Optimal least squares deterministic parameter estimation from a class of block-circulant-with-circulant-block linear model

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Optimal Least Squares Deterministic Parameter Estimation from a

Class of Block-Circulant-with-Circulant-Block Linear Model

Jwo-Yuh Wu and Ta-Sung Lee

Department of Communication Engineering, National Chiao Tung University,

1001, Ta-Hsueh Road, Hsinchu, Taiwan

Tel: 886-35-712121, ext. 54549; Fax: 886-35-710116

Emails: jywu@cc.nctu.edu.tw; tslee@mail.nctu.edu.tw

Abstract- This paper investigates the least-squares (LS) estimation of unknown deterministic parameters from a standard linear model characterized by a class of block-circulant-with-circulant-block (BCCB) matrix. We propose a method for designing the BCCB system matrix coefficients to minimize the mean square error incurred by the LS estimate, under certain equality and inequality constraints. By exploiting the eigenvalue characteristic of BCCB matrices, precise analysis is undertaken to derive a closed-form solution. The considered optimization problem arises in the study of blind channel estimation for single-carrier block transmission with cyclic prefix; the presented analysis reveals several key features associated with the BCCB family, and shows an original investigation of the BCCB matrix structure for facilitating linear optimal parameter estimation.

Index Terms: Least squares; parameter estimation; circulant matrix; block circulant matrix with circulant blocks; blind channel estimation.

I. INTRODUCTION

The estimation of linearly mixed parameters subject to additive white Gaussian measurement noise has been addressed in diverse fields in science and engineering. Mathematically, this problem is formulated through the linear model

y=Ax v , (1.1) + where y is the observed data vector, x is the unknown signal of interest, A is a known matrix of full column rank, and

2

( ,Tv k)

v & 0 I , where I denotes the k kk × identity matrix

for some k. There have been many criteria for reliably estimating x based on (1), depending on the priori knowledge known about x [8], [14]. When x is treated as deterministic, one popular solution scheme is the least-squares (LS) estimate, namely,

ˆ :LS

(

H

)

1 H

=

x A A A y . (1.2) Despite its simplicity, the LS solution is attractive for it produces the optimal linear unbiased estimate under white noise assumption [8], [12]. To assess the performance of x ,ˆLS

one commonly used metric is the mean square error (MSE), namely, E

{

ˆxLSx2₂

}

, where E{}¸ is the expectation

operator. By equivalently rewriting ˆx in (1.2) as LS

ˆLS

(

H

)

1 H

= +

x x A A A v (1.3) and since v &( ,0Tv2I), it is straightforward to show

E

{

ˆ_LS ₂2

}

=T_v2Tr ¡

(

H

)

1¯°

¢ ±

x x A A , (1.4) where Tr[ ]¸ denotes the trace. The performance of the LS solution thus depends crucially on the matrix A . In many situations it is plausible to judiciously choose A to improve

the utmost solution reliability, e.g., in training based channel estimation the matrix A contains the pilot symbols and are designed to minimize the MSE in (1.4) under a total transmit power constraint [9], [11].

In this paper, we focus on a special class of linear model, wherein the matrix A is described by

A=QT , (1.5) in which T contains an arbitrary column subset of the

2 2

N ×N identity matrix, Q is a block-circulant-with- circulant-blocks (BCCB) matrix [3, p-184] of dimension

2 2 N ×N : 2 2 2 2 2 1 2 1 2 2 3 2 2 2 2 2 3 2 2 2 2 2 2 1 2 (0) (1) ( 2) ( 1) ( 1) (0) ( 3) ( 2) : (2) (3) (0) (1) (1) (2) ( 1) (0) N N N N N N N N N N p p p N p N p N p p N p N p p p p p p p N p ¯ ¡ ° ¡ ° ¡ ° ¡ ° ¡ ° = ¡ ° ¡ ° ¡ ° ¡ ° ¡ ° ¡ ° ¡ ° ¢ ± I J J J J I J J Q J J I J J J J I " " # # " # # " " (1.6) where p n( ), 0bnbN , are some positive real numbers, 1 and J  N N× is the circulant permutation matrix defined by 1 ( 1) 1 ( 1) 1 1 : N N N × × ¯ ¡ ° = ¡ ° ¡ ° ¢ ± 0 J I 0 . (1.7) We will seek for the optimal p n( ) which minimizes

