# Complex Neutrosophic Soft Matrices Framework: An Application in Signal Processing

Authors: Madad Khan; Saima Anis; Kainat Bibi; Sohail Iqbal; Florentin Smarandache
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce the concept of complex neutrosophic soft matrices. We define some basic operations including complement, union, and intersection on these matrices. We extend the concept of complex neutrosophic soft sets to complex neutrosophic soft matrices and prove related properties. Moreover, we develop an algorithm using complex neutrosophic soft matrices and apply it in signal processing.


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## Body

## 1. Introduction

The models of real-life problems in almost every field of science like mathematics, physics, operations research, medical sciences, engineering, computer science, artificial intelligence, and management sciences are mostly full of complexities. Many theories have been developed to overcome these uncertainties; one among those theories is fuzzy set theory. Zadeh was the first who gave the concept of a fuzzy set in 1965 [1]. Fuzzy sets are the generalizations or extensions of crisps sets.In order to add the concept of nonmembership term to the definition of fuzzy set, the concept of an intuitionistic fuzzy set was introduced by Atanassov in 1986 [2], where he added the concept of nonmembership term to the definition of fuzzy set. The intuitionistic fuzzy set is characterized by a membership function μ and a nonmembership function υ with ranges [0, 1]. The intuitionistic fuzzy set is the generalization of a fuzzy set. An intuitionistic fuzzy set can be applied in several fields including modeling, medical diagnosis, and decision-making. [3] Molodtsov introduced the concept of a soft set in 1999 and developed the fundamental results related to this theory. Basic operations including complement, union, and intersection are also defined on this set. Molodtsov used soft sets for applications in games, probability, and operational theories [3–6]. In 2018, Smarandache generalized the soft set to the hypersoft set by transforming the classical uniargument function F into a multiargument function [7]. Maji et al. [8] introduced the concept of fuzzy soft sets by combining soft sets and fuzzy sets and applied them in decision-making problems [9]. In [10], Cagman and Enginolu used soft matrix theory for applications in decision-making problems.The concept of neutrosophy was introduced by Smarandache [11] in 1998. A neutrosophic set is characterized by a truth membership function T, an indeterminacy function I, and a falsity membership function F. A neutrosophic set is a mathematical framework which generalizes the concept of a classical set, fuzzy set, intuitionistic fuzzy set, and interval valued fuzzy set [12]. In [13], Nabeeh introduced a method that can promote a personal selection process by integrating the neutrosophic analytical hierarchy process to show the proper solution among distinct options with order preference technique similar to an ideal solution (TOPSIS). In [14], Baset introduced a concept of a neutrosophy technique called type 2 neutrosophic numbers. By combining type 2 neutrosophic numbers and TOPSIS, they suggested a novel method T2NN-TOPSIS which has a lot of applications in group decision-making. They worked on a multicriteria group decision-making technique of the analytical network process method and Visekriterijusmska Optmzacija I Kommpromisno Resenje method under neutrosophic environment that deals high-order imprecision and incomplete information [15].The largest number set is a complex set which is introduced by Gauss in 1795 and is the extension of a real number set. According to same fashion, a complex fuzzy set is extension to a fuzzy set as here the range set is extended from interval0,1 to a closed disc of radius one in complex plane. The degree of membership a complex fuzzy set is not restricted to a value in 0,1; it is extended to a complex value lies in a disc of radius one in the complex plane.Complex fuzzy sets are not simply a linear extension of conventional fuzzy sets; complex fuzzy sets allow a natural extension of fuzzy set theory to problems that are either very difficult or impossible to address with one-dimensional grades of membership. It is an obvious fact that uncertainty, indeterminacy, inconsistency, and incompleteness in data are periodic in nature. In order to address this difficulty, in 2002, Daniel Ramot was the first who gave the concept of a complex fuzzy set. The concept of a complex neutrosophic set was introduced in [16].The complex fuzzy setC is described as membership function, with range in closed unit disc in the complex plane. The complex-valued membership function ϕsx is defined as ϕsx=tsxei.