In this article we try to propose a new model of an application-specific integrated circuit (ASIC) for calculation of the error probability in the trans-ionospheric channel. The new ASIC based on using of residue number system and this model investigates advisability of developing such hardware. While travelling through the ionosphere the signals of the space communication systems are subject to modifications in amplitude, phase and polarization. Thats which leads to intermittent reception, high error rate and weak noise immunity. In prospect by using of the analytical error probability, we can set to transmitter frequency and timings parameters of a signal to do preemptive correction and avoid weak noise immunity.
The problem of determining the noise immunity of trans-ionospheric channel is a very important one because in the modern world space communications plays a major role in many economic activities. Its also important in applications of military and scientific fields. While travelling through the ionosphere the signals of space communication systems are subject to modifications in amplitude, phase and polarization [Shev15]. That is which leads to intermittent reception, high error rate and weak noise immunity. This negative influence increased especially in periods of ionospheric disturbance and more effects on trans-ionospheric channel. In papers [Pash06] considered, that to increase noise immunity of communication systems, correction of timings and frequency of signal must be calculated in real time, but in modern conventional systems it is almost impossible.
It is known that the propagation time of the signal from geostationary orbit is theoretically not be less than 240 ms, and consider that signal processing and switching, can make delay above 400 ms [Yon97]. Delay is critical in real-time systems and presents problems for a latency-sensitive applications such as voice communication, and under the described conditions, pause between voices will be at least 500 ms, by the way for a public telephone allowable delay is considered to 150-250 ms [Cis10]. In case of ionospheric disturbances, noise immunity of system will reduced dramatically and communication session will be lost.
Prospective communications systems have to monitor ionosphere state continuously in real time mode by passive listening of radio tracks and have to set correct frequency and timing parameters of signal before the conversation begins. Thus can avoid communication lost, especially in case of ionospheric disturbances. However, an insertion of corrections in a real-time mode introduces high demands on calculator speed.
Considering of the total delays of - propagation, processing, calculations, and corrections, required that an overall delay was minimal, and the only reserve of its reduction is hidden in computation process. It is known that cost-effective way to improve calculation speed is a parallel calculations.
In this paper described a model of an application-specific integrated circuit (ASIC) for an application in modern space communication systems. Scheme of the ASIC based on natural mathematical parallelism of residue number system (RNS). 2
Noise immunity plays a major role in communication systems, which characterizes endurance from many negative factors such as interference, Rayleigh fading, frequency-selective fading etc. Analysis of space communication field determines that as the noise immunity factor may be the error probability in received signal. Most systems require that value of the error probability would not be greater than permissible Per ≤ Peral = 10−5 [Gar09, Jac14, Sre16]. Currently this factor evaluated by statistical methods by counting an errors in received data symbols. Such an approach has several disadvantages, such as sufficiently large time intervals with data acquisition and hence a large inertia of the processing of statistical data, which leads to the fact that the data are not relevant in conditions of ionosphere changes and spacecraft motion.
In the paper [Pash06] proposed that the ionospheric fading are characterized by Rayleigh and Rice distributions, and the most interesting equations, describes a frequency-selective fading (FSF) and intersymbol interference (ISI) of signal. Also proposed an analytical evaluation method:
Per = ψ(h2, γ2, ηc, ηm) where h2 is the relationship of signal noise ratio in antenna, γ2 is the coefficient of Rayleigh fading characteristic, ηc = ψ(F0/Fk), ηm = ψ(1/TsFk) coefficients of energy fading in condition of frequency-selective fading (FSF) and intersymbol interference (ISI) of signal respectively, duration of signal Ts, spectral width F0, coherence bandwidth Fk.
Coefficients of energy fading described [Pash06]: ηc ηm = = 1 + 1 1 erf (πTsFk) − π√π TsFk exp −(πTsFk)2 .
The disadvantage of this method is the high latency computing of the coefficients ηc, ηm, due to the complexity of calculation of elementary and special functions.