MSE: Tv2Tr

(

H H

)

1

¯

= ¡ °

¢ T Q QT ±, (1.8) subject to the following two design constraints

1 2 0 ( ) N n p n N = =

, (1.9) min ( )p n 2p for some E 0< < . (1.10) E 1

Such a problem arises in the study of blind channel estimation for single-carrier transmission with cyclic prefix [5], [16] and also the related multi-antenna system with space-time block coding [2], [17]. By exploiting the eigen-structure of the BCCB matrix Q, in this paper we propose a method for constructing a closed-form optimal p n( ). Block circulant matrices (not necessarily with circulant blocks) have found important applications in computational reduction [4], [13], [15], and in the study of spectral distribution for Toeplitz matrices [1]. The presented study in this paper brings out the nice features of the block circulant family in optimal linear parameter estimation.

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II. PROPOSED OPTIMAL SOLUTION

A. Design Approach

Minimization of the cost function of the form (1.8) has been considered in [9], [11]. The reported solution approach therein is via the following inequality: since T Q QT are positive H H definite, it follows

(

)

1 1 , H H H H i i i Tr ¡ ¯°p _¡_¢ ¯_°_± ¢T Q QT ±

T Q QT , (2.1) and equality holds whenever T Q QTH H is diagonal [10, p-1041]. If equality (1.9) is the only design concern, it is easy to check that the impulse sequence

p m( )2=N , and p n( )2= for n0 vm, (2.2) where 0bmbN is fixed but arbitrary, diagonalizes 1

H H

T Q QT and is thus the minimizing solution. However, given the additional threshold requirement (1.10), one cannot rely on this principle for finding a solution since, subject to the BCCB structure of Q and p n( )2> , it is impossible to 0

choose p n( ) to render T Q QTH H diagonal. Another possible solution scheme would be via numerical search techniques. However, as the cost function in (1.8) are non-convex in p n( ), there do not seem to have efficient methods for finding the global optimum. In what follows, we propose an alternative strategy to address the considered optimization problem. Our approach is grounded on a key fact shown in the next lemma, which as we show below will facilitate the exploitation of the BCCB property of Q to derive a closed-form solution.

Lemma 2.1: Let M be a square nonsingular matrix, and M be constructed from M by deleting an arbitrary subset of its columns. Then

Tr¡

(

H

)

1¯°bTr ¡

(

H

)

1¯°

¢ M M ± ¢ M M ±.

[Proof]: Without loss of generality we assume M is split as d

¯

= ¡_¢ °_±

M M M , in which M contains the columns to be _d deleted; otherwise we can multiply M from the right by a permutation matrix to put it in this partition. It thus follows

H H H d H d H H H d d d d ¯ ¯ ¡ ° _¯ ¡ ° =¡ °_¡_¢ _°_±=¡ ° ¡ ° ¡ ° ¡ ° ¡ ° ¢ ± ¢ ± M M M M M M M M M M M M M M . (2.3)

Since M is nonsingular, M M is positive definite. By the H inversion lemma for block matrix [8, p-572], we have

(

)

(

)

(

)

(

)

(

)

1 1 1 1 1 H H H H H d d d d H H H H d d d d = ¯ ¡ × ° ¡ ° ¡ ° ¡ ° ¡ × ° ¡ ° ¢ ± M M M M M M M M M M M M M M M M M M (2.4) in which the notation “×” stands for the block off-diagonal submatrices irrelevant to the proof procedures. From (2.4), we have

(

)

(

)

(

)

(

)

1 1 1 1 1 H H H H H d d d d H H H H d d d d Tr Tr Tr ¯ ¯₌ _¡ _° ¡ ° _¡ _° ¢ ± _¢ _± ¯ ¡ ° + ¡ ° ¢ ± M M M M M M M M M M M M M M M M M M (2.5)

Since

(

M MH

)

1 is positive definite, so are its principle submatrices and (2.5) implies

(

H

)

1

(

H H

(

H

)

1 H

)

