CC BY
3DOI: 10.24412/2181 -144X-2024-3-28-33
Jumaev O.A., Ismoilov M.T., Rakhimov A.K., Rashidov G. H.
MATHEMATICAL DESCRIPTION AND ALGORITHMS FOR THE IMPLEMENTATION OF DIGITAL FILTERS
Jumaev OA Ф000-0001-5657-8176] Ismoilov MT 2^000-0002-6469-0085]
Rakhimov A.K. 3[0009-0004-6290-4977], Rashidov G. H. 4[0009-0009-2410-4975]
1Navoi State University of Mining and Technologies, Professor, department of metrology, standardization
and certification, DSc in technical sciences, 2Navoi State University of Mining and Technologies, Associate professor, department of metrology, standardization and certification, PhD in technical sciences, 3Navoi State University of Mining and Technologies, assistant, department of metrology, standardization and
certification,
4Navoi State University of Mining and Technologies, independent researcher.
Annotatsiya. Ushbu maqolada cheksiz va chekli impuls xarakteristikasiga ega raqamli filtrlarni ishlab chiqish uchun umumlashtirilgan model taqdim etilgan, shuningdek, ularni axborotni qayta ishlashning elektron tizimlarida amalga oshirish imkoniyatlari muhokama qilingan. Bunday filtrlarning kompleks chastotali koordinatalardagispektraltasvirlariga alohida e'tibor qaratilgan. Keltirilgan algoritmlarraqamli filtrlaryordamida berilgan chastotada signalning oniy spektrining haqiqiy va mavhum qismlarini samarali hisoblash imkonini beradi. Laplas almashtirishi asosida filtrlarning statsionar va nostatsionar rejimlarda ishlashini tahlil qilishning taklif etilgan usuli qismlarga ajratish teoremasini qo'llash kabi an'anaviy yondashuvlarga nisbatan ancha kam hisoblash resurslarini talab qiladi.
Kalit so'zlar: filter, Laplas almashtirishi, matematik model, impulsli funksiya, vazn funksiyasi.
Аннотация. В данной статье представлена обобщенная модель для разработки цифровых фильтров с бесконечной и конечной импульсной характеристикой а также обсуждаются их возможные реализации в электронных системах обработки информации. Особое внимание уделено спектральным представлениям таких фильтров в координатах комплексной частоты. Приведенные алгоритмы позволяют с помощью цифровых фильтров эффективно вычислять вещественную и мнимую части мгновенного спектра сигнала на заданной частоте. Предложенный метод анализа работы фильтров в стационарных и нестационарных режимах на основе преобразования Лапласа требует значительно меньших вычислительных ресурсов по сравнению с традиционными подходами, такими как применение теоремы разложения.
Ключевые слова: фильтр, преобразование Лапласа, математическая модель, импульсная функция, весовая функция.
Annotation. This article presents a generalized model for developing infinite and finite impulse characteristics digital filters, as well as discusses their possible implementation in electronic information processing systems. Particular attention is paid to the spectral representations of such filters in complex frequency coordinates. The presented algorithms allow for the efficient calculation of the real and imaginary parts of the instantaneous signal spectrum at a given frequency using digital filters. The proposed method for analyzing filter performance in steady-state and transient modes based on the Laplace transform requires significantly fewer computational resources compared to traditional approaches, such as the application of the expansion theorem. Key words: filter, Laplace transform, mathematical model, pulse function, weight function.
Introduction
Using summarized mathematical models for input signals and frequency filters, combined with spectral portrayals of the Laplace transform, grants for the advancement of elementary and effective methods for analyzing the quality of filter performance and, more broadly, information processing devices in measurement systems. Using these methods, we will be able to analyze linear systems and model transient processes in the measurement system. Another advantage of these methods is that they allow us to assess the effectiveness of digital filters.
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Furthermore, the use of spectral representations of the Laplace transform enables the integration of the spatial state method for synthesizing spectral density and filter frequencies.
