In this paper, we introduce the notion of amplitude interval-valued complex Pythagorean fuzzy sets (AIVCPFSs). The motivation for this extension is the utility of interval-valued complex fuzzy sets in membership and non-membership degree which can express the two dimensional ambiguous information as well as the interaction among any set of parameters when they are in the form of interval-valued. The principle of AIVCPFS is a mixture of the two separated theories such as interval-valued complex fuzzy set and complex Pythagorean fuzzy set which covers the truth grade (TG) and falsity grade (FG) in the form of the complex number whose real part is the sub-interval of the unit interval. We discuss some set-theoretic operations and laws of the AIVCPFSs. We study some particular examples and basic results of these operations and laws. We use AIVCPFSs in signals and systems because its behavior is similar to a Fourier transform in certain cases. Moreover, we develop a new algorithm using AIVCPFSs for applications in signals and systems by which we identify a reference signal out of the large number of signals detected by a digital receiver. We use the inverse discrete Fourier transform for the membership and non-membership functions of AIVCPFSs for incoming signals and a reference signal. Thus a method for measuring the resembling values of two signals is provided by which we can identify the reference signal.
A fuzzy set is well-known to have numerous applications in applied sciences, engineering, and computer science. The concept of a fuzzy set [40] is similar to that of probability and has values in the closed interval [0,1]. In fact, because there are no restrictions on choices, this notion is better suited for modeling in all fields of science. The models of real-world problems contain ambiguities that cannot be resolved using crisp sets. Furthermore, fuzzy sets can be applied in medicine, business, and related health sciences.
The concept of fuzzy set (FS) has been utilized in different areas, but the concept of FS has limited applications. Because if a person faced information in the form of truth grade (TG) and false grade (FG), then the concept of FS can not be used in such problems. To overcome this deficiency, Atanassov [7] introduced the notion of intuitionistic FS (IFS). IFS gives the information’s in the form of TG and FG against the value which is taken from the set of attributes with a rule that is the sum duplet is limited to the unit interval. After their successful utilization, certain researchers have employed it in the natural environment of different areas. For example, Atanassov [8] developed the interval-valued IFS and their application’s; Garg and Rani [19] explored similarity measures based on the transformed right-angle tringles among IFSs, Ejegwa and Onyeke [15] intuitionistic fuzzy statistical correlation algorithm, Aydin and Enginoglu [10] proposed interval-valued intuitionistic fuzzy parametrized interval-valued intuitionistic fuzzy soft sets, and Huang et al. [24] developed the complete ranking method for interval-valued IFSs.
Although the IFSs were very potent to deal with uncertain opinions but they fail to perform under the situations when sum of membership and non-memberships exceeds 1. To overcome this deficiency, a Pythagorean FS (PFS) was introduced by Yager [39]. PFS gives the information’s in the form of TG and FG against the value which is taken from the set of attributes with a rule that is the sum of the squares of duplet is limited to the unit interval. Many researchers utilized the PFS in different areas of science. For example, Garg [20] explored interval-valued PFSs and their application’s, Gao et al. [17] developed the quantum Pythagorean fuzzy evidence theory, Zulqarnain et al. [42] initiated the TOPSIS method for Pythagorean fuzzy hyper-soft sets, and Calik [12] initiated the AHP and TOPSIS method for PFSs and discussed their application in green supplier chain management. Wei [38] investigated the multiple attribute decision-making problems with Pythagorean fuzzy information. Peng et al. [31] discussed a novel score function and distance measure for interval-valued Pythagorean fuzzy number. A product ranking method that combines feature–opinion pairs mining and interval-valued Pythagorean fuzzy (IVPF) sets was proposed by Fu in [16].
In many real-life problems, decision-making is considered as a powerful tool to manipulate the data involving imprecise and vague information. To fix the mathematical problems containing more generalized datasets, an emerging model called q-rung orthopair fuzzy soft sets offers a comprehensive framework for a number of multi-attribute decision-making (MADM) situations but this model is not capable to deal effectively with situations having bipolar soft data. Ali et al. [4] introduced a novel hybrid model under the name of q-rung orthopair fuzzy bipolar soft set (q-ROFBSS, henceforth), an efficient bipolar soft generalization of q-rung orthopair fuzzy set model. They [5] introduced the theory of an innovative hybrid model called the fuzzy bipolar soft expert sets, as a natural extension of two existing models (including fuzzy soft expert sets and fuzzy bipolar soft sets). Moreover, Ali et al. [6] proposed a novel hybrid model, namely, the Fermatean fuzzy bipolar soft set (FFBSS, in short) model as a general extension of two powerful existing models, that is, fuzzy bipolar soft set and Pythagorean fuzzy bipolar soft set models. Kamci et al. [25] proposed a new effective tool to address complex multicriteria decision-making problems where all decision-making information are provided by decision makers in both the trapezoidal neutrosophic environment and the bipolar neutrosophic environment. The notion of an interval-valued picture hesitant fuzzy set was proposed by Kamci in [26] and he discussed the application of interval-valued picture hesitant fuzzy set in decision-making problems. Petchimuthu et al. [32] introduced the mean operators and normalized fuzzy weighted mean operators of the fuzzy soft matrices and established two algorithms using these operators to deal multicriteria group decision making (MCGDM) problems. Abraham [1] formulated adaptive nature-inspired computational models combining different knowledge representation schemes, decision-making models, and learning strategies to solve a computational task. Ashtiani et al. [3] proposed the interval-valued fuzzy TOPSIS method for solving MCDM problems.