1 d d d d Tr Tr ¯ _{¯ p} _¡ _° ¡ ° ¢ M M ± ¡¢M M M M M M M M °±. (2.6) Using the matrix inversion lemma [8, p-571], inequality (2.6) can be further expanded into

( )

(

)

( ) ( )

( )

1 1 1 1 1 1 1 1 1 1 1 1 H H H H H d d d d H H H H H H H H H d d d d d d H H H H H H H H H d d d d d d Tr Tr Tr Tr Tr ¯ ¯ _¡ ¬ _° ¡ ° p _¡ _° ¡ ° ® ¢ ± ¡_¢ °_± _¬ ¯ ¡ ° = ¡ + _® ° ¡ ° ¢ ± ¯ _¬ ¡ ° = _¡ _°+ _® ¢ ± M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M ¡M M M M M M M M M M M M M M M M1¯° ¡ ° ¡ ° ¢ ± (2.7) Since

(

H H

(

H

)

1 H

)

1 d d d d M M M M M M M M is a principle

submatrix of

(

M MH

)

1 (cf. (2.4)), it is positive definite and so is

(

H

)

1 H

(

H H

(

H

)

1 H

)

1 H

(

H

)

1 d d d d d d M M M M M M M M M M M M M M M M .

The result then follows from (2.7). As QT contains a column subset of Q , Lemma 2.1 asserts that Tr _¡

(

H H

)

1¯_° ¢ T Q QT ± is upper bounded by

(

)

1 H Tr _¡ ¯_° ¢ Q Q ±. This thus suggests a suboptimal, but would be more simple and efficient, way of designing p n( ): we can simply choose p n( )

to minimize

J :=Tr _¡

(

H

)

1¯_°

¢Q Q ±, (2.8) since Tr _¡

(

H H

)

1¯_°

¢T Q QT ± would in turn be kept small. The main advantage of the proposed design formulation is that we can directly take advantage of the BCCB structure of Q to derive a closed-form solution. Toward this end, we shall first express the cost function J in (2.8) in a more tractable form. Since Tr ¡

(

H

)

1¯°

¢Q Q ± is the sum of the

2

N eigenvalues associated with

(

Q QH

)

1 [6, p-42], we propose to rewrite J in terms of such spectral characteristics; this is specified next and will lay the foundation for subsequent analytic design.

B. Eigen-Structure of BCCB Matrix

The eigen-property of the BCCB matrix family has been studied in [3]. A distinctive feature of the BCCB matrices is that they are diagonalizable by FFT based operations. This is pinned down via the following lemma; the result will be used for explicitly computing the eigenvalues of the matrix Q .

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We will hereafter denote by BCCB_{N N}_, the set of all

2 2

N ×N block circulant matrices with circulant blocks, each characterized by N circulant matrices of dimension N× ,N

F the N N× FFT matrix, and the Kronecker product [7, p-243].

Lemma 2.2 [3, p-185]: If XBCCB_{N N}_, , then X can be diagonalized by F F . More precisely, let

{

C₀,",C_N₁

}

be the set of N× circulant matrices on the top row block of X ,N

and let - be the diagonal matrix containing the eigenvalues n

of C . Then we have n ( )

(

)

1 1 1 0 N n n N n 8 - = _¬ =

® X F F F F , (2.9)

with 8_N :=diag

{

_¢_¡ 1 X X2 "XN1¯_°_±T

}

, X: exp( 2 / )= j Q N . Conversely, any matrix of the form (F F )-

(

F1F1

)

for some diagonal - belongs to BCCB_{N N}_, . Based on Lemma 2.2, we can determine the eigenvalues of the matrix Q in (1.6). Observe that, despite its BCCB structure, the circulant block submatrices of Q are further characterized by J : this will lead to very elegant eigenvalue characteristics. n Roughly speaking, if we define the vector

p: [ (0)= p 2 p(1)2 " p N( 1) ]2T  , (2.10) N then the N2 eigenvalues of Q is completely determined by the N eigenvalues associated with the N×N circulant matrix with p as the first row. More precisely, we have the T following theorem (see Appendix A for a proof).

