# A Novel DOA Estimation Algorithm Using Array Rotation Technique

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## Abstract

**:**

## 1. Introduction

## 2. The MUSIC Algorithm of 2-D UCA

_{i}(t), i = 1, 2, …, D (D < M), where t is the time variable, as shown in Figure 1. The D sources are assumed to be from far-field with azimuth θ

_{i}and elevation φ

_{i}, i = 1, 2, …, D. Assume the radius of UCA is r and the noise is additive white Gaussian noise (AWGN).

**A**(θ,φ) = [

**a**(θ

_{1},φ

_{1}),

**a**(θ

_{2},φ

_{2}), ⋯,

**a**(θ

_{D},φ

_{D})]

**a**(θ

_{i},φ

_{i}) = [a

_{1}(θ

_{i},φ

_{i}),a

_{2}(θ

_{i},φ

_{i}), ⋯, a

_{M}(θ

_{i},φ

_{i})]

^{T}

_{k}(θ

_{i},φ

_{ki}) = exp(− jωτ

_{i}) = exp(− j dr cos (2π(k − 1)/ M − θ

_{i}) cos φ

_{i})

**X**(t)=

**AS**(t)+

**N**(t)

**X**(t) is M × 1 array output vector;

**A**is M × D array steering matrix;

**S**(t) is D × 1 signal vector;

**N**(t) is M × 1 noise vector.

**X**(t),

**S**(t) and

**N**(t) are abbreviated as

**X**,

**S**and

**N**, respectively, and the array covariance matrix of

**X**can be written as

**R**= E[

**XX**

^{H}]=

**APA**

^{H}+σ

^{2}

**I**

**P**= E[

**SS**

^{H}] = diag[P

_{1}, P

_{2}, …, P

_{i}, …, P

_{D}]; P

_{i}is the power of the ith signal; σ

^{2}is noise power;

**I**is M × 1 identity matrix. E[

**•**] and [

**•**]

^{H}denote the statistical expectation and the Hermitian transpose, respectively. In real systems, the covariance matrix

**R**can be estimated from a finite set of sample snapshots as

**∑**= diag{λ

_{1}≥ λ

_{2}, ⋯, λ

_{M}}; λ

_{i}and

**v**

_{i}are the eigenvalue and corresponding eigenvector of , respectively. In ideal conditions, λ

_{1}≥ λ

_{2}⋯ ≥ λ

_{D}≥ λ

_{D }

_{+1}= ⋯ = λ

_{M }= σ

^{2}. Assume that the number of incident signals D is known, can be described as

**U**

_{S}= [

**v**

_{1},

**v**

_{2}, ⋯,

**v**

_{D}] and

**U**

_{N}= [

**v**

_{D}

_{+1},

**v**

_{D}

_{+2}, ⋯,

**v**

_{M}] are the signal subspace and noise subspace, respectively;

**∑**

_{S}= diag{λ

_{1, }⋯, λ

_{D}} and

**∑**

_{N}= diag{λ

_{D+}

_{1}, ⋯, λ

_{M}} are diagonal matrices related to the signal and noise power, respectively. Because the D-incident sources spanning the signal space are orthogonal to the noise space, so the MUSIC algorithm can estimate the DOA as

_{MUSIC}(θ,φ) is expected to show a large positive value if (θ,φ) is a true DOA, because

**a**

^{H}(θ,φ) v

_{i}= 0, i = D +1, ⋯, M. Here, the signal is processed before the data is demodulated, where the carrier phase information is maintained through the sampling.

## 3. The Proposed Algorithm

#### 3.1. The Rotation Array Structure of Proposed Algorithm

_{Z}, v

_{Z}= ω

_{Z}r = 2πƒ

_{z}r, where ω

_{Z}is rotational angular frequency; ƒ

_{z}is rotation frequency. The rotation period is T. After time Δt (Δt < T), the elements 1 and 2 are rotating to the position 1ʹ and 2ʹ, respectively. Assume the array radius is r; l is the incident signal direction; lʹ is the projection of l onto the XOY plane; θ and φ are azimuth and elevation, respectively.

- (1)
- While the baseline 1–2 is rotating, the baseline is certainly vertical to Z-axis with absolute uniform velocity;
- (2)
- Select 2M elements at uniformly-time interval to make them form the virtual UCA within T/2 period;
- (3)
- The signals remain static during the measurement time.