FSs, IFSs, and PFSs are unable to handle imprecise, inconsistent, and incomplete periodic information. These theories apply to different disciplines of research, but both have one major flaw: the inability to model two-dimensional events. To address this issue, Ramot et al. [33] proposed a complex fuzzy set. The CFS is the extension of a FS, whose range set is extended from closed interval [0, 1] to a disc of radius one in a complex plane. It is defined as follows:
The term γS (x) is called the amplitude term of the grade of membership belongs to [0, 1] and ωS (x) is called the phase term, which is a real-valued function. Thus μS (x) is a complex-valued function that maps each input value to a unique value in a unit circle in a complex plane. The phase term of the CFS model is vital in identifying the features of the model. This term differentiates a CFS model from all other models in the literature. The ability of a CFS to represent two-dimensional phenomena makes it superior for managing data that is ambiguous and intuitive in time-periodic phenomena. Ramot et al. suggested that regular issues or recurring problem phenomena be more appropriately modeled by using the phase component of CFS memberships, such as expressing two-country economics values on each other over time. One of the desired applications of CFSs, according to Dick, is to represent phenomena with comparatively periodic behavior [33]. A CFS is very similar to a Fourier transform; in fact, it is a specific form of the Fourier transform by considering a range of the Fourier transforms to a complex unit disc. The Fourier transform is useful in a variety of domains, including signals and systems, communication, astronomy, geology, and optics. Therefore, like a Fourier transform, a CFS can be used in certain models. In [33], Ramot et al. proposed a CFS-based algorithm for identifying an unknown signal received by a digital receiver with the reference signal. Zhang et al. in [41] modified the method introduced in [33]. Further, Ali and Smarandache in [2] introduced the notion of a neutrosophic complex set and apply it to signal processing. Dai et al. [13] proposed the notion of an interval-valued complex fuzzy set (IVCFS) and developed some series of distance measures between IVCFSs by using Hamming and Euclidean metrics. From the definition of the IVCFS, the principles of interval-valued complex fuzzy logic was developed by Greenfield et al. in [23]. Nasir et al. [30] introduced the innovative concept of the interval valued complex fuzzy relations (IVCFRs) and discussed their applications in medical diagnosis.
Alkouri et al. [9] introduced notion of the CIFS, which is based on two complex functions that represent the membership and non-membership grade of an object. Ma et al. [28] developed a CFS-based method to solve problems having multiple periodic factors. Garg and Rani [18] proposed some aggregation operators for CIFSs and utilized those operators in multiattribute decision makings (MADM). Some distance measures for complex intuitionistic fuzzy soft sets (CIFSSs) are developed by Kumar and Bajaj in [27], whereas the theory of power aggregation operators for CIFSs was proposed by Rani and Garg [34] which was further utilized in MADM. Singh et al. [36] proposed a method for drawing the interval-valued complex lattice and its navigation at user-required complex granules with demonstration. Moreover, Grag and Rani [21] presented the concept of the complex interval-valued intuitionistic fuzzy (CIVIF) set, their algebraic operations and their corresponding aggregation operators. Ullah et al. [37] introduced the concept of complex Pythagorean fuzzy sets. They discussed the very basic concepts of complex Pythagorean fuzzy sets and studied their properties. Moreover, some distance measures for complex Pythagorean fuzzy sets are developed and their properties are investigated. After their successful utilization, certain researchers have employed it in the natural environment of separated areas.
Amplitude interval-valued complex Pythagorean fuzzy set
In this paper, we introduce the concept of AIVCPFS, which is a generalization of CPFSs. The AIVCPFS CP over a non-empty set U is a structure CP = {〈x, ɛCP (x), ΘCP (x) 〉 | x ∈ U}, where and ΘCP (x) are defined as:
where belong to D[0,1] (set of all sub intervals of [0,1])
with and are real-valued functions belong to [0,1] such that Simply, we will denote it as .
Note that the functions and ΘCP (x) denote the membership and non-membership grades of AIVCPFS. The main goals of this manuscript are discussed below.
(i). To propose the novel approach of AIVCPFSs.
(ii). To explore certain set-theoretic operations, basic results and laws of the AIVCPFSs.
(iii). To discuss the applications of AIVCPFSs in signals and systems.
Why we need this new model?
