QRD-based Precoder Selection for Maximum-likelihood MIMO Detection

2010 IEEE 21 st Intemational Symposium on Personal Indoor and Mobile Radio Communications

QRD-based Precoder Selection for

Maximum-likelihood MIMO Detection

Chun-Tao Lin, and Wen-Rong Wu Department of Electrical Engineering,

National Chiao Thng University, Hsinchu, Taiwan, 300, R.O.c. E-mail: [email protected]@faculty.nctu.edu.tw

Abstract-Precoding is an effective method to improve the transmission quality in multiple-input multiple-output (MIMO) systems. In a real-world system, the precoder is selected from a codebook, and its index is fed back to the transmitter. For a maximum-likelihood (ML) receiver, the criterion for precoder selection is equivalent to maximizing the minimum distance of the received signal constellation. The derivation of the optimum solution, however, may be of high computational complexity due to the requirement of the exhaustive search. To reduce the computational complexity, a suboptimum solution based on singular value decomposition (SVD) has been proposed in literature. In this paper, we propose using a QR decomposition (QRD) based method for precoder selection. To further improve the system performance, we also propose an enhanced QRD­ based selection method. With Givens rotations, the computational complexity of the enhanced QRD-based method can be effectively reduced. Finally, we combine precoding with receive antenna selection, and use the proposed QRD-based methods to solve this joint optimization problem. Simulation results show that the proposed approaches can significantly improve the system performance.

I. INTRODUCTION

Spatial multiplexing is a promising method to achieve high spectra efficiencies in multiple-input multiple-output (MIMO) systems [1]. The drawback of the spatial multiplexing scheme is that the error rate performance is greatly affected by channel fading [2]. One way to alleviate the performance loss is to adopt the precoding technique at the transmitter, where the transmit symbol vector is multiplied by a precoding matrix before signal transmission. The main problem for precoding is that the precoding matrix must be fed back to the transmitter, and it is not possible to use infinite precision for the matrix. In practice, a finite-set codebook, which is pre-developed and available at both the transmitter and the receiver, is used for conducting precoding. For one MIMO channel, a precoder is selected from the codebook at the receiver, and then the index of the selected precoder is fed back to the transmitter. This is a simple yet effective approach in real-world precoding [3]. Thus, how to choose the precoder from a codebook becomes an important issue.

It is well-known that in precoding, different receiver struc­ tures may require different selection criteria. For linear re­ ceivers, several precoder selection methods have been pro­ posed in [3], including post signal-to-noise ratio (SNR) max­ imization and mean-square-error (MSE) minimization. In this paper, we focus on the maximum-likelihood (ML) receiver.

Note that, under high SNR, the error rate performance of an ML receiver strongly depends on the minimum distance of the received signal constellation, referred to as free distance. Therefore, we can choose the precoder that reshapes the MIMO channel to have the largest free distance. However, it is difficult to evaluate the free distance of a MIMO channel. This is because an exhaustive search is usually required, and the computational complexity can be very high. Thus, a suboptimum solution based on singular value decomposition (SVD) was then proposed in [3]. Instead of maximizing the free distance itself, the SVD-based method maximizes the lower bound of the free distance. Recently, another lower bound for the free distance via QR decomposition (QRD) was developed in [4]. It has been theoretically proved [5] that the QRD-based lower bound is tighter than the SVD-based one. In this paper, we propose using the QRD-based lower bound as a precoder selection criterion. To further improve the system performance, we also propose a method, referred to as enhanced QRD-based method, to tighten the lower bound. With Givens rotations, the computational complexity of the enhanced QRD-based method can be effectively reduced.

Except for precoding, antenna selection is also a common approach to improving the transmission quality in MIMO systems [6], [7]. It is simple to conduct antenna selection, and the computational complexity is very low. Antenna selection can be combined with precoding. With this scheme, the performance can be improved while the additional complexity is limited. For ML receivers, the objective in either precoding or antenna selection is to maximize the free distance. Thus, we can perform these two schemes jointly and optimize the joint selection with our proposed QRD-based methods. Note that antenna selection can be conducted at either the transmitter or the receiver. In this paper, we only consider receive antenna selection [8] since the overhead of required feedback bits will not be increased, or the system can achieve the target performance with less feedback bits. Simulation results show that the proposed methods outperform the conventional SVD­ based method and the joint precoder/antenna selection scheme improve the system performance even further.

