform distribution (by means of π-Test) can be evaluated. The results of computer simulations confirm the effectiveness of the proposed approach. In addition, the described testing procedure is suitable for the testing of PRNGs of any type, since it uses just a random sequence with statistically independent variable. But the latter can belong to any PRNG and thus can be reduced to the Gaussian type sequence on the basis of the Central Limit Theorem Probability.

In the paper [1], the use of the Lorentz transformation in the ordinary differential equations was considered which describe behavior and properties of time-dependent systems in inertial reference frames (IRF). The similar approach is possible with respect to the problems of mathematical physics, where, in addition to the time variable, spatial coordinates are used in which physical processes are taking place. As well-known, these processes are described by partial differential equations with initial and boundary conditions [2]. In the present paper, the substituting of Lorentz ratios (LR) in these equations is considered and the consequences resulting from this are discussed.

Lorentz transformations

The Lorentz transformations belong to the mathematical apparatus of the Special relativity and describe relationship between two IRF one of which is fixed and the other moves relative to the first with a constant speed. The core of these transformations is Lorentz factor (LF) [3]:

        γ =      (1)

where

    v – the speed of the moving IRF

    c – the speed of the light in vacuum.

In the following, two specific LR will be used:

1) In the time domain:

        Δt´= γ·Δt        (2)

where

    Δt – the time interval in the fixed IRF

    Δt´- the time interval in the moving IRF,

and

2) In the spatial domain:

        Δr´= Δr / γ                (3)

where

    Δr – the length interval in the fixed IRF

    Δr´- the length interval in the moving IRF

In the following, all the notations with acute accent (´) will be related to the moving IRF.

Equations of mathematical physics

Generally, the apparatus of mathematical physics is presented by partial differential equations with initial and/or boundary conditions. The latter is formulating in accordance with a specific area of application (oscillations, electrodynamics, etc.). In the general case, such equations can be represented specifically in two-dimensional Euclidean space as [2]:

F(x,y,u, ,…, ) = 0          (4)

where u (x, y) is required differentiable function.

As implementation of (4), in mathematical physics distinguish hyperbolic, parabolic and elliptic types of equations. The first two types are presented accordingly with:

1. The wave equation:

 = a²    (5)

that describes oscillatory processes in physical agents,

2. The heat propagation equation:

 = b²        (6)

that describes the processes of heat propagation in physical agents.

Below, these equations are studied under using LR with displacement into the moving IRF.

Substituting of Lorentz Transformations

Note the differential entries in equations (5-6) can be taken in account as small increments. Then regarding (2-3) in accordance with the principle represented in [4] we obtain:

        ∂t = ∂t´/ γ                                    (7)

        ∂x = γ·∂x´                                    (8)

        ∂y = γ·∂y´                                    (9)

So in the moving IRF, the equations (5) and (6) can be written in the following form:

1) of the wave equation:

 = a´ ²         (10)

where

a´= a / γ²                                    (11)

2) of the heat propagation equation:

 = b´ ²               (12)

where

b´ = b / γ^3/2                    (13)

The expressions (10-13) show the principle, which can be used for determination of coefficients of partial differential equation of any order by use of Lorentz transformation.

Heat propagation equation in the moving IRF

Consider equation (6) in connection with a classical problem of heat distribution with the initial condition [2]:

        u(x,0) = δ(x)                                    (14)

where δ(x) is the Dirac delta function.

The standard solution to this problem is given in terms of the core of the equation (6).

        Φ(x,t) = exp(-x²/4b² t)                            (15)

In the moving IRF, this expression will use the value b´ from (13):

        Φ(x,t) =  exp(-x²/4b´ ² t)                        (16)    

It means that the core of the heat propagation equation depends parametrically on LF.

In the Fig. 1, as an indication of this dependence, the curves of the function (16) for various values of LF (specifically by t = 1 and b = 0.75) are presented.