ηsx that assigns a complex value of membership to any x in U (universal set) such that tsx and ηsx both are real-valued with tsx is fuzzy set and i=−1, where tsx is called amplitude term and ηsx is called phase term.Physically the complex fuzzy set is used for representing the complex fuzzy solar activity (solar maximum and solar minimum) through the measurement of sunspot number and is also used in signal processing. The complex neutrosophic set is the generalization of a complex fuzzy set and a neutrosophic set. The complex neutrosophic set is characterized by complex-valued truth membership function, complex-valued indeterminate function, and complex-valued falsehood function. In short, a complex neutrosophic set is more generalized because it is not only the generalization of all the current frameworks but also describes the information in a complete and comprehensive way.A fuzzy set with its generalizations, like intuitionistic fuzzy sets, interval valued fuzzy sets, and cubic sets, represents uncertainties in models of the one-dimensional phenomenon while a complex fuzzy set is the only generalization of a fuzzy set which deals with the models of real-life problems with the two-dimensional and periodic phenomenon. A complex fuzzy set is more applicable because of its nature and can be used more widely in all branches of sciences. Since it is similar to that of a Fourier transform, more explicitly it is a particular sort of Fourier transform with the only restriction on the range which is a complex unit disc. A Fourier transform is used in signals and systems; that is, a Fourier transform is the mathematical tool for representing both continuous and discrete signals. Taking advantage of a complex fuzzy set, being a specific form of Fourier transform, it can be used to represent signals in a particular region of consideration. A neutrosophic set is the generalization of a fuzzy set which deals with the problems containing uncertainties of truthfulness, falsehood, and neutrality. The complex neutrosophic set has three major parts, that is, truth, intermediate, and falsehood membership functions. The truth membership function is totally the same as that of a complex fuzzy set while intermediate and falsehood membership functions are the new additions to it. Thus, a complex neutrosophic set can be applied more widely compared with other fuzzy sets.In the vast area of science and technology, matrices play an important role. Classical matrix theory cannot solve all models of the daily life problems. In order to overcome these difficulties, Yang and Ji in [17] initiated a matrix representation of a fuzzy soft set and successfully applied the proposed notion of a fuzzy soft matrix in certain decision-making problems.This work is basically the extension of the work of Ramot et al. [18], Alkouri and Saleh [19], Cai [20, 21], and Zhang et al. [22] to neutrosophic sets. Here, in this paper, we extend the concept by defining the complex neutrosophic fuzzy soft set and then the complex neutrosophic fuzzy soft matrix (CNFSM). Further, we discuss some basic operations on CNFSM and finally we develop an algorithm using these matrices and apply it in signal processing.Soft matrices are widely used in signals and systems, decision-making problems, and medical diagnosis. This article has two aims. In the first part, we present theoretical foundations of the complex neutrosophic fuzzy soft matrices. These theoretical foundations provide basic notions and operations on complex neutrosophic soft matrices such as complex neutrosophic fuzzy soft zero matrix, complex neutrosophic fuzzy soft universal matrix, complex neutrosophic fuzzy soft submatrices, union of complex neutrosophic fuzzy soft matrices, intersection of complex neutrosophic fuzzy soft matrices, and complement of complex neutrosophic fuzzy soft matrices. Then, we introduce some fundamental results and discuss main strategies for applications of this concept in signals and systems, as well as a coherent discussion of the theory of complex neutrosophic fuzzy soft matrices. The aim of these new concepts is to provide a modern method with mathematical procedure to identify a reference signal out of large number of signals received by a digital receiver. The complex neutrosophic fuzzy soft matrix is the generalization of the fuzzy soft matrix, complex fuzzy soft matrix, and Pythagorean fuzzy soft matrix. The degree of membership function, nonmembership function, and phase terms are all applied to each entry of the matrix which give more fruitful results for a better choice in signals and systems along with other fields such as decision-making problems, medical diagnosis, and pattern recognition. These applied contexts provide solid evidence of the wide applications of the complex neutrosophic fuzzy soft matrices approach to signals and systems and decision-making problems.