The remainder of this paper is organized as follows. Section II outlines the system and signal model we use. Section III gives the SVD-based and proposed QRD-based selection criteria, and Section IV describes the joint optimization for precoder and antenna selection. Section V provides simulation

results evaluating the performance of the proposed algorithms. Finally, we draw conclusions in Section VI.

II. SYSTEM AND SIGNAL MODELS

Consider a wireless MIMO system with

Nt

transmit anten­ nas and

Nr

receive antennas, as described in Fig. l. Let

H

denote an

Nr

x

Nt (Nt � Nr)

channel matrix. We assume that

H

is available to the receiver, but not to the transmitter. For precoder selection, the receiver first chooses a precoder

F p

from the codebook F. Note that the codebook consists of a finite set of precoding matrices, i.e., F =

{F

1,

F

2,

... ,F B},

where

B

is the size of the codebook, and P E

{I,··· ,B}.

Assume this codebook is known at both the transmitter and the receiver. Via a feedback channel, the receiver sends this index p to the transmitter. Finally, the transmitter uses the corresponding precoder

F p

for precoding. We assume each precoding matrix in F has unit-norm columns so that the total transmit power is constrained. For spatial multiplexing, the input symbols are multiplexed into an

M

x

1

symbol vector

s,

and then multiplied by an

Nt

x

M (Nt � Nr � M)

precoder

F p

before transmission. Thus, the received signal vector can be expressed as

y

=

HFps+n

(1)

where

n

is the

Nr

x

1

Gaussian noise vector with the covariance a2IN�. The ML estimate of the transmit vector

S

can be expressed as

s

=

min

Ily

-

HF p

Si

11

2 (2)

"iES

where S is the set of all possible transmitted symbol vectors. Note that when a colored noise is considered in the system model, we have to conduct the whitening process at the receiver so that the minimum distance criterion in (2) can be used for signal detection.

Tx

Rx

Code book 1+---1

Fig. 1. System model for a limited-feedback precoding MIMO system.

III. PRECODER SELECTION CRITERIA

The free distance, which dominates the error rate perfor­ mance of ML detection for high SNR regimes, is defined as

dJ2

min -

- min (3)

Let

(Si - Sj)

denote the difference vector, where i

:I j.

Thus, for a given channel realization

H,

the optimum precoder selection criterion is to choose

F p

E F such that the free distance is maximized. From (3), we observe that the optimum solution is found by the exhaustive search over all possible difference vectors. Such numerical search, however, may be prohibitive when a large number of transmit bit-streams and a high-order QAM are adopted. In practice, we consider a suboptimum solution in which a lower bound of the free distance is maximized.

A.

SVD-based selection criterion

Assume that

HF p

is an

Nr

x

M

full column rank matrix, and its SVD is given as

HF p

=

U A V*, where * represent

the operation of Hermitian transpose, U is an

Nr

x

Nr

unitary matrix, V is an

M

x

M

unitary matrix, and A

=

[diag

(

Al, A2, . . . , AM

)

OMX(N�-M)r is an

Nr

x

M

matrix. The non-zero entries of A are the singular values of

HF p.

Based on the Rayleigh-Rits theorem, a lower bound for the free distance using SVD was derived in [6]. It is shown that

(4) where AM is the minimum singular value of the matrix

HF p,

and

cP.nin

is the minimum distance between any two distinct transmit symbol vectors. Note that

cJ2min

is a deterministic value for a fixed QAM modulation size. Therefore, the lower bound only depends on the minimum singular value of

HF p.

In [3], Heath

et al.

proposed the use of (4) to solve the precoder selection problem. The SVD-based precoder selection method can then be described as follows. With a given channel matrix

H,

conduct SVD for each

HF p.