Fig. 1 The heat distribution in the rod in dependence on LF

Note the curves in Fig. 1 are similar to a Gaussian random process [5]. Under this analogy, we can assert that an increase of LF and thus the speed of the moving IRF leads to the decrease in the dispersion of heat distribution. Presumably, this statement is also valid in a more general formulation. However, proof of this is outside the scope of this article.

Wave equation in the moving IRF

As a specific case of a wave equation, consider the oscillations of a string with the fixed ends, while this string is directed along the vector of the moving IRF. Such a formulation of the problem leads to the equation (5) with boundary conditions. The solution to this equation for a string of the length r is normally represented with Fourier series, where the first element corresponds to the frequency of fundamental tone of the oscillation. This frequency is determined as:

        Ω = πa / r                                    (17)

in the fixed IRF, and

        Ω` = πa`/ r`                                    (18)

in the moving IRF.

From (18) in regard to (3) and (11) we obtain:

        Ω` = (πa/γ²) / (r/γ)= πa/ γr                            (19)

Finally, the equations (17) and (19) imply an important ratio:

        Ω` = Ω / γ                                    (20)

Since γ > 1, it leads to the assertion that in the moving IRF the oscillation frequencies of a string decrease by a factor of γ, compared to the fixed IRF.

Some consequences, arising from this phenomenon are discussed below on specific examples.

Discussion

One of the consequences of the Special relativity is the so called “redshift” of the light spectrum [3]. In the paper [4], this effect was also stated in respect of oscillation generated by electric circuits. At the same time, in solution to the partial differential equation for a string with fixed ends, should be distinguished another interesting feature which is connected not only with the redshift only, but actually with the compression of the frequency spectrum.

Consider in the fixed IRF the set of strings with the fundamental tone frequencies Ω ₁ < Ω ₂ <…. <
Ω_n .
Then in the moving IRF we obtain:

        Ω`<sub>m</sub> - Ω`_k = Ω_m/γ – Ω_k/γ = (Ω_m – Ω_k)/γ                    (21)

where m > k .

Thus, in the moving IRF, the intervals between fixed frequencies are reduced by the factor of γ compared with the fixed IRF. Note this kind of spectrum compression should not be confused, for example, with the psychoacoustic compression in the spectral area which is aimed to reduce the flow rate while maintaining the quality of a playback [6].

Next, we can imagine a guitar that is “tuned” in the fixed IRF in accordance with the intervals of the harmonic series [7]. Thereafter, this guitar is displacing into the moving IRF. With that action, as a result of relativistic compression of the spectrum (defined above, however with all other things being equal) the guitar will be “detuned”. The degree of such a detuning depends on LF, which in turn is connected with the speed of the moving IRF. For example, by the redshift on 1 Hz the speed of the moving IRF should be estimated as about 6.7% of the speed of light in a vacuum.

Conclusion

Above, the use of Lorentz transformations in some problems of mathematical physics was considered. Thus, with substituting of Lorentz ratios into partial differential equations, the equation coefficients will parametrically depend on the Lorentz factor value and thus on the relative speed of inertial reference frame in which physical processes are taking place. Further, this dependence is expressed in solution of partial differential equations. It is shown that for the heat propagation equation the relativistic decrease in the dispersion of heat occurs. In the case of a string oscillation equation, the oscillation fundamental frequency decreases by analogy with the redshift of the light spectrum. At the same time relativistic compression of the frequency spectrum occurs, which leads to a reduction in the intervals between the fixed frequencies, for example, in case of harmonic series.

Finally, although only second order differential equations were considered above, the represented approach can easily be extended to higher spatial order.

Ц(x,t) = exp(-x²/4b² t)

The Lorentz invariance refers to the property of mathematical relations for physical processes or functions to retain their form by transition from one inertial reference frame (IRF) to another [1]. Without loss of generality, it can be assumed that all IRF are collinear. Therefore, in this article will be used Lorentz transformation of the kind [2]:

τ‘ = λτ                                                                                   (1)

λ=                                                                                     

where:

τ , τ‘ – the time intervals in the fixed and moving IRF‘s, respectively

v – IRF relative speed

c – light speed in vacuum

The Lorentz invariance property referring to a random process was already considered in [3]. In this paper, it will be generalized to the class of self-similar random processes (hereinafter, SSR processes) [4].