## 2. Preliminaries

Here, we begin with a numerical example of a complex neutrosophic set which is already defined above.Example 1.
LetX=x1,x2,x3 be a universe of discourse. Then, the complex neutrosophic set S in X is given as(1)S=0.6ej0.3,ejπ/2,0.3ej0.6x1+0.4ej0,0.9ejπ/4,0.4ejπ/4x2+0.5ej2π/3,0.2ej0.2,0.7ejπ/3x3.Definition 1.
(fuzzy set (FS) [1]). Fuzzy set is defined by an arbitrary mapping from a nonempty set X to the unit interval 0,1, i.e., f:X⟶0,1. The set of all fuzzy subsets of X is denoted by FX, i.e., FX=f:fis a function from Xinto0,1.
Soft set theory is a generalization of fuzzy set theory, which was proposed by Molodtsov in 1999.Definition 2.
(soft set (SS) [3]). Let U be the universal set, E be the set of parameters, and A⊆E and PU be the power set of U, then a soft set FA is defined by a mapping.
fA:E⟶PU such that fAx=ϕ if x∉A.
In other words, we can say that soft setFA over U is the parameterized family of subsets of U, that is, FA=x,fAx:x∈E,fAx∈PU.Definition 3.
(fuzzy soft set (FSS) [8]). Let U be the universe of discourse, E be the set of parameters, and A⊆E, then a fuzzy soft set GA is defined by a mapping: gA:E⟶P′U where P′U is the collection of all fuzzy subsets of U, such that gAx=ϕ if x∉A.
In other words, we can say that fuzzy soft setGA over U is the parameterized family of fuzzy subsets of U, that is, GA=x,gAx:x∈E,gAx∈P′U.Definition 4.
(intuitionistic fuzzy set (IFS) [2]). An intuitionistic fuzzy set I on a nonempty set U (universal set) is defined by the set of triplets given by(2)I=x,μIx,γIx:x∈U.
Here,μIx and γIx both are functions from U to 0,1 as μIx:U⟶0,1 and γIx:U⟶0,1. Here, μIx represents the degree of membership and γIx represents the degree of nonmembership of each element x∈U to the set I, respectively, also 0≤μIx+γIx≤2, for all x∈U.Definition 5.
(complex fuzzy set (CFS) [18]). The complex fuzzy set S on universe of discourse X is described as complex-valued membership function μSx that assigns value of membership of the form rsxejwsx to any element x∈X, where j=−1, μSx involves two real-valued rsx and wsx, with rsx∈0,1.
Mathematically,S=x,μsx:x∈X.Definition 6.
(complex intuitionistic fuzzy set (CIFS) [19]). The complex intuitionistic fuzzy set CI on a nonempty set U (universal set) is defined by the set of triplets given by CI=x,μCIx,γCIx:x∈U. Here, μCIx=rCIxejwCIx and γCIx=lCIxejmCIx both are functions from U to closed unit disc in the complex plane and also μCIx represents the degree of membership and γCIx represents the degree of nonmembership of each element x∈U to the set CI, respectively, and also 0≤rCIx+lCIx≤2, for all x∈U.Definition 7.
(complex neutrosophic fuzzy set (CNFS) [16]). The complex neutrosophicfuzzy set N on a nonempty set U (universal set) is defined by the set as N=x,TNx,INx,FNx:x∈U. Here, TNx=rNxejwNx, INx=lNxejmNx, and FNx=pNxejqNx are the complex-valued functions from U to the closed unit disc in the complex plane where TNx describes complex-valued truth membership function, INx describes complex-valued indeterminate membership function, and FNx describes complex-valued falsehood membership function of each element x∈U to the set N, respectively, and also 0≤rNx+lNx+pNx≤3, for all x∈U.