The error function (the Gauss error function) is a special function defined as: erf (x) = 2 Z x √ π 0 et2 dt this integral cannot be evaluated in closed form in terms of elementary functions, which lead than no path to direct computing. Perhaps integral form can be represented as a Taylor series expansion, however, accuracy quite small; in interval of x > 4 it takes a series of 54 degrees for the accuracy of given task. Speed of convergence of such series decreases to the edges of erf (x).
An analysis showed that the calculators based on different processing methods have different performance and hardware costs and their implementation is ambiguous. Considering of high cost and complexity of developing practical hardware, we can conclude for need in a computer modeling of calculator, as necessary comparison on speed and hardware costs. For example, serial adder has a greater delay but has a small hardware costs. On the other hand, a parallel adder or carry-lookahead adder is substantially less computation delay but much higher hardware costs. For the full carry-lookahead adder, costs are comparable with costs of implementing of the entire device.
Calculators based on theory of conventional and modular arithmetic are not equal for basic principles and its hard to compare. Lets consider the main differences between conventional numeric system (CNS) and residue number system (RNS) in the construction of hardware calculators:
- In the RNS are not determined operations of division and comparison, and we need in using of complicated methods.
- Some operations such as multiplication in RNS requires costly resource scaling, on other hand in CNS scaling is performed by trivial bit shift.
- After a series of the order change operations such a multiplication, in the RNS in order to avoid overflow should be built calculators with significantly overweight range and must use basis extension methods.
Therefore, the comparison of calculators in terms of word size is not correct, since the fundamental differences between numeric systems. Word size should be selected by the desired task, accuracy and numeric range. 3
Suppose for the problem statements, we need to evaluate the noise immunity factor of trans-ionospheric channel and provide the calculation of the coefficients of energy loss (2, 3) for calculating of the error probability (1) of signal.
We need: 1. to carry out approximation of error function (4) with a given accuracy in the presentation of elementary functions in RNS,
2. to develop a mathematical model of calculator of noise immunity factor of trans-ionospheric communication channels. Calculator must be based on the representation of elementary functions in the RNS, 3. to make comparisons and evaluate hardware costs. 4 4.1
Error Function Approximation
In this paper, we used a Chebyshev polynomial, to reduce calculation delays in evaluation of the error function.
Polynomial approximation is a preferred way for direct computing of the functions evaluation in way of using of
RNS. Polynomial calculation provides choosing of accuracy on a predetermined range. To reduce approximation
range we can use odd property of the error function erf (−x) = −erf (x). The error function has no singularities
and asymptotically approaches to 1, if required accuracy achieved at the edges, range should be limited. erf (
T = l · t (6) At the moment, there is no analytical expression of evaluation of delays and hardware costs of traditional elements such as: adder, multiplier, divider, calculating elementary functions.
General block diagram of calculating of noise immunity factor of satellite communication channels shown in figure 2.
Length l or the path of propagation delay or gate delay expressed as the number of sequentially connected gates, shown in table 2 and 3 (for RNS). Hardware costs or number of gates for implementation of units, shown in table 2 and 3 (for RNS). where ls length for convetional adder, cs it hardware costs, csum numbers of elements of full 4-bit adder, ce cost of carry scheme, ccarry cost of carry.
Conventional matrix multiplier: l = c = 12 · (A − 1) + 1,
A · Csum32 − 1, lm cm = = 4 · A − 7 6 · A2 − 5 · A ld cd = =
A · (lsum32 + 1)
Csum32 + 1 + Ccary + 4 · A lf cf = = lsum32 + 3 + S, (Csum32 + A + 1) · S, lsum csum = = ls + lt, X Ci + 2mi+1 + 1, where A is a word size, lsum32 length of adder, Csum32 is a cost of an adder, Ccarry is a cost of carry bit.
Conventional elementary function calculator, represented by a parallel-connected comparators with common bus: where lsum32 is a length of an adder, Csum32 is costs of an adder, A is a word size.
Modular adder implemented by the scheme of carry-lookahead adder with tables of residues. It is compromise scheme in hardware costs and calculation speed. It formula is: where A is a word size (number of bits), Csum32 is a hardware costs of an adder.
