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The performance of traditional direction of arrival (DOA) estimation algorithm based on uniform circular array (UCA) is constrained by the array aperture. Furthermore, the array requires more antenna elements than targets, which will increase the size and weight of the device and cause higher energy loss. In order to solve these issues, a novel low energy algorithm utilizing array base-line rotation for multiple targets estimation is proposed. By rotating two elements and setting a fixed time delay, even the number of elements is selected to form a virtual UCA. Then, the received data of signals will be sampled at multiple positions, which improves the array elements utilization greatly. 2D-DOA estimation of the rotation array is accomplished via multiple signal classification (MUSIC) algorithms. Finally, the Cramer-Rao bound (CRB) is derived and simulation results verified the effectiveness of the proposed algorithm with high resolution and estimation accuracy performance. Besides, because of the significant reduction of array elements number, the array antennas system is much simpler and less complex than traditional array.

With the development of array signal processing, the direction of arrival (DOA) estimation algorithms are used widely in radar, sonar, atmosphere, communication and so on. Since the 1980s, many high resolution 2-dim (2-D) DOA algorithms were proposed in succession. Among them, the multiple signal classification (MUSIC) algorithm [

In this paper, a novel rotation MUSIC (R-MUSIC) algorithm for static targets based on the array rotating technique is proposed. This algorithm can obtain any even number of elements by rotating only two elements, so that it can receive the signal data at multiple positions. Besides, the R-MUSIC algorithm can break the constraints that the number of antenna elements must be more than the number of incident signals. Most importantly, the array system will be much simpler than the traditional array. Computer simulations verified the effectiveness and superior performance of the proposed method.

The remainder of this paper is organized as follows.

Consider a UCA with _{i}_{i}_{i}

Uniform circular array diagram.

Consider the reference point is 0, the ideal steering matrix can be obtained from the array geometry as
_{1},_{1}), _{2},_{2}), ⋯, _{D}_{D}_{i}_{i}_{1}(_{i}_{i}_{2}(_{i}_{i}_{M}_{i}_{i}^{T}_{k}_{i}_{ki}_{i}_{i}_{i}

Equation (4) can be written in matrix form as
^{H}]=^{H}+σ^{2}^{H}] = diag[_{1}, _{2}, _{i}_{D}_{i}^{2} is noise power; ^{H} denote the statistical expectation and the Hermitian transpose, respectively. In real systems, the covariance matrix _{1} ≥ _{2}, ⋯, _{M}_{i}_{i}_{1} ≥ _{2} ⋯ ≥ _{D} _{D }_{+1} = ⋯ = _{M }^{2}. Assume that the number of incident signals _{S} = [_{1}, _{2}, ⋯, _{D}_{N} = [_{D}_{+1}, _{D}_{+2}, ⋯, _{M}_{S} = diag{_{1, }⋯, _{D}_{N} = diag{_{D+}_{1}, ⋯, _{M}_{MUSIC} (^{H} (_{i}

As shown in _{Z}_{Z}_{Z}r_{z}r_{Z}_{z}

Rotation array structure diagram.

In order to derive the proposed algorithm, some assumptions are clarified at first,

While the baseline 1–2 is rotating, the baseline is certainly vertical to

Select 2

The signals remain static during the measurement time.

At time _{m}_{m}_{m}_{1 }_{1 }_{mi}_{1m} is
_{1m} = _{mi }_{∆}

Then Equation (4) is modified as

Assuming the signal sources remain static, the element rotation will cause the Doppler frequency shift. As shown in _{z}_{z}_{z}t

According to the Doppler frequency formula _{d} = _{d} can be written as
_{d} = _{z}_{z}t_{z}r_{z}t_{d})

Let _{tm}_{tm}_{m}_{1} = [_{11}, _{12}, ⋯, _{1M}]^{T} is _{1} = [_{1}(_{1}), _{1}(_{2}), ⋯, _{1}(_{M}^{T} is _{1} can be written as
_{1} = [_{1}(_{1}, _{1}), _{1}(_{2}, _{2}), ⋯, _{1}(_{D}_{D}_{1}(_{j}_{j}_{11}(_{1}, _{1}), _{12}(_{2}, _{2}), ⋯, _{1M }_{D}_{D}^{T}_{1m}_{i}_{i}_{mi}

Denote _{1} = diag[_{t1}_{t2}_{tM}_{1} = _{1}_{1}_{1}

In the same way, the vector received by element 2 within

Similarly, denoting _{2 }= diag[_{t1}_{t2}_{tM}_{tm}_{m}_{2} = _{2}_{2}_{2}

When the array elements are rotating more than _{1 }with _{2}, _{1 }and _{2}

Using Equations (7), (9) and (10), MUSIC spectrum function will be obtained.

