TST MUSIC Algorithm

Note that the MUSIC algorithms can not distinguish the delays or DOA if they are very close. In lieu of this, we can make use of the space-time characteristics of the multi-ray channel by arranging the MUSIC algorithms employed in a tree-structured manner. We can then utilize the estimated estimated delays or DOAs to help us separate the incoming rays into some groups so that the DOAs and delays can be more precisely determined in the next stage. To achieve this, this section addresses a temporal filtering process and a spatial beamforming process, which are a refinement of those addressed in [12], to separate the incoming rays into several groups according to the estimated delays and DOAs in the previous stages.

First, we consider a temporal filtering process, which can separate the rays into several groups based on the estimated delays. To explain this, we assume that the received data only contains one user with J rays with delays τ

l

, l = 1 · · · J . For simplicity, if we neglect the effect caused by the previous bit, (2.1) can be written as

If we want to remove the ray with delay θ

i

, we can consider the following filtering matrices {U

^t _i

} given by

U

^t _i

= I − ˜g

⁰

(τ

i

)˜g

⁰

(τ )

^T _i

, l= 1 · · · J (3.5)

õ

Figure 3.3: The T-ESPRIT algirhtm flow chart.

where we have used (2.1) and the fact that ˜g(τ

i

)

^T

U

^t _i

= 0. As such, the component corresponding to the ray with delay τ

i

is removed. Note that the components of the temporal vectors of the other rays now become ˜g(τ

l

)

^T

− ˜g(τ

l

)

^T

˜g

⁰

(τ

i

)˜g

⁰

(τ

i

)

^T

rather than ˜g(τ

l

)

^T

, as all rays are not orthogonal to each other. Therefore, if we intend to filter out more than one rays from the data, we need to use the Gram-Schmidt orthogonalization procedure to obtain a new set of temporal vectors given by ˜g

⁰

(τ

l

) as:

˜

g

⁰

(τ

l

) = g(τ˜

l

) −

^{P l−1} _j=1

c

_lj

g˜

⁰

(τ

j

)

k˜g(τ

l

) −

^{P l−1} _j=1

c

_lj

g˜

⁰

(τ

j

)k (3.7) where k · k denotes the Frobenius norm, and c

lj

= ˜g(τ

l

)

^T

· ˜g

⁰

(τ

j

). It then follows that if we want to filter out all rays except the j

^th

ray in the received data, the data after filtering can be readily shown to be

X

^j _m

= X

m

·

J

Y

l=1;l6=j

U

^t _l

(3.8)

The above discussion is also applicable to the spatial filtering, which can be invoked to separate the incoming the incoming rays into several groups based on the estimated DOA’s.

According to the above discussion, the proposed tree-structured MUSIC is to use two T-MUSIC algorithms and one S-MUSIC algorithm aletrnatively in conjunction with a temporal filtering process and a spatial beamforming process to enhance the estimation accuracy of the MUSIC algorithms. The overall procesures can be summarized as follows[14]:

Step 1: Rough Delay Estimation:

From the received data, we can estimate the temporal covariance matrix by Rˆ

^t

= 1 where q is the number of the total group delays of all users.

Step 2: Temporal Filtering:

Exploit (3.5) to determine the temporal filtering matrices {U

^t _i

} for i = 1, 2, . . . , q, and use (3.8) to obtain X

^j _m

for j = 1, 2, . . . , q, which are the data after the temporal filtering.

Step 3: DOA Estimation:

From each X

^j _m

, we can estimate the spatial covariance matrices by Rˆ

^s _j

= 1 tempo-ral covariance matrices. Then apply the T-MUSIC algorithm again but with different temporal array manifolds, (

^{Q q} _i=1;i6=j

U

^t _i

)

^T

g(τ ), to estimate the delays˜ {ˆτ

_1,1 ⁰

, . . . ,τˆ

_1,p ⁰

1

, . . . ,τˆ

_q,1 ⁰

, . . . ,τˆ

_q,p ⁰ _q

}. Finally, the pair of θ

i,j

and ˆτ

_i,j ⁰

for i = 1, 2, . . . , q, j = 1, 2, . . . , p

q

are the resulting DOA-delay estimates.

τ

Figure 3.4: The evolution of the signal contents for the estimation of the parame-ters of ray one.

The rationale of the TST-MUSIC is to incorporate three 1-D MUSIC (S-MUSIC and T-(S-MUSIC) algorithms with beamforming techniques and the filtering techniques to group, isolate, and then estimate and pair the 2-D parameters of a fading channel. To simplify the algorithm description, we first assume that there are only three rays present in the system.

As shown in Fig. 3.4(a), three rays are characterized by their temporal-spatial coordinates on the DOA-delay plane. Note that ray 1 and ray 2 possess close time delays (τ

1

≈ τ

2

), but diverse DOAs (θ

1

< θ

₂

); while ray 1 and ray 3 are close in the DOAs (θ

1

≈ θ

3

), but with far apart delays (τ

1

< τ

₃

). The tree structure of the TST-MUSIC algorithm for this scenario is illustrated in Fig. 3.5.

In addition, corresponding to Fig. 3.4, Fig. 3.5 shows the evolution of the data contents as the parameters of ray 1 are estimated in the tree structure.

The TSMUSIC treats those temporally-close rays, which flatten the T-MUSIC spectrum, as a group. Therefore, ray 1 and ray 2 in Fig. 3.4(a) are regarded as one group, while ray 3 is considered another group. By applying the T-MUSIC to the rows of X

t

, the resulting group delays are estimated, denoted by ˆt

₁

and ˆt

₂

. Based on the group delay estimates, ˆt

₁

and ˆt

₂

, we define the temporal

 

Figure 3.5: The tree structure of the TST-MUSIC in solving the scenario shown in Fig. 3.4.

filtering matrices U

^t _i

as

U

^t ₁

= I − ˜g(ˆt

₁

) · ˜g(ˆt

₁

)

^H

and U

^t ₂

= I − ˜g(ˆt

₂

) · ˜g(ˆt

₂

)

^H

.

