#### 3.1. Space–Time Covariance Matrix

In practice, it is almost impossible to count an ideal covariance matrix. Hence, in algorithm execution process, covariance matrix must be estimated by a limited

N snapshots sample, which is given as:

We assume that all the incoming signals are narrowband sources, therefore the following holds:

where

m∈N, and

z = exp(

j2

πf_{c}/

f_{s}) is the time delay factor. Moreover,

f_{s} denotes the sampling pulse repetition frequency, and

f_{c} denotes the target carrier frequency. Here, we set

Q time delay taps and only one sampling pulses between adjacent two time delay taps. The received data vector operated by STP can be written as:

where ⊗ is Kronecker product,

**I**_{Q}_{×Q} denotes the

Q ×

Q identity matrix and

**Z** denotes the

Q × 1matrix expressed as

According to the aforementioned assumptions, the received signal covariance matrix by STP can be calculated by:

where

**I**_{QL×QL} is the Q

L × Q

L identity matrix, and

**∑**_{N} and

**∑**_{F} are diagonal matrices,

#### 3.2. FF Sources DOA and MCCs Estimation

By implementing eigenvalue decomposition (EVD) of

**R**_{ST}, the following equation holds:

where,

**Λ**_{s} is the diagonal matrix which contains the

K largest eigenvalues. At the same time,

**U**_{n} is the Q

L × (Q

L −

K) eigenvectors matrix which spans the noise subspace of

**R**_{ST}, and

**U**_{s} is the Q

L ×

K signal eigenvectors matrix of

**R**_{ST} which spans the signal subspace.

As the fact that the MCM is a column full rank Toeplitz matrix, the authors can construct a MUSIC spectrum search function to estimate DOAs and ranges parameters, which is expressed as:

From Formula (23), it is obvious that the computational cost of this equation is unbearable with the unknown **C**. Even if **C** is accurately estimated as a prior, it requires a 3D spectrum search to estimate and match the DOA and range parameters pair as well. In order to decrease the huge computational cost, the DOA estimation of FF sources must be decoupled from mixed sources estimates and MCM.

Referring to the discussion of mutual coupling problem with (ULA) in [

5],

**Ca**(

θ,

φ,

r) in UCA could be reformulated as:

where

**c** is the vector of

P + 1 nonzero MCCs, which expressed as:

**B**(

θ,

φ,

r) =

**B**_{1} +

**B**_{2} is composed of two

L × (

P + 1) matrices, and they are defined as:

where, {∙}

_{p}_{,q} represents the element corresponding to the

p-th row and

q-th column of the matrix, [∙]

_{p} represents the element corresponding to

p-th element of steering vector, and mod(

p,

q) denotes the modulus after dividing

p by

q.

It is obvious that

**U**_{s} and the combination of (

**I**_{Q}_{×Q}⊗

**C**)(

**Z**⊗

**A**_{N}) and (

**I**_{Q}_{×Q}⊗

**C**)(

**Z**⊗

**A**_{F}) can span the same signal subspace, and the signal subspace is orthogonal to the noise subspace spanned by

**U**_{n}. Therefore, the following equations hold:

Therefore, based on Formula (24)–(29), the FF sources’ DOAs can be estimated by the following spectrum search function:

where

**W**(

θ,

φ) is defined as:

Note that

**c** ≠

**0** and

**W**(

θ,

φ) is a Hermite and nonnegative definite matrix. Based on the principle of RERA [

29],

**c**^{H}**W**(

θ,

φ)

**c** would be zero only when

**W**(

θ,

φ) is a singular matrix. In other words, the determinant of

**W**(

θ,

φ) would be equal to zero only when both parameters

φ and

θ are equal to any FF source’ DOA expressed by

φ_{k} and

θ_{k} (

k = 1 +

K_{1},…,

K). Consequently, the FF sources’ DOAs could be estimated accurately by searching the

K_{2} highest spectrum peaks through the following function:

where, det[∙] signifies the determinant of a matrix. From equation (32), it is easy seen that

p_{F}(

θ,

φ) is independently separated from equation (30). On the other hand, since the smallest eigenvalue of

**W**(

θ,

φ) is also equal to zero when parameters

φ and

θ are equal to

φ_{k} and

θ_{k} (

k = 1 +

K_{1},…,

K), the DOAs can also be estimated from the following spatial spectrum searching:

where,

λ_{min}[∙] denotes the smallest eigenvalue of a matrix. Therefore, the computational cost of estimating FF sources’ DOAs through (32) or (33) is effectively reduced compared with the multi-dimensional MUSIC spectrum search function. It is because of that, that only a 2D spectrum search process is required.

