Designs of Electronic Devices using Combinatorial Optimization

VOLODYMYR RIZNYK

Department of Computer Aided Control Systems,

Lviv Polytechnic National University,

79013, Lviv-13, Stepan Bandera Str., 12,

UKRAINE

Abstract: - This paper involves design techniques of electronic devices by combinatorial optimization for

improving the quality indices of electronic devices or systems concerning performance reliability,

transformation speed, positioning precision, and functionality, using proposals based on remarkable

properties and structural perfection of combinatorial configurations, such as difference sets and “Golomb

rulers”. These design techniques make it possible to configure electronic devices or systems with fewer

elements than at present while maintaining or improving on functionality and the other significant operating

characteristics of the devices.

Key-Words: - Combinatorial configuration, Golomb ruler, Ideal Ring Bundle, optimization, sequential circuit,

code-to-resistance decoding matrix, digital-analog code-to-voltage converter, self-correcting

code.

Received: March 13, 2023. Revised: October 12, 2023. Accepted: November 27, 2023. Published: December 31, 2023.

1 Introduction

The Electronics have wide world recognition with

connection rapid development in the field of

microprocessor and microelectronic technologies

within the last few years. Modern electronics is the

science of the interaction of fundamental and novel

advanced methods for creating electronic devices to

convert electromagnetic energy, mainly for the

transmission, processing, and storage of

information. Circuits constructed from multiple

discrete electronic components instead of a

packaged IC would typically be extremely high-

speed low-resolution power-hungry types, as used

in military radar systems. There are problems with

improving the quality indices of electronic and

mechatronic devices concerning configure code

control systems with fewer elements than at present

while maintaining functionality, and resolving the

ability of the system. S. Golomb developed the idea

of using the advantages of multi-bit shift registers

with a balanced number of 0 and 1, or 00, 01, 10,

11, revealing in them the absence of

autocorrelation, which made it possible to improve

encoding systems - decoding signals with

correction of errors using sequences generated by

shift registers. S. Golomb used versions of these

sequences (Reed-Solomon codes) to encode video

images of Mars, in CDMA cell phones (Code

Division Multiple Access) with multiple access and

code separation of communication channels. In

1956, he joined the Glenn L. Martin Company,

which later became a defense contractor, [1].

Combinatorial optimization of radio and electronic

devices involves techniques for enhancement

quality indexes of devices and technologies with

non-uniform structures (e.g., linear antennas) based

on the theory of combinatorial configurations such

as difference sets and “Golomb rulers“. The

Golomb ruler was named for Solomon W.

Golomb. In mathematics, a Golomb ruler is a set of

marks at integer positions along a ruler such that no

two pairs of marks are the same distance apart.

There is no requirement that a Golomb ruler be

able to measure all distances up to its length, but if

it does, it is called a perfect Golomb ruler. It has

been proved that no perfect Golomb ruler exists for

five or more marks, [2]. This paper deals with

techniques for enhanced positioning precision and

discrete resolution of electronic systems, using

combinatorial theory [3] and novel mathematical

principle, based on the Galois fields theory [4],

perfect difference sets [5], and a new conceptual

model of electronic devices or systems based on

the Ideal Ring Bundles (IRBs). The notion is tied

closely to concepts proper of computational

intelligence under toroidal reference systems, [6].

Ongoing advances improve the link between

computational and physical elements, increasing

the reliability and functionality of electronic

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systems using intelligent mechanisms, using design

techniques based on remarkable properties and

structural perfection of IRBs. Of very importance in

the science is the role of mathematical models for

the synthesis and optimization of electronic

devices, using a novel design based on

combinatorial sequencing theory, namely the

concept of Ideal Ring Bundles (IRBs) which can be

used for the synthesis of the devices or circuits, for

improving such quality indices as reliability,

precision, speed, resolving ability, and

functionality.

2 Ideal Ordered Combinatorial

Constructions

The “ordered chain” approach to the study of

elements and events is known to be of widespread

applicability and has been extremely effective

when applied to the problem of finding the

optimum ordered arrangement of structural

elements in a distributed technological system.