Methods
In general, the input signals of the information processing and communication devices of measurement systems can be described as controlled object. As for analog filters, the mathematical model of the weight (impulse) function for digital IFR filters should be conferred, as a set of discrete attenuating oscillatory components.
Km(z) = km z?mf+zclf (1)
z uimz+u0m
Mathematical description of a digital IFR filter as a weight function
g(k) = Iim=i3m(k) (2)
9m(X) = kme amkTCOs(WmkT - 4m) (3)
And the corresponding transfer function looks like
K(z)=im=iKm(z) (4)
Here, the following remark is appropriate:
c2m = cos )sin(WmT) - C0s(4m)C0s(WmT)],
dim Tm.cos(wmT) d0m ^rn>^m & m >zm 6pm ,zm 6pm .
In these formulas (1-4), zm and pm are used differently to stand for the poles of the m-th component of the system and the transfer function than the poles of the images of the nth component of the signal zn and pn in order to avert confusion when using them together [1,2,3,4]. Based on the formulas (1-4), we obtain the following instrument for the algorithm of the m-th component of the FIR filter
ym(k) = a2mx(k) + aimx(k -1) + dimy(k -1)- domy(k - 2) (5)
where a2m = kmc2m> a1m = kmc1m.
When using signal processors containing course for aggregate complex numbers with accumulation, the generalized IFR filter algorithm can be written in a more compact form
ym(k) = GmX(k) + Zmym(k - 1) (6)
To appliance the algorithm (6) in practice using conventional microprocessors, It is important to specify the action to be performed. y(k) = Re(y(k)).
There may be a different way to find the result. We divide expression 6 into two parts, which are real and imaginary parts. ym(k) ycm(k) =
Gcmx(k) + zcmycm(k ^ zsmysm(k ^ ysm(k) = Gsmx(k) + zcmysm(k 1) + zsmycm(k ^
Compared to the algorithm (4), a little more computational operations are required, but simultaneously two filters are implemented [5,6,7,8]. The compact form of the generalized impulse function of the KIH filter can be represented as follows [9, 10, 11].
g(k) = Re(GTZ(q, k) - GTZ(Q Ck - N)) (8)
From the expression (2-12), it follows that to form a mathematical description of the digital filter, components of the impulse function G = [Gm]M = [km6~^m]M and
G' = [Gm]M, complex frequency vector q = [pm]M = [-am + jwm]M and vector T = [Tm]M, whose elements determine the duration. (length) of the filter pulse function components, Q'= diag(q), T- discretization step. For G' = diag(G)eQT or Gm = GmepmTm, the impulse function (7) corresponds to the IFR filter, and for G' = 0 - to the filter with an infinite impulse characteristic (FIR filter). When using the traditional (non-recursive) algorithm for implementing the IFR filter, for the length of the weight function
[12,13,14]. Therefore, the higher the discretization frequency with a constant filter length, the higher the computational costs required to implement the FIR filter. To reduce computational costs, recursive algorithms for implementing IFR filters should be used, the possibility of which was first noted in the work [15,16,17].
Discussion
The recursive algorithms of the FIR-filters are directly derived from the proposed mathematical model (7) of the filters. We will present the expressions for the algorithm of the m-th component of the IFR filter with a generalized weight function in complex form
Em(z) = GmT
Z-Zr
- GmT
Z-Zr
Z-Nm = zT
Gm—G'Z
Z-Zr
(9)
From the last expression, we find the filter algorithm
ym(k) = TGmx(k) - TG^x(k - Nm) + zmym(k - 1)
(10)
The structural diagram of the FIR filter (10) is shown in Fig.1. It follows from the obtained expression for the FIR filter algorithm that the similar FIR filter algorithm differs only in the presence of the G{x(k — Nt) component, i.e. the FIR filter (highlighted by a punctuation in Fig. 1) in the approach under consideration is a private case of FIR filters.
