Amplitude interval-valued complex Pythagorean fuzzy sets with applications in signals processing

Abstract

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.

Keywords

Amplitude interval-valued complex Pythagorean fuzzy set Complex fuzzy set Fuzzy set Inverse discrete Fourier transform

1 Introduction

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: $μ_{S} (x) = γ_{S} (x) e^{i ω_{S} (x)} .$

The term γ_S (x) is called the amplitude term of the grade of membership belongs to [0, 1] and ω_S (x) is called the phase term, which is a real-valued function. Thus μ_S (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 C_P over a non-empty set U is a structure C_P = {〈x, ɛ_{C
_P} (x), Θ_{C
_P} (x) 〉 | x ∈ U}, where $ɛ_{C_{P}} (x)$ and Θ_{C
_P} (x) are defined as: $ϵ_{C_{P}} (x) = a_{C_{P}} (x) e^{i 2 π u_{C_{P}} (x)} and Θ_{C_{P}} (x) {=b}_{C_{P}} (x) e^{i2 π v_{C_{P}} (x)}$ where $i = \sqrt{- 1}, a_{C_{P}} (x) = [a^{-}, a^{+}] a n d b_{C_{P}} (x) = [b^{-}, b^{+}]$ belong to D[0,1] (set of all sub intervals of [0,1]) with $0 \leq {(a^{-})}^{2} + {(b^{-})}^{2} \leq 1, 0 \leq {(a^{+})}^{2} + {(b^{+})}^{2} \leq 1$ and $u_{C_{P}} (x), v_{C_{P}} (x)$ are real-valued functions belong to [0,1] such that $0 \leq {(u_{C_{P}})}^{2} + {(v_{C_{P}})}^{2} \leq 1.$ Simply, we will denote it as $C_{P} = 〈 ϵ_{C_{P}}, Θ_{C_{P}} 〉$ .

Note that the functions $ɛ_{C_{P}} (x)$ and Θ_{C
_P} (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 a_{C
_P} (x) and b_{C
_P} (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.

2 Interval valued fuzzy sets (i-v fuzzy sets)

Now we will recall the notions of i-v fuzzy sets.

An interval number is $\overset{˘}{a} = [a^{-}, a^{+}],$ where 0 ≤ a^- ≤ a⁺ ≤ 1 . Let D [0, 1] denotes the family of all closed subintervals of [0, 1] , i.e., $D [0, 1] = {u = [u^{-}, u^{+}] : 0 \leq u^{-} \leq u^{+} \leq 1} .$

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⁺ .

(iv) . α + [u^-, u⁺] = [α + u^-, α + u⁺] for any scalar α ∈ [0, 1] .

(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 $A = {(x, μ_{A} (x))}, μ_{A} (x) : X ⟶ D [0, 1]},$ where D [0, 1] denote the family of all closed intervals contained in the interval [0, 1] .

The membership function μ_A (x) can be written as: $μ_{A} (x) = [μ_{A}^{L} (x), μ_{A}^{U} (x)],$ which is a subset of [0, 1] .

Definition 2. [22] Let A and B be two i-v fuzzy sets on X, and μ_A (x) and μ_B (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 $\begin{matrix} A \cup B & = {(x, μ_{A \cup B}^{-} (x))} \\ μ_{A \cup B}^{-} (x) & = [μ_{A \cup B}^{L} (x), μ_{A \cup B}^{U} (x)], x \in X, \end{matrix}$

where $\begin{matrix} μ_{A \cup B}^{L} (x) & = max [μ_{A}^{L} (x), μ_{B}^{L} (x)] \\ μ_{A \cup B}^{U} (x) & = max [μ_{A}^{U} (x), μ_{B}^{U} (x)] . \end{matrix}$

Definition 3. [22] Let A and B be two i-v fuzzy sets on X, and μ_A (x) and μ_B (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 $\begin{matrix} A \cap B & = {(x, μ_{A \cap B}^{-} (x))} \\ μ_{A \cap B}^{-} (x) & = [μ_{A \cap B}^{L} (x), μ_{A \cap B}^{U} (x)], x \in X, \end{matrix}$

where $\begin{matrix} μ_{A \cap B}^{L} (x) & = min [μ_{A}^{L} (x), μ_{B}^{L} (x)] \\ μ_{A \cap B}^{U} (x) & = min [μ_{A}^{U} (x), μ_{B}^{U} (x)] . \end{matrix}$

Definition 4. [23] An interval-valued complex fuzzy set over a universe of discourse X is defined by a membership function $μ_{A} (x) = r_{A} (x) . e^{i ϱ_{A} (x)},$ where r_A (x) ∈ I^[0,1] is the interval-valued membership magnitude, $i = \sqrt{- 1}$ and ϱ_A (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 $ℂ$ is defined as $ℂ = {(x, T_{ℂ} (x), F_{ℂ} (x) / x \in X},$ where $T_{ℂ} : U ⟶ {z_{1} :$ $z_{1} \in ℂ, | z_{1} | \leq 1}$ and $F_{ℂ} : U ⟶ {z_{2} :$ $z_{2} \in ℂ, | z_{2} | \leq 1},$ such that $T_{ℂ} = z_{1} = a_{1} + {ib}_{1}$ and $F_{ℂ} = z_{2} = a_{2} + {ib}_{2}$ provided that 0 ≤ |z₁|² + |z₂|² ≤ 1 or $T_{ℂ} (x) = ħ_{ℂ} (x) . e^{i 2 π ϖ_{ℂ} (x)}$ and $F_{ℂ} (x) = ℓ_{ℂ} (x) . e^{i 2 π ϑ_{ℂ} (x)}$ satisfying the conditions: $0 \leq (ħ_{ℂ} (x))^{2} + (ℓ_{ℂ} (x))^{2} \leq 1$ and $0 \leq (ϖ_{ℂ} (x))^{2} + (ϑ_{ℂ} (x))^{2} \leq 1 .$

3 Amplitude interval-valued complex pythagorean fuzzy Sets

In this section, we introduce a new concept of the AIVCPFS which is the generalization of CPFS.

Definition 6. The AIVCPFS C_P of a non-empty set U is a structure C_P = {〈x, $ɛ_{C_{P}} (x),$ Θ_{C
_P} (x) 〉 | x ∈ U}, where $ɛ_{C_{P}} (x)$ and Θ_{C
_P} (x) are defined as: $ɛ_{C_{P}} (x) = a_{C_{P}} (x) e^{i 2 π u_{C_{P}} (x)} Θ_{C_{P}} (x) = b_{C_{P}} (x) e^{i 2 π v_{C_{P}} (x)}$ where $i = \sqrt{- 1},$ a_{C
_P} (x) = [a^-, a⁺] and b_{C
_P} (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 u_{C
_P} (x), v_{C
_P} (x) are real valued functions belong to [0, 1] such that 0 ≤ (u_{C
_P}) ² + (v_{C
_P}) ² ≤ 1 . Simply we will denote it as C_P = 〈ɛ_{C
_P}, Θ_{C
_P}〉.

Note that the function $ɛ_{C_{P}} (x)$ and Θ_{C
_P} (x), denotes the membership and non-membership grades of the amplitude interval-valued complex Pythagorean fuzzy sets.

3.1 Some operations on amplitude interval-valued complex pythagorean Fuzzy Sets

Definition 7. Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉 and C_{P
₂} = 〈a_{C
_{P
₂}} (x) e^{i2πu_{C
_P
₂}(x)}, b_{C
_{P
₂}} (x) e^{i2πv_{C
_P
₂}(x)}〉 their grade values, respectively. The union of these two AIVCPFSs C_{P
₁} and C_{P
₂}, denoted C_{P
₁} ∪ C_{P
₂}, is specified by a function $\begin{matrix} C_{P_{1}} \cup C_{P_{2}} \\ = 〈 a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)}, b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \cup \\ 〈 a_{C_{P_{2}}} (x) e^{i 2 π u_{C_{P_{2}}} (x)}, b_{C_{P_{2}}} (x) e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ = 〈 [a_{1}^{-}, a_{1}^{+}] e^{i 2 π u_{C_{P_{1}}} (x)}, [b_{1}^{-}, b_{1}^{+}] e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \cup \\ 〈 [a_{2}^{-}, a_{2}^{+}] e^{i 2 π u_{C_{P_{2}}} (x)}, [b_{2}^{-}, a_{2}^{+}] e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ = 〈 \begin{matrix} [c] c [max (a_{1}^{-}, a_{2}^{-}), max (a_{1}^{+}, a_{2}^{+})] e^{i max {2 π u_{C_{P_{1}}} (x), 2 π u_{C_{P_{2}}} (x)}, \\ [min (b_{1}^{-}, b_{2}^{-}), min (b_{1}^{+}, b_{2}^{+})] e^{i min {2 π v_{C_{P_{1}}} (x), 2 π v_{C_{P_{2}}} (x)} \end{matrix} 〉 \end{matrix}$