Theorem 2.3: Let F be the N N× FFT matrix; also, associated with the vector p in (2.10) we define the polynomial

_{( ) :} ₍₀₎2 ₍₁₎2 1 ₍ ₁₎2 (N1)

z ₌p ₊p z _{+ +}p N z

p " . (2.11)

Then the N2 eigenvalues of the matrix Q defined in (1.6) are exactly given by the N replicas of the N-tuple

{

p(1) ( ), , pX ", p(XN1)

}

. Based on Theorem 2.3, it follows immediately that the eigenvalues of Q Q are N replicas of the N-tuple H

{

p(1)2, , p( )X 2 ", p(XN1)2

}

; the objective function J in (2.8) is thus

(

)

1 1 2 0| ( ) | N H k k N J Tr X = ¯ = ¡ °= ¢Q Q ±

p . (2.12) Equation (2.12) rewrites Tr ¡

(

H

)

1¯° ¢Q Q ± in terms of the “frequency responses” p( )Xn ’s in a rather simple way: it is just a scaled sum of | ( ) |pXk 2 over 0b bk N . The 1 derivation of the optimal p n is based on equation (2.12) and ( )

is shown below.

C. Optimal Solution

The first step toward a solution is to transform the two constraints (1.9) and (1.10) in terms of p( )Xn . With (1.9) and (2.11), it is easy to check that, for k = ,0

1 0 2 0 ( ) (1) N ( ) n p n N X = = =

= p p . (2.13) The following lemma provides an upper bound on p( )Xk for 1b bk N ; the result is crucial for deriving the optimal 1 solution (see Appendix B for a proof).

Lemma 2.4: For any p n satisfying (1.9) and (1.10), we have ( ) p( )Xk bN(1E) for all 1b bk N . (2.14) 1 With (2.13) and (2.14), the optimal p n is shown in the next ( )

theorem.

Theorem 2.5: The optimal p n minimizing ( ) Tr ¡

(

H

)

1¯° ¢Q Q ±, subject to constraints (1.9) and (1.10), is given by the following the two-level form solution: for a fixed but arbitrary

0bmbN ,1 2 ( ) ( 1) p m =N N E, and p n( )2= for nE vm, (2.15) leading to _min 1 ( 1)₂ (1 ) N J N N E = + . (2.16)

[Proof]: We claim that i)Tr ¡

(

H

)

1¯°pJ_min

¢Q Q ± for any p n( ) satisfying (1.9) and (1.10), and ii) equality is attained by the

two-level scheme (2.15); the theorem thus follows. To show claim i), we observe form (2.12) and (2.13) that

(

)

1 1 2 0 1 1 2 2 2 1 1 | ( ) | 1 | (1) | | ( ) | | ( ) | N H k k N N k k k k N Tr N N N N X X X = = = _{¯ =} ¡ ° ¢ ± = + = +

Q Q p p p p (2.17) From (2.14), it follows ( ) 2 ₂ 1 ₂ (1 ) k N X E p p , 1b bk N , (2.18) 1 With (2.17) and (2.18), we have

(

)

1 1 2 2 1 1 1 1 (1 ) | ( ) | N H k k N N Tr N X N N E = _{¯ = +} _p ₊ ¡ ° ¢Q Q ±

p , (2.19)

which proves claim i). To show claim ii), it is noted that the

two-level solution (2.15) yields, for any kv ,0

{

}

{

}

{

}

, 1 2 0 1 0 ( ) ( ) ( 1) (1 ) (1 ) N k kn km kn n n m N km kn km n p n N N N N X X E X E X E X E X E X = v = = = + = + =

p (2.20) where the last equality in (2.20) follows since

1 0 0 N kn n X = =

for any kv . Equations (2.13) and (2.20) 0 show p(1)=N and p( )Xk =N(1E) for 1b bk N ;1 hence the two-level scheme (2.15) attains J_min in (2.16).

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Recall that the impulse sequence (2.2) is optimal whenever (1.9) is the only design concern. When an additional threshold on the magnitude of p n is imposed as in (1.10), it turns out ( ) that the best choice is the “impulse-like” two-level solution (2.15). With (2.16), the minimal J_min is seen to decrease whenever E is decreased. It is noted that two-level solution (2.15) minimizes Tr _¡

(

H

)

1¯_°

¢Q Q ±, but its optimality with respect to Tr _¡

(

H H

)

1¯_°

¢T Q QT ± appears intractable to verify. Our simulation results indicate that it indeed seems to be the minimizing solution, irrespective of the choices of T.