Self-Similar Random Processes

Consider a nontrivial random process defined by the ℋ(τ). This process is self-similar under condition, that for a given numbers g >1 (scale factor) and h (characteristic index) the relation

                   ℋ(τ) ≝ g^h·ℋ(gτ)                                                                    (2)

is satisfied [4].

Relation (2) expresses the property of the SSR process, according to which this process reproduces itself at any selected time interval to be extended with scale factor g.

It will be shown that for any specified SSR process the corresponding value of h can be determined.

Lorentz Invariance and Self-Similar Random Processes

In the framework of the mathematical representation, the condition of Lorentz invariance for the random process ℋ(τ) is of the kind:

ℋ(τ) ≝ µ·ℋ(τ`)                                                                     (3)

where µ is a unique real number.

Consider relation (3) in terms of autocorrelation. Then equality in (3) formally expresses the fact that autocorrelation function of the random process retains its form by transition to the moving IRF. In this regard, such a random process will be interpreted as Lorentz-invariant [1].

At the same time, if ℋ(τ) has been represented as SSR process, taking into account (1) relation (3) can be rewritten in the form:

ℋ(τ) ≝ µ·ℋ(λτ)                                                                     (4)

Comparing (2) and (4) one can obtain:

ℋ(τ) ≝ λ^h·ℋ(λτ)                                                                     (5)

The relation above is identical with (2) up to substitution λ = g . Thus, in order the random process ℋ(τ) to be Lorentz-invariant, it must be an SSR process.

Fig.1 Lorentz invariance of SSR process with τ and τ` set by mean of Lorentz transformation.

Instances of Lorentz-Invariant SSR processes

By the instances of the random processes considered below, these are considered by use of autocorrelation functions. Moreover, in regard to the Lorentz relation (1) below will be always assumed g = λ.

1. White noise process.

This process has the autocorrelation function of the kind [5]:

S_w(τ) = δ(τ)                                                                            (6)

where δ(τ) is Dirac function.

Then, taking into account (1), one can obtain:

S_w(τ`) = δ(λτ) = δ(τ        )                                                                (7)

Thus, the autocorrelation function of the white noise retains obviously its form, and it could be considered:

ℋ_w(τ)≝ λ^oℋ_w(λτ) = ℋ_w(λτ)                                                     (8)

Therefore, according to (3) the white noise as SSR process is Lorentz-invariant with characteristic index h = 0.

2. Pink noise process.

This process is characterized by the constant spectral density (without loss of generality, it will be taken equal to 1) at -d ≤ ω ≤ d , and zero like spectral density in the rest of the frequency range. The autocorrelation function of the process (using Fourier integral) is of the kind [5]:

S_p(τ) =                                                     (9)

Then in the moving IRF taking (1) into account it could be considered:

S_p(τ`) =   =                     (10)

where:  ϖ = λω and ᵭ = d/λ.

Thus, the autocorrelation function of the pink noise retains obviously its form, and it could be considered:

ℋ_p(τ) =  λ^-1·ℋ_p(λτ)                                                                          (11)

Therefore, according to (3) the pink noise as SSR process is Lorentz-invariant with characteristic index h = -1.

3. Brownian process.

Brownian process is well-known random Wiener process with self-similarity (thus, SSR process) which in terms of  relation (2) is of the kind [6]:

ℋ_b(τ) ≝ g^-½·ℋ_b(gτ)                                                                (12)

Here, in regard to (1) we assume g = λ and then obtain:

ℋ_b(τ`) ≝ λ^-½·ℋ_b(λτ)                                                               (13)

Therefore, in regard to (2):

ℋ_b(τ) ≝ µ·ℋ_b(τ`)                                                                   (14)

where µ= λ^-½.