Example 1. Let $\begin{matrix} C_{P_{1}} & = 〈 \begin{matrix} [c] c (a, [0.1, 0.3] e^{i 2 π (0.5)}, [0.4, 0.9] e^{i 2 π (0)}), \\ (b, [0.4, 0.6] e^{i 2 π (0.3)}, [0.5, 0.8] e^{i 2 π (0.5)}), \\ (c, [0.2, 0.7] e^{i 2 π (0.1)}, [0.4, 0.5] e^{i 2 π (0.25)}) \end{matrix} 〉, \\ C_{P_{2}} & = 〈 \begin{matrix} [c] c (a, [0.3, 0.6] e^{i 2 π (0)}, [0.5, 0.8] e^{i 2 π (0.5)}), \\ (b, [0.1, 0.3] e^{i 2 π (0.25)}, [0.6, 0.7] e^{i 2 π (0.5)}), \\ (c, [0.3, 0.6] e^{i 2 π (0.125)}, [0.4, 0.7] e^{i 2 π (0.1)}) . \end{matrix} 〉, \end{matrix}$

then $C_{P_{1}} \cup C_{P_{2}} = 〈 \begin{matrix} [c] c (a, [0.3, 0.6] e^{i 2 π (0.5)}, [0.4, 0.8] e^{i 2 π (0)}), \\ (b, [0.4, 0.6] e^{i 2 π (0.3)}, [0.5, 0.7] e^{i 2 π (0.5)}), \\ (c, [0.3, 0.7] e^{i 2 π (0.125)}, [0.4, 0.5] e^{i 2 π (0.1)}) . \end{matrix} 〉 .$

Definition 8. Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉 and C_{P
₂} = 〈a_{C
_{P
₂}} (x) e^{i2πu_{C
_P
₂}(x)}, b_{C
_{P
₂}} (x) e^{i2πv_{C
_P
₂}(x)}〉 their grade values, respectively. The intersection of these two AIVCPFSs C_{P
₁} and C_{P
₂}, denoted C_{P
₁} ∪ C_{P
₂}, is specified by a function $\begin{matrix} C_{P_{1}} \cap C_{P_{2}} \\ = 〈 a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)}, b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \cap \\ 〈 a_{C_{P_{2}}} (x) e^{i 2 π u_{C_{P_{2}}} (x)}, b_{C_{P_{2}}} (x) e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ = 〈 [a_{1}^{-}, a_{1}^{+}] e^{i 2 π u_{C_{P_{1}}} (x)}, [b_{1}^{-}, b_{1}^{+}] e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \cap \\ 〈 [a_{2}^{-}, a_{2}^{+}] e^{i 2 π u_{C_{P_{2}}} (x)}, [b_{2}^{-}, a_{2}^{+}] e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ = 〈 \begin{matrix} [c] c [min (a_{1}^{-}, a_{2}^{-}), min (a_{1}^{+}, a_{2}^{+})] e^{i min {2 π u_{C_{P_{1}}} (x), 2 π u_{C_{P_{2}}} (x)}, \\ [max (b_{1}^{-}, b_{2}^{-}), max (b_{1}^{+}, b_{2}^{+})] e^{i max {2 π v_{C_{P_{1}}} (x), 2 π v_{C_{P_{2}}} (x)} \end{matrix} 〉 \end{matrix}$

Example 2. Consider C_{P
₁} and C_{P
₂} in Example 1. Then, the intersection is given as: $\begin{matrix} C_{P_{1}} \cap C_{P_{2}} = \\ 〈 \begin{matrix} [c] c (x, [0.1, 0.3] e^{i 2 π (0)}, [0.5, 0.9] e^{i 2 π (0.5)}), \\ (b, [0.1, 0.3] e^{i 2 π (0.25)}, [0.6, 0.8] e^{i 2 π (0.5)}), \\ (c, [0.2, 0.6] e^{i 2 π (0.1)}, [0.4, 0.7] e^{i 2 π (0.25)}) \end{matrix} 〉 . \end{matrix}$

Definition 9. Let C_{P
₁} be the AIVCPFS on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉 its grade value. The amplitude interval-valued complex Pythagorean fuzzy complement of C_{P
₁}, denoted $C_{P_{1}}^{c}$ is specified by a function $C_{P_{1}}^{c} = 〈 b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)}, a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)} 〉 .$

Example 3. Consider C_{P
₁} in Example 1. Then, the complement of C_{P
₁} is given as: $C_{P_{1}}^{c} = 〈 \begin{matrix} [c] c (a, [0.4, 0.9] e^{i 2 π (0)}), [0.1, 0.3] e^{i 2 π (0.5)}) \\ (b, [0.5, 0.8] e^{i 2 π (0.5)}), [0.4, 0.6] e^{i 2 π (0.3)} \\ (c, [0.4, 0.5] e^{i 2 π (0.25)}), [0.2, 0.7] e^{i 2 π (0.1)} \end{matrix} 〉 .$

Definition 10. Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉 and C_{P
₂} = 〈a_{C
_{P
₂}} (x) e^{i2πu_{C
_P
₂}(x)}, b_{C
_{P
₂}} (x) e^{i2πv_{C
_P
₂}(x)}〉 their grade values, respectively. The product of these two AIVCPFSs C_{P
₁} and C_{P
₂}, denoted C_{P
₁} • C_{P
₂}, is specified by a function $\begin{matrix} C_{P_{1}} • C_{P_{2}} \\ = 〈 a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)}, b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)} 〉 • \\ 〈 a_{C_{P_{2}}} (x) e^{i 2 π u_{C_{P_{2}}} (x)}, b_{C_{P_{2}}} (x) e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ = 〈 [a_{1}^{-}, a_{1}^{+}] e^{i 2 π u_{C_{P_{1}}} (x)}, [b_{1}^{-}, b_{1}^{+}] e^{i 2 π v_{C_{P_{1}}} (x)} 〉 • \\ 〈 [a_{2}^{-}, a_{2}^{+}] e^{i 2 π u_{C_{P_{2}}} (x)}, [b_{2}^{-}, a_{2}^{+}] e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ = 〈 \begin{matrix} [c] c [a_{1}^{-} \times a_{2}^{-}, a_{1}^{+} \times a_{2}^{+}] e^{i 2 π {u_{C_{P_{1}}} (x) . u_{C_{P_{2}}} (x)}}, \\ [b_{1}^{-} \times b_{2}^{-}, b_{1}^{+} \times b_{2}^{+}] e^{i 2 π {v_{C_{P_{1}}} (x) . v_{C_{P_{2}}} (x)}} \end{matrix} 〉 . \end{matrix}$

Example 4. Consider C_{P
₁} and C_{P
₂} in Example 1. Then, the product is $\begin{matrix} C_{P_{1}} • C_{P_{2}} = \\ 〈 \begin{matrix} [c] c (a, [0.03, 0.18] e^{i 2 π (0)}, [0.2, 0.72] e^{i 2 π (0)}), \\ (b, [0.04, 0.18] e^{i 2 π (0.075)}, [0.3, 0.56] e^{i 2 π (0.25)}), \\ (c, [0.06, 0.42] e^{i 2 π (0.0125)}, [0.16, 0.35] e^{i 2 π (0.025)}) \end{matrix} 〉 . \end{matrix}$

Definition 11. Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉 and C_{P
₂} = 〈a_{C
_{P
₂}} (x) e^{i2πu_{C
_P
₂}(x)}, b_{C
_{P
₂}} (x) e^{i2πv_{C
_P
₂}(x)}〉 their grade values, respectively. The simple difference of these two AIVCPFSs C_{P
₁} and C_{P
₂}, denoted C_{P
₁} \ C_{P
₂}, is specified by a function $\begin{matrix} C_{P_{1}} \ C_{P_{2}} & = C_{P_{1}} \cap C_{P_{2}}^{c} \\ = 〈 a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)}, b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \cap \\ 〈 b_{C_{P_{2}}} (x) e^{i 2 π v_{C_{P_{2}}} (x)}, a_{C_{P_{2}}} (x) e^{i 2 π u_{C_{P_{2}}} (x)} 〉, \end{matrix}$ where " c " represents the complement of the AIVCPFSs.

Example 5. Consider C_{P
₁} and C_{P
₂} in Example 1. Then, the simple difference is $\begin{matrix} C_{P_{1}} \ C_{P_{2}} = \\ 〈 \begin{matrix} [c] c (a, [0.1, 0.3] e^{i 2 π (0.5)}, [0.4, 0.9] e^{i 2 π (0)}), \\ (b, [0.4, 0.6] e^{i 2 π (0.6)}, [0.5, 0.8] e^{i 2 π (0.5)}), \\ (c, [0.2, 0.7] e^{i 2 π (0.1)}, [0.4, 0.7] e^{i 2 π (0.125)}) \end{matrix} 〉 . \end{matrix}$

Definition 12. Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉 and C_{P
₂} = 〈a_{C
_{P
₂}} (x) e^{i2πu_{C
_P
₂}(x)}, b_{C
_{P
₂}} (x) e^{i2πv_{C
_P
₂}(x)}〉 their grade values, respectively. The symmetric difference of these two AIVCPFSs C_{P
₁} and C_{P
₂}, denoted C_{P
₁}ΔC_{P
₂}, is specified by a function $C_{P_{1}} △ C_{P_{2}} = (C_{P_{1}} \cap C_{P_{2}}^{c}) \cup (C_{P_{1}}^{c} \cap C_{P_{2}}),$ where " c " represents the complement of the AIVCPFSs.

Proposition 1. The AIVCPFSs C_{P
₁} and C_{P
₂} over a crisp set U, satisfy De-Morgan Laws.

