Introduction

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.

1. Сучилин В.А. Relativistic approach to signals and systems // Современные научные исследования и инновации. 2017. № 11 [Электронный ресурс. URL: http://web.snauka.ru/issues/2017/11/84761
2. Прохоров А. М. Физическая энциклопедия. Рипол Классик, 1988. 704 с.
3. Лэм Г. Аналоговые  и цифровые  фильтры. Расчет  и реализация. М.: Мир, 1982. 592 с.
4. Сучилин В.А. Функции с экстремальной концентрацией в теории фильтров. «Электронное моделирование», Киев. – 1991, т. 13, №4
5. Suchilin V.A. Eigenfilter as a discrete window for power spectra estimation. Proc 1992 Int. Conf. Com. Technology Vol 1.
6. Сучилин В.А. Рекурсивные цифровые фильтры на основе функций с двойной ортогональностью. Изв. Вузов «Радиоэлектроника», Киев. – 1987, №12
7. Гантмахер Ф. Р. Теория матриц. – М.: Наука, 1966. 576 с.
8. Parks T., McClellan J. Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase. IEEE Transactions on Circuit Theory (Vol 19, Issue 2, March 1972 )
9. Algazi V., Suk Minsoo. On the frequency weighted least-square design of finite duration filters. IEEE Transactions on Circuits and Systems (Vol. 22, Issue 12, Dec 1975)
10. Сучилин В.А. Квазиоптимальные цифровые фильтры с импульсной характеристикой конечной длины. Изв. Вузов «Радиоэлектроника», Киев. – 1989, №3
11. Сучилин В.А. Синтез нерекурсивных цифровых фильтров по критерию энергетического баланса. Изв. Вузов «Радиоэлектроника», Киев. – 1987, №3

Все статьи автора «Сучилин Владимир Александрович»