Constrained TST MUSIC Algorithm

Note that the projection matrices in the TST algorithm are based on the delays or DOAs estimated in the previous stage. The parameter estimation error in turn propagates to the next group partition stage and thus inevitably degrades the overall performance, as the algorithm proceeds in a tree-structured manner.

In light of this setback, this project attempts to determine the temporal filters and spatial beamformers based on some statistical averages by minimizing the average filtered or beamforming output power. In other words, in the tem-poral filtering stage, rather than resorting to the projection matrices in the TST algorithm , we now determine a set of temporal filters w

_j ^t

’s which minimizes the output power. More specifically, the spatial filter for the group corresponding to the delay ˆt

_j

is given by

E[||X

m

w

_j ^t

||

²

] under the constraints g(ˆ˜ t

_k

)

^T

w

_j ^t

= δ

k−j

(3.16) for k = 1, · · · , 2p, where || · || denotes the 2-norm and the constraints in (3.16) are imposed to ensure that the desired ray will remain intact while eliminating the other signals when passing through the temporal filters, as required in the partition of the incoming rays addressed above. (3.16) can be readily expressed as a linearly constrained quadratic optimization problem given by

min

w ^t

j

w

_j ^tH

R

^t

w

^t _j

under C

^t

w

^t _j

= e

^t _j

(3.17)

where R

^t

is the temporal covariance matrix, e

^t _j

is a 2p × 1 vector with 1 on the j

^th

position and zeros elsewhere, and

C

^t

= [˜g(ˆt

₁

), · · · , ˜g(ˆt

_2p

)]

^T

(3.18) The solution of (3.17) can be readily shown as

w

^t _j

= (R

^t

)

⁻¹

C

^tH

(C

^t

(R

^t

)

⁻¹

C

^tH

)

⁻¹

e

^t _j

(3.19) The partitioned data matrix in (3.8) is now replaced by

X

m,j

= X

m

· w

^t _j

(3.20)

As for the spatial beamforming process, we need to filter out the groups with different delays in the temporal domain, but still possess the temporal information to estimate the delays. Therefore, we use X

⁰ _m,j

in (3.8) instead of X

m,j

to proceed to the next step. As the above, to carry out the spatial beamforming process based on the available a(ˆθ

_j,k

) in j

^th

group, we determine the spatial filters w

^s _j,k

’s by minimizing the beamforming output power given by

E[||X

⁰ _m,j

w

_j,k ^s

||

²

] under a(ˆθ

_j,i

)

^H

w

^s _j,k

= δ

i−k

(3.21) where the constraints are again imposed to ensure that the desired ray can remain intact after the spatial beamforming process. (3.21), along the same line as the above, can be posed as

min

w ^s

j,k

w

^sH _j,k

R

^s _j

w

^s _j,k

under C

^s _j

w

^s _j,k

= e

^s _k

(3.22) for k = 1, · · · , q

j

, where e

^s _k

is a q

j

× 1 vector with 1 on the j

^th

position and zeros elsewhere, and

C

^s _j

= [a(ˆθ

_j,1

), · · · , a(ˆθ

_{j,q j}

)]

^H

(3.23) and the solution of (3.22) can be readily shown as

w

^s _j,k

= (R

^s _j

)

⁻¹

C

^sH _j

(C

^s _j

(R

^s _j

)

⁻¹

C

^sH _j

)

⁻¹

e

^s _k

(3.24) Finally, the data, after constrained beamforming, becomes

X

_m,j,k

= (w

_j,k ^s

)

^H

· X

⁰ _m,j

(3.25)

As such, the overall structure of the constrained TST is described as follows:

Step 1: Rough Delay Estimation:

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

^t

= 1

P M

M

X

m=1

X

^H _m

· X

m

. (3.26)

where M is the total number of symbols. We then apply the T-MUSIC to ˆR

^t

to estimate the group delays {ˆt

₁

, . . . , ˆt

_p

}, where p is the number of the total group delays of all users.

Step 2: Constrained Temporal Filtering:

Determine the temporal filters {w

_j ^t

} for j = 1, 2, . . . , p, by (3.19) and use (3.20) to obtain X

m,j

for j = 1, 2, . . . , p, which are the partitioned data after the constrained temporal filtering.

Step 3: DOA Estimation:

From each X

m,j

, we can estimate the corresponding spatial covariance

Step 4: Constrained Spatial Beamforming and Projection:

Repeat (3.27) to obtain ˆR

^s _j

, but with X

m,j

being replaced by X

⁰ _m,j

in (3.8).

Determine the spatial filters {w

^s _j,k

}, 1 ≤ j ≤ p, 1 ≤ k ≤ q

j

by (3.24) and use (3.25) to obtain X

m,j,k

for all j’s and k’s, which are the data containing only the ray with DOA ˆθ

_j,k

.

Step 5: Delay Estimation:

Repeat (3.26) with partitioned data X

m,j,k

to find the new temporal covariance matrices. Apply the T-MUSIC algorithm again but with different temporal array manifolds, (

^{Q 2p} _i=1;i6=j

U

^t _i

)

^T

g˜(τ ), to estimate the delays {ˆτ

_j,1

, . . . ,τˆ

_{j,q j}

}, 1 ≤ j ≤ p.

(ˆθ

_j,k

,τˆ

_j,k

),1 ≤ j ≤ p, 1 ≤ k ≤ q

j

, are the resulting DOA-delay estimates.

The above algorithm only considers the joint DOA-delay estimates for the incoming rays. If we are interested in the rays associated with each user, we can note that the delays in step 1 are estimated by searching the array manifolds of

˜

g(.). Because ˜g(.) depends on the spreading codes, it implies that when the group delays are estimated, each of these can be simultaneously associated with a specific user.

We determine the arithmetic operations required by the proposed approach referred as constrained TST algorithm. Note that in general the length of the total samples is greater than the number of the antennas and the size of temporal vector employed, i.e., M  P , M  N , the computations required by the are therefore dictated by (a) estimation of the first and second temporal covariance matrices, which require M N

²

P and KM N

²

multiplications, respectively; (b) es-timation of the spatial covariance matrices, which require pM P

²

multiplications, respectively; (c) temporal and spatial filtering processes, which require pM N P and KM N P multiplications, respectively, and the filtering process by the projection matrices, which require pM N

²

P multiplications. As a whole, the total number of multiplications required is about M P N (pN + N + p + K) + pM P

²

+ KM N

²

.

The proposed algorithm makes use of one spatial (S)-MUSIC and two tem-poral (T)-MUSIC algorithms alternatively to estimate the DOAs and the group delays, respectively. As such, the incoming rays are grouped, isolated, and then estimated, and the pairing of the estimated DOAs and delays are automatically de-termined. In addition, a constrained temporal filtering process and a constrained spatial beamforming process are conducted. Such a constrained filtering approach can effectively partition the incoming rays and suppress the propagation error in

the tree-structured scheme, thus in turn enhancing the overall performance. The pairing of the estimated DOAs and delays is also automatically determined. The number of antennas required by the algorithm can be much less than that of the incoming rays. The proposed algorithm can either provide superior performance or call for substantially lower computational complexity. Simulations are conducted to verify the relevant results compared with previous works.

Fig. 3.6 shows the structure of the constrained TST algorithm.

%

Figure 3.6: The structure of the proposed Constrained TST-MUSIC

在文檔中 發展軟體無線電技術---智慧型天線系統/技術之研發---子計劃Ⅴ:低複雜度多使用者解調器之設計與應用(I) (頁 20-23)