Thus, in view of (3) one can consider the Brownian process as Lorentz-invariant with characteristic index h= -½.

Conclusion

Above, the Lorentz invariance condition of random processes to be self-similar is formulated. It is shown that the random processes, such as white and pink noise as well as Brownian process, correspond to this condition. Thus, the physical invariants of the special theory of relativity can be supplemented by the considered representatives of the random processes. At the same time, not limited to these, any process which corresponds to the Lorentz invariance condition considered here may be the subject of further reviews.

The gravitational constant is among the fundamental physical constants associated with universal physical laws and theories. At the same time, a special feature of the gravitational constant is that its value does not follow from any mathematical relationship. That is why the accuracy feature was steadily the crucial point of each measuring of the gravitational constant [1]. The aim of this article is to show, that in regard to the accuracy achieved up-to-date, the row of values of the gravitational constant can be considered as measurement spread. This approach makes the accuracy in frames of
exponential representation of the gravitational constant considered not higher than two decimal places of the mantissa. In the fallowing we will proceed exactly from this hypothesis.

Brief history

The gravitational constant G first appeared in the law formulated by Newton in 1666 and therefore is also called Newtonian constant of gravitation. The first measurement of the gravitational constant was carried out in 1798 by Henry Cavendish (G = 6.754⋅10^-11). In its experiments the torsion balance appliance was used designed by geologist John Mitchell [2].

The complexity of the measurement of the gravitational constant bases on the fact that the force of gravity is too small to be measured with the guaranteed accuracy. Ат the same time the biggest problem is the influence of the earth’s gravity which must be eliminated when measuring. In turn, this induces the problem of the obtaining of exact value of the Earth density. Therefore, the further experiments have required more sophisticated equipment that appeared only much later.

Stochastic simulation

Lately, measurement of the gravitational constant were carried out repeatedly especially in the period of the last forty years. It seemed this made it possible to improve the accuracy of measurements, nevertheless it did not lead to the final result. Below, in the Tab. 1 the results of the measurement of the gravitational constant registered by CODATA (The Committee on Data for Science and Technology [3]) are given in the chronological order.

As these results have been obtained by different researcher teams we can assume that they are independent in statistical sense. Let us consider a hypothesis that the spread in the decimal places after the base value of 6.67 has a stochastic nature. In order to examine the spread of the measurement data, we have used the stochastic simulation approach [4]. The results of this simulation are also given in the Tab. 1.

Tab. 1 The measurement of the gravitational constant.

The data of the columns in the Tab. 1 are visualized in the Fig. 1.

Fig. 1 The values of the gravitational constant vs. the stochastic simulation.

This chart is of interest in relation to the striking similarity of measurement data and the results of the simulation. The spread of the data in regard to the base value 6.67 is shown in the Tab. 2 and Tab. 3.

Tab. 2 The measurement data spread

Tab. 3 The stochastic simulation data spread

Now, should be the similarity of these two data series formally proved, we will have right to assume that the data spread after the second decimal place is similar to the uniform distribution.

Let choose the statistical measure of the similarity of the two sets in the form of correlation factor [5]:

        υ_x,y = Cov(X,Y)/(σ_x·σ_y)                            (1)

Then for the data in the Tab.1 (as a single simulation series) we obtain the well suited value υ_x,y ≈ 0.65. Anyway, by the multiserial simulation the average correlation factor is ῡ_x,y ≈ 0.25 that in view of the small size of the sets should be considered as a valid confirmation of our hypothesis formulated above [5].

Discussion

In regard to the Tab. 1 the gravitational constant can be also represented in the form:

G = 6.67·10^-11+ η                                 (2)

where: η < 10^-13 - the measurement spread.