## 3. Complex Neutrosophic Fuzzy Soft Matrix Theory

In this section, we introduced a new concept of complex neutrosophic fuzzy soft matrices. We defined the operations of union, intersection, compliment, and submatrices. We defined zero and universal matrices. Moreover, we proved some related results.Definition 8.
(complex neutrosophic fuzzy soft matrix (CNFSM)). Consider a universal setU=u1,u2,u3,…,um and set of parameters E=e1,e2,e3,…,en such that A⊆E and cA,A be a complex neutrosophic fuzzy soft set over U,E. Then, the CNFSS cA,A in matrix form is represented by Am×n=aijm×n or A.=aij where i=1,2,3,…,m and j=1,2,3,…,n.(3)Here,aij=μjui=μjTui,μjIui,μjFui,ifej∈A,0,0,0ifej∉A.
Now,μjTui,μjIui,μjFui represents degrees of membership of truth, intermediate, and falsehood on ui. Throughout this paper, we will use the abbreviation CNFSMm×n for complex neutrosophic fuzzy soft matrix over U. Following is the example of a complex neutrosophic fuzzy soft matrix.Example 2.
LetU=u1,u2,u3 be a universal set representing the three firms, E=e1costly,e2beautiful,e3luxurious be the parameters set, and A=e1,e2⊆E. Then, CNFSS cA,A over the universal set U is given by(4)cA,A=cAe1=u1,0.3ejπ,0.6ejπ/2,ejπ,u2,0.7ejπ/4,0.8ejπ/4,0.5ejπ/2,u3,0.9ejπ,0.1ejπ/6,0.2ejπ/2,cAe2=u1,0.1ejπ/3,0.2ejπ/6,0.1ejπ,u2,0.3ejπ/2,0.9ejπ/2,0.9ejπ/4,u3,0.5ejπ/3,0.5ejπ,0.6ejπ/3.
Here,(5)0.3ejπ=0.3cosπ+jsinπ=0.3−1+0=−0.30.3ejπ=−0.3=0.30.66ejπ/2=0.6cosπ2+jsinπ2=0.60+j=0.6j0.6ejπ/2=0.6j=0.36=0.6ejπ=cosπ+jsinπ=−1+0=−1ejπ=−1=1,0.7ejπ/4=0.7cosπ4+jsinπ4=0.712+j12=0.70.707+j0.707=0.494+j0.4940.7ejπ/4=0.494+j0.494=0.244+0.244=0.69,0.8ejπ/4=0.8cosπ4+jsinπ4=0.812+j12=0.80.707+j0.707=0.5656+j0.5656,|0.8ejπ/4|=|0.5656+j0.5656|=0.319+0.319=0.790.50.5ejπ/2=0.5cosπ2+jsinπ2=0.5j0.5ejπ/2=0.5j=0.25=0.5,0.9ejπ=0.9cosπ+jsinπ=0.9−1=−0.90.9ejπ=−0.9=0.90.1ejπ/6=0.1cosπ6+jsinπ6=0.10.866+j0.5=0.0866+j0.050.1ejπ/6=|0.0866+j0.05|=0.0074+0.0025=0.099,0.2ejπ/2=0.2cosπ2+jsinπ2=0.20+j=0.2j0.2ejπ/2=0.2j=0.04=0.2,0.1ejπ/3=0.1cosπ3+jsinπ3=0.10.5+j0.866=0.05+j0.08660.1ejπ/3=0.05+j0.0866=0.0025+0.0074=0.090.2ejπ/6=0.2cosπ6+jsinπ6=0.20.866+j0.5=0.1732+j0.10.2ejπ/6=0.1732+j0.1=0.029+0.01=0.19,0.1ejπ=0.1cosπ+jsinπ=0.1−1+0=−0.10.1ejπ=−