2.1 Sums On Ordered-Chain Sequence

Let us calculate all Sn sums of the terms in the

numerical n-stage chain sequence of distinct

positive integers Cn = {K1, K2, . . ., Kn}, where we

require all terms in each sum to be consecutive

elements of the sequence. The maximum such sum

is the sum K1+ K2+. . .+ Kn = T of all n elements;

and the maximum number of distinct sums is:

Sn = 1 + 2 + . . . + n = n (n+1)/2 (1)

2.2 Sums On Ordered-Ring Sequence

If we regard the chain sequence Cn as being cyclic,

so that Kn is followed by K1 , we call this a ring

sequence. A sum of consecutive terms in the ring

sequence can have any of the n terms as its starting

point, and can be of any length (number of terms)

from 1 to n – 1. In addition, there is the sum T of all

n terms, which is the same independent of the

starting point. Hence the maximum number of

distinct sums S(n) of consecutive terms of the ring

sequence is given by:

S(n) = n (n –1) + 1 (2)

Comparing equations (1) and (2), we see that

the number of sums S(n) for consecutive terms in

the ring topology is nearly double the number of

sums Sn in the daisy-chain topology, for the same

sequence Cn of n terms.

Definition. An n-stage ring sequence Cn = {K1,

K2, . . ., Kn} of natural numbers for which the set of

all S(n) circular sums consists of the numbers from

1 to S(n) = n·(n–1) + 1, each number occurring

exactly R- times is called an “Ideal Ring Bundle”

(IRB), [6].

Here is an example of an IRB with n=5 and

S(n) = 21, namely {1, 3, 10, 2, 5}. To see this, we

observe:

1=1 6=5+1 11=2+5+1+3

16=1+3+10+2

2=2 7=2+5 12=10+2 17=10+2+5

3=3 8=2+5+1 13=3+10

18=10+2+5+1

4=1+3 9=5+1+3 14=1+3+10

19=5+1+3+10

5=5 10=10 15=3 =10 =2

20=3+10+2+5

21=1+3+10+2+5

Note that if we allow summing over more than

one complete revolution around the ring, we can

obtain all positive integers as such sums. Thus:

22=1+3+10+2+5+1, 23=2+5+1+3+10+2, etc.

Next, we consider a more general type of IRB,

where the S(n) ring-sums of consecutive terms give

us each integer value from 1 to M, for some integer

M, exactly R times, as well as the value of M +1

(the sum of all n numbers) exactly once. Here we

see that: M= n (n - 1) / R (3)

An example with n = 4 and R = 2, so that M =

6, is the ring sequence (1, 1, 2, 3), for which the

sums of consecutive terms are:

1, 1, 2, 3;

1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 1 = 4;

1 + 1 + 2 = 4, 1 + 2 + 3 = 6, 2 + 3 + 1 = 6,

3 + 1 + 1 = 5;

1 + 1 + 2 + 3 = 7.

We see that each “circular sum” from 1 to 6

occurs exactly twice (R=2). We say that this IRB

has the parameters n=4, R=2.

3 Design of Electronic Devices by

Combinatorial Optimization

3.1 Code-to-Resistance Decoding Matrix

Let's consider an example of constructing a code-

to-resistance decoding matrix that can be used to

synthesize digital functional units for various

purposes (Figure 1).

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Fig. 1: Code-to-resistance decoding matrix

The matrix (Figure 1) is made in the form of four

(n=4) resistors R1… …R4 in turn interconnected in

series with diodes D1... D4. The values of

resistances are selected according to the cyclic ratio

1:3:2:7. Two groups of outputs P and Q are

connected to the analog output of the digital code-

resistance converter "code-resistance" through the

code-controlled keys (pj =1,2,3,4) and (qj =1,2,3,4).

The ratio of least resistance r is equal to S(n) = (n2

– n + 1) = 13-th part of the total resistance of

resistors and opened diodes connected in series,

where n is the number of keys in each group of the

matrix. When the keys are locked in the digits of

the same name, the total resistance of the digital

converter is equal to zero. For other code

combinations, the resistance of the conversion is

equal to the appropriate part from the total

resistance of the ring circuit connected in series

elements, and each code combination (pj, qj)

corresponds to a new value of resistance from r to

n·(n–1)=4·3=12r with discreteness

r=(R1+R2+R3+R4)/13. If the code-to-resistance

decoding matrix is designed with the “+” sign of

the P side, depending resistance value of the digital

converter from code (pj, qj) is found in Table 1.