Although, the existing models of fuzzy set theory are highly potent to deal with ambiguous information but their limitations highlighted the need of a superior model which can serve as a vigorous tool to handle obscure information. The FSs, IFSs and PFSs lack an important aspect of the information, i. e., the inability to discuss two-dimensional phenomena. This limitation of the existing model leads to the establishment of new models, namely, CFSs, CIFSs and CPFSs. On the other hand, the modern extension of FSs for two dimensional uncertain data, including CFSs, CIFSs and CPFSs, are not able to to address the relationship among any number of attributes when they are in the form of interval-valued. These facts motivate us to build up a fruitful and practical model which can address the two dimensional information as well as the interaction among any set of parameters when they are in the form of interval-valued.
The AIVCPFSs change the earliest notion of the CPFSs by maintaining that, it is essential to add the complex interval number to the expression of membership function and non-membership function. The addition of complex-valued interval numbers as TG and complex-valued interval numbers as FG made it distinct from CPFSs. In TG and FG, the terms aCP (x) and bCP (x) are any interval numbers, it shows more probability of occurrence, that is, it does not show the exact value of occurrence, but it shows the occurrence in the interval D [0, 1] . So the AIVCPFSs are more faithful than a PFS and CPFS.
The paper is organized as follows. A formal definition, set theoretic operations and laws of the AIVCPFSs are provided in Section 3. Some particular examples and basic results of these novel concepts are also presented in this section. The distance functions and α-equalities of AIVCPFSs are discussed in section 4. Section 5 is concerned with applications of AIVCPFSs. In section 6, the comparison analysis of the application is provided. In section 7, we discussed the conclusion of this manuscript.
Interval valued fuzzy sets (i-v fuzzy sets)
Now we will recall the notions of i-v fuzzy sets.
An interval number is where 0 ≤ a- ≤ a+ ≤ 1 . Let D [0, 1] denotes the family of all closed subintervals of [0, 1] , i.e.,
Let u = [u-, u+] and v = [v-, v+] be two elements of D [0, 1] . Then we define the operations " ≤ " , " ≥ " , " = " "scalar addition" and "scalar multiplication" as follows
(i) . u ≤ v if and only if u- ≤ v- and u+ ≤ v+ .
(ii) . u ≥ v if and only if u- ≥ v- and u+ ≥ v+ .
(iii) . u = v if and only if u- = v- and u+ = v+ .
(v) . α • [u-, u+] = [α • u-, α • u+] for any scalar α ∈ [0, 1] , where denotes usual multiplication.
Definition 1. [22] An i-v fuzzy set A defined on the X is given by
where D [0, 1] denote the family of all closed intervals contained in the interval [0, 1] .
The membership function μA (x) can be written as:
which is a subset of [0, 1] .
Definition 2. [22] Let A and B be two i-v fuzzy sets on X, and μA (x) and μB (x) their grade values, respectively. The union of these two i-v fuzzy sets A and B, denoted A ∪ B, is specified by a function
where
Definition 3. [22] Let A and B be two i-v fuzzy sets on X, and μA (x) and μB (x) their grade values, respectively. The intersection of these two i-v fuzzy sets A and B, denoted A ∩ B, is specified by a function
where
Definition 4. [23] An interval-valued complex fuzzy set over a universe of discourse X is defined by a membership function
where rA (x) ∈ I[0,1] is the interval-valued membership magnitude, and ϱA (x) is the membership phase.
Note that I[0,1] is the set of all closed subintervals of [0, 1] .
Definition 5. [37] A CPFS is defined as where and such that and provided that 0 ≤ |z1|2 + |z2|2 ≤ 1 or and satisfying the conditions: and
In this section, we introduce a new concept of the AIVCPFS which is the generalization of CPFS.
Definition 6. The AIVCPFS CP of a non-empty set U is a structure CP = {〈x, ΘCP (x) 〉 | x ∈ U}, where and ΘCP (x) are defined as:
where aCP (x) = [a-, a+] and bCP (x) = [b-, b+] belong to D [0, 1](set of all sub intervals of [0, 1]) with 0 ≤ (a-) 2 + (b-) 2 ≤ 1, 0 ≤ (a+) 2 + (b+) 2 ≤ 1 and uCP (x), vCP (x) are real valued functions belong to [0, 1] such that 0 ≤ (uCP) 2 + (vCP) 2 ≤ 1 . Simply we will denote it as CP = 〈ɛCP, ΘCP〉.
Note that the function and ΘCP (x), denotes the membership and non-membership grades of the amplitude interval-valued complex Pythagorean fuzzy sets.
