This paper presents a novel scheme for joint frequency and direction of arrival (DOA) estimation, that pairs frequencies and DOAs automatically without additional computations. First, when the property of the Kronecker product is used in the received array signal of the multiple-delay output model, the frequency-angle steering vector can be reconstructed as the product of the frequency steering vector and the angle steering vector. The frequency of the incoming signal is then obtained by searching for the minimal eigenvalue among the smallest eigenvalues that depend on the frequency parameters but are irrelevant to the DOAs. Subsequently, the DOA related to the selected frequency is acquired through some operations on the minimal eigenvector according to the Rayleigh–Ritz theorem, which realizes the natural pairing of frequencies and DOAs. Furthermore, the proposed method can not only distinguish multiple sources, but also effectively deal with other arrays. The effectiveness and superiority of the proposed algorithm are further analyzed by simulations.

Frequency estimation and DOA estimation for an antenna array are the basic but key problems in the area of array signal processing [

Using multiple-delay outputs, the authors in [

To pair the frequency and angle estimations automatically and avoid potential pairing mistakes, relative to the multiple-delay output model used in [

An outline of this paper is as follows. Section 2 describes the data model. Section 3 addresses the algorithmic issues encountered with some remarks. Section 4 shows the simulation results, which are followed by conclusions in Section 5.

_{M}

Consider an arbitrary array of

where

where

The received signal of array antennas can be shown in matrix form as

where

The angle steering vector is

Afterwards, the delayed signal for

where

Therefore, the delayed signal for

where

According to

Define the frequency steering vector

where

From _{k}_{k}_{xx}^{H}. Using the eigenvalue decomposition of _{xx}_{n}_{xx}_{k}

However, to obtain all source parameters

By the property of the Kronecker product, the frequency-angle steering vector can be reformulated in the following form

From _{k}

Inserting

Define

where

Since

Notice that

If

then, _{k}

and

where

To solve for

in which

For an unambiguous uniform linear array, the interspacing between two adjacent elements _{q}_{k}

It should be pointed out that _{k}_{k}

The proposed method can also address the case in which multiple signals have the same frequency but different DOAs by distinguishing the number of minimal eigenvalues with respect to _{k}

Because the frequency is obtained via an eigenvalue and the angle is obtained via an eigenvector, this method is named the EVEV algorithm.

It is worth noting that to obtain the frequency _{k}_{k}

By the eigenvalue decomposition of _{xx}_{s}_{s}_{1} (the first _{s}_{2} (the last _{s}

Then we can obtain the eigenvalues _{k}

Subsequently, based on the Rayleigh–Ritz theorem, _{k}

In this section, we analyze the computational complexities of the EVEV and ESEV algorithms and compare them with that of the ESES method [_{xx}_{n}_{xx}

It can be concluded that the EVEV method has the largest computational complexity among these three methods if the number of search grids

From these aforementioned derivations, some remarks can be made:

_{k}_{k}

_{k}

The performances of the proposed methods are evaluated with some Monte Carlo simulations. In the following experiments, assume that _{s}_{Z}

_{Z}_{Z}_{Z}_{Z}

Frequency (Hz) | ESES | EVEV | ESEV | ||
---|---|---|---|---|---|

Angle 1 | Angle2 | Angle 3 | Angle | Angle | |

200,000 | 46.0 | 20.0 | – | 20.0 | 20.0 |

400,000 | 69.7 | 61.9 | 50.0 | 50.0 | 50.0 |

800,000 | 80.0 | 76.4 | 71.3 | 80.0 | 80.0 |

_{Z}_{Z}_{Z}

From

This paper has considered the problem of joint frequency and DOA estimation in array signal processing. By applying the Rayleigh–Ritz theorem to the multiple-delay output model, the proposed methods can achieve automatically paired frequency and DOA estimations without an exhaustive two-dimensional search. The investigations, which were demonstrated in theory and through the use of extensive simulations, have shown that the proposed methods have better estimation performances than those of the ESES and ESEV methods, making a good compromise between estimation accuracy and computational complexity compared with EVEV and ESES. Furthermore, the proposed two methods can always resolve no fewer sources than can ESES. In addition, these methods can also be used for other arrays, including but not limited to nonuniform linear arrays. Future research might be expected to extend the applications of these methods to situations in the presence of multiple coherent sources, array gain-phase errors or mutual coupling effects among the examined sensors.

The authors acknowledge the help of their colleagues and families.