Choose the precoder

F p

E

F whose

HF p

provides the largest AM.

With this criterion, only computing the minimum singular value AM of each

HF p

is required, and the computational complexity can be reduced dramatically. However, the problem for the SVD-based method is that the lower bound (4) may not be tight enough for evaluating the free distance.

B.

QRD-based selection criterion

With QRD, we can factorize the matrix

HF p

in the form of

HF p

= QR

, where

Q

is an

Nr

x

M

column-wise orthonormal matrix and

R

is an

M

x

M

upper triangular matrix with positive real-valued diagonal entries as

(

R

�

,l

R=

_.

o o

)

Via this decomposition, we can have another lower bound for the free distance as

(5) where

[R]min

is the minimum diagonal value of

R.

Thus, we can have the QRD-based selection criterion described as follows. With a given channel matrix

H,

conduct QRD for

each

HF p.

Choose the precoder

F p

E :F whose

HF p

provides the largest

[R]min.

With the Cholesky factorization, it has been shown [5] that the lower bound achieved with QRD is tighter than that with SVD; that is

[R]min 2:: AM.

(6)

In other words, (5) will be more accurate when evaluating the free distance. We hereby provide another proof for (6). The main idea is treating the QRD as a special case of the generalized triangular decomposition (GID) [9] and use the corresponding properties.

Definition

1: Let

g =

(a1, a2, . . . , am)

and Q

=

(b1, b2,

• • •

, bm)

be two positive, real-valued sequences satis­

fying

and

b1 2:: b2 2:: . . . 2:: bm·

We say that

g

majorizes Q in the product sense [10], [11] if

I I

II

ak 2::

II bk

k=l

k=l

for all l

= 1,2,

... , m, and with equality when l

=

m.

Proposition

1: The inequality in (6) holds for any

M

x

M

full rank matrix

HF p.

Proof.

With QRD

HF p

=

QR,

we can express

R

as

R

=

Q*HFp.

(7)

Since the singular values �

=

( A1, A2, . . . , AM)

of

HF p

are invariant under the unitary transformation, it follows that

HF p

and

R

provide the same singular values. Note that

R

is an upper triangular matrix, which means its diagonal elements

r. =

(r1, r2, . . . , r M

) are exactly the eigenvalues of

R.

Arrange both sequences � and

r.

in a decreasing order. By a theorem in [12], we can have that � majorizes

r.,

which is equivalent to the following inequality

I I

II

Ak 2::

II

rk

k=l

k=l

(8) for alll

= 1,2, . . . , M,

and with equality when l

= M.

Thus, we can conclude that

[R]min 2:: AM,

which completes the proof.

C.

Enhanced QRD-based selection criterion

The tightness of the QRD-based lower bound in (5) may degrade for large

M

even though (6) still holds. In this subsection, we propose a simple method to enlarge

[R]min

for each

HF p

so that the bound provided by QRD is even tighter. Assume that the same QAM modulation is adopted

for each transmit bit-stream. Under this assumption, the ML detection criterion in (2) can be written as

s

=

min

Ily

-

HF pPP*Si 112

8iES

=

min

Ily

-

H/S� 112

(9)

8�ES

where

P

is a permutation matrix,

H'

=

HF pP,

and

si =

P*Si.

The key observation that leads to the enhanced QRD­ based method is that the solution of the ML detection in (9) remains the same since

si

is a vector obtained with an element ordering of

Si.

Note that

H'

is a matrix obtained with a column ordering of

HF p,

and the QRD of

H'

will give a different

[R]min.

This fact motivates us to adopt the permutation method to further tighten the QRD-based lower bound. Also note that the permutation method proposed here cannot be adopted in the SVD-based scheme since the singular values of

HF p

are independent of columns permutation. The details for the enhanced-QRD method can be summarized as follows. For a given

HF p,

we can obtain

M!

matrices with column permutations denoted as

H1

=

HFpP1,H2

=

HF pP 2, . . . , HM!