Table 1. Dependence on resistance value of the

digital converter from code (pj, qj) if the matrix is

designed with the “+” sign of the P side

12r

9 r

7 r

10 r

11r

If the code-to-resistance decoding matrix is

designed with the “+” sign of the Q side, depending

resistance value of the digital converter from code

(pj, qj) is found in Table 2.

Table 2. Dependence on resistance value of the

digital converter from code (pj, qj) if the matrix is

designed with the “+” sign of the Q side

4 r

6 r

12r

3 r

10r

11r

When one of the keys of group P and one of

the keys of group Q are closed in disparate digits,

the resistance of the digital converter becomes

equal to the total resistance of one or more resistors

connected in series, depending on the numbers of

the keys to be closed. In this case, each code

combination corresponds to a new resistance value.

For example, if there are four (n= 4) resistors and

four diodes in the decoding matrix, it is possible to

implement 12 resistance values with the same

increment step, using only two code-controlled

keys through which current flows. Such a decoding

matrix makes it possible to increase the reliability

of digital converters by reducing the number of

current-flowed keys, and the code implemented. In

turn, in this case, simplifies the detection of single

errors, because the presence of more than one unit

in at least one of the n-bit combinations, or the

presence of all zeros in at least one of the code

combinations, indicates an error. If the operating

point of the diodes is set at the linear section of the

voltage-ampere characteristic, then the value of the

resistances of the open diodes remains constant,

which can be taken into account. Correctly selected

operating mode and load invariability are ensured

by circuits, which include DC amplifiers. In the

general case, only two current carrying keys are

enough to configure a code-to-resistance decoding

matrix with any needed step of discreteness r=

(R1+R2+…+Rn)/ (n2 – n + 1) and extended operating

range using combinatorial optimization based on

the IRB-configuration with appropriate parameters.

3.2 Digital-analog Code-To-Voltage

Converter

Digital-to-analog converters (DACs) in industrial

applications are often used as the controlling

element. Digital technology has revolutionized the

way most of the equipment works. Data is

converted into binary code and then reassembled

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back into its original form at the reception point.

Scheme of digital-analog code-to-voltage converter

with minimal number of code-controlled keys

through which current flows, given in Figure 2. The

device contains two groups of control busbars P and

Q with four (n=4) paired code-controlled keys in

group P, and four controlled keys in group Q.

Fig. 2: Scheme of digital-analog code-to-voltage

converter

Signals corresponding to the code (pj, qj) make

it possible to vary the output voltage from 0 to 13u

with a step of discreteness u = U1/13

correspondently to Table 3.

Table 3. Dependence on the output voltage of the

digital-analog code-to-voltage converter from code

(pj, qj)

12u

10u

11u

To see Table 3, we can observe, that the four

(n= 4) resistors and four diodes in the decoding

matrix make it possible to implement output

voltage from 0 to 13u with the increment step u =

U1/13. In the general case, only three current

carrying controlled keys amply to configure a

digital-analog code-to-voltage converters with any

needed step of discreteness u = U1/ S(n) = U1/(n2 –

n + 1) using combinatorial optimization by

selecting IRB with appropriate parameters. Such

digital-analog code-to-voltage converter makes it

possible to increase the reliability of digital

converters by reducing the number of current-

flowed keys in a circuit alive. If the operating point

of the diodes is set at the linear section of the

voltage-ampere characteristic, then the value of the

resistances of the open diodes remains constant,

which can be taken into account.

3.3 Devices of Forming “Monolithic Codes”

A device of forming monolithic codes designed on

the IRB {1,3,2,7} with parameters n=4, R=1

depicted in the scheme (Figure 3).

Fig. 3: A device for forming monolithic codes

designed on the IRB {1,3,2,7} with parameters

n=4, R=1

In the scheme, Figure 3 all logic elements 1 are

normally open (NO) contacts, 2 - logic elements

OR, while 3 and 4 are code-controlled keys of group

P, and Q correspondently, 5- common power wire.