Some operations on amplitude interval-valued complex pythagorean Fuzzy Sets
Definition 7. Let CP1 and CP2 be two AIVCPFSs on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉 and CP2 = 〈aCP2 (x) ei2πuCP2(x), bCP2 (x) ei2πvCP2(x)〉 their grade values, respectively. The union of these two AIVCPFSs CP1 and CP2, denoted CP1 ∪ CP2, is specified by a function
Example 1. Let
then
Definition 8. Let CP1 and CP2 be two AIVCPFSs on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉 and CP2 = 〈aCP2 (x) ei2πuCP2(x), bCP2 (x) ei2πvCP2(x)〉 their grade values, respectively. The intersection of these two AIVCPFSs CP1 and CP2, denoted CP1 ∪ CP2, is specified by a function
Example 2. Consider CP1 and CP2 in Example 1. Then, the intersection is given as:
Definition 9. Let CP1 be the AIVCPFS on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉 its grade value. The amplitude interval-valued complex Pythagorean fuzzy complement of CP1, denoted is specified by a function
Example 3. Consider CP1 in Example 1. Then, the complement of CP1 is given as:
Definition 10. Let CP1 and CP2 be two AIVCPFSs on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉 and CP2 = 〈aCP2 (x) ei2πuCP2(x), bCP2 (x) ei2πvCP2(x)〉 their grade values, respectively. The product of these two AIVCPFSs CP1 and CP2, denoted CP1 • CP2, is specified by a function
Example 4. Consider CP1 and CP2 in Example 1. Then, the product is
Definition 11. Let CP1 and CP2 be two AIVCPFSs on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉 and CP2 = 〈aCP2 (x) ei2πuCP2(x), bCP2 (x) ei2πvCP2(x)〉 their grade values, respectively. The simple difference of these two AIVCPFSs CP1 and CP2, denoted CP1 \ CP2, is specified by a function
where " c " represents the complement of the AIVCPFSs.
Example 5. Consider CP1 and CP2 in Example 1. Then, the simple difference is
Definition 12. Let CP1 and CP2 be two AIVCPFSs on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉 and CP2 = 〈aCP2 (x) ei2πuCP2(x), bCP2 (x) ei2πvCP2(x)〉 their grade values, respectively. The symmetric difference of these two AIVCPFSs CP1 and CP2, denoted CP1ΔCP2, is specified by a function
where " c " represents the complement of the AIVCPFSs.
Proposition 1.The AIVCPFSs CP1 and CP2 over a crisp set U, satisfy De-Morgan Laws.
Proof. (i) . Let CP1 and CP2 be two AIVCPFSs on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉, and CP2 = 〈aCP2 (x) ei2πuCP2(x), bCP2 (x) ei2πvCP2(x)〉 their membership functions, respectively. We suppose Also assume that
and
∀x ∈ U . Thus
Now
Also
Therefore from (1) and (2), we have
(ii) .
It is easy to prove. □
Proposition 2.The AIVCPFS CP1, satisfy involution law, that is,
Proof. Let CP1 be the AIVCPFS on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉, is their membership function then,
Proposition 3.The union and intersection of any finite number of AIVCPFSs are always an AIVCPFS.
Proof. Let CP1, CP2, . . . , CPn be any n AIVCPFSs and
denote their membership functions, respectively.
Let Also, let
∀x ∈ U . Now
which is also an AIVCPFS.
(ii) . It is easy to prove.□
Proposition 4.The AIVCPFSs CP1 and CP2 over a crisp set U, satisfy the equivalence formula
Proof. It is easy to prove. □
Theorem 1.If CP1 and CP2 are AIVCPFSs on U then,
(i) . (Equivalence Formula)
(ii) . (Symmetrical Difference Formula).
Theorem 2.If CP1, CP2 and CP3 are three AIVCPFSs on U then,
Distance functions and α-equalities of amplitude interval-valued complex pythagorean fuzzy sets
In this section, we introduced a distance function and α-equalities using AIVCPFSs.
Definition 13. A distance function of the AIVCPFSs is a function dCP : CP (U) × CP (U) ⟶ [0, 1], such that for any AIVCPFSs CP1, CP2 and CP3, the following axioms hold
Let CP1 and CP2 be two AIVCPFSs on U, and CP1 = 〈aCP1 (x) ei2πuCP1(x), bCP1 (x) ei2πvCP1(x)〉, and CP2 = 〈aCP2 (x) ei2πuCP2(x), bCP2 (x) ei2πvCP2(x)〉 their membership functions, respectively. The distance function of AIVCPFSs is specified by a function
where l denotes the length.
Example 6. Let CP1 and CP2 be two AIVCPFSs on U = {ϰ1, ϰ2, ϰ3} then,
It is easy to see that
Thus
Using the above definitions we arrived to a crucial theorem of this section with a rather simple proof.
Theorem 3.The function dCP is a distance function of AIVCPFSs on U.
The conditions (2) and (3) are straight forward. To prove (4), we have
Definition 14. Let CP1 and CP2 be two AIVCPFSs on U . Then, CP1 and CP2 are said to be α-equal (CP1 = αCP2) if and only if d (CP1, CP2) ≤1 - α or α ≤ 1 - d (CP1, CP2).
(vi) (α1 ∗ α2) ∗ α3 = α1 ∗ (α1 ∗ α2), for all α1, α2, α3 ∈ [0, 1].