Otherwise, the equality it can be rewritten in a more elegant way:

        G = σ·c^-1                                    (3)

where:

c – speed of the light in free space

0.0200129 ≤ σ ≤ 0.0222551

For example, by σ = 0.02 and c = 299.792.458 we obtain exactly:

        G = 6.671281903963·10^-11

On the other hand, the CODATA recommended value of gravitational constant is G = 6.67408⋅10⁻¹¹ [3]. In this case σ ≈ 0.0200084. In addition, the equality (3) can be used for obtaining a modified form of the Planck units [6].

Tab. 4 The modified form of the Planck units (ᵬ = (ℏ/σ)^1/2, ᵭ = (ℏ·σ)^1/2, k - the Boltzmann
constant).

The modified form of the Planck units in the Tab. 4 can be considered as more preferable because of it uses only the true universal constant which is the speed of light in free space [3].

In addition, the equality (3) leads to the fact that the gravitational constant loses its status of fundamental constant since it is expressed through the other fundamental constant i.e. the light speed in free space.

Conclusion

Above, the results of determination of the gravitational constant obtained over a long period are considered. By means of stochastic simulation it is shown that the spread of the gravitational constant data can be described with the uniform distribution. A formal statement for determination of gravitational constant and the associated modified representation of the Planck units are derived.

The author does not in any way suspect the results obtained earlier through complex and apparently expensive experiments. At the same time for such experiments in the future, it is worth to ask oneself about the really meaningful accuracy of the gravitational constant, which from our point of view is virtually limited to the second decimal place of the mantissa of exponential data record. However, the accuracy of the gravitational constant can undoubtedly be improved by increasing the accuracy of the tools used for its measurement.

In a broad sense, energy homeostasis ensures the maintenance of the dynamic balance of the system under disturbing influences, for example, in connection with changes in external conditions [1]. Dynamic models of homeostasis at the level of individual bio-systems were considered in [2] – [3]. It has also been suggested that some extreme principle is at the core of homeostasis at any level. [4]. In physics, this one is the universal principle of least action [5]. It is shown below that basing only on this principle it is possible to formulate a mathematical state-space model of energy homeostasis. Therefore, we call it as the principle model of energy homeostasis.

The principle model’s idea of energy homeostasis

In the most general formulation, the principle of least action reduces to the fact that the behavior of the system in question should provide a minimum of the Hamilton functional. Further, the use of this principle leads to the second-order Lagrange equation [5]:

  –  = 0       (1)

where:
u and v - vectors whose components are state-space variables q_i(t) and q̇_i(t)
W(u;v) – energy function

We will proceed from the fact that the function of the energy state can be represented as a positive definite quadratic form:

W(u;v) = 0.5 YSY^T , >0       (2)

where:
Y - vector of state-space variables
S – real symmetric matrix whose elements are determined by the parameters of
homeostasis.
So in case of two variables Y = {q(t),q̇(t)} and:

S =       (3)

Substituting (3) into (2), we obtain the explicit expression for the energy state function:

W(q(t),q̇(t)) = 0.5 (a∙q(t)²+2c∙q(t)∙q̇(t)+b∙q̇(t)²)        (4)

Further, substitution of (4) in (1) leads to differential equation:

h∙q̈(t) + g∙q̇(t) = 0     (5)

where: h = b – c and g = c – a.

For a nontrivial solution of this equation, disturbing influences function F(q(t),q̇(t)) should be introduced into the right-hand side of (5). Then the ordinary equation (5) takes the form:

h∙q̈(t) + g∙q̇(t) = F(q(t),q̇(t))       (6)

Note, equation (6) is a generalization of the models formulated for some specific bio-systems [2]-[3].

Here, to make clear some possible solutions we consider two main particular cases of the principle model of energy homeostasis depending on the specific representation of the function F(q(t),q̇(t)).

Linear case

In order to represent the function F (q (t), q̇ (t)) in closed form in case of small disturbing influences, we restrict ourselves to the first terms of the Taylor series expansion near the balance point of homeostasis. So in this linear case, omitting the intermediate calculations, we obtain:

F(q(t),q̇(t)) = α+ β∙q(t) + γ∙q̇(t)          (7)

where: б, в and г – coefficients of Taylor series.