Conventional multiplier, Walles tree: where A is a word size.
Conventional divider (implemented by multiple ticks scheme in reason of high costs of sequential implementation) where ls is a length of carry-lookahead adder, lt is a delay of a table, Ci - costs of i-adder, mi is a word size of i-modulus.
Modular multiplier implemented by a scheme of ”a fast RNS galois field multiplier” [Rad90]: where lsum is a length of modular adder, mi is a word size of i-modulus, Cmi is a cost of modular adder of i-modulus.
Base extension implemented by a Garner’s method[Gra16]: lmul cmul = = where lext is a length of base extension, lmul is a length of modular multiplier, lsum is a length of modular adder, cext is a costs of a base extension, cmul is a costs of a modular multiplier, csum is a cost of modular adder, ctab is a cost of conversion table, Am is a total word size of RNS.
Elementary functions implemented by method of [Gra15] and it formula is: lfunc cfunc = = lscl + 2, cscl +
X 2ai + ai) where lscl is length of modular scaling, cscl is a costs of modular scaling, ai is a word size of i-modulus,
Conversion from CNS to RNS is implemented by method of adding by the tree with 8 adders: where lsum is a length of modular adder, csum is a costs of modular adder.
Calculus of energy coefficients ηc, ηm in CNS: ltr ctr = = 11 · lsum, (8 · 4 + 4 + 2 + 1) · csum, lηc cηc lηm cηm = = = = 3 · ls + 7 · lm + 2 · lf , 3 · cs + 7 · cm + 2 · cf , ls + 8 · lm + 2 · lf , cs + 8 · cm + 2 · cf , where ls is a length of adder, lm is a length of multiplier, lf is a length of function, cs is a costs of adder, cm is a costs of multiplier, cf is a costs of function.
Calculus of energy coefficients ηc, ηm in RNS is implemented with modular scaling for avoid of overflow and it formula is: lηc cηc lηm cηm = = = = 3 · lsum + 7 · lmul + 1 · lscal + 2 · lfunc + 2 · lscal, 3 · csum + 7 · cmul + 3 · cscal + 2 · cfunc, 1 · lsum + 8 · lmul + 1 · lscal + 2 · lfunc + 2 · lscal, 1 · csum + 8 · cmul + 3 · cscal + 2 · cfunc, where lsum is a length of modular adder, lmul is a length of modular multiplier, lscal is a length of modular scaling, lfunc is a length of modular funnction, csum is costs of modular adder, cmul is costs of mofular multiplier, cscal is a costs of modular scaling, cfunc is a costs of modular function.
Length l of propagation delay and hardware costs of coefficients of energy loses calculator is shown in figure 3 in sections a and b. Section c and d is for models of noise immunity factor calculator.
Implementation schemes of Per calculator given in figure 4, that relevant with figure 3 (propagation delay in sections c and hardware cost in d). 5
1. Taken an approximation of error function by using of Chebyshev polynomial with required accuracy 10−7.
2. Developed new mathematical models of the error probability calculator of trans-ionospheric channel. Models based on conventional and RNS representation of elementary functions.
3. By the modeling of noise immunity factor calculator, established that application of the theory of modular arithmetic improves computational speed. Hybrid scheme (RNS with CNS) provides increasing of computational efficiency is more than 2 times, while increasing in hardware costs by 93%. This result explained by the high efficiency with using of modular structures in the given task.
4. On the basis of inductive inferences derived formula, that defines the length of the propagation path of signal in computation and hardware costs for CNS and RNS.
5. The analysis showed that calculator implemented by using of conventional numeric system has the lowest hardware cost ( Fig. 3d position 1). However, calculating speed of that calculator is relatively small (Fig. 3c position 1). Practical interest is represented by a calculator with the lowest length of propagation delay (Fig. 3c position 5), constructed according to the hybrid scheme (by using CNS and RNS). The increase in performance is due to the increase in hardware costs, compared to the scheme that uses the traditional approach. [Mit08] Introductory Digital Systems Laboratory. MIT, DEPARTMENT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE, 6.111, 2008.