Proof. It is easy. □
Theorem 5.If CP1 and CP2 be two AIVCPFSs and dCP is a distance function then,
(i) CP1 = 0CP2 ⇔ dCP (CP1, CP2) ≤1 .
(ii) CP1 = 1CP2 ⇔ dCP (CP1, CP2) =0 .
(iii) CP1 = αCP2 ⇔ CP2 = αCP1 .
(vi) CP1 = α1CP2 and α2 ≤ α1 ⇒ CP1 = α2CP2 .
(v) If CP1 = αiCP2 then,
(vi) For all CP1, CP2 there exists a unique α such that CP1 = αCP2 and if CP1 = α′CP2 then, α′ ≤ α.
Proof. (i) . Suppose that CP1 = 0CP2 then dCP (CP1, CP2) ≤1 - 0 and hence dCP (CP1, CP2) ≤1 .
Conversely suppose that dCP (CP1, CP2) ≤1 . It can be written as dCP (CP1, CP2) ≤1 - 0 . This implies that CP1 = 0CP2 .
(ii) . Suppose that CP1 = 1CP2 then dCP (CP1, CP2) ≤1 - 1 ⇒ dCP (CP1, CP2) ≤0 . But dCP is a distance function so dCP (CP1, CP2) ≥0 . Thus dCP (CP1, CP2) =0
Conversely suppose that dCP (CP1, CP2) =0 . It can be written as dCP (CP1, CP2) =1 - 1 . This implies that CP1 = 1CP2 .
(iii) . Assume that CP1 = αCP2 then by definition of α-equal dCP (CP1, CP2) =1 - α . Since, dCP is a distance function so dCP (CP1, CP2) = dCP (CP2, CP1) and hence, dCP (CP2, CP1) =1 - α . Thus CP2 = αCP1 .
(iv) . Let CP1 = α1CP2 then, by definition of α-equal dCP (CP1, CP2) =1 - α1 . Since, for α2 ≤ α1 the inequality also holds, that is, dCP (CP1, CP2) =1 - α2. Thus CP1 = α2CP2 .
(v) Let CP1 = αiCP2 then,
Thus it is easy to see that
Therefore,
Hence .
(vi) Let α = 1 - dCP (CP1, CP2) . Then, CP1 = αCP2. If CP1 = α′CP2, we have 1 - α = dCP (CP1, CP2) ≤1 - α′ . Consequently α′ ≤ α . Now suppose there exists α1 and α2 which simultaneously satisfy the required properties, then α1 ≤ α2 and α2 ≤ α1 . This implies that α1 = α2 and hence α is unique. □
Theorem 6.If CP1 = α1CP2 and CP2 = α2CP3, then CP1 = αCP3, where α = α1 ∗ α2 .
Proof. Since CP1 = α1CP2 and CP2 = α2CP3, we have
and
Therefore,
and
Consequently, we have
and dCP (CP1, CP3) form definition 10. Therefore
Thus, CP1 = αCP3 . □
Theorem 7.If CP1 = αCP2, then
Proof. Consider
Therefore, □
Application in signal processing
Now, we will discuss a real-life application of an AIVCPFS in signals and systems. Since, a radar system detects the signals of other aircraft, ships, or other objects and the speed and direction in which they travel by sending out a pulse of high-frequency electromagnetic waves. Like a radar system, the AIVCPFSs model are used to identify a particular signal of interest out of a large number of signals received by a digital receiver. The AIVCPFS has two major parts that are complex-valued interval number as truth membership function and complex-valued interval number as falsehood membership function. In truth membership function and false membership function, the terms aCP (x) and bCP (x) are any interval numbers, it shows more probability of occurrence and non-occurrence, respectively, that is, it does not show the exact value of occurrence and non-occurrence, but it shows the probability of occurrence and non occurence in the interval D [0, 1] . So, the AIVCPFSs model are more significant in case of rain and other forms of precipitation which can cause echo signals that mask the desired target echoes.
We discuss the following definitions of inverse discrete Fourier transform, Discrete and inverse discrete Fourier transform matrix.
Definition 15. [35] The DFT for {x′ (N) :1 ≤ N ≤ N - 1} given by as a matrix in product form.
but the IDFT is given by
Definition 16. The Mth inverse discrete Fourier transform (IDFT) coefficient of a length M sequence {x (M)} is defined as
where x′ (q) has different values [35].
Since the Fourier transform applies only to continuous signals of time, analyzing discrete signals in the frequency domain requires that we first modify the Fourier transform equations so they are structurally compatible with the digital samples of a continuous signal. The IDFT is the discrete-time version of the inverse Fourier transform. As for the Fourier transform and inverse Fourier transform, the DFT and IDFT represent a Fourier transform pair in the discrete domain. The DFT allows one to convert a set of digital time samples to its frequency domain representation. In contrast, the IDFT can be used to invert the DFT samples, allowing one to reconstruct the signal samples xn (m) directly from its frequency domain form, x′ (q). These two equations are thus interchangeable, since either conveys all of the signal information.