Considering equation (5), it leads to differential equation of the second order

h∙q̈(t) + (g-γ) q̇(t) – β∙q(t) – α = 0          (8)

Using the substitution s(t) = – (б+в·q(t)), finally obtain:

ā∙s̈(t) + c̄∙ṡ(t) + s(t) = 0            (9)

where: aЇ = –h/в and cЇ = (г – g)/в.

In particular aЇ = 0, and equation (9) reduces to the differential equation of the first order:

c̄∙ṡ(t) + s(t) = 0        (10)

Note, analogs of equations (9) and (10) are, respectively, the oscillation and aperiodic structural units, which correspond to the cybernetic interpretation of homeostasis as a dynamic system with feedback [6].

Non-linear case

In general, the function on the right-hand side of equation (1) can be represented with Taylor series with terms of the second and higher orders or with a specific non-linear expression. As example of the latter, consider the disturbing influences function:

F(q(t),q̇(t)) = g∙q̇(t)q(t)² – q(t)      (11)

Then, using (11) in the right-hand side of equation (5) we obtain:

h∙q̈(t) + g∙(1-q(t)²)∙q̇(t) + q(t) = 0       (12)

Actually, this equation describes the well-known Van der Pol oscillator, which is frequently used as a model of the processes in electrical or biological systems [7].

This oscillator is characterized by limit cycles and bifurcation points. However, its oscillations are harmonic at g = 0, and with an increase of |g| deviate more and more from harmonic oscillations approaching ultimately the relaxation oscillations. In addition, in the light of dependence of this nonlinear model on the choice of an inertial reference frame, it points to a new, relativistic aspect of homeostasis study [8].

Discussion

A possible problem of the presented principle model of energy homeostasis is that the state variables here are not directly related to the actual parameters of homeostasis. Therefore, the practical use of this model should be preceded by an assessment of its parameters. Here we confine ourselves to pointing to methods of identification theory that can be used for such an assessment [9].

As shown above, by small disturbing influences, the principle model of energy homeostasis is similar to linear dynamic systems. Therefore, to assess the stability of the energy homeostasis, one can use the known criteria of dynamic systems [6]. In a more general case, the principle model is nonlinear and, with the more significant disturbing influences, leads to limit cycles. Thus, it allows one to study energy homeostasis coincided with external influences of a cyclic type [4].

Conclusions

Above the principle model of energy homeostasis, which only is based on the principle of least action was considered. This model shows that by small disturbing influences, the homeostasis behaves like structural units of dynamic system of oscillation or aperiodic type. Based on the analogy, an approach to assessing the stability of energy homeostasis is proposed. In general this model is nonlinear and by more significant disturbing influences shows the limit cycles or bifurcation points. In addition, in the light of dependence of this nonlinear model on the choice of an inertial reference frame, it points to a new, relativistic aspect of homeostasis study.

The features for the perception of audio information by human ear are well-known [1]. One of these features is the inability of the human hearing to distinguish rather weak sounds in the presence of a more intense tone nearby. In psychoacoustics, it is denoted as the frequency masking. In the frequency domain, this appears when two harmonic oscillations are simultaneously perceived in the restricted frequency range called as the masking area [2]. Actually, the choice of this area should be associated with the limitations of the resolution of the auditory perception, which is the ability to distinguish between two individual but close tone pitches. Empirically, in the range of less than 1000 Hz, the normal human hearing perceives a frequency deviation less than 3 Hz (up to 1.5 Hz). Moreover, above 1000 Hz, it can be estimated as follows [2]:

            ŝ ≈ 0.0035 f   ,           (Ɐ f > 1 kHz)             (1)

In general, it is senseless to talk about the frequency masking, if some tones are indistinguishable to human hearing. Therefore, the interval [f-s;f+s] can be considered as the genuine masking area.