Proof. (i) . Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉, and C_{P
₂} = 〈a_{C
_{P
₂}} (x) e^{i2πu_{C
_P
₂}(x)}, b_{C
_{P
₂}} (x) e^{i2πv_{C
_P
₂}(x)}〉 their membership functions, respectively. We suppose $a_{C_{P_{1}}} (x) = [a_{1}^{-}, a_{1}^{+}],$ $a_{C_{P_{2}}} (x) = [a_{2}^{-}, a_{2}^{+}],$ $b_{C_{P_{1}}} (x) = [b_{1}^{-}, b_{1}^{+}],$ $b_{C_{P_{2}}} (x) = [b_{2}^{-}, b_{2}^{+}] .$ Also assume that $\begin{matrix} [a_{1}^{-}, a_{1}^{+}] & = [max (a_{1}^{-}, a_{2}^{-}), max (a_{1}^{+}, a_{2}^{+})], \\ [b_{2}^{-}, b_{2}^{+}] & = [min (b_{1}^{-}, b_{2}^{-}), min (b_{1}^{+}, b_{2}^{+})], \end{matrix}$ and $u_{C_{P_{1}}} (x) \leq v_{C_{P_{1}}} (x), u_{C_{P_{2}}} (x) \leq v_{C_{P_{2}}} (x),$ ∀x ∈ U . Thus $\begin{matrix} C_{P_{1}} \cup C_{P_{2}} & = 〈 a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)}, b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \cup \\ 〈 a_{C_{P_{2}}} (x) e^{i 2 π u_{C_{P_{2}}} (x)}, b_{C_{P_{2}}} (x) e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ = 〈 [a_{1}^{-}, a_{1}^{+}] e^{i 2 π u_{C_{P_{2}}} (x)}, [b_{2}^{-}, b_{2}^{+}] e^{i 2 π v_{C_{P_{1}}} (x)} 〉 . \end{matrix}$

Now $\begin{matrix} (1) (C_{P_{1}} \cup C_{P_{2}})^{c} = 〈 [b_{2}^{-}, b_{2}^{+}] e^{i 2 π v_{C_{P_{1}}} (x)}, [a_{1}^{-}, a_{1}^{+}] e^{i 2 π u_{C_{P_{2}}} (x)} 〉 . \end{matrix}$

Also $\begin{matrix} (2) C_{P_{1}}^{c} \cap C_{P_{2}}^{c} = 〈 [b_{2}^{-}, b_{2}^{+}] e^{i 2 π v_{C_{P_{1}}} (x)}, [a_{1}^{-}, a_{1}^{+}] e^{i 2 π u_{C_{P_{2}}} (x)} 〉 . \end{matrix}$

Therefore from (1) and (2), we have $(C_{P_{1}} \cup C_{P_{2}})^{c} = C_{P_{1}}^{c} \cap C_{P_{2}}^{c} .$

(ii) . $(C_{P_{1}} \cap C_{P_{2}})^{c} = C_{P_{1}}^{c} \cup C_{P_{2}}^{c} .$

It is easy to prove. □

Proposition 2. The AIVCPFS C_{P
₁}, satisfy involution law, that is, $(C_{P_{1}}^{c})^{c} = C_{P_{1}} .$

Proof. Let C_{P
₁} be the AIVCPFS on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_P
₁}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_P
₁}(x)}〉, is their membership function then, $\begin{matrix} C_{P_{1}}^{c} & = 〈 b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)}, a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)} 〉 \\ (C_{P_{1}}^{c})^{c} & = 〈 a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)}, b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \\ = C_{P_{1}} . □ \end{matrix}$

Proposition 3. The union and intersection of any finite number of AIVCPFSs are always an AIVCPFS.

Proof. Let C_{P
₁}, C_{P
₂}, . . . , C_{P
_n} be any n AIVCPFSs and $\begin{matrix} C_{P_{1}} & = 〈 a_{C_{P_{1}}} (x) e^{i 2 π u_{C_{P_{1}}} (x)}, b_{C_{P_{1}}} (x) e^{i 2 π v_{C_{P_{1}}} (x)} 〉 \\ C_{P_{2}} & = 〈 a_{C_{P_{2}}} (x) e^{i 2 π u_{C_{P_{2}}} (x)}, b_{C_{P_{2}}} (x) e^{i 2 π v_{C_{P_{2}}} (x)} 〉 \\ . \\ . \\ . \\ C_{P_{n}} & = 〈 a_{C_{P_{n}}} (x) e^{i 2 π u_{C_{P_{n}}} (x)}, b_{C_{P_{n}}} (x) e^{i 2 π v_{C_{P_{n}}} (x)} 〉, \end{matrix}$ denote their membership functions, respectively.

Let $a_{C_{P_{1}}} (x) = [a_{1}^{-}, a_{1}^{+}],$ $a_{C_{P_{2}}} (x) = [a_{2}^{-}, a_{2}^{+}], . . .,$ $a_{C_{P_{n}}} (x) = [a_{n}^{-}, a_{n}^{+}],$ $b_{C_{P_{1}}} (x) = [b_{1}^{-}, b_{1}^{+}],$ $b_{C_{P_{2}}} (x) = [b_{2}^{-}, b_{2}^{+}], . . .,$ $b_{C_{P_{n}}} (x) = [b_{n}^{-}, b_{n}^{+}] .$ Also, let $\begin{matrix} a_{C_{P}} (x) & = [a^{-}, a^{+}] = [max (a_{1}^{-}, a_{2}^{-}, . . ., a_{n}^{-}), max (a_{1}^{+}, a_{2}^{+}, . . ., a_{n}^{+})], \\ b_{C_{P}} (x) & = [b^{-}, b^{+}] = [min (b_{1}^{-}, b_{2}^{-}, . . ., b_{n}^{-}), min (b_{1}^{+}, b_{2}^{+}, . . ., b_{n}^{+})], \\ u_{C_{P}} (x) & = max (u_{C_{P_{1}}} (x), u_{C_{P_{2}}} (x), . . ., u_{C_{P_{n}}} (x)), \\ v_{C_{P}} (x) & = min (v_{C_{P_{1}}} (x), v_{C_{P_{2}}} (x), . . ., v_{C_{P_{n}}} (x)), \end{matrix}$ ∀x ∈ U . Now $\begin{array}{l} C_{P_{1}} \cup C_{P_{2}} \cup ... \cup C_{P_{n}} = 〈 [\max (a_{1}^{-}, a_{2}^{-}, ..., a_{n}^{-}), \max (a_{1}^{+}, a_{2}^{+}, ..., a_{2}^{+})] \\ e^{i 2 π \max (u_{C_{P_{1}}} (x), u_{C_{P_{2}}} (x), ..., u_{C_{P_{n}}} (x))}, [\min (b_{1}^{-}, b_{2}^{-}, ..., b_{n}^{-}), \min (b_{1}^{+}, b_{2}^{+}, ..., b_{n}^{+})] \\ e^{i 2 π \min (v_{C_{P_{1}}} (x), v_{C_{P_{2}}} (x), ..., v_{C_{P_{n}}} (x))} 〉 \\ = 〈 [a^{-}, a^{+}] e^{i 2 π u_{C_{P}} (x)}, [b^{-}, b^{+}] e^{i 2 π \min v_{C_{P}} (x)} 〉 \\ = 〈 a_{C_{P}} (x) e^{i 2 π u_{C_{P}} (x)}, b_{C_{P}} (x) e^{i 2 π \min v_{C_{P}} (x)} 〉 \\ = C_{P} \end{array}$ which is also an AIVCPFS.

(ii) . It is easy to prove.□

Proposition 4. The AIVCPFSs C_{P
₁} and C_{P
₂} over a crisp set U, satisfy the equivalence formula $(C_{P_{1}}^{c} \cup C_{P_{2}}) \cap (C_{P_{1}} \cup C_{P_{2}}^{c}) = (C_{P_{1}}^{c} \cap C_{P_{2}}^{c}) \cup (C_{P_{1}} \cap C_{P_{2}}) .$

Proof. It is easy to prove. □

Theorem 1. If C_{P
₁} and C_{P
₂} are AIVCPFSs on U then,

(i) . $(C_{P_{1}}^{c} \cup C_{P_{2}}) \cap (C_{P_{1}} \cup C_{P_{2}}^{c}) = (C_{P_{1}}^{c} \cap C_{P_{2}}^{c}) \cup (C_{P_{1}} \cap C_{P_{2}}) .$ (Equivalence Formula)

(ii) . $(C_{P_{1}}^{c} \cap C_{P_{2}}) \cup (C_{P_{1}} \cap C_{P_{2}}^{c}) = (C_{P_{1}}^{c} \cup C_{P_{2}}^{c}) \cap (C_{P_{1}} \cup C_{P_{2}}) .$ (Symmetrical Difference Formula).

Theorem 2. If C_{P
₁}, C_{P
₂} and C_{P
₃} are three AIVCPFSs on U then,

(i) . C_{P
₁} ∪ C_{P
₂} = C_{P
₂} ∪ C_{P
₁} .

(ii) . C_{P
₁} ∩ C_{P
₂} = C_{P
₂} ∩ C_{P
₁} .

(iii) . C_{P
₁} ∪ (C_{P
₂} ∪ C_{P
₃}) = (C_{P
₁} ∪ C_{P
₂}) ∪ C_{P
₃} .

(iv) . C_{P
₁} ∩ (C_{P
₂} ∩ C_{P
₃}) = (C_{P
₁} ∩ C_{P
₂}) ∩ C_{P
₃} .