Note that U

^t ₁

(or U

^t ₂

) is also the complement projection matrix of ˜g(ˆt

₁

) (or ˜g(ˆt

₂

)) with ˜g(ˆt

₁

)

^H

· U

^t ₁

= 0

^T

(or ˜g(ˆt

₂

)

^H

· U

^t ₂

= 0

^T

). In the 3-ray scenario shown in Fig.

3.4(a), we have τ

1

≈ τ

2

≈ ˆt

₁

< τ

₃

≈ ˆt

₂

, which implies that ||˜g(τ

1

)

^T

·U

^t ₁

|| ≈ ||˜g(τ

2

)

^T

· U

^t ₁

|| ≈ 0  ||˜g(τ

₃

)

^T

· U

^t ₁

|| and ||˜g(τ

₁

)

^T

· U

^t ₂

|| ≈ ||˜g(τ

₂

)

^T

· U

^t ₂

||  ||˜g(τ

₃

)

^T

· U

^t ₂

|| ≈ 0, where the notation || · || denotes the 2-norm of a vector. With these facts, the TST-MUSIC post-multiplies U

^t _i

to X

t

, which is referred to as the temporal filtering process to separate the rays with delays τ

1

, τ

₂

from the ray with delay τ

3

. As a result, two group matrices, denoted as X

1

and X

2

, are thus generated as

X

₁

= X

t

· U

^t ₂

where “≈” in (3.12) and (3.13) means that the residue signals from ray 3 and from ray 1 and 2, respectively, are neglected. Discussions about the magnitude of the

neglected residue signal will be given at the end of this section. The performance degradation caused by this neglect of residue signal is also discussed in Section 4. It is shown in Appendix B that the transformed noise matrices in (3.12) and (3.13) are still temporally and spatially white within the projected subspace.

Note that the two dominant rays contained in (3.12) have their DOAs θ

2

>

θ

₁

. The DOA estimates ˆθ

₁

and ˆθ

₂

can thus be accurately obtained by applying the S-MUSIC to X

₁

. Similarly, ˆθ

₃

is obtained by applying the S-MUSIC to X

₂

. Note that ˆθ

₁

≈ ˆθ

₃

, but the signal of ray 1 and that of ray 3 are separated into two different signal groups before the S-MUSIC is applied. Therefore, θ

1

and θ

3

can also be accurately estimated with the help of the temporal filtering following the first T-MUSIC. Also note that, right after estimating {θ

k

} in the S-MUSIC, the estimated array vectors {a(ˆθ

_k

)} are determined and will be used in the spatial beamforming described below.

To further divide each group matrix into several single-ray matrices, the spatial beamforming matrices U

^s _i

can be defined as

U

^s ₁

= I − a(ˆθ

₁

) · a(ˆθ

₁

)

^H

, U

^s ₂

= I − a(ˆθ

₂

) · a(ˆθ

₂

)

^H

for X

₁

, and

U

^s ₃

= I − a(ˆθ

₃

) · a(ˆθ

₃

)

^H

for X

2

, respectively. Note that U

^s _i

nulls the signal from ray i as U

^s ₁

· a(ˆθ

₁

) = 0, U

^s ₂

· a(ˆθ

₂

) = 0 and U

^s ₃

· a(ˆθ

₃

) = 0.

Similar to the temporal filtering process, the TST-MUSIC algorithm pre-multiplies X

₁

by U

^s ₁

and U

^s ₂

, which is referred to as the spatial beamforming process, to null the corresponding ray. It follows that two single-ray matrices, X

_1,1

and X

_1,2

are formed, respectively, as

X

_1,1

= U

^s ₂

· X

1

≈ β

1

· U

^s ₂

a(θ

1

) · ˜g(τ

1

)

^T

U

^t ₂

+ U

^s ₂

· N · U

^t ₂

(3.14) and

X

_1,2

= U

^s ₁

· X

1

≈ β

₂

· U

^s ₁

a(θ

₂

) · ˜g(τ

₂

)

^T

U

^t ₂

+ U

^s ₁

· N · U

^t ₂

. (3.15)

Again, the residues of ray 2 and ray 1 are neglected in (3.14) and (3.15), respectively. The noise matrices in (3.14) and (3.15) are again still temporally and spatially white within the projected subspace. In (3.14) and (3.15), the single-ray structure of X

1,1

and X

1,2

implies that the two rays with close ray delays, ray 1 and ray 2, are separated into different sub-groups by the spatial beamforming process. As a result, by applying the T-MUSIC algorithm to X

1,1

and X

1,2

, ray delay estimates ˆτ

₁

and ˆτ

₂

can then be accurately estimated, respectively. It also follows that the pairing of (ˆτ

₁

, ˆθ

₁

) and (ˆτ

₂

, ˆθ

₂

) is automatically achieved. Note

that, in the process of the second T-MUSIC algorithm, the temporal array vector should be U

^t ₁

· ˜g(τ ) as shown in (3.14) and (3.15) rather than ˜g(τ ). Furthermore, since the group-delay information ˆt

₁

is known, the searching region shrinks to the vicinity of ˆt

₁

instead of the whole τ axis. For the other branch of the signal with respect to X

₂

, only single ray is found. Therefore, no spatial beamforming is needed, that is X

2,1

= X

2

and ˆτ

₃

= ˆt

₂

.

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