In the initial state keys of both these groups are

unlocked, all elements 1 opened, ring chain is de-

energized. When one of the keys of group P and one

of the keys of group Q are closed, informative

signal “1” enters through selected logic element 2

(OR) into the ring structure, filling the “1” line

corresponding numbers connected in series these

elements. At this moment signal “1” appears on the

output of one or more elements 2 simultaneously,

and on the correspondent outputs X1… X4 of the

device forms a monolithic code combination with

digit code weights X1=”1”, X2=”3”, X3=”2”, X4=”7”.

Forming monolithic code combinations according to

digit code weights X1=”1”, X2=”3”, X3=”2”, X4=”7”

is in Table 4.

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Table 4. Monolithic code combinations according

to digit code weights X1=”1”, X2=”3”, X3=”2”,

X4=”7”

Number

Code

Here we observe that each allowed code

combination forms as a ring sequence of no more

than one packets of connected in series symbols

“1”, as well as “0”.

Creating monolithic code combinations

depending on code (pj, qj) based on the IRB

{1,3,2,7} is in Table 5.

Table 5. Monolithic code combinations depending

on code (pj, qj) based on the IRB {1,3,2,7}

Monolithic code forms code combinations as a

ring sequence of solid connected symbols “1” as

well as “0” for identifying the correct code words at

the receiving end using the majoritarian approach

to detect and correct errors. This property makes it

possible to use self-correcting signals providing

concurrently faster than classic code transmission

in a noise channel.

4 Discussion

As is evident, the conceptual model of electronic

devices or systems based on the Ideal Ring Bundles

(IRBs) demonstrates the advantages of the

underlying innovative combinatorial methodology,

and values of structural elements (resistances, digit

code weights, etc.) are selected according to the

cyclic ratio of consecutive terms of the ratio. The

mutually unambiguous compliance with a set of

indexed elements of a set of binary code

combinations formed by this device has been

achieved in the system. In turn, it was possible due

to reducing the natural redundancy in the system.

Instead, Table 1 and Table 2 give examples of the

code-to-resistance decoding matrix, depending

resistance value of the digital converter according

to the cyclic ratio 1:3:2:7 organized from the IRB

with parameters n=4, S=13, R=1.

Reasoning along similarly, in the next tables

related designing digital-analog code-to-voltage

converter (Table 3), forming monolithic code

combinations according to digit code weights

X1=”1”, X2=”3”, X3=”2”, X4=”7”, based on the IRB

{1,3,2,7} (Table 4), and creating monolithic code

combinations depending code (pj, qj) (Tables 5).

Equations (2) and (3) formalize a large class of

one- and multi-dimensional optimized binary IRB-

codes from non-redundant vector codes of high

performance to coding self-correcting vector data

signals with faster than classic codes transmission

of multidimensional information by noise

communication channels. One of them is an

optimum binary monolithic star- code for

processing two- or multidimensional vector signals

under a toroidal coordinate system, [6]. Digital-to-

analog converters (DACs) typically comprise a

digital encoder and a set of current steering

switches, capacitive cells, other elements such as

resistor strings, or a combination of these, bringing

about mismatches among the components that

make up the DAC introduce nonlinear error. This

technique, referred to as dynamic element matching

(DEM), eliminates component mismatches as a

distortion-limiting factor in many practical cases by

making the system linear on average by virtue of its

shuffling scheme, [7], [8], [9], [10]. Proposed in

[7], DAC can achieve comparable performance as

the conventional ones with two fewer MSB bits

used for DEM, which simplifies the DEM block

significantly by reducing the MUX count. Paper

[8], analyzes a calibration technique that

circumyents the need for RZ pulse shaping by

adaptively measuring and canceling ISI over the

DAC’s first Nyquist band. As demonstrated in the

paper [9], it is possible to devise algorithms that

cancel errors in the Nyquist band of interest. The

results apply to most multi-bit DAC architectures

and all types of DEM known to the authors, and

they reduce to previously published continuous-

time DEM DAC results in the absence of ISI. The

paper [10], describes two techniques that are

inherently more linear than prior-art DACs, namely

the virtual-ground-switched resistor DAC and the

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zapped virtual-ground-switched dual return-to-open

DAC. Flicker noise can be eliminated by chopping,

but one needs to pay careful attention to minimize

chopping artifacts.