Seen from the analysis above, the key advantages of the proposed method is, by sampling the received signal date at a uniform time interval while the array antennas are rotating, more than two antenna elements can be obtained and it can estimate more than two DOAs. Besides, because the number of antennas is much smaller, the array system is greatly simplified than the traditional array and it will become much easier to calibrate with channel phase errors in practice.

Consider the rotation frequency is ƒ_{z}_{z}_{s }

Assume the snapshots is

As a result, if (28) is satisfied, the stability of the algorithm could be ensured. Consider the sampling frequency of a receiver is 50 MHz, the elements number is 8, the snapshots number _{z}

Cramer-Rao bound (CRB) gives a lower bound of unbiased parameter estimation. In this section, some assumptions are considered to hold throughout this section first: (1) The number of selected array elements should be greater than that of signals (^{H}(^{2}^{T}(_{S} = E{^{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 [

Define _{S},
^{T},^{T}]^{T}_{1}, ⋯, _{D}^{T}, _{1}, ⋯, _{D}^{T}. The CRB of the angle parameters is defined as
^{T}]^{-1}

The 2_{θθ }_{φφ}_{θ}_{φ}

Then the matrix _{θθ}_{φφ}_{φθ}_{θφ}

This section demonstrates the performance of the proposed method via numerical simulation. In all simulation examples, we use the rotation array structure shown in _{z}

In this part, the resolution is defined as
_{m}_{1} + _{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.

In order to verify the effectiveness of the proposed method, some spatial spectrum figures are shown in this section. Assume there are several incoherent signals impinging on the array, the signal frequencies are 6 GHz, SNR (signal to noise ratio) is 20 dB and the snapshots are 100.

Two signal DOAs are (90°, 81°) and (90°, 85°), respectively, and both the signal frequencies are 6 GHz.

Spatial spectrum with multiple signals. (

Successful resolution probability

Successful resolution probability

Two signal DOAs are (90°, 81°) and (90°, 85°), respectively, both the signal frequencies are 6 GHz.

RMSE

RMSE

The influence of channel mismatch errors on the resolution probability of the eight-element R-MUSIC and five-element MUSIC is shown in

It can be seen from the

Successful resolution probability

The RMSE

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

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 |

There are two inherent incident signals with DOAs are (90°, 81°) and (90°, 85°) respectively, both the signal frequencies are 6 GHz. The snapshots number is _{z}_{Z}. However, in fact, the rotation velocity will be greater or smaller than the theory rotation frequency ƒ_{z}_{z}

As shown in _{z}_{z}

Successful resolution probability

The performance of classical MUSIC algorithm based on UCA is constrained by array aperture greatly, and most DOA estimation algorithms demand that the number of elements be larger than that of incident signals. Focusing on this problem, a novel multiple DOAs estimation algorithm based on rotation array is proposed in this paper, which has lower energy loss and complexity. Computer simulations verify the effectiveness of the proposed method, and the number of incident signals that the algorithm could estimate is more than that of the elements. Besides, the proposed array model could be used for any algorithm based on UCA. In the future work, we will focus on the application of the proposed algorithm [

This work was supported by the National Natural Science Foundation of China (Grant No 61101161). Guangjie Han’s work is supported by “Natural Science Foundation of Jiangsu Province of China, No.BK20131137”. This work has been partially supported by Instituto de Telecomunicações, Next Generation Networks and Applications Group (NetGNA), Covilhã Delegation, by National Funding from the FCT—Fundação para a Ciência e a Tecnologia through the Pest-OE/EEI/LA0008/2013 Project.

The authors declare no conflict of interest.