(v) . C_{P
₁} ∩ (C_{P
₂} ∪ C_{P
₃}) = (C_{P
₁} ∩ C_{P
₂}) ∪ (C_{P
₁} ∩ C_{P
₃}) .

(vi) . C_{P
₁} ∪ (C_{P
₂} ∩ C_{P
₃}) = (C_{P
₁} ∪ C_{P
₂}) ∩ (C_{P
₁} ∪ C_{P
₃}) .

4 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 d_{C
_P} : C_P (U) × C_P (U) ⟶ [0, 1], such that for any AIVCPFSs C_{P
₁}, C_{P
₂} and C_{P
₃}, the following axioms hold

1. 0 ≤ d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1,

2 . d_{C
_P} (C_{P
₁}, C_{P
₂}) =0 if and only if C_{P
₁} = C_{P
₂},

3. d_{C
_P} (C_{P
₁}, C_{P
₂}) = d_{C
_P} (C_{P
₂}, C_{P
₁}),

4. d_{C
_P} (C_{P
₁}, C_{P
₃}) ≤ d_{C
_P} (C_{P
₁}, C_{P
₂}) + d_{C
_P} (C_{P
₂}, C_{P
₃}) for all C_{P
₁}, C_{P
₂}, C_{P
₃} ∈ C_P (U) .

Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U, and C_{P
₁} = 〈a_{C
_{P
₁}} (x) e^{i2πu_{C
_{P
₁}}(x)}, b_{C
_{P
₁}} (x) e^{i2πv_{C
_{P
₁}}(x)}〉, and C_{P
₂} = 〈a_{C
_{P
₂}} (x) e^{i2πu_{C
_{P
₂}}(x)}, b_{C
_{P
₂}} (x) e^{i2πv_{C
_{P
₂}}(x)}〉 their membership functions, respectively. The distance function of AIVCPFSs is specified by a function $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) = \\ max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} \end{matrix}$ where l denotes the length.

Example 6. Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U = {ϰ₁, ϰ₂, ϰ₃} then, $\begin{matrix} C_{P_{1}} = \\ 〈 \begin{matrix} [c] c (ϰ_{1}, [0.3, 0.7] e^{i 2 π (0.5)}, [0.5, 0.6] e^{i 2 π (0.1)}), \\ (ϰ_{2}, [0.4, 0.6] e^{i 2 π (0.5)}, [0.2, 0.8] e^{i 2 π (0.6)}), \\ (ϰ_{3}, [0, 0.8] e^{i 2 π (0.8)}, [0.2, 0.4] e^{i 2 π (0.4)}) \end{matrix} 〉 \\ C_{P_{2}} = \\ 〈 \begin{matrix} [c] c (ϰ_{1}, [0.1, 0.7] e^{i 2 π (0.5)}, [0.3, 0.6] e^{i 2 π (0.8)}), \\ (ϰ_{2}, [0.4, 0.8] e^{i 2 π (0.2)}, [0.1, 0.5] e^{i 2 π (0.6)}), \\ (ϰ_{3}, [0.4, 0.6] e^{i 2 π (0.7)}, [0.5, 0.7] e^{i 2 π (0.6)}) . \end{matrix} 〉 \end{matrix}$

It is easy to see that $\begin{matrix} sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) | \\ = 0.6, sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) | = 0.2 \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) | & = 0.3, \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | = 0.7 . \end{matrix}$

Thus $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) = \\ max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} \\ = 0.7 \end{matrix}$

Using the above definitions we arrived to a crucial theorem of this section with a rather simple proof.

Theorem 3. The function d_{C
_P} is a distance function of AIVCPFSs on U.

Proof. The condition 0 ≤ d_{C
_P} (C_{P
₁}, C_{P
₂}) obviously holds true. Next, consider $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} \\ = max {1, 1, \frac{1}{2 π} (2 π), \frac{1}{2 π} (2 π)} \\ = max {1,, 1, 1, 1} = 1 . \end{matrix}$

Therefore, 0 ≤ d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1 .

The conditions (2) and (3) are straight forward. To prove (4), we have $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{3}}) \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{3}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{3}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{3}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{3}}} (x) | \end{matrix}} \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) + l (a_{C_{P_{2}}} (x)) - l (a_{C_{P_{3}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) + l (b_{C_{P_{2}}} (x)) - l (b_{C_{P_{3}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) + 2 π u_{C_{P_{2}}} (x) - 2 π u_{C_{P_{3}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) + 2 π v_{C_{P_{2}}} (x) - 2 π v_{C_{P_{3}}} (x) | \end{matrix}} \\ \leq max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) | + \\ sup_{x \in U} | l (a_{C_{P_{2}}} (x)) - l (a_{C_{P_{3}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) | + \\ sup_{x \in U} | l (b_{C_{P_{2}}} (x)) - l (b_{C_{P_{3}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) | + \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{2}}} (x) - 2 π u_{C_{P_{3}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | + \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{2}}} (x) - 2 π v_{C_{P_{3}}} (x) | \end{matrix}} \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} + \\ max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{2}}} (x)) - l (a_{C_{P_{3}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{2}}} (x)) - l (b_{C_{P_{3}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{2}}} (x) - 2 π u_{C_{P_{3}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{2}}} (x) - 2 π v_{C_{P_{3}}} (x) | \end{matrix}} \\ = d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) + d_{C_{P}} (C_{P_{2}}, C_{P_{3}}) \\ d_{C_{P}} (C_{P_{1}}, C_{P_{3}}) \\ \leq d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) + d_{C_{P}} (C_{P_{2}}, C_{P_{3}}) . □ \end{matrix}$

Definition 14. Let C_{P
₁} and C_{P
₂} be two AIVCPFSs on U . Then, C_{P
₁} and C_{P
₂} are said to be α-equal (C_{P
₁} = αC_{P
₂}) if and only if d (C_{P
₁}, C_{P
₂}) ≤1 - α or α ≤ 1 - d (C_{P
₁}, C_{P
₂}).

Theorem 4. Let α₁ ∗ α₂ = max(0, α₁ + α₂ - 1), 0 ≤ α₁, α₂ ≤ 1. The

(i) 0 ∗ α₁ = 0,

(ii) 1 ∗ α₁ = α₁, for all α₁ ∈ [0, 1]

(iii) 0 ≤ α₁ ∗ α₂ ≤ 1, for all α₁, α₂ ∈ [0, 1]

(iv) $α_{1} \leq α_{1}^{'} \Rightarrow α_{1} * α_{2} \leq α_{1}^{'} * α_{2}$ , for all $α_{1}, α_{1}^{'}, α_{2} \in [0, 1]$

(v) α₁ ∗ α₂ = α₂ ∗ α₁, for all α₁, α₂ ∈ [0, 1]

(vi) (α₁ ∗ α₂) ∗ α₃ = α₁ ∗ (α₁ ∗ α₂), for all α₁, α₂, α₃ ∈ [0, 1].

Proof. It is easy. □

Theorem 5. If C_{P
₁} and C_{P
₂} be two AIVCPFSs and d_{C
_P} is a distance function then,

(i) C_{P
₁} = 0C_{P
₂} ⇔ d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1 .

(ii) C_{P
₁} = 1C_{P
₂} ⇔ d_{C
_P} (C_{P
₁}, C_{P
₂}) =0 .

(iii) C_{P
₁} = αC_{P
₂} ⇔ C_{P
₂} = αC_{P
₁} .

(vi) C_{P
₁} = α₁C_{P
₂} and α₂ ≤ α₁ ⇒ C_{P
₁} = α₂C_{P
₂} .

(v) If C_{P
₁} = α_iC_{P
₂} then, $C_{P_{1}} = sup_{i \in I} δ_{α} C_{P_{2}} .$

(vi) For all C_{P
₁}, C_{P
₂} there exists a unique α such that C_{P
₁} = αC_{P
₂} and if C_{P
₁} = α′C_{P
₂} then, α′ ≤ α.

Proof. (i) . Suppose that C_{P
₁} = 0C_{P
₂} then d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1 - 0 and hence d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1 .

Conversely suppose that d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1 . It can be written as d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1 - 0 . This implies that C_{P
₁} = 0C_{P
₂} .

(ii) . Suppose that C_{P
₁} = 1C_{P
₂} then d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤1 - 1 ⇒ d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤0 . But d_{C
_P} is a distance function so d_{C
_P} (C_{P
₁}, C_{P
₂}) ≥0 . Thus d_{C
_P} (C_{P
₁}, C_{P
₂}) =0

Conversely suppose that d_{C
_P} (C_{P
₁}, C_{P
₂}) =0 . It can be written as d_{C
_P} (C_{P
₁}, C_{P
₂}) =1 - 1 . This implies that C_{P
₁} = 1C_{P
₂} .

(iii) . Assume that C_{P
₁} = αC_{P
₂} then by definition of α-equal d_{C
_P} (C_{P
₁}, C_{P
₂}) =1 - α . Since, d_{C
_P} is a distance function so d_{C
_P} (C_{P
₁}, C_{P
₂}) = d_{C
_P} (C_{P
₂}, C_{P
₁}) and hence, d_{C
_P} (C_{P
₂}, C_{P
₁}) =1 - α . Thus C_{P
₂} = αC_{P
₁} .