0.1=0.1,0.3ejπ/2=0.3cosπ2+jsinπ2=0.3j0.3ejπ/2=0.3j=0.09=0.30.9ejπ/2=0.9cosπ/2+jsinπ2=0.9j0.9ejπ/2=0.9j=0.81=0.9,0.9ejπ/4=0.9cosπ4+jsinπ4=0.912+j12=0.90.707+j0.707=0.636+j0.6360.9ejπ/4=|0.636+j0.636|=0.404+0.404=0.898,0.5ejπ/3=0.5cosπ3+jsinπ3=0.50.5+j0.866=0.25+j0.4330.5ejπ/3=0.25+j0.433=0.0625+0.187=0.4990.5ejπ=0.5cosπ+jsinπ=0.5−1=−0.50.5ejπ=−0.5=0.5,0.6ejπ/3=0.6cosπ3+jsinπ3=0.60.5+j0.866=0.3+j0.5190.6ejπ/3=0.3+j0.519=0.09+0.269=0.599.
Now, the abovementioned CNFSScA,A in matrix form is given by(6)A=0.3,0.6,10.09,0.19,0.10,0,00.69,0.79,0.50.3,0.9,0.8980,0,00.9,0.099,0.20.499,0.5,0.5990,0,0.Definition 9.
(complex neutrosophic fuzzy soft zero natrix). Letaij∈CNFSMm×n, then aij is called complex neutrosophic fuzzy soft zero matrix if aij,rij,lij=0,0,0, for all i and j, and is denoted by 0.Example 3.
(7)0=0,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,0.Definition 10 (complex neutrosophic fuzzy soft universal matrix).
Letaij∈CNFSMm×n, then aij is called complex neutrosophic fuzzy soft universal matrix if aij,rij,lij=1,1,1, for all i and j, and is represented by 1.(8)1=1,1,11,1,11,1,11,1,11,1,11,1,11,1,11,1,11,1,1.Definition 11 (complex neutrosophic fuzzy soft submatrices).
LetAm×n and Bm×n be two CNFSMs, then(i)
Am×n is a CNFS submatrix of Bm×n and is denoted by Am×n⊑Bm×n if aij=aij,aij′,aij″≼¯bij=bij,bij′,bij″, that is, aij≼¯bij,aij″≼¯bij′,aij″≼¯bij″, for all aij∈Am×n,bij∈Bm×n(ii)
Am×n is a proper CNFS submatrix of Bm×n and is denoted by Am×n⊏Bm×n if aij=aij,aij′,aij″≺bij=bij,bij′,bij″, that is, aij≺bij,aij′≺bij′,aij″≺bij″, for all aij∈Am×n, bij∈Bm×n and for at least one entry aij≺bij, that is, aij≺bij,aij′≺bij′,aij″≺bij″(iii)
TwoCNFSMsAm×n and Bm×mn are equal and are denoted by Am×n=Bm×n, if aij=aij,aij′,aij″=bij=bij,bij′,bij″, that is. aij=bij,aij′=bij′,aij″=bij″, for all aij∈Am×n, bij∈Bm×nExample 4.
Let(9)A2×2=0.2,0.4,0.10.1,0.5,0.20.3,0.7,0.30.5,0.4,0.4,B2×2=0.2,0.4,0.10.3,0.7,0.90.3,0.7,0.30.7,0.5,0.7.
So, we can write thatA2×2⊏B2×2. Moreover, A⊏B.Definition 12.
(union/intersection and compliment of complex neutrosophic fuzzy soft matrices).
LetAm×n and Bm×n be two CNFSM, then the CNFSMCm×n is called(i)