=

HF pP M!,

where

P n

is a permutation matrix corresponding to a specific permutation pattern

n.

Let their QRDs be expressed as

(10) where

n = 1, . . . , M!.

From the above permutation method,

M!

different minimum diagonal entries for a given

HF p

can be obtained, and we can choose the maximum one, denoted by

[R]min,maz,

as the minimum diagonal entry of

HF p.

Therefore, the enhanced QRD-based method is given as follows. With a given channel matrix

H,

use the permutation method to compute the

[R]min,maz

for each

HF p.

Then, choose the precoder

F p

E :F whose

HF p

provides the largest

[R]min,maz.

The permutation method we propose can tighten the lower bound in (5), but the computational complexity will be increased due to the extra

(M

-

1)

QRD operations. To

reduce the complexity, we can use Givens rotations [10], [13] for computing the QRD of each

Hn.

First, we assume that

H1

=

Q1R1

is available via a complete QRD, and

H2

is another matrix different from

H1

by exchanging two neighbor columns. We then seek to obtain

R2

of

H2

without another complete QRD. Denote P as a permutation matrix that exchanges two specific neighbor columns of

H1.

We then have (11) where

R1

is a near upper triangular matrix. Now, all we have to do is to transfer

R1

into an upper triangular matrix. Since P only exchanges two neighbor columns of Rl. we can upper­ trianglize

R1

by applying a simple Givens rotation matrix

G1,

that is,

G1R1

=

T,

where

T

is an upper triangular matrix. Thus we can rewrite (11) as

where

Q2

=

QI Gi

is a unitary matrix. Since the QRD

of a full column rank matrix is unique [10], we know that

Q2T

in (12) is the QRD of

H2,

and

T

is equal to

R2.

In other words, we obtain

R2

by simply left-multiplying a Givens rotation matrix on

RI

rather than by performing a complete QRD on

H2.

Therefore, we can dramatically reduce the computational complexity of the enhanced QRD-based scheme. Fig. 2 illustrates (for

M

=

3)

how each

Rn

can

be derived with Givens rotations.

82 =[h2 hi h31 83 =[hl h3 h21

1

1

84 =[h2 _{h3 hl1} 85 = [h3 _{hi h21}

1

86 =[h3 h2 hl1

Fig. 2. The ordering of computing each Rn of HF p. M = 3

D. Capacity-based selection criterion

The criterion of capacity maximization is also widely con­ sidered in the precoding problem [31. With a given equivalent channel matrix

HF p,

the capacity can be expressed as

C =

log2 det(IM

+

�

F;H*HF p)

(13)

where p is the average SNR per receive antenna,

detO

denotes the determinant, and

1M

is an

M

x

M

identity matrix. Therefore, the capacity-based method can be given as follows. Compute channel capacity using (13) for each

HF p.

Choose the precoder

F p

E :F whose

HF p

provides the largest C.

The capacity-based selection criterion is derived from a general capacity formula, which is independent of the receiver structure. Thus, it may not provide the guaranteed performance improvement for some channel realizations. Also, the permu­ tation method in our enhanced QRD-based scheme cannot be adopted in the capacity-based method since the channel capacity C is invariant under the column permutation of the matrix

HFp.

E. Complexity comparisons

One way to quantify the complexity of the matrix compu­ tation is to count the number of floating operations (FLOPS). Several efficient algorithms for conducting QRD and SVD are given in [131. In general, SVD requires more FLOPS than QRD does. As a result, the QRD-based selection scheme not only has better performance, but also requires lower compu­ tational complexity. For the enhanced QRD-based scheme, the computational complexity of performing QRD on all

Hn

(for a given

HF p)

is

0(M!M3).

As mentioned, we can reduce the complexity via Givens rotations, in which only one complete QRD and

(M!

-

1)

upper-triangulization

operations are required. Each upper-triangulization operation only needs

0(3

x

42)

FLOPS. Thus, the overall computational complexity is reduced from

0(M!M3)

to

0(M3 + 48(M!