(iv) . Let C_{P
₁} = α₁C_{P
₂} then, by definition of α-equal d_{C
_P} (C_{P
₁}, C_{P
₂}) =1 - α₁ . Since, for α₂ ≤ α₁ the inequality also holds, that is, d_{C
_P} (C_{P
₁}, C_{P
₂}) =1 - α₂. Thus C_{P
₁} = α₂C_{P
₂} .

(v) Let C_{P
₁} = α_iC_{P
₂} then, $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} \\ \leq 1 - α_{i}, for all i \in I . \end{matrix}$ Thus it is easy to see that $\begin{matrix} sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) | & \leq 1 - sup_{i \in I} α_{i} \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) | & \leq 1 - sup_{i \in I} α_{i} \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) | & \leq 1 - sup_{i \in I} α_{i} \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | & \leq 1 - sup_{i \in I} α_{i} \end{matrix}$ Therefore, $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} \\ \leq 1 - sup_{i \in I} α_{i} . \end{matrix}$ Hence $C_{P_{1}} = sup_{i \in I} δ_{α} C_{P_{2}}$ .

(vi) Let α = 1 - d_{C
_P} (C_{P
₁}, C_{P
₂}) . Then, C_{P
₁} = αC_{P
₂}. If C_{P
₁} = α′C_{P
₂}, we have 1 - α = d_{C
_P} (C_{P
₁}, C_{P
₂}) ≤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 C_{P
₁} = α₁C_{P
₂} and C_{P
₂} = α₂C_{P
₃}, then C_{P
₁} = αC_{P
₃}, where α = α₁ ∗ α₂ .

Proof. Since C_{P
₁} = α₁C_{P
₂} and C_{P
₂} = α₂C_{P
₃}, we have $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} \\ \leq 1 - α_{1} . \end{matrix}$

and $\begin{matrix} d_{C_{P}} (C_{P_{2}}, C_{P_{3}}) \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{2}}} (x)) - l (a_{C_{P_{3}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{2}}} (x)) - l (b_{C_{P_{3}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{2}}} (x) - 2 π u_{C_{P_{3}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{2}}} (x) - 2 π v_{C_{P_{3}}} (x) | \end{matrix}} \\ \leq 1 - α_{2} . \end{matrix}$

Therefore, $\begin{matrix} sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) | & \leq 1 - α_{1} \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) | & \leq 1 - α_{1} \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) | & \leq 1 - α_{1} \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | & \leq 1 - α_{1} \end{matrix}$

and $\begin{matrix} sup_{x \in U} | l (a_{C_{P_{2}}} (x)) - l (a_{C_{P_{3}}} (x)) | & \leq 1 - α_{2} \\ sup_{x \in U} | l (b_{C_{P_{2}}} (x)) - l (b_{C_{P_{3}}} (x)) | & \leq 1 - α_{2} \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{2}}} (x) - 2 π u_{C_{P_{3}}} (x) | & \leq 1 - α_{2} \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{2}}} (x) - 2 π v_{C_{P_{3}}} (x) | & \leq 1 - α_{2} . \end{matrix}$

Consequently, we have $\begin{matrix} d_{C_{P}} (C_{P_{1}}, C_{P_{3}}) = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) + l (a_{C_{P_{2}}} (x)) - l (a_{C_{P_{3}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) + l (b_{C_{P_{2}}} (x)) - l (b_{C_{P_{3}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) + 2 π u_{C_{P_{2}}} (x) - 2 π u_{C_{P_{3}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) + 2 π v_{C_{P_{2}}} (x) - 2 π v_{C_{P_{3}}} (x) | . \end{matrix}} \\ \leq max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) | + sup_{x \in U} | l (a_{C_{P_{2}}} (x)) - l (a_{C_{P_{3}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) | + sup_{x \in U} | l (b_{C_{P_{2}}} (x)) - l (b_{C_{P_{3}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) | + sup_{x \in U} | 2 π u_{C_{P_{2}}} (x) - 2 π u_{C_{P_{3}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | + sup_{x \in U} | 2 π v_{C_{P_{2}}} (x) - 2 π v_{C_{P_{3}}} (x) | . \end{matrix}} \\ \leq max {(1 - α_{1}) + (1 - α_{2}), (1 - α_{1}) + (1 - α_{2}), (1 - α_{1}) + \\ (1 - α_{2}), (1 - α_{1}) + (1 - α_{2})} \\ = (1 - α_{1}) + (1 - α_{2}) = 1 - (α_{1} + α_{2} - 1) \end{matrix}$

and d_{C
_P} (C_{P
₁}, C_{P
₃}) form definition 10. Therefore $d_{C_{P}} (C_{P_{1}}, C_{P_{3}}) \leq 1 - (α_{1} * α_{2}) = 1 - α .$

Thus, C_{P
₁} = αC_{P
₃} . □

Theorem 7. If C_{P
₁} = αC_{P
₂}, then $C_{P_{1}}^{c} = α C_{P_{2}}^{c} .$

Proof. Consider $\begin{matrix} d_{C_{P}} (C_{P_{1}}^{c}, C_{P_{2}}^{c}) & = max {\begin{matrix} [c] c sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) | \end{matrix}} \\ = max {\begin{matrix} [c] c sup_{x \in U} | l (a_{C_{P_{1}}} (x)) - l (a_{C_{P_{2}}} (x)) |, \\ sup_{x \in U} | l (b_{C_{P_{1}}} (x)) - l (b_{C_{P_{2}}} (x)) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π u_{C_{P_{1}}} (x) - 2 π u_{C_{P_{2}}} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | 2 π v_{C_{P_{1}}} (x) - 2 π v_{C_{P_{2}}} (x) | \end{matrix}} \\ = d_{C_{P}} (C_{P_{1}}, C_{P_{2}}) \leq 1 - α . \end{matrix}$

Therefore, $C_{P_{1}}^{c} = α C_{P_{2}}^{c} .$ □

5 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 a_{C
_P} (x) and b_{C
_P} (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. $\begin{matrix} [\begin{matrix} x^{'} (0) \\ x^{'} (1) \\ x^{'} (2) \\ . \\ . \\ . \\ x^{'} (M - 1) \end{matrix}] = \\ [\begin{matrix} 1 & 1 & 1 & . & . & . & 1 \\ 1 & e^{i (- \frac{2 π}{M})} & e^{i (- \frac{2 π}{M} 2)} & . & . & . & e^{i (- \frac{2 π}{M} (M - 1))} \\ 1 & e^{i (- \frac{2 π}{M} 2)} & e^{i (- \frac{2 π}{M} 4)} & . & . & . & e^{i (- \frac{2 π}{M} 2 (M - 1))} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & e^{i (- \frac{2 π}{M} (M - 1))} & e^{i (- \frac{2 π}{M} 2 (M - 1))} & . & . & . & e^{i (- \frac{2 π}{M} (M - 1)^{2})} \end{matrix}] [\begin{matrix} x (0) \\ x (1) \\ x (2) \\ . \\ . \\ . \\ x (M - 1) \end{matrix}] \end{matrix}$

but the IDFT is given by $\begin{matrix} [\begin{matrix} x (0) \\ x (1) \\ x (2) \\ . \\ . \\ . \\ x (M - 1) \end{matrix}] \\ = \frac{1}{M} [\begin{matrix} 1 & 1 & 1 & . & . & . & 1 \\ 1 & e^{i \frac{2 π}{M}} & e^{i \frac{2 π}{M} 2} & . & . & . & e^{i \frac{2 π}{M} (M - 1)} \\ 1 & e^{i \frac{2 π}{M} 2} & e^{i \frac{2 π}{M} 4} & . & . & . & e^{i \frac{2 π}{M} 2 (M - 1)} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & e^{i \frac{2 π}{M} (M - 1)} & e^{i \frac{2 π}{M} 2 (M - 1)} & . & . & . & e^{i \frac{2 π}{M} (M - 1)^{2}} \end{matrix}] [\begin{matrix} x^{'} (0) \\ x^{'} (1) \\ x^{'} (2) \\ . \\ . \\ . \\ x^{'} (M - 1) \end{matrix}] \end{matrix}$

Definition 16. The Mth inverse discrete Fourier transform (IDFT) coefficient of a length M sequence {x (M)} is defined as $\begin{matrix} x_{n} (m) = \frac{1}{M} \overset{M - 1}{\sum_{q = 0}} x^{'} (q) e^{i \frac{2 π}{M} mq}, \\ m, q \in {0, 1, 2, . . ., M - 1} \end{matrix}$ 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 x_n (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, $U_{1}^{*}, U_{2}^{*}, U_{3}^{*}, . . ., U_{m}^{*}$ have been received by a particular receiver. Each of these signals is noted at M different times (samples). Let U_n (m) be the n - th (0 ≤ m ≤ M - 1) signal. The inverse discrete Fourier transform of U_n (m) is: $\begin{matrix} U_{n}^{*} (m) = 〈 \frac{1}{M} \overset{M - 1}{\sum_{q = 0}} U [q] e^{i \frac{2 π}{M} mq}, \frac{1}{M} \overset{M - 1}{\sum_{q = 0}} U^{'} [q] e^{i \frac{2 π}{M} mq} 〉; \\ m, q = 0, 1, 2, . . ., M - 1 \end{matrix}$ (1)