Union ofAm×n and Bm×n and is denoted by Am×n⊔Bm×n if Cm×n=maxAm×n,Bm×n, for all i and j, that is, cij=maxaij,bij,minaij′,bij′,minaij″,bij″ where cij=cij,cij′,cij″(ii)
Intersection ofAm×n and Bm×n is denoted by Am×n⊓Bm×n if Cm×n=minAm×n,Bm×n, for all i and j, that is, cij=minaij,bij,maxaij′,bij′,maxaij″,bij″, where cij=cij,cij′,cij″(iii)
Complement ofAm×n is denoted by Am×n′ if Cm×n=1−Am×n, for all i and j, that is, cij=1−aij,1−aij′,1−aij″, where cij=cij,cij′,cij″Example 5.
Assume that(10)A2×2=0.3,0.6,10.65,0,0.60.3,0.9,00.8,0.7,0.9,B2×2=0.49,0.5,0.40.2,0,0.30.1,0.9,0.30,0,0,then,A2×2⊔B2×2=0.49,0.5,0.40.65,0,0.30.3,0.9,00.8,0,0,A2×2⊓B2×2=0.3,0.6,10.2,0,0.60.1,0.9,0.30,0.7,0.9,A2×2′=0.7,0.4,00.35,1,0.40.7,0.1,10.2,0.3,0.1.Proposition 1.
LetAm×n be a CNFSM, then(11)iAm×n′′=Am×n,ii0′=1.Proof.
It follows from definition.Proposition 2.
LetAm×n, Bm×n, and Cm×n be three CNFSMs, then(12)iAm×n=Bm×nandBm×n=Cm×n⟹Am×n=Cm×n,iiAm×n⊑Bm×nandBm×n⊑Am×n⟹Am×n=Bm×n.Proof.
It follows from definition.Proposition 3.
LetAm×n and Bm×n be two CNFSMs, then(13)Am×n⊑Bm×nandBm×n⊑Cm×n⟹Am×n⊑Cm×n.Proof.
It follows from definition.Proposition 4.
LetAm×n and Bm×n be two CNFSMs, then(14)iAm×n⊔Bm×n=Bm×n⊔Am×n,iiAm×n⊓Bm×n=Bm×n⊓Am×n,iiiAm×n⊔Bm×n⊔Cm×n=Am×n⊔Bm×n⊔Cm×n,ivAm×n⊓Bm×n⊓Cm×n=Am×n⊓Bm×n⊓Cm×n,vAm×n⊔Bm×n⊓Cm×n=Am×n⊔Bm×n⊓Am×n⊔Cm×n,viA⊓Bm×n⊔Cm×n=Am×n⊓Bm×n⊔Am×n⊓Cm×n.Proof.
(15)iAm×n⊔Bm×n=maxAm×n,Bm×n=maxBm×n,Am×n=Bm×n⊔Am×n,iiAm×n⊓Bm×n=minAm×n,Bm×n=minBm×n,Am×n=Bm×n⊓Am×n,iiiAm×n⊔Bm×n⊔Cm×n=maxAm×n⊔Bm×n,Cm×n=maxmaxAm×n,Bm×n,Cm×n=maxAm×n,maxBm×n,Cm×n=maxAm×n,Bm×n⊔Cm×n=Am×n⊔Bm×n⊔Cm×n,ivAm×n⊓Bm×n⊓Cm×n=minAm×n⊓Bm×n,Cm×n=minminAm×n,Bm×n,Cm×n=minAm×n,minBm×n,Cm×n=minAm×n,Bm×n⊓Cm×n=Am×n⊓Bm×n⊓Cm×n.vAm×n⊔Bm×n⊓Cm×n=maxAm×n,Bm×n⊓Cm×n=maxAm×n,minBm×n,Cm×n=minmaxAm×n,Bm×n,maxAm×n,Cm×n=minAm×n⊔Bm×n,Am×n⊔Cm×n=Am×n⊔Bm×n⊓Am×n⊔Cm×n,viAm×n⊓Bm×n⊔Cm×n=minAm×n,Bm×n⊔Cm×n=minAm×n,maxBm×n,Cm×n=maxminAm×n,Bm×n,minAm×n,Cm×n=maxAm×n⊓Bm×n,Am×n⊓Cm×n=Am×n⊓Bm×n⊔Am×n⊓Cm×n.Proposition 5.
LetAm×n and Bm×n be two CNFSMs, then the De-Morgan laws are valid: (16)iAm×n⊔Bm×n′=Am×n′⊓Bm×n′iiAm×n⊓Bm×n′=Am×n′⊔Bm×n′.Proof.
(17)iAm×n⊔Bm×n′=maxAm×n,Bm×n′=1−maxAm×n,Bm×n=min1−Am×n,1−Bm×n=Am×n′⊓Bm×n′,Am×n⊓Bm×n′=minAm×n,Bm×n′=1−minAm×n,Bm×n=max1−Am×n,1−Bm×n=Am×n′⊔Bm×n′.