It is noteworthy that the proposed algorithm works in a similar way as that defined in [

28,

29]. However, the algorithm in [

28] only solves the problem of pure FF signals, whereas the proposed algorithm aims to deal with the problem of mixed FF and NF signals. Moreover, the algorithm in [

29] solves the mixed sources problem using a ULA. However, our work explicitly addresses the mixed sources problem under mutual coupling effect, using a UCA.

Substituting the estimated

φ_{k} and

θ_{k} (

k = 1 +

K_{1},…,

K) and then perform eigendecomposition of

**W**(

θ_{k},

φ_{k}), we can obtain the smallest eigenvalue

λ_{min} and the corresponding eigenvector

**v**_{min}, where

**v**_{min} has been normalized by its first element. Finally, we can obtain the non-zero MCCs by:

According to the symmetric Toeplitz structure modeled as Formula (6), the MCM can be reconstructed after all MCCs have been calculated. Therefore, the reconstructed MCM could be used to eliminate the mutual coupling effects in the following signal processing process.

#### 3.3. NF Sources DOA Estimation

This section begins with the discussion of MCM. It is noteworthy that the MCM of an UCA is different to that in an ULA, and it has some special properties based on its special array structure.

Remark 1:

The MCM

**C** is a symmetric Toeplitz matrix, which holds:

and its inverse matrix is also a symmetric Toeplitz matrix, which holds:

Remark 2:

With

**C** being a complex symmetric Toeplitz matrix,

**CC**^{H} is a real symmetric Toeplitz matrix (see

Appendix A), which holds:

and the inverse matrix of

**CC**^{H} is also a real symmetric Toeplitz matrix, which holds:

As known, the estimation accuracy of NF sources will be improved as the SNR increases. If we can eliminate the noise, the SNR of NF will be approximately equal to infinity. Fortunately, the special properties of the MCM and array structure in UCA can be used to eliminate the noise and FF components effectively. Consequently, we can deal with the NF sources estimation problem only in a space domain with a high enough SNR.

According to Formula (17) and (19), the received signal covariance matrix only in space domain is expressed as:

After estimating and reconstructing mutual coupling matrix, the mutual coupling effects can be eliminated effectively. Therefore, we can get:

Based on the symmetric property of UCA configuration with an even number of sensors, the following holds:

Due to the fact that

**R**_{F} is a Hermitian matrix but

**R**_{N} only holds a Hermitian structure, we can obtain the following:

where

**J** is the

L ×

L particular transformational matrix which can be defined as:

where,

**I**_{M}_{×M} denotes the

M ×

M identity matrix. Consequently, the property differences between

**R**_{F} and

**R**_{N} can be utilized to implement the FF sources components elimination in the covariance matrix. Then, the differencing matrix can be expressed as:

where,

**A**_{D} is the virtual steering vectors expressed as:

Based on (13), (41) and (43), the following holds (see

Appendix B)

where ⊙ denotes Hadamard product, and

**D** is a rotation factor matrix defined as:

And then, by implementing the EVD of

**R**_{D}, the following holds:

where,

**Δ**_{s} is a diagonal matrix which contains 2

K_{1} non-zero eigenvalues, and

γ_{0} is the zero eigenvalue,

**V**_{s} is the

L × 2

K_{1} eigenvectors matrix spanning the signal subspace of

**R**_{D}, and

**V**_{n} is the

L × (

L − 2

K_{1}) eigenvectors matrix spanning the noise subspace. It is obvious that the following holds:

Consequently, we can construct a diagonal matrix:

and the NF sources’ DOAs could be estimated accurately by searching the

K_{1} highest spectrum peaks through the following function:

where

**Q** denotes an arbitrary

L × 2

K_{1} full column rank matrix. The typical value of

**Q** is set as that composed of arbitrary 2

K_{1} columns of the

L ×

L identity matrix. Conversely, since

**Q**^{H}{

**J****V**_{s}−

**Φ**(

θ,

φ)

**V**_{s}} contains zero eigenvalue only when

φ and

θ are equal to

φ_{k} and

θ_{k} (

k = 1,…,

K_{1}), the DOAs can also be estimated from the following spatial spectrum searching:

In this subsection, the covariance differencing operation in UCA eliminate the FF components and noise effectively based on the structure differences between the NF and FF sources covariance matrices. As a conclusion, the proposed algorithm successfully avoids the 3D spectrum searching and sources classification processing with the help of covariance differencing.