In equation (1) , $U [q] e^{i \frac{2 π}{M} mq} = [a^{-}, a^{+}] e^{i \frac{2 π}{M} mq}$ shows the membership function of amplitude interval-valued complex Pythagorean fuzzy sets and $U^{'} [q] e^{i \frac{2 π}{M} mq}$ 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 $U [q] e^{i \frac{2 π}{M} mq}$ shows the degree of occurrence of the receiving signals and the non-membership function $U^{'} [q] e^{i \frac{2 π}{M} mq}$ 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: $\begin{matrix} r^{*} (m) = 〈 \frac{1}{M} \overset{M - 1}{\sum_{q = 0}} R [q] e^{i \frac{2 π}{M} mq}, \frac{1}{M} \overset{M - 1}{\sum_{q = 0}} R^{'} [q] e^{i \frac{2 π}{M} mq} 〉; \\ m, q = 0, 1, 2, . . ., M - 1 . \end{matrix}$ (2)

The inverse discrete Fourier transform of the signal $U_{n}^{*} (m)$ 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 $U_{n}^{*} (m) .$ For this, we first expande $U_{n}^{*} (m) = 〈 \frac{1}{M} {\sum_{q = 0}}^{M - 1} U [q] e^{i \frac{2 π}{M} mq}, \frac{1}{M} {\sum_{q = 0}}^{M - 1} U^{'} [q] e^{i \frac{2 π}{M} mq} 〉$ for q = 0, 1, 2, . . . , M - 1, we get $U_{n}^{*} (m) = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i \frac{2 π}{M} m (0)} + U [1] e^{i \frac{2 π}{M} m (1)} + U [2] e^{i \frac{2 π}{M} m (2)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} m (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i \frac{2 π}{M} m (0)} + U^{'} [1] e^{i \frac{2 π}{M} m (1)} + U^{'} [2] e^{i \frac{2 π}{M} m (2)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} m (M - 1)} \end{matrix}] \end{matrix} 〉 .$ (3)

From equation (3) we get M-samples by putting m = 0, 1, 2, 3, . . . , M - 1 .

For m = 0, we have $U_{n}^{*} (0) = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i . 0} + U [1] e^{i \frac{2 π}{M} (0) (1)} + U [2] e^{i \frac{2 π}{M} (0) (2)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} (0) (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i . 0} + U^{'} [1] e^{i \frac{2 π}{M} (0) (1)} + U^{'} [2] e^{i \frac{2 π}{M} (0) (2)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} (0) (M - 1)} \end{matrix}] \end{matrix} 〉,$ $U_{n}^{*} (0) = 〈 \begin{matrix} [c] c \frac{1}{M} [U [0] e^{i . 0} + U [1] e^{i 0} + U [2] e^{i . 0} + . . . + U [M - 1] e^{i . 0}], \\ \frac{1}{M} [U^{'} [0] e^{i . 0} + U^{'} [1] e^{i . 0} + U^{'} [2] e^{i . 0} + . . . + U^{'} [M - 1] e^{i . 0}] \end{matrix} 〉 .$ (4)

Where $U_{n}^{*} (0),$ denotes the 1st sample of the $U_{n}^{*} (m)$ signal.

For m = 1, we have $U_{n}^{*} (1) = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i . 0} + U [1] e^{i \frac{2 π}{M} (1) (1)} + U [2] e^{i \frac{2 π}{M} (1) (2)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} (1) (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i . 0} + U^{'} [1] e^{i \frac{2 π}{M} (1) (1)} + U^{'} [2] e^{i \frac{2 π}{M} (1) (2)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} (1) (M - 1)} \end{matrix}] \end{matrix} 〉,$ $U_{n}^{*} (1) = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i . 0} + U [1] e^{i \frac{2 π}{M} (1)} + U [2] e^{i \frac{2 π}{M} (2)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i . 0} + U^{'} [1] e^{i \frac{2 π}{M} (1)} + U^{'} [2] e^{i \frac{2 π}{M} (2)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} (M - 1)} \end{matrix}] \end{matrix} 〉 .$ (5)

Where $U_{n}^{*} (1),$ denotes the second sample of the $U_{n}^{*} (m)$ signal.

For m = 2, we have $U_{n}^{*} (2) = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i . 0} + U [1] e^{i \frac{2 π}{M} (2) (1)} + U [2] e^{i \frac{2 π}{M} (2) (2)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} (2) (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i . 0} + U^{'} [1] e^{i \frac{2 π}{M} (2) (1)} + U^{'} [2] e^{i \frac{2 π}{M} (2) (2)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} (2) (M - 1)} \end{matrix}] \end{matrix} 〉,$ $U_{n}^{*} (2) = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i . 0} + U [1] e^{i \frac{2 π}{M} (2)} + U [2] e^{i \frac{2 π}{M} (4)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} 2 (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i . 0} + U^{'} [1] e^{i \frac{2 π}{M} (2)} + U^{'} [2] e^{i \frac{2 π}{M} (4)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} 2 (M - 1)} \end{matrix}] \end{matrix} 〉 .$ (6)

Equation (6) , shows the third sample of the $U_{n}^{*} (m)$ signal.

Continuing this process, for m = M - 1, we have $\begin{matrix} U_{n}^{*} (M - 1) = \\ 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i . 0} + U [1] e^{i \frac{2 π}{M} (M - 1) (1)} + U [2] e^{i \frac{2 π}{M} (M - 1) (2)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} (M - 1) (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i . 0} + U^{'} [1] e^{i \frac{2 π}{M} (M - 1) (1)} + U^{'} [2] e^{i \frac{2 π}{M} (M - 1) (2)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} (M - 1) (M - 1)} \end{matrix}] \end{matrix} 〉, \end{matrix}$ (7) $\begin{matrix} u_{n}^{*} (M - 1) = \\ 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c U [0] e^{i . 0} + U [1] e^{i \frac{2 π}{M} (M - 1)} + U [2] e^{i \frac{2 π}{M} 2 (M - 1)} \\ + . . . + U [M - 1] e^{i \frac{2 π}{M} (M - 1)^{2}} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c U^{'} [0] e^{i . 0} + U^{'} [1] e^{i \frac{2 π}{M} (M - 1)} + U^{'} [2] e^{i \frac{2 π}{M} 2 (M - 1)} \\ + . . . + U^{'} [M - 1] e^{i \frac{2 π}{M} (M - 1)^{2}} \end{matrix}] \end{matrix} 〉 . \end{matrix}$

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 . $r^{*} (0) = 〈 \begin{matrix} [c] c \frac{1}{M} [R [0] e^{i . 0} + R [1] e^{i 0} + R [2] e^{i . 0} + . . . + R [M - 1] e^{i . 0}], \\ \frac{1}{M} [R^{'} [0] e^{i . 0} + R^{'} [1] e^{i . 0} + R^{'} [2] e^{i . 0} + . . . + R^{'} [M - 1] e^{i . 0}] \end{matrix} 〉 .$ (8) $r^{*} (1) = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c R [0] e^{i . 0} + R [1] e^{i \frac{2 π}{M} (1)} + R [2] e^{i \frac{2 π}{M} (2)} \\ + . . . + R [M - 1] e^{i \frac{2 π}{M} (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c R^{'} [0] e^{i . 0} + R^{'} [1] e^{i \frac{2 π}{M} (1)} + R^{'} [2] e^{i \frac{2 π}{M} (2)} \\ + . . . + R^{'} [M - 1] e^{i \frac{2 π}{M} (M - 1)} \end{matrix}] \end{matrix} 〉 .$ (9) $\begin{matrix} . \\ . \\ . \\ r^{*} (M - 1) & = 〈 \begin{matrix} [c] c \frac{1}{M} [\begin{matrix} [c] c R [0] e^{i . 0} + R [1] e^{i \frac{2 π}{M} (M - 1) (1)} + R [2] e^{i \frac{2 π}{M} (M - 1) (2)} \\ + . . . + R [M - 1] e^{i \frac{2 π}{M} (M - 1) (M - 1)} \end{matrix}], \\ \frac{1}{M} [\begin{matrix} [c] c R^{'} [0] e^{i . 0} + R^{'} [1] e^{i \frac{2 π}{M} (M - 1) (1)} + R^{'} [2] e^{i \frac{2 π}{M} (M - 1) (2)} \\ + . . . + R^{'} [M - 1] e^{i \frac{2 π}{M} (M - 1) (M - 1)} \end{matrix}] \end{matrix} 〉 . \end{matrix}$ (10)

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,

$U^{*} [q] = \frac{l (U [q]) + l (U^{'} [q])}{2}, R^{*} [q] = \frac{l (R [q]) + l (R^{'} [q])}{2},$

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

$\begin{matrix} U_{n}^{*} (0) \\ = 〈 \frac{1}{M} [U^{*} [0] e^{i . 0} + U^{*} [1] e^{i 0} + U^{*} [2] e^{i . 0} + . . . + U^{*} [M - 1] e^{i . 0}] 〉 . \end{matrix}$ (11) $U_{n}^{*} (1) = 〈 \frac{1}{M} [\begin{matrix} [c] c U^{*} [0] e^{i . 0} + U^{*} [1] e^{i \frac{π}{M}} + U^{*} [2] e^{i \frac{2 π}{M}} \\ + . . . + U^{*} [M - 1] e^{i \frac{π}{M} (M - 1)} \end{matrix}] 〉 .$ (12) $U_{n}^{*} (2) = 〈 \frac{1}{M} [\begin{matrix} [c] c U^{*} [0] e^{i . 0} + U^{*} [1] e^{i \frac{2 π}{M}} + U^{*} [2] e^{i \frac{2 π}{M} (2)} \\ + . . . + U^{*} [M - 1] e^{i \frac{2 π}{M} (M - 1)} \end{matrix}] 〉 .$ (13) $U_{n}^{*} (M - 1) = 〈 \frac{1}{M} [\begin{matrix} [c] c U^{*} [0] e^{i . 0} + U^{*} [1] e^{i \frac{π}{M} (M - 1)} + U^{*} [2] e^{i \frac{2 π}{M} (M - 1)} \\ + . . . + U^{*} [M - 1] e^{i \frac{π}{M} (M - 1)^{2}} \end{matrix}] 〉 .$ (14)

