# The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Definition**

**1.**

**Theorem**

**1.**

- (i)
**0**$\perp A$ for all $A\in {F}^{C}\left(U\right)$,- (ii)
- if $A\perp B$ then $B\perp A$,
- (iii)
- if $A\perp A$ then $A=$
**0**.

**Proof.**

**Remark**

**1.**

**0**denote the complex fuzzy set where the membership vectors of each element of the universal set is the zero vector.

**Problem**

**1.**

**Problem**

**2.**

## 3. Results

**Definition**

**2.**

**Theorem**

**2.**

- (i)
- If $A\perp B$ then $Ref\left(A\right)\perp Ref\left(B\right)$.
- (ii)
- If $A\perp B$ then $Ro{t}_{\theta}\left(A\right)\perp Ro{t}_{\theta}\left(B\right)$ for any θ radians.

**Proof.**

#### 3.1. Complex Fuzzy Complement

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

#### 3.2. Complex Fuzzy Union

**Theorem**

**4.**

**Proof.**

**Example**

**2.**

**Theorem**

**5.**

**Proof.**

**Example**

**3.**

**Corollary**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Example**

**4.**

#### 3.3. Complex Fuzzy Intersection

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Corollary**

**10.**

**Proof.**

- (i)
- $A\perp B,\Rightarrow A\circ C\perp B\circ C$: Sum (See Theorems 4 and 7);
- (ii)
- $A\perp B,A\perp C\Rightarrow A\perp B\circ C$: Max, Min and Winner Take All (See Theorems 5 and 8);

#### 3.4. Complex Fuzzy Inference

Premise: | X is ${A}^{*}$; |

Rule: | IF X is A, THEN Y is B; |

Consequence: | Y is ${B}^{*}$ (denote $CFI(A,B;{A}^{*})$). |

**Definition**

**3.**

**Theorem**

**11.**

**Proof.**

**Example**

**5.**

**Example**

**6.**

#### 3.5. Example Application

- Step 1
- Normalize the amplitudes of all Fourier coefficients. Let ${\mathbf{A}}_{l}=({A}_{l,1},{A}_{l,2},...,{A}_{l,N})$ be the vector of amplitudes of ${S}_{l}$’s Fourier coefficients, ($1\le l\le L$). Let ${\mathbf{A}}_{R}=({A}_{R,1},{A}_{R,2},...,{A}_{R,N})$ be the vector of amplitudes of ${S}_{R}$’s Fourier coefficients, ($1\le l\le L$). Let ${\mathbf{B}}_{l}$ be the normalized vector $1/norm\left({\mathbf{A}}_{l}\right)\xb7{\mathbf{A}}_{l}$, where $norm\left({\mathbf{A}}_{l}\right)=\sqrt{{\Sigma}_{n=1}^{N}{\left({A}_{l,n}\right)}^{2}}$. Let ${\mathbf{B}}_{R}$ be the normalized vector $1/norm\left({\mathbf{A}}_{R}\right)\xb7{\mathbf{A}}_{R}$. Then ${\mathbf{B}}_{l}=({B}_{l,1},{B}_{l,2},...,{B}_{l,N})$ is the vector of normalized amplitudes of ${S}_{l}$’s Fourier coefficients. ${\mathbf{B}}_{R}=({A}_{R,1},{A}_{R,2},...,{A}_{R,N})$ is the vector of normalized amplitudes of R’s Fourier coefficients.
- Step 2
- Composition the t samples ${S}_{l}\left(t\right)$, $1\le t\le N$ for each signal ${S}_{l}$ ($1\le l\le L$). Define new complex fuzzy sets as:$$\begin{array}{c}\hfill {B}_{l}{e}^{{a}_{l}}={B}_{l,1}\oplus {B}_{l,2}\oplus \cdots \oplus {B}_{l,n}{e}^{{a}_{l,1}\otimes {a}_{l,2}\otimes \cdots \otimes {a}_{l,N}}\end{array}$$Similarly, define a new complex set as:$$\begin{array}{c}\hfill {B}_{R}{e}^{{a}_{R}}={B}_{R,1}\oplus {B}_{R,2}\oplus \cdots \oplus {B}_{R,n}{e}^{{a}_{R,1}\otimes {a}_{R,2}\otimes \cdots \otimes {a}_{R,N}}\end{array}$$
- Step 3
- For each ${B}_{l}$ ($1\le l\le L$), define its in-phase and quadrature terms, respectively, as:$$\begin{array}{c}\hfill {I}_{l}={B}_{l}\mathrm{cos}\left({a}_{l}\right)\end{array}$$$$\begin{array}{c}\hfill {Q}_{l}={B}_{l}\mathrm{sin}\left({a}_{l}\right).\end{array}$$Similarly, define R’s in-phase and quadrature terms, respectively, as:$$\begin{array}{c}\hfill {I}_{R}={B}_{R}\mathrm{cos}\left({a}_{R}\right)\end{array}$$$$\begin{array}{c}\hfill {Q}_{R}={B}_{R}\mathrm{sin}\left({a}_{R}\right).\end{array}$$
- Step 4
- Calculate the distance between ${S}_{l}$ ($1\le l\le L$) and R:$$\begin{array}{c}\hfill d({S}_{l},R)=\frac{1}{2}\mathrm{max}\left(\right|{I}_{R}-{I}_{l}|,|{Q}_{R}-{Q}_{l}\left|\right)\end{array}$$
- Step 5
- In order to conclude if ${S}_{l}$ may be identified as R, compare $1-d({S}_{l},R)$ to a threshold $\delta $. If $1-d({S}_{l},R)$ exceeds the threshold, identify ${S}_{l}$ as R.

**Example**

**7.**

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Hu, B.; Bi, L.; Dai, S.
The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection. *Symmetry* **2017**, *9*, 175.
https://doi.org/10.3390/sym9090175

**AMA Style**

Hu B, Bi L, Dai S.
The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection. *Symmetry*. 2017; 9(9):175.
https://doi.org/10.3390/sym9090175

**Chicago/Turabian Style**

Hu, Bo, Lvqing Bi, and Songsong Dai.
2017. "The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection" *Symmetry* 9, no. 9: 175.
https://doi.org/10.3390/sym9090175