We take a particular case that is x′ (q) = [a-, a+] belongs to the set D [0, 1] (set of all sub intervals of [0, 1]).
In the following, we develop an algorithm using the AIVCPFSs in signals and systems for the identification of a particular signal received by a particular receiver.
Let m be different electromagnetic signals, have been received by a particular receiver. Each of these signals is noted at M different times (samples). Let Un (m) be the n - th (0 ≤ m ≤ M - 1) signal. The inverse discrete Fourier transform of Un (m) is:
In equation (1) , shows the membership function of amplitude interval-valued complex Pythagorean fuzzy sets and shows the non-membership function of AIVCPFSs.
Note that, if the electromagnetic signals are reflected especially by materials of considerable electrical conductivity such as most metals, seawater, wet ground, etc. In such a case, the membership function shows the degree of occurrence of the receiving signals and the non-membership function shows the degree of non-occurrence receiving signals.
We use the AIVCPFSs in signals and systems using new kinds of matrices to identify a particular signal out of large signals detected by a digital receiver. For this, we have a reference signal r∗ . This reference signal r∗ is noted M times (samples). The IDFT of the reference signal r∗ is:
The inverse discrete Fourier transform of the signal and r∗ (m) represent discrete signals in discrete-time version.
To compare the similarity of the received signals with the reference signal, we apply the following method.
Algorithm
Step 1.
Compute the time domain samples of the signal For this, we first expande for q = 0, 1, 2, . . . , M - 1, we get
From equation (3) we get M-samples by putting m = 0, 1, 2, 3, . . . , M - 1 .
For m = 0, we have
Where denotes the 1st sample of the signal.
For m = 1, we have
Where denotes the second sample of the signal.
For m = 2, we have
Equation (6) , shows the third sample of the signal.
Continuing this process, for m = M - 1, we have
A similar argument repeats for the reference signal r∗ (m), we get the following M-samples of the reference signal r∗ by putting m = 0, 1, 2, 3, . . . , M - 1 .
Step 2.
To compute the original form of the inverse discrete Fourier transform of the n-th signal and the reference signal, we find the arithmetic means of the length of amplitude terms, that is,
for q = 0, 1, 2, . . . , M - 1, and l denotes the length of interval.
Similarly, compute the arithmetic mean of the phase terms then, equation (4)-(10), implies that
Similarly,
Now equations (11) - (18) show the original form of the one-dimensional inverse discrete Fourier transform.
Note that these samples are also time domain.
Step 3.
Now we develop the matrix form for these M time domain samples of the signal um (M) and the reference signal Θ′ (M) using definition [35], that is we have
and
In the above equation, the first matrix on the right-hand side is formed from the values of phase term called phase matrix, while the second matrix is formed from the values of amplitude term is called amplitude matrix and M denote the number of samples of the signal.
Step 4.
Multiply these two matrices and dividing by the number of samples M of the signal. We get all the values in the disc of radius one in a complex plane. As the order does not hold for complex numbers, so we take absolute of these M time domain samples of the signal and the reference signal r∗ (m), that is
These two matrices are called absolute value matrix.
Step 5.
Now we take the maximum value from the absolute value matrix of the time domain signal and the time domain reference signal r∗ (m) . If these two values are nearly the same, then the signal identifies a reference signal.
Example 7. Assume that four different electromagnetic signals, , , and from four different aircrafts A1, A2, A3, and A4, have been received by a radar system. Each of these time domain signals is sampled four times. Let r∗ (m) be the reference signal. The inverse discrete Fourier transform of the signal m, n = 0, 1, 2, 3 is:
where
Also
where
Following steps 1 and 2 in the above algorithms, we get
and
First of all, we find the values of the sample of reference signal r∗ (m) . For this, we have
Now the absolute value matrix of the reference signal is
The maximum value is 0.53 .
Now for the signal of A1 aircraft; m = 0, 1, 2, 3, we have
Now the absolute value matrix of the signal is
Here the maximum value is
Now for the signal of A2 aircraft; m = 0, 1, 2, 3, we have
Now
Now the absolute value matrix of the signal is
Here the maximum value is .
Now for the signal of A3 aircraft; m = 0, 1, 2, 3, we have
Now the absolute value matrix of the signal is
Here the maximum value is .
Now for the signal of A4 aircraft; m = 0, 1, 2, 3, we have
Now the absolute value matrix of the signal is
Here the maximum value is .
Now from the following table of maximum values
Since the absolute value of is nearly same the absolute value of the reference signal. Thus the time domain signal identifies as a reference signal.
Thus, we conclude that the other three aircraft A1, A2, and A4 are from other countries except for A3.