Similarly, $r^{*} (0) = 〈 \frac{1}{M} [R^{*} [0] e^{i . 0} + R^{*} [1] e^{i 0} + R^{*} [2] e^{i . 0} + . . . + R^{*} [M - 1] e^{i . 0}] 〉 .$ (15) $r^{*} (1) = 〈 \frac{1}{M} [\begin{matrix} [c] c R^{*} [0] e^{i . 0} + R^{*} [1] e^{i \frac{π}{M}} + R^{*} [2] e^{i \frac{2 π}{M}} \\ + . . . + U^{*} [M - 1] e^{i \frac{π}{M} (M - 1)} \end{matrix}] 〉 .$ (16) $r^{*} (2) = 〈 \frac{1}{M} [\begin{matrix} [c] c R^{*} [0] e^{i . 0} + R^{*} [1] e^{i \frac{2 π}{M}} + R^{*} [2] e^{i \frac{2 π}{M} (2)} \\ + . . . + R^{*} [M - 1] e^{i \frac{2 π}{M} (M - 1)} \end{matrix}] 〉 .$ (17) $r^{*} (M - 1) = 〈 \frac{1}{M} [\begin{matrix} [c] c R^{*} [0] e^{i . 0} + R^{*} [1] e^{i \frac{π}{M} (M - 1)} + R^{*} [2] e^{i \frac{2 π}{M} (M - 1)} \\ + . . . + R^{*} [M - 1] e^{i \frac{π}{M} (M - 1)^{2}} \end{matrix}] 〉 .$ (18)

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 u_m (M) and the reference signal Θ′ (M) using definition [35], that is we have

$[\begin{matrix} U_{n}^{*} (0) \\ U_{n}^{*} (1) \\ U_{n}^{*} (2) \\ . \\ . \\ . \\ U_{n}^{*} (M - 1) \end{matrix}] = \frac{1}{M} [\begin{matrix} 1 & 1 & 1 & . & . & . & 1 \\ 1 & e^{i \frac{π}{M}} & e^{i \frac{2 π}{M}} & . & . & . & e^{i \frac{π}{M} (M - 1)} \\ 1 & e^{i \frac{2 π}{M}} & e^{i \frac{2 π}{M} 2} & . & . & . & e^{i \frac{2 π}{M} (M - 1)} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & e^{i \frac{π}{M} (M - 1)} & e^{i \frac{2 π}{M} (M - 1)} & . & . & . & e^{i \frac{π}{M} (M - 1)^{2}} \end{matrix}] [\begin{matrix} U^{*} [0] \\ U^{*} [1] \\ U^{*} [2] \\ . \\ . \\ . \\ U^{*} [M - 1] \end{matrix}]$

and $[\begin{matrix} r^{*} (0) \\ r^{*} (1) \\ r^{*} (2) \\ . \\ . \\ . \\ r^{*} (M - 1) \end{matrix}] = \frac{1}{M} [\begin{matrix} 1 & 1 & 1 & . & . & . & 1 \\ 1 & e^{i \frac{π}{M}} & e^{i \frac{2 π}{M}} & . & . & . & e^{i \frac{π}{M} (M - 1)} \\ 1 & e^{i \frac{2 π}{M}} & e^{i \frac{2 π}{M} 2} & . & . & . & e^{i \frac{2 π}{M} (M - 1)} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & e^{i \frac{π}{M} (M - 1)} & e^{i \frac{2 π}{M} (M - 1)} & . & . & . & e^{i \frac{π}{M} (M - 1)^{2}} \end{matrix}] [\begin{matrix} R^{*} [0] \\ R^{*} [1] \\ R^{*} [2] \\ . \\ . \\ . \\ R^{*} [M - 1] \end{matrix}]$

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 $U_{n}^{*} (m)$ and the reference signal r^∗ (m), that is $[\begin{matrix} ∣ U_{n}^{*} (0) ∣ \\ ∣ U_{n}^{*} (1) ∣ \\ ∣ U_{n}^{*} (2) ∣ \\ . \\ . \\ . \\ ∣ U_{n}^{*} (M - 1) ∣ \end{matrix}] and [\begin{matrix} ∣ r^{*} (0) ∣ \\ ∣ r^{*} (1) ∣ \\ ∣ r^{*} (2) ∣ \\ . \\ . \\ . \\ ∣ r^{*} (m - 1) ∣ \end{matrix}] .$

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 $U_{n}^{*} (m)$ and the time domain reference signal r^∗ (m) . If these two values are nearly the same, then the signal $U_{n}^{*} (m)$ identifies a reference signal.

Example 7. Assume that four different electromagnetic signals, $U_{1}^{*} (m)$ , $U_{2}^{*} (m),$ $U_{3}^{*} (m)$ , and $U_{4}^{*} (m)$ 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 $U_{n}^{*} (m);$ m, n = 0, 1, 2, 3 is: $U_{n}^{*} (m) = 〈 \frac{1}{4} \overset{3}{\sum_{q = 0}} U [q] e^{i \frac{2 π}{4} mq}, \frac{1}{4} \overset{3}{\sum_{q = 0}} U^{'} [q] e^{i \frac{2 π}{4} mq} 〉; m, q = 0, 1, 2, 3$ (1)

where $U_{n} [q] \in D [0, 1] .$

Also $\begin{matrix} r^{*} (m) = 〈 \frac{1}{4} \overset{3}{\sum_{q = 0}} R [q] e^{i \frac{2 π}{4} mq}, \frac{1}{4} \overset{3}{\sum_{q = 0}} R^{'} [q] e^{i \frac{2 π}{4} mq} 〉; \\ m, q = 0, 1, 2, 3 \end{matrix}$ (2)

where $R [q]), R^{'} [q] \in D [0, 1] .$

Following steps 1 and 2 in the above algorithms, we get $\begin{matrix} (\begin{matrix} U_{n}^{*} (0) \\ U_{n}^{*} (1) \\ U_{n}^{*} (2) \\ U_{n}^{*} (3) \end{matrix}) & = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & e^{i \frac{π}{4}} & e^{i π} & e^{i \frac{3 π}{2}} \\ 1 & e^{i π} & e^{i 2 π} & e^{i 3 π} \\ 1 & e^{i \frac{3 π}{2}} & e^{i 3 π} & e^{i \frac{9 π}{2}} \end{matrix}) (\begin{matrix} U^{*} [0] \\ U^{*} [1] \\ U^{*} [1] \\ U^{*} [1] \end{matrix}) \\ = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0.707 + i 0.707 & - 1 + 0 . i & 0 - i \\ 1 & - 1 + 0 . i & 1 + 0 . i & - 1 + 0 . i \\ 1 & 0 - i & - 1 + 0 . i & 0 + i \end{matrix}) (\begin{matrix} U^{*} [0] \\ U^{*} [1] \\ U^{*} [1] \\ U^{*} [1] \end{matrix}) \end{matrix}$

and $(\begin{matrix} r^{*} (0) \\ r^{*} (1) \\ r^{*} (2) \\ r^{*} (3) \end{matrix}) = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0.707 + i 0.707 & - 1 + 0 . i & 0 - i \\ 1 & - 1 + 0 . i & 1 + 0 . i & - 1 + 0 . i \\ 1 & 0 - i & - 1 + 0 . i & 0 + i \end{matrix}) (\begin{matrix} R^{*} [0] \\ R^{*} [1] \\ R^{*} [1] \\ R^{*} [1] \end{matrix})$

First of all, we find the values of the sample of reference signal r^∗ (m) . For this, we have $R^{*} [q] = {\begin{matrix} [c] c 0.5; q = 0 \\ 1; q = 1 \\ 0.6; q = 2 \\ 0; q = 3 \end{matrix}$

$\begin{matrix} (\begin{matrix} r^{*} (0) \\ r^{*} (1) \\ r^{*} (2) \\ r^{*} (3) \end{matrix}) & = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0.707 + i 0.707 & - 1 + 0 . i & 0 - i \\ 1 & - 1 + 0 . i & 1 + 0 . i & - 1 + 0 . i \\ 1 & 0 - i & - 1 + 0 . i & 0 + i \end{matrix}) (\begin{matrix} 0.5 \\ 1 \\ 0.6 \\ 0 \end{matrix}) \\ = \frac{1}{4} (\begin{matrix} 2.1 \\ 1.81 + i 0.707 \\ 0.1 \\ - 0.1 - i \end{matrix}) \end{matrix}$

Now the absolute value matrix of the reference signal is $[\begin{matrix} ∣ r^{*} (0) ∣ \\ ∣ r^{*} (1) ∣ \\ ∣ r^{*} (2) ∣ \\ ∣ r^{*} (3) ∣ \end{matrix}] = [\begin{matrix} 0.53 \\ 0.49 \\ 0.03 \\ 0.25 \end{matrix}] .$

The maximum value is 0.53 .