-1))

�

0(M3 + M!).

As for the capacity-based method,

the computational complexity is

0(M3),

which mainly arises from computing the determinant and the matrix multiplication

F;H*HF p

in (13). Note that there is an additional overhead for the capacity-based method since the variance of the chan­ nel noise is required.

IV. JOINT PRECODER AND ANTENNA SELECTION

Antenna selection is a simple yet effective method to enhance the diversity gain in a wireless MIMO system. It has been shown that with ML detection, the optimum antenna subset is the one giving the largest free distance. We hereby propose a scheme that combines transmit precoding with an­ tenna selection. Note that antenna selection can be conducted at either the transmitter or the receiver side. In this paper, we only consider the receive antenna selection. The advantage of receive antenna selection is that the feedback overhead will not be increased. The system model for joint precoder and receive antenna selection is shown in Fig. 3.

Tx

MIMO Channel

Rx

Codebook 1+---1

Fig. 3. System model for joint precoder and receive antenna selection in a MIMO system.

Assume that

Nt � Nr

>

M.

It means we have

(�)

receive antenna subsets to choose. According to some criterion, the receiver jointly determines the optimum precoder

F p

E :F , and the receive antenna subset indicated by the index q. Note that, via the feedback channel, only the index p will be sent back to the transmitter since the antenna selection is performed at the receiver side. The received signal in (1) can be rewritten as

(14) where

Hq

is the channel matrix corresponding to the selected receive antenna subset. The ML detection can then be rewritten as

min (15)

Thus, among

B

(�)

possible combinations, we choose a pair of

HqF p

that can give the largest free distance. The joint

precoder and receive antenna selection problem can be viewed as a two-dimensional optimization problem as shown in Fig. 4. As mentioned, the optimum solution needs an exhaustive search over all possible difference vectors in (IS), which requires high computational complexity. Thus, we can use the suboptimum solution described in Section III for this joint optimization problem. The inequalities (4) and (5) can be rewritten as

(16) and

(17) respectively. Since

[R]min � AM,

we expect that (17) will be a better criterion for the optimization problem. Furthermore, the enhanced QRD-based method can also be used for further performance improvement. Similarly, the computational complexity of the enhanced QRD-based method can be reduced by applying Givens rotations.

F 2 F I Free Distance #2 Free Distance # I s bs u et#1 Free Distance #(8+2) Free Distance #(8+ _I) subset #2 · . . · . . · . . � ______________ �y� ______________ J Antenna Selection ,

Fig. 4. Two-dimensional search for the optimum Hq F p in joint optimization

problems.

V. SIMULATION RESULTS

In this section, we report simulation results demonstrating the effectiveness of the proposed algorithms. In simulations, we consider a flat-fading MIMO channel, of which the entries are assumed to be Li.d complex Gaussian random variables with zero mean and unit variance. The QPSK modulation is assumed at the transmitter while the ML detection is conducted at the receiver. The codebooks we use in the simulations are obtained from [14].

Fig. 5 shows the bit error rate (BER) performance of precoding. Here,

Nt

=

6, Nr

=

M

= 3, and

B

=

64.

As

we can see, the scheme without precoding (3 x 3) suffers from the performance loss in channel fading. For precoding, the QRD-based method indeed outperforms the SVD-based method, and the enhanced QRD-based method achieves the best performance, about

2

dB better than the SVD-based method. Besides, the performance of capacity-based method is slightly better the SVD-based method but worse than the proposed methods. Note that the computational complexity of the capacity-based method is higher since the receiver needs to estimate the variance of channel noise.

-+-No Precoding (3x3 ML) . 10-' -+-SVD-based __ Capacity-based --e-ORO-based -Enhanced ORO-based ... ... .. 10-6�==::;:::::==:::I::="---'-______ L-____ --'-____ �� 4 6 8 10 12 Average SNR(d8) 14 16 Fig. 5. BER performance comparison for precoder selection with Nt =

6, Nr = 3, M = 3, and B = 64