## 4. Complex Neutrosophic Fuzzy Soft Decision-Making Method

Now, we are going to discuss real-life applications of newly definedCNFSMm×n. We will show how our theoretical concepts and results can be applied to the real-life phenomenon. Specifically, we will show that CNFSMm×n explains how to get a better and clear signal for identification with a given reference signal. Before moving towards the algorithm, we will define the fuzzy soft (FS) max-min decision-making method (FSMmDM) by using FS max-min decision function and also define here the optimum FS on universal set U.Definition 13.
(fuzzy soft (FS) max-min decision-making function [10]). Let cip∈SMm×n2, Ik=p:thereexisti,cip≠0,k−1n<p≤kn, for all k∈I=1,2,3,…,n. Then, soft max-min decision function, denoted Mm, is defined as follows:(18)Mm:SMm×n2⟶SMmm×1,Mmcip=maxk∈Itk,where(19)tk=minp∈Ikcip,ifIk≠,0,ifIk=.
The one column soft matrixMmcip is called max-min soft decision-making matrix.Definition 14 (see [10]).
LetU=u1,u2,…,um be a universal set and Mmcip=di1. Then, a subset of U can be obtained by using di1 as in the following way optdi1U=ui:ui∈U,di1=1, which is called an optimum set on U.


### 4.1. Decision-Making Algorithm

step 1.
Suppose thatM different signals S1t′,S2t′,…,SMt′ are detected and sampled by a receiver and let U=S1t′,S2t′,…,SMt′. Each of these signals is sampled N times. Let Smr′ denote the r/th sample 1≤r′≤N of the mth signal 1≤m≤M. Now, we know that each signal has its Fourier transform. So, each received signal can be expressed as summation of its Fourier components as(20)Smr′=1N∑n=1NCm,nei2πn−1r′−1/N, thenSmr′=1N∑n=1NCm,n⋅ei2πn−1r′−1/N,where Cm,n1≤n≤N represents complex Fourier coefficients of Sm. The above expression can also be rewritten as follows:Smr′=1/N∑n=1NBm,n⋅ei2πn−1r′−1+Nβm,n/N, where Cm,n=Bm,neiβm,n, with Bm,n,βm,n real-valued and Bm,n≥0, for all n, where 1≤n≤N.step 2.
The above given signals are expressed as in matrix form asA=Smr′N×M, that is, express N samples of each signal (total M signals) in columns:(21)A=S1T1,S1I1,S1F1S2T1,S2I1,S2F1...SMT1,SMI1,SMF1S1T2,S1I2,S1F2S2T2,S2I2,S2F2...SMT2,SMI2,SMF2......S1TN,S1IN⋅S1FNS2TN,S2IN,S2FN...SMTN,SMIN,SMFN.step 3.
Similarly, we will construct another matrix by the signalsSm∗r.(22)B=S1∗T1,S1∗I1,S1∗F1S2∗T1,S2∗I1,S2∗F1...SM∗T1,SM∗I1,SM∗F1S1∗T2,S1∗I2,S1∗F2S2∗T2,S2∗I2,S2∗F2...SM∗T2,SM∗I2,SM∗F2......S1∗TN,S1∗IN,S1∗FNS2∗TN,S2∗IN,S2∗FN...SM∗TN,SM∗IN,SM∗FN.step 4.
Multiply matricesA and B using usual multiplication of matrices. In this multiplication, the truth value of the entry of the first matrix will be multiplied by the truth value of the entry of the second matrix. The intermediate and false values of the entries are multiplied similarly.step 5.
The complex neutrosophic fuzzy soft max-min decision-making matrix (CNFSMmDM) is found by taking minimum of truth, intermediate memberships, and maximum of falsehood membership values of each column, and we will get a column matrixdi1, where 1≤i≤M.step 6.
An optimum setoptMmABU on U is found, that is,(23)max|SjTi|,maxSjIui,minSjFui, for1≤j≤Mand1≤i≤N.


## 4.1. Decision-Making Algorithm

step 1.
Suppose thatM different signals S1t′,S2t′,…,SMt′ are detected and sampled by a receiver and let U=S1t′,S2t′,…,SMt′. Each of these signals is sampled N times. Let Smr′ denote the r/th sample 1≤r′≤N of the mth signal 1≤m≤M. Now, we know that each signal has its Fourier transform. So, each received signal can be expressed as summation of its Fourier components as(20)Smr′=1N∑n=1NCm,nei2πn−1r′−1/N, thenSmr′=1N∑n=1NCm,n⋅ei2πn−1r′−1/N,where Cm,n1≤n≤N represents complex Fourier coefficients of Sm. The above expression can also be rewritten as follows:Smr′=1/N∑n=1NBm,n⋅ei2πn−1r′−1+Nβm,n/N, where Cm,n=Bm,neiβm,n, with Bm,n,βm,n real-valued and Bm,n≥0, for all n, where 1≤n≤N.step 2.
The above given signals are expressed as in matrix form asA=Smr′N×M, that is, express N samples of each signal (total M signals) in columns:(21)A=S1T1,S1I1,S1F1S2T1,S2I1,S2F1...SMT1,SMI1,SMF1S1T2,S1I2,S1F2S2T2,S2I2,S2F2...SMT2,SMI2,SMF2......S1TN,S1IN⋅S1FNS2TN,S2IN,S2FN...SMTN,SMIN,SMFN.step 3.
Similarly, we will construct another matrix by the signalsSm∗r.(22)B=S1∗T1,S1∗I1,S1∗F1S2∗T1,S2∗I1,S2∗F1...SM∗T1,SM∗I1,SM∗F1S1∗T2,S1∗I2,S1∗F2S2∗T2,S2∗I2,S2∗F2...SM∗T2,SM∗I2,SM∗F2......S1∗TN,S1∗IN,S1∗FNS2∗TN,S2∗IN,S2∗FN...SM∗TN,SM∗IN,SM∗FN.step 4.
Multiply matricesA and B using usual multiplication of matrices. In this multiplication, the truth value of the entry of the first matrix will be multiplied by the truth value of the entry of the second matrix. The intermediate and false values of the entries are multiplied similarly.step 5.
The complex neutrosophic fuzzy soft max-min decision-making matrix (CNFSMmDM) is found by taking minimum of truth, intermediate memberships, and maximum of falsehood membership values of each column, and we will get a column matrixdi1, where 1≤i≤M.step 6.
An optimum setoptMmABU on U is found, that is,(23)max|SjTi|,maxSjIui,minSjFui, for1≤j≤Mand1≤i≤N.