5 Conclusion and Outlook

Designs of electronic devices using the remarkable

properties and structural perfection of IRBs provide

an ability to reproduce nearly double the number of

combinatorial varieties the number in the Golomb

rulers, so long as an ability to configure electronic

devices with a limited number of elements and

bonds, while maintaining or improving on

resolving ability and the other operating

characteristics of the devices. Therefore, a code-to-

resistance decoding matrix designed from an IRB

makes it possible to increase the reliability of

digital converters by reducing the number of

current-flowed keys. Moreover, in this case

simplifies the detection of single errors. A digital-

analog code-to-voltage converter makes it possible

to increase the reliability of digital converters by

reducing the number of current-flowed keys. If the

operating point of the diodes is set at the linear

section of the voltage-ampere characteristic, then

the value of the resistances of the open diodes

remains constant, which can be taken into account.

Optimized monolithic code provides detecting and

self-correcting signals faster than classic codes. The

underlying design techniques provide configure

codes that have been defined as the optimized

binary weighed ring monolithic codes with a priory

any needed step of discreteness and extended

operating range, forming a large class of

performance self- correcting coded signals with

faster than classic codes transmission of

information by noise communication channels.

These design techniques make it possible to

configure electronic devices or systems with fewer

elements than at present, while maintaining or

improving on functionality and the other significant

operating characteristics of the devices. Prospect

for further research is the development of

combinatorial techniques in electronic devices and

systems for improving such quality indices as

reliability, transmission speed, positioning

precision, and ability to reproduce the maximum

number of combinatorial varieties in the system

with a limited number of elements and bonds, using

remarkable properties and structural perfection of

IRBs. The Ideal Ring Bundles provide, essentially,

a new conceptual model of electronic devices.

Moreover, the optimization has been embedded in

the underlying combinatorial models for direct

applications in electronic engineering to offer

ample scope for progress in sciences, technology,

and commerce.

References:

[1] Golomb, B.; Gong, G.; Hales, A., eds.

(2023). The Wisdom of Solomon: The Genius

and Legacy of Solomon Golomb. World

Scientific, https://doi.org/10.1142/12211.

[2] Shearer, J. B., (19 February 1998). "Table of

lengths of shortest known Golomb rulers".

IBM. Archived from the original on 25 June

2017.

[3] M. Jr. Hall, “Combinatorial Theory,” 2nd

Edition, Wiley-Interscience, 1998, 464 p.

[4] Rotman J. Galois Theory. 1998. 106 p.

DOI: 10.1007/978-1-4612-0617-0.

[5] E.H. Moore, H.S. Pollatsek, Difference Sets:

Connecting Algebra, Combinatorics, and

Geometry, AMS, 2013.

[6] V. V.Riznyk, “Formalization coding methods

of information under toroidal coordinate

systems”, Radio Electronics, Computer

Science, Control. 2021. № 2, pp.144-153.

DOI.10.15588/1607-3274-2021-2-15.

[7] Wei Mao, Yongfu Li, Chun-Huat Heng and

Yong Lian, "High dynamic performance

current-steering DAC design with nested-

segment structure", IEEE Trans. Very Large

Scale Integr. (VLSI) Syst., vol. 26, no. 5, pp.

995-999, May 2018.

DOI: 10.1109/TVLSI.2018.2791462.

[8] D. Kong and I. Galton, "Adaptive

cancellation of static and dynamic mismatch

error in continuous-time DACs", IEEE

Trans. Circuits Syst. I Reg. Papers, vol. 65,

no. 2, pp. 421-433, Feb. 2018.

DOI: 10.1109/TCSI.2017.2737986.

[9] J. Remple and I. Galton, "The effects of

inter-symbol interference in dynamic element

matching DACs", IEEE Trans. Circuits Syst.

I Reg. Papers, vol. 64, no. 1, pp. 14-23, Jan.

2017. DOI: 10.1109/TCSI.2016.2601026.

[10] Enrique Alvarez-Fontecilla; Paul S.

Wilkins; Steven C. Rose, “Understanding

high-resolution dynamic element matching

DACs (feature),” IEEE Circuits and Systems

Magazine, vol.23, Issue 4, Fourthquarter

2023. DOI: 10.1109/MCAS.2023.3325504.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at

all stages from the formulation of the problem to

the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflicts of interest to declare.

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(Attribution 4.0 International, CC BY 4.0)

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