III. CONCLUSION

In this paper we investigate the optimal LS estimation from a class of BCCB linear model, which is encountered in our recent study in blind channel estimation problems. We show a method for designing the system matrix coefficients to minimize MSE under certain equality and inequality constraints. The proposed approach minimizes an upper bound on MSE, and exploits the BCCB system matrix structure as well as the associated spectral characteristics. The frequency-domain-based formulation in terms of eigenvalues nicely tackles the inequality constraint, allows precise analysis procedures, and eventually leads to an appealing simple closed-form solution. We will try to generalize the results to less restricted families like block circulant matrices or matrices with circulant blocks; this would find potential applications in channel estimation for MIMO cyclic prefix based single- (multi-) carrier block transmission.

APPENDIX A: PROOF OF THEOREM 2.3

The matrix Q in (1.6) is characterized by the N circulant matrices

{

p(0)2I_N, , p(1)2J ", p N( 1)2JN1

}

on its top row block. Let un N be the vector containing the eigenvalues of the matrix p n( )2Jn, 0bnbN . We then have [3, 1 p-73], for 0bnbN ,1

u_n = N¸F r1 T_n , (A.1) where r denotes the first row of _n p n( )2Jn. By definition of J in (1.7), it can be deduced that, for 1bnbN ,1

2 0 =p(0) T1

r e and ( )2 T ₁ n =p n N+ n

r e , (A.2)

where e denotes the l th unit standard vector in _l N. From (A.1) and (A.2), it follows that, for 1bnbN ,1

u₀= N p¸ (0)2f and ₀ un = N p n¸ ( )2fN n , (A.3)

where

f_n := N1¸[1 Xn X2n " X(N2)n X(N1)n T] (A.4) is the (n+1)th column of F1,0bnbN . By Lemma 1 2.1, the eigenvalues of Q are given by the diagonal entries of the matrix 1 0 N n n N n 8 - =

, where -n =diag{ }un . Since

{ }

n

n N N diag

8 = ¸ f (this follows by definition of 8_N and from (A.4)), the eigenvalues of Q can simply be computed as entries of the vector

1 0 N n n n N =

¸

f u . From (A.3) and (A.4), it can be seen that

f₀u₀=p(0) [2fT₀ fT₀ " f₀T T] , (A.5) and, for 1bnbN1, 2 ( 1) ( ) [ T n T N n T ]T n n p n N n X N n X N n = f u f f " f . (A.6)

From (A.5) and (A.6), the vector N ¸

N_n₌₀1f_n u_n can be computed as 1 2 2 0 ₁ 1 1 1 2 2 0 1 1 1 1 2 ( 1) 2 0 1 (0) ( ) (0) ( ) (0) ( ) N N n n N n N N n n N N N N n N n n p p n p p n N N p p n X X = = = ¯ ¡ + ° ¡ ° _¯ ¡ ° _¡ _° ¡ ° _¡ _° ¡ ₊ ° _¡ _° ¡ ° ₌ _¡ _° ¡ ° _¡ _° ¡ ° _¡ _° ¡ ° _¡ _° ¡ ° _¡ _° ¡ ° _¡_¢ _°_± ¡ ₊ ° ¡ ° ¡ ° ¢ ±

f f F p F p f f F p f f # # ( 8 ( 8 ( (A.7) where p is given in (2.10) and ₍N N× _{is the Hankel}

matrix with [1 0 0]" T as the first column and [0 1 0 0]" as the last row. It can be checked by definition that

1 n _{( )}T n 1 N 8 ₌ F J F and hence 1 n _{( )}T n 1 N N¸F8 (p= J ¸ N¸F(p,0d dn N1. (A.8) From (A.7) and (A.8) we can see that, for 2 ib bN , the entries of the ith N-dimensional block of

1 0 N n n n N = ¸

f u are simply a permuted version of those in the first one, namely,

1

N¸F(p. As a result, the N2 eigenvalues of Q thus assume N distinct values only. Since (= F2 [3, p-33], we have N ¸F1(p= N¸Fp. The assertion thus follows by definition of F.