The use of a comprehensive form for recording weight and system functions, as well as a filter algorithm, allows for a more compact recording of the Fir filter's mathematical description. When implementing the recursive FIR filter algorithm (10) using signal processors with the support of complex multiplication operations with accumulation, only three multiplication operations with accumulation are required, unlike the traditional non-recursive FIR filter algorithm,
Let's divide the algorithm (10) into two algorithms that correspond to its real and imaginary parts
ycm(k) = TGcmX(k) — TGCmX(k — Nm) + zcm.ycmik — ^ — zsm.ysmik — ^ ysm(k) = TGsmX(k) — TGSmX(k — Nm) + Zcmysm(k — ^ + zsm.ycmik — ^
(11)
Fig. 1. The structural diagram of the FIR filter algorithm.
The aforementioned algorithms allow for the determination of the instantaneous signal spectrum at a given complex frequency by using two FIR filters. Another recursive implementation algorithm for the m-th component from the expression for the systemic function. Individual cases of the considered filters are those that perform instantaneous spectral analysis using the Fourier transform (am=0), including filters that implement algorithms for moving averages (am=0, wm=0).
z
z
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S2>
To analyze IFR filters, analysis methods widely used in electrical engineering and automatic control theory can be used, including methods of direct differential equation solution, operator method, state space method.
For the analysis of filters, whose input signals can be described using expression (1), it is proposed to use methods of analysis It is necessary to determine the terminology used in the analysis of frequency filters. The term "established value of the output signal" is correct for input influences in the form of a set of sinusoidal components. If the input signal is attenuated or final, it is better to use a more general term - forced component.
Figure 2 shows the transfer function of the filter, through which it becomes possible to determine and analyze the frequency characteristics.
At the same time, the symbolic form of the input signal recording, the weight function, and the output signal (complex amplitudes of the input signal, the forced component, the weight function, and the free component) is shown above, and the transition from symbolic images to original ones is shown below. The expressions reaction is:
Yn = XnK(Pn),Ym = GmX(Pm).
The physical meaning of the signal spectrum x(t) and the frequency characteristic can be formulated as follows.
x(t) = Re(XTept)
g(t) = Re(GTeqt)
G
x
K(p)
X(p)
y = diag(G)X(q)
Y = diag(X)K(p)
ynp(t) = Re(YTept)
yCB(t) = Re(yTeqt)
Fig. 2. Determining the components formed at the output of the filter
The functions X(p) and X'(p), presented in Fig. 3, have the following dependence on the input signal component images:
X(p) = X(p)/e-vt\X'(p) = X'(p)/e-^t'i. The calculation of the FIR filter reaction for the state of change in the characteristics of the input signal is shown in Fig. 3. Which y^ - n is the n th column of the y matrix.
Y=[GmX(pm)n]M:N, Y' = [G mX (pm)n] M,N
p. q X(p)
g(t)=Re(GT e qt) X'(p) ycB(t) = Re (Znï^eq(t-tnï - SnY' (n)Teq(t-tn)
Fig. 3. Defining compulsory and free components
The aforementioned FIR filter analysis method can be effectively used in the practice of developing and researching modern measuring devices and measuring system automation devices. If only a transfer function is given for the filter, it is important, to find the complex amplitudes of the impulse function. In this case, we assume that all the poles of the transfer function K(p) of the FIR filter are known, and it is necessary to find the complex amplitudes Gm = kmej* .
To determine km and $m for two complex-paired roots, the following expressions are proposed
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km = \Н{рт)\№т,фт = n/2 - arg(#(pm)) where H(pm) = (p - pm)(p - p'-m+1)K(p)\v=Pm, pm and pm+1 - complexed roots.
If the root is real ( pm = -a. be used
(11)
m, Gm = km ), then the following well-known expression can
km = (Р-Рт)к(р)\
V=Pm
(12)
Conclusion
The mathematical model of the impulse (weight) function for the absolute majority of physically realized frequency filters with infinite impulse characteristics (FIR filters) can be represented as a set of attenuating oscillatory components.
Unlike the division of a complex linear system into elementary links of the first and second orders accepted in the theory of automatic control, the analysis of complex systems is carried out based on the weight and momentum functions, not on the transfer function. This allows us to use the general principle of classifying signals and systems by analyzing time and frequency characteristics.
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