Comparison
The CFS has many applications, particularly in signal processing and image restoration as it represents the particular form of a Fourier series. Here we presented the application of AIVCPFSs in signals and systems. In this practical application, one of the main issues is that how to choose a suitable model. We examined this idea in-depth and used the AIVCPFSs in signals and systems by introducing an algorithm using the new kind of matrix. In this application, we identified a reference signal out of large interest signals detected by a digital receiver.
Ramot et al. in [33] introduced an algorithm to identify the unknown signal received by the digital receiver with reference signal R. The example is demonstrated as following:
Assume L different electromagnetic or speech signals, S1 (t), S2 (t), . . . , SL (t), have been detected and sampled by a digital receiver. Each received signal is sampled N times. Let Sl (k) denote the k - th sample for the l - th signal. Now, the (discrete) Fourier transforms of the received signals may be obtained, and each signal represented as the sum of its Fourier components
where with Al,n, αl,n are real-valued.
The Fourier coefficient of the reference signal R be CR,n, where 1 ≤ n ≤ N . Thus
where with AR,n, αR,n are real-valued.
Calculating a measure of the similarity between two signals is possible by comparing their Fourier transforms (see the method in [33]), we get
where
Note that μSl,R denotes the total grade of similarity and μSl,R (n) is a complex grade of membership.
If threshold frequency is 0.8 then, |μS3,R| > μthreshold . Thus S3 (k) is the reference signal.
In this algorithm, the authors actually tried to find the highest resemblance with the known signal R. To determine the relationship among any number of attributes when they are in the form of interval-valued, then the existing theory has been failed. Moreover, the membership functions and non-membership functions play a vital role in complicated problems. In our proposed method, we used membership functions and non-membership functions to develop an algorithm that gives the resemblance value of unknown signals among all unknown signals received by the digital receiver. We compared it with the known signal R and observed that both the values are nearly the same. Thus, we identified one unknown signal as a reference signal R among the several signals detected by the receiver. The model presented in this paper for identifying a reference signal using AIVCPFSs is a unique method than the methods previously developed. Here, we used a discrete Fourier transform matrix to develop an algorithm for further use in signal processing. Moreover, through this model, we determined the fuzzy values of each signal separately, that detected by a digital receiver. In fact, we compared these values of different signals and the reference signal and we easily identified the reference signal. Moreover, it is seen clearly that how the detected signals matched with the reference signal. However our designed model is different from other models present in literature and is not a perfect one, it stuck with a deficiency of theoretical support. The concept of the matrix for AIVCPSs may be useful for applications. Therefore it will be significant for future work.
Conclusions
A new type of set, the AIVCPFS, was presented in this paper. The principle of AIVCPFS is a valuable procedure to manage inconsistent and awkward information genuine life troubles. The principle of AIVCPFS is a mixture of the two separated theories such as the CFS and interval-valued Pythagorean fuzzy set which covers the TG and FG in the form of the complex number whose real parts are the sub-interval of the unit interval. A comprehensive study of the mathematical properties of AIVCPFSs was presented. The study began by considering the basic set-theoretic operations of complement, union, intersection, simple product, simple difference, and symmetric difference. The distance function of AIVCPFSs was also considered. We presented some basic results and examples of these operations. Moreover, we have used an AIVCPFS in signals and systems. We have introduced a new method to develop a new kind of matrix using an AIVCPFS, through which we identified a reference signal among several signals detected by the digital receiver. In the future, the same method can be used for continuous data of signals using a continuous Fourier transform. This method can also be applied for the identification of signals in geology, communication, astronomy, and optics. Moreover, this work and further study of CFSs will give a new direction of applications in different fields of science and engineering.
Conflict of Interest
The authors declare that they have no conflict of interests.
Footnotes
Acknowledgments
This work is financially supported by the Higher Education Commission of Pakistan (Grant No: 7750/Federal/NRPU/R&D/HEC/2017).
Authorship Contributions
All authors contributed equally.
References
1.
AbrahamA., Hybrid approaches for approximate reasoning, Journal of Intelligent & Fuzzy Systems23(2, 3) (2012), 41–42.
AshtianiB., HaghighiradF., MakuiA., Ali MontazerG., Extension of fuzzy TOPSIS method based on interval-valued fuzzy sets, Applied Soft Computing9(2) (2009), 457–461.
4.
AliG., AlolaiyanH., PamuμcarD.,
AsifM. and
LateefN., A Novel MADM Framework under q-Rung Orthopair Fuzzy Bipolar Soft Sets, Mathematics9(17) (2021), 2163.
5.
AliG., MuhiuddinG., AdeelA., Zain Ul AbidinM., Ranking effectiveness of COVID-19 tests Using fuzzy bipolar soft expert sets, Mathematical Problems in Engineering (2021).
AydinT., EnginoğluS., Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets and their application in decision-making, Journal of Ambient Intelligence and Humanized Computing12(1) (2021), 1541–1558.
11.
CaiY.K., δ-Equalities of fuzzy sets, Fuzzy Sets Syst76(1) (1995), 97–112.