APPENDIX B: PROOF OF LEMMA 2.4

The assertion relies on the following key observation: any given p n( ) satisfying (1.9) and (1.10) can be constructed by “squeezing” the peak value at n=m of the two-level solution (2.15) so that the ground values at other n’s are “raised” to the prescribed levels. More precisely, let p n( ) be an admissible sequence such that E<p n( )2<N(N1)E for n! , where the index set ! is a subset of {0,",N1 \} { }m . Then p n can be expressed as ( )

( )2 ( 1) _n n p m N N E =

% ! , (B.1) 2 ( ) n

p n = + %E for n! , and p n( )2=E for n! , (B.2)

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where %_n > models the excessive power over the ground 0 level E for n ! . The sequence of the form (B.1) and (B.2) satisfies the constraints (1.9) and (1.10); in particular, since

2

( )

p m p is required, we can infer from (B.1) that E _n (1 ) n N E % b

! . (B.3) We assume for the moment that m= ; as one will see, the 0 result for the 1bmbN case easily follows. Associated 1 with p n( ) in (B.1) and (B.2), we have, for 1b bk N ,1

(

)

1 2 0 ( ) ( ) ( 1) N k kn n kn kn n n n n n p n N N X X E E X EX = = ¯ ¡ ° =_¡ % _°+ + % + ¡ ° ¢ ±

p ! ! ! 1 0 (1 ) ( 1) 0 kn kn kn n n n n N kn n N E X E EX EX X

E

= ¯ ¯ ¡ ° ¡ ° =_¡ + % _°+ +_¡ + _° ¡ ° ¡ ° ¢ ± ¢ ± =

=

! ! ! (1 ) ( kn 1) . n n N E X ¯ ¡ ° =_¡ + % _° ¡ ° ¢

! ± (B.4) Define the nonnegative number

: (1 ) n n d N E =

% ! , (B.5) Since Xkn =cosnR_kjsinnR_k, where R_k : 2 /= Qk N, and with (B.5), it follows from (B.4) that

2 2 2 ( )k cos sin n k n k n n d n n X R R ¯ ¯ ¡ ° ¡ ° = +_¡ % _° +_¡ % _° ¡ ° ¡ ° ¢

± ¢

± p ! ! 2 2 2 2 cos cos sin . n k n k n n n k n d d n n n R R R ¬ ¬ = + % + % ® ® _¬ + % ®

! ! ! (B.6) Observe that 2 2 2 , cos sin

2 l m(cos cos sin sin )

m l n k n k n n n n n l k m k l k m k n n n n n n n n n R R R R R R ¬ ¬ _% ₊ _% ® ® = % + % % +

! ! ! ! 2 , 2 2 , 2 cos( ) 2 , l m l m l m l m n n n l m k n n n n n n n n n n n n n R = % + % % _¬ b % + % % = % ®

! ! ! ! ! (B.7) and that 2 ncos k 2 n n n d nR d ¬ ¬ _% _b _%

! ®

! ® . (B.8) From (B.7) and (B.8), p( )Xk 2 in (B.6) is upper bounded as

2 2 ₂ ( )k 2 n n d d X b + % ¬+ % ¬

®

® p 2 2 2 n (1 ) , n d N E _¬ = + % =

! ® (B.9) in which the last equality follows from the definition of d in (B.5). This thus proves the lemma, under the assumption

0

m= in (B.1). For 1bmbN , equation (B.4) is then 1 accordingly modified as ( ) ( )k (1 ) ( k m n 1) mn n n N X E X X ¯ ¡ ° =_¡ + % _° ¡ ° ¢

± p ! . (B.10) By going through the same procedures as in (B.5)~(B.8) the conclusion (B.9) will follow.

ACKNOWLEDGMENT

This work is sponsored by the National Science Council of Taiwan under joint grants NSC 95-2752-E-002-009, NSC-95- 2623-7-009-008-D, and the MoE ATU Program.

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Optimal least squares deterministic parameter estimation from a class of block-circulant-with-circulant-block linear model