_{m}, the elements 1 and 2 start to sample the received signal data, so t

_{m}can be expressed as

_{m}= t

_{1 }+ (m − 1)∆τ

_{1 }= 0, m = 1,2, ⋯,M ; ∆τ is the time delay which is needed for selecting two neighboring elements, 0 < ∆τ < T / 2. As we know, the wave path difference between the reference point O and the mth element for the ith signal is τ

_{mi}. In the rotation array model, the element m is obtained by rotating element 1 through time (m − 1)∆τ, so the total wave path difference τ

_{1m}is

_{1m}= τ

_{mi }+ (m−1)

_{∆}τ

_{z}in the source incident direction Ɩʹ, we obtain

_{z}sin(θ − ω

_{z}t) cos φ

_{d}= v•ƒ /c, ƒ

_{d}can be written as

_{d}= v

_{z}sin(θ − ω

_{z}t) cos φ •ƒ /c

= 2πƒ

_{z}r sin(θ − ω

_{z}t) cos φ •ƒ / c

_{d})

_{tm}= exp(− j ω(m − 1)∆τ), the vector received by element 1 within T/2 period can be expressed by

_{tm}is the phase difference of element 1 at time t

_{m}relative to the initial position. Equation (17) can be further simplified as

**X**

_{1}= [x

_{11}, x

_{12}, ⋯, x

_{1M}]

^{T}is M × 1 array output matrix;

**N**

_{1}= [n

_{1}(t

_{1}), n

_{1}(t

_{2}), ⋯, n

_{1}(t

_{M})]

^{T}is M × 1 noise matrix. The steering matrix

**A**

_{1}can be written as

**A**

_{1}= [

**a**

_{1}(θ

_{1}, φ

_{1}),

**a**

_{1}(θ

_{2}, φ

_{2}), ⋯,

**a**

_{1}(θ

_{D}, φ

_{D})]

**a**

_{1}(θ

_{j}, φ

_{j}) = [a

_{11}(θ

_{1}, φ

_{1}), a

_{12}(θ

_{2}, φ

_{2}), ⋯, a

_{1M }(θ

_{D}, φ

_{D})]

^{T}

_{1m}(θ

_{i},φ

_{i}) = exp(− j ωτ

_{mi}); ; m = 1,2⋯, M; i = 1,2, ⋯, D.

**Φ**

_{1}= diag[ϕ

_{t1},ϕ

_{t2}, ⋯, ϕ

_{tM}], (17) can be expressed by

**X**

_{1}=

**Φ**

_{1}

**A**

_{1}

**S**+

**N**

_{1}

**Φ**

_{2 }= diag[ϕ

_{t1}, ϕ

_{t2}, ⋯, ϕ

_{tM}]; ϕ

_{tm}is the phase difference of element 2 at time t

_{m}relative to the initial position; therefore, the array output matrix of element 2 within T/2 is modeled as

**X**

_{2}=

**Φ**

_{2}

**A**

_{2}

**S**+

**N**

_{2}

**X**

_{1 }with

**X**

_{2},

**X**can be represented by

**X**

_{1 }and

**X**

_{2}

#### 3.2. How to Choose Array Rotation Velocity

_{z}, and ƒ

_{z}= 1/T, ƒ

_{s }is the sampling frequency of the receiver. Select 2M antenna elements within T/2 period while array rotating. Because the characteristics of element 1 are the same as element 2, we only need to analyze element 1 in this subsection. As we know, the received signal data is sampled while the array is rotating, which will cause a tiny phase difference between every two sampled data. In order to ensure the stability of sampled data, ∆τ should satisfy the following equation

_{z}≪ 31 250 Hz. If the antenna rotation frequency is much smaller than 31,250 Hz, the stability of the sampled data will be guaranteed.

## 4. The Cramer-Rao Bound

^{H}(t)} = σ

^{2}I, E{n(t)n

^{T}(t)} = 0; (3) The signal covariance matrix R

_{S}= E{

**SS**

^{H}} is positive definite. Furthermore, the signals and noise are uncorrelated for all time. Under these conditions, we derive the Cramer-Rao bound (CRB) formula for the algorithm proposed in this paper. The derivation process approximates the method proposed in [24].