## 5. Applications

Step 1.
Assume thatu1,u2, and u3 be any three signals received by a digital receiver from any source. Each signal is a triplet of numbers. The first number of triplet represents the truth value, second represents the intermediate value, and the third represents the false value corresponding to each signal. Now, each of these signals is sampled three times. Let R be the given known reference signal. Each signal is compared with the reference signal in order to get the high degree of resemblance with the reference signal R. Now, we obtain the matrix A by setting the signals along column and their three times sampling along row. Similarly, we will obtain the matrix B.step 2.
MatricesA and B are given by(24)A=0.7,0.4,0.50.6,0.7,10.8,1,0.70.8,0.5,0.30.2,0,0.90.5,0.8,0.40.4,0,0.80.8,0.4,0.60,0.3,0.9.step 3.
(25)B=0.4,0.4,00.6,0.7,0.40.1,0.3,00.3,0.7,0.70.4,0.9,0.40.1,0.6,0.40.2,0.4,0.50.4,0.5,0.30.8,0.5,0.8.step 4.
Now, we will calculate the product of above defined matrices by usual multiplication of matrices. In this multiplication, the truth value of the entry of the first matrix will be multiplied by the truth value of the entry of the second matrix. Similarly, the intermediate and false values of the entries are multiplied.(26)AB=0.62,0.69,0.420.98,0.96,0.450.77,0.59,0.60.48,0.52,0.830.76,0.75,0.60.5,0.55,0.680.4,0.4,0.870.56,0.51,0.830.12,0.39,0.96.step 5.
We calculateCNFSMmDMAB=di1, for all i=1,2,3, where di1 is defined as di1=mintk1=mint11,t21,t31 for all k=1,2,3.(27)d11=mintk1=mint11,t21,t31=min0.62,0.69,0.42,0.48,0.52,0.83,0.4,0.4,0.87=0.4,0.4,0.42,d21=mintk2=mint12,t22,t32=min0.98,0.96,0.45,0.76,0.75,0.6,0.56,0.51,0.83=0.56,0.51,0.45,d31=mintk3=mint13,t23,t33=min0.77,0.59,0.6,0.5,0.55,0.68,0.12,0.39,0.96=0.12,0.39,0.6.
We obtainCNFSMmDM as follows:(28)CNFSMmDMAB=di1=0.4,0.4,0.420.56,0.51,0.450.12,0.39,0.6.Step 6.
Finally, we find out an optimum set onU as follows: optMmABU=u2. So, the signal which is identified as a reference signal is the signal u2.


## 6. Conclusion

This paper consists of CNFSM and different types of complex neutrosophic soft matrices with examples. We introduced some new operations on complex neutrosophic fuzzy soft matrices and explore related properties. Further, we constructed a complex neutrosophic soft decision-making algorithm with the help of these matrices and used it in signal processing. We hope that our finding will help in enhancing the study on complex neutrosophic soft theory and will open a new direction for applications especially in decision sciences. In future, we will define some new operations on complex neutrosophic fuzzy soft sets and will introduce some new algorithms for signals and other related decision-making in social sciences. Specifically, we will use complex fuzzy sets and complex neutrosophic fuzzy sets in signal processing for modeling of continuous signals.



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