Now for the signal $U_{1}^{*} (m)$ of A₁ aircraft; m = 0, 1, 2, 3, we have

$U^{*} [q] = {\begin{matrix} [c] c 0.8; q = 0 \\ 0.3; q = 1 \\ 0.5; q = 2 \\ 1; q = 3 \end{matrix}$ $(\begin{matrix} U_{1}^{*} (0) \\ U_{1}^{*} (1) \\ U_{1}^{*} (2) \\ U_{1}^{*} (3) \end{matrix}) = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0.707 + i 0.707 & - 1 + 0 . i & 0 - i \\ 1 & - 1 + 0 . i & 1 + 0 . i & - 1 + 0 . i \\ 1 & 0 - i & - 1 + 0 . i & 0 + i \end{matrix}) (\begin{matrix} 0.8 \\ 0.3 \\ 0.5 \\ 1 \end{matrix})$

$(\begin{matrix} U_{1}^{*} (0) \\ U_{1}^{*} (1) \\ U_{1}^{*} (2) \\ U_{1}^{*} (3) \end{matrix}) = \frac{1}{4} (\begin{matrix} 2.6 \\ 0.51 - 0.79 i \\ 0 \\ 0.3 + 0.7 i \end{matrix})$

Now the absolute value matrix of the signal $U_{1}^{*} (m)$ is $(\begin{matrix} | U_{1}^{*} (0) | \\ | U_{1}^{*} (1) | \\ | U_{1}^{*} (2) | \\ | U_{1}^{*} (3) | \end{matrix}) = \frac{1}{4} (\begin{matrix} 0.65 \\ 0.28 \\ 0 \\ 0.19 \end{matrix})$

Here the maximum value is $| U_{1}^{*} (0) | = 0.65 .$

Now for the signal $U_{2}^{*} (m)$ of A₂ aircraft; m = 0, 1, 2, 3, we have $U^{*} [q] = {\begin{matrix} [c] c 0.7; q = 0 \\ 1; q = 1 \\ 0.3; q = 2 \\ 0.9; q = 3 \end{matrix}$

Now $\begin{matrix} (\begin{matrix} U_{2}^{*} (0) \\ U_{2}^{*} (1) \\ U_{2}^{*} (2) \\ U_{2}^{*} (3) \end{matrix}) & = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0.707 + i 0.707 & - 1 + 0 . i & 0 - i \\ 1 & - 1 + 0 . i & 1 + 0 . i & - 1 + 0 . i \\ 1 & 0 - i & - 1 + 0 . i & 0 + i \end{matrix}) (\begin{matrix} 0.7 \\ 1 \\ 0.3 \\ 0.9 \end{matrix}) \\ = \frac{1}{4} (\begin{matrix} 2.9 \\ 1.12 - 0.19 i \\ - 0.9 \\ 0.4 - 0.1 i \end{matrix}) \end{matrix}$

Now the absolute value matrix of the signal $U_{2}^{*} (m)$ is $(\begin{matrix} | U_{2}^{*} (0) | \\ | U_{2}^{*} (1) | \\ | U_{2}^{*} (2) | \\ | U_{2}^{*} (3) | \end{matrix}) = (\begin{matrix} 0.73 \\ 0.28 \\ 0.23 \\ 0.1 \end{matrix})$

Here the maximum value is $| U_{2}^{*} (0) | = 0.73$ .

Now for the signal $U_{3}^{*} (m)$ of A₃ aircraft; m = 0, 1, 2, 3, we have

$U^{*} [q] = {\begin{matrix} [c] c 0; q = 0 \\ 0; q = 1 \\ 1; q = 2 \\ 1; q = 3 \end{matrix}$

$\begin{matrix} (\begin{matrix} U_{3}^{*} (0) \\ U_{3}^{*} (1) \\ U_{3}^{*} (2) \\ U_{3}^{*} (3) \end{matrix}) & = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0.707 + i 0.707 & - 1 + 0 . i & 0 - i \\ 1 & - 1 + 0 . i & 1 + 0 . i & - 1 + 0 . i \\ 1 & 0 - i & - 1 + 0 . i & 0 + i \end{matrix}) (\begin{matrix} 0 \\ 0 \\ 1 \\ 1 \end{matrix}) \\ = \frac{1}{4} (\begin{matrix} 2 \\ 1 - i \\ 0 \\ - 1 + i \end{matrix}) \end{matrix}$

Now the absolute value matrix of the signal $U_{3}^{*} (m)$ is $(\begin{matrix} | U_{3}^{*} (0) | \\ | U_{3}^{*} (1) | \\ | U_{3}^{*} (2) | \\ | U_{3}^{*} (3) | \end{matrix}) = (\begin{matrix} 0.5 \\ 0.35 \\ 0 \\ 0.35 \end{matrix})$

Here the maximum value is $| U_{3}^{*} (0) | = 0.5$ .

Now for the signal $U_{4}^{*} (m)$ of A₄ aircraft; m = 0, 1, 2, 3, we have $U^{*} [q] = {\begin{matrix} [c] c 1; q = 0 \\ 0; q = 1 \\ 0; q = 2 \\ 0; q = 3 \end{matrix}$

$\begin{matrix} (\begin{matrix} U_{4}^{*} (0) \\ U_{4}^{*} (1) \\ U_{4}^{*} (2) \\ U_{4}^{*} (3) \end{matrix}) & = \frac{1}{4} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0.707 + i 0.707 & - 1 + 0 . i & 0 - i \\ 1 & - 1 + 0 . i & 1 + 0 . i & - 1 + 0 . i \\ 1 & 0 - i & - 1 + 0 . i & 0 + i \end{matrix}) (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) \\ = \frac{1}{4} (\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}) \end{matrix}$

Now the absolute value matrix of the signal $U_{4}^{*} (m)$ is $(\begin{matrix} | U_{4}^{*} (0) | \\ | U_{4}^{*} (1) | \\ | U_{4}^{*} (2) | \\ | U_{4}^{*} (3) | \end{matrix}) = (\begin{matrix} 0.25 \\ 0.25 \\ 0.25 \\ 0.25 \end{matrix})$

Here the maximum value is $| U_{4}^{*} (3) | = 0.25$ .

Now from the following table of maximum values $\begin{matrix} | r * (m) | & 0.53 \\ | U_{1}^{*} (M) | & 0.65 \\ | U_{2}^{*} (M) | & 0.73 \\ | U_{3}^{*} (M) | & 0.5 \\ | U_{4}^{*} (M) | & 0.25 \end{matrix}$ Since the absolute value of $U_{3}^{*} (M)$ is nearly same the absolute value of the reference signal. Thus the time domain signal $U_{3}^{*} (m)$ identifies as a reference signal.

Thus, we conclude that the other three aircraft A₁, A₂, and A₄ are from other countries except for A₃.

6 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), . . . , S_L (t), have been detected and sampled by a digital receiver. Each received signal is sampled N times. Let S_l (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 $S_{l} (k) = \frac{1}{N} \underset{n = 1}{\sum^{N}} A_{l, n} . e^{i \frac{2 π (n - 1) (k - 1) + α_{l, n}}{N}}$ where $C_{l, n} = A_{l, n} . e^{i \frac{α_{l, n}}{N}},$ with A_l,n, α_l,n are real-valued.

The Fourier coefficient of the reference signal R be C_R,n, where 1 ≤ n ≤ N . Thus $R (k) = \frac{1}{N} \underset{n = 1}{\sum^{N}} A_{R, n} . e^{i \frac{2 π (n - 1) (k - 1) + α_{R, n}}{N}}$ where $C_{R, n} = A_{R, n} . e^{i \frac{α_{R, n}}{N}},$ with A_R,n, α_R,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 $\begin{matrix} | μ_{S_{1, R}} | & 0.67 \\ | μ_{S_{2, R}} | & 0.45 \\ | μ_{S_{3, R}} | & 0.9 \\ | μ_{S_{4, R}} | & 0.63 \end{matrix}$ where $μ_{S_{l, R}} = \frac{\sum_{M} μ_{S_{l, R}} (n)}{m},$ Note that μ_{S
_l,R} denotes the total grade of similarity and μ_{S
_l,R} (n) is a complex grade of membership.

If threshold frequency is 0.8 then, |μ_{S
_3,R}| > μ_threshold . Thus S₃ (k) is the reference signal. $\begin{matrix} \Pr o p o s e d M e t h o d & | U_{1}^{*} (M) | & | U_{2}^{*} (M) | & | U_{3}^{*} (M) | & | U_{4}^{*} (M) | \\ Signal Values & 0.65 & 0.73 & 0.5 & 0.25 \\ Identify Signal & 3 rd Signal (U_{3}^{*} (k)) & - & - & - \\ Threshold Frquency & \times & - & - & - \\ R a m o t M e t h o d & | μ_{S_{1, R}} | & | μ_{S_{2, R}} | & | μ_{S_{3, R}} | & | μ_{S_{4, R}} | \\ Signal Values & 0.67 & 0.45 & 0.9 & 0.63 \\ Identify Signal & 3 rd Signal (S_{3} (k)) & - & - & - \\ Threshold Frquency & 0.8 & - & - & - \end{matrix} \begin{matrix} | r * (m) | \\ 0.53 \\ - \\ - \\ - \\ - \end{matrix}$

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.

7 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.

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