**u**is the angle parameters vector contained in the signal covariance matrix

**R**

_{S},

**u**= [

**θ**

^{T},

**φ**

^{T}]

^{T}

**θ**= [θ

_{1}, ⋯, θ

_{D}]

^{T},

**φ**= [φ

_{1}, ⋯, φ

_{D}]

^{T}. The CRB of the angle parameters is defined as

**u**) ≥ CRB

**u**) = E[(

**û - u**)(

**û - u**)

^{T}]

**F**

^{-1}

**u**is given by

**F**

_{θθ }is the block matrix of azimuth estimator and

**F**

_{φφ}is the block matrix of elevation estimator. The m, n elements of

**F**is represented as

**G**= [

**G**

_{θ},

**G**

_{φ}]

**F**

_{θθ},

**F**

_{φφ},

**F**

_{φθ},

**F**

_{θφ}, is represented as

## 5. Simulation Examples

_{z}is 15 Hz, and the noise background is AWGN. Select M = 8 and M = 10 elements for forming the UCA to estimate DOAs. To verify the performance of the R-MUSIC algorithm, some comparison simulations of MUSIC algorithm with five elements UCA and eight elements UCA are carried as well.

#### 5.1. Resolution Performance Simulation

_{m}= (θ

_{1}+ θ

_{2})/ 2. When the right-hand side of (30) is smaller than the left-hand side, the two angles can be distinguished; while the right-hand sides of (30) is great than the left-hand sides, then the two angles cannot be distinguished. Furthermore, the successful resolution probability is defined as the ratio of successful test numbers to the total test numbers.

#### 5.1.1. The Spatial Spectrum of the R-MUSIC Algorithm

#### 5.1.2. The Resolution Probability versus SNR

**Figure 3.**Spatial spectrum with multiple signals. (

**a**) Two signal DOAs are (90°, 70°) and (90°, 80°); (

**b**) Three DOAs are (60°, 30°), ( 90°, 60°) and ( 120°, 80°); (

**c**) Four DOAs are (60°, 40°), (90°, 80°), (120°, 60°) and (140°, 40°).

#### 5.2. Estimation Accuracy Performance Simulation

#### 5.3. Channel Mismatch Errors Simulation

Channel mismatch errors [degree] | 5-elment UCA-MUSIC | 8-element R-MUSIC |
---|---|---|

0 | 0.1414 | 0.2739 |

5 | 0.2191 | 0.2162 |

10 | 0.4817 | 0.4000 |

15 | 0.5441 | 0.4427 |

20 | 0.6132 | 0.4336 |

#### 5.4. Resolution Performance versus Rotation Frequency Errors

_{z}= 15H

_{Z}. However, in fact, the rotation velocity will be greater or smaller than the theory rotation frequency ƒ

_{z}. Now, assume the actual rotation frequency is ƒʹ, and let ƒʹ range from 14.3 to 15.5 Hz by the step 0.1 Hz in this simulation. So, the rotation frequency error is ∆ƒ = ƒʹ − ƒ

_{z}. shows the successful resolution probability versus rotation frequency error when SNR = 4 dB, 8dB, 10 dB and 20 dB, successively.

_{z}and the SNR>8 dB, the successful resolution probability of the R-MUSIC is 100%; However, when the actual rotation frequency ƒʹ is larger or smaller than ƒ

_{z}, the successful resolution probability will decrease. The larger the |∆ƒ| is, the lower the successful resolution probability is. On the other hand, when SNR = 4 dB, the successful rotation frequency (>60%) ranges from 15 − 0.3 ~ 15 + 0.3 Hz; When SNR = 10 dB, the successful rotation frequency ranges from 15−0.4 ~ 15 + 0.4 Hz; When SNR = 20 dB, the successful rotation frequency ranges from 15 − 0.5 ~ 15 + 0.4 Hz; that is to say, with increases in SNR, the rotation array system can tolerate wider rotation frequency error range. However, when ∆ƒ is larger than a certain value, even increasing SNR would not improve resolution. That is because according to Equation (15), a larger rotation frequency error will cause a Doppler frequency shift, which has a great influence on the estimation and resolution performance of the proposed algorithm.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Lan, X.; Wan, L.; Han, G.; Rodrigues, J.J.P.C.
A Novel DOA Estimation Algorithm Using Array Rotation Technique. *Future Internet* **2014**, *6*, 155-170.
https://doi.org/10.3390/fi6010155

**AMA Style**

Lan X, Wan L, Han G, Rodrigues JJPC.
A Novel DOA Estimation Algorithm Using Array Rotation Technique. *Future Internet*. 2014; 6(1):155-170.
https://doi.org/10.3390/fi6010155

**Chicago/Turabian Style**

Lan, Xiaoyu, Liangtian Wan, Guangjie Han, and Joel J. P. C. Rodrigues.
2014. "A Novel DOA Estimation Algorithm Using Array Rotation Technique" *Future Internet* 6, no. 1: 155-170.
https://doi.org/10.3390/fi6010155