The use of the frequency masking with DFT

To use of the frequency masking, the digitalized audio signal is transformed in the frequency domain, which is performed by means of DFT [3]-[4]. The latter recalculates N consecutive samples of the digital signal {x_n} into N pairs of coefficients of the complex spectrum S(k) characterizing the representation of the signal in the frequency domain:

Re [S(k)] =         (2)

Im [S(k)] = 

Further, based on these entities, the amplitude spectrum of the signal is formed. Due to the central symmetry of the amplitude spectrum (i.e. spectrogram), only the first N/2 values are used for the analysis:

|A(k)| =  , k = 1…Ñ           (3)

where Ñ = (N+1)/2 by odd N, and  Ñ = N/2 by even N.

Thus the DFT allows getting N samples of a signal spectrum in the range from zero to half of the sampling rate. Such spectrum, in contrast to the “continuous” spectrum of the signal, is discrete and, as N increases, more and more approaches the real spectrum. In digital audio compression, since N is usually finite when using the DFT it is also reasonable to take into account the spectral resolution, which is expressed by the equality:

ȗ = F_s / N       (4)

With the audio spectrogram, the minimum size of the masking area must be formed in view of the actual spectral resolution, since the use of the frequency masking should be based on evaluating values for two adjacent frequencies of the spectrogram. That means that the reasonable use of the frequency masking is possible only when the following criterion is met:

ȗ ≤ ŝ       (5)

Let us define the spectrogram frequency as cut-off frequency, when after this one the criterion (5) is no longer satisfied. In view of (1) and (4), that can be simply expressed as:

f_c ≈ F_s (0.0035 N)^-1          (6)

Thus, the cut-off frequency is directly proportional to the sampling frequency of the audio signal and inversely proportional to the number of samples of DFT.

Example of cut-off frequencies

In the audio compression systems (i.e. audio encoder MP3 [6] or digital broadcasting [4]), the implementation of DFT is generally basing on 1024-point FFT [7]. For this case, the results of applying (6) to the sampling rates used in audio compression systems are shown in the following table.

Table 1. Cut-off frequencies by 1024-point FFT

Obviously, by the e.g. doubled power of FFT (i.e. 2048-point FFT) the cut-off frequencies in Table 1 will be increased by two times. That leads to the entire range of frequencies perceived by the human hearing while using the each of the sampling rate of the Table 1.

Discussion

From the results in Table 1, it follows that the area of the reasonable use of the frequency masking in audio compression systems depends on the spectral resolution. As an example, consider the spectrogram (Fig. 1), where 1024-point FFT is used.

The frequency masking for this spectrogram by the sampling rate of 32 kHz only can be applied at the range up to the red bar in Fig 1 i.e. 9.0 kHz. For the other two sampling rates, these are 12.5 kHz (green bar) and 13.5 kHz (blue bar) accordingly. Fortunately, the average human hearing is unable to perceive frequencies above 12 kHz [2]. This actually prevents the occurrence of audible artifacts at the top of the frequency range which is perceivable by average human hearing, though noticeable to sophisticated music listeners. At the same time, the criterion (5) would be met at all three sampling rates at all frequency range, when the power of FFT is equal to 2048 or more by the audio compression.

Conclusions

As shown above, by using some psychoacoustics effects in audio compression systems it is necessary to take into account both spectral resolution and the resolution of auditory perception. In this regard, a specific frequency can be specified for each sampling rate, which actually determines the frequency range within which these effects may be reasonably applied without noticeable distortion of the sound information. Hereby, the crucial thing is to specify a correct relation between the masking area, on the one hand, and the spectral resolution, on the other. In audio compression systems, the spectral resolution is determined under consideration both the sampling rate and the power of FFT. It is shown that with the use of the 1024-point FFT the frequency masking reasonably works just within the frequency range which is perceived by average human hearing. However, from the point of view of music expert requirements, the use of e.g. 2048-point FFT is more preferable.