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.

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.