12.
ÇalikA., A novel Pythagorean fuzzy AHP and fuzzy TOPSIS methodology for green supplier selection in the Industry 4.0 era, Soft Computing25(3) (2021), 2253–2265.
13.
DaiS., BiL., HuB., Distance measures between the interval-valued complex fuzzy sets, Mathematics7(6) (2019), 549.
EjegwaP.A., OnyekeI.C., Intuitionistic fuzzy statistical correlation algorithm with applications to multicriteria-based decision-making processes, International Journal of Intelligent Systems36(3) (2021), 1386–1407.
16.
FuX., OuyangT., YangZ., LiuS., A product ranking method combining the features, Opinion pairs mining and interval-valued Pythagorean fuzzy sets, Applied Soft Computing97 (2020), 106803.
17.
GaoX., PanL., DengY., Quantum pythagorean fuzzy evidence theory (qpfet): A negation of quantum mass function view, IEEE Transactions on Fuzzy Systems (2021).
18.
GargH., RaniD., Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arabian Journal for Science and Engineering44(3) (2019), 2679–2698.
19.
GargH., RaniD., Novel similarity measure based on the transformed right-angled triangles between intuitionistic fuzzy sets and its applications, Cognitive Computation13(2) (2021), 447–465.
20.
GargH., A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent & Fuzzy Systems31(1) (2016), 529–540.
21.
GargH., RaniD., Complex interval-valued intuitionistic fuzzy sets and their aggregation operators, Fundamenta Informaticae164(1) (2019), 61–101.
22.
GorzałczanyM.B., A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems21(1) (1987), 1–17.
23.
GreenfieldS., ChiclanaF., DickS., Interval-valued complex fuzzy logic, In 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), (2016), 2014–2019.
24.
HuangW., ZhangF., XuS., A complete ranking method for interval-valued intuitionistic fuzzy numbers and its applications to multicriteria decision making, Soft Computing25(3) (2021), 2513–2520.
25.
KamaciH., GargH., PetchimuthuS., Bipolar trapezoidal neutrosophic sets and their Dombi operators with applications in multicriteria decision making, Soft Computing (2021), 1–24.
26.
KamaciH., PetchimuthuS., AkçetinE., Dynamic aggregation operators and Einstein operations based on interval-valued picture hesitant fuzzy information and their applications in multi-period decision making, Computational and Applied Mathematics40(4) (2021), 1–52.
27.
KumarT., BajajR.K., On complex intuitionistic fuzzy soft sets with distance measures and entropies, Journal of Mathematics (2014).
28.
MaJ., ZhangG., LuJ., A method for multiple periodic factor prediction problems using complex fuzzy sets, IEEE Trans Fuzzy Syst20(1) (2012), 32–45.
29.
MaiersJ., SherifY.S., Applications of fuzzy set theory, IEEE Transactions on Systems, Man and Cybernetics15(1) (1985), 175–189.
30.
NasirA., JanN., GumaeiA., KhanS.U., Medical diagnosis and life span of sufferer using interval valued complex fuzzy relations, IEEE Access9 (2021), 93764–93780.
31.
PengX., LiW., Algorithms for interval-valued pythagorean fuzzy sets in emergency decision making based on multiparametric similarity measures and WDBA, Ieee Access7 (2019), 7419–7441.
32.
PetchimuthuS., GargH., KamaciH., AtagünA.O., The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM, Computational and Applied Mathematics39(2) (2020), 1–32.
RaniD., GargH., Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision–making, Expert Syst35(6) (2018), 12325.
35.
SelesnickI.W., SchullerG., The Discrete Fourier Transform, 2nd chapter of the book The transform and data compression Handbook, editted by K.R. Rao and P. C. Yip, CRC Press, Boca Raton, (2001).
36.
SinghP.K., SelvachandranG., KumarC.A., Interval-valued complex fuzzy concept lattice and its granular decomposition, In Recent developments in machine learning and data analytics, Springer, Singapore, (2019), 275–283.
37.
UllahK., MahmoodT., AliZ., JanN., On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition, Complex & Intelligent Systems6(1) (2020), 15–27.
38.
WeiG., Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making, Journal of Intelligent & Fuzzy Systems33(4) (2017), 2119–2132.
39.
YagerR.R., Pythagorean fuzzy subsets, In 2013 joint
IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS) (pp. 57–61). IEEE..
ZhangG., DillonS.T., CaiY.K., MaJ., LuJ., Operation properties and δ-equalities of complex fuzzy sets, Int. J. Approx. Reason.50 (2009), 1227–1249.
42.
ZulqarnainR.M., SiddiqueI., JaradF., AliR., AbdeljawadT., Development of TOPSIS Technique under Pythagorean Fuzzy Hypersoft Environment Based on Correlation Coefficient and Its Application towards the Selection of Antivirus Mask in COVID-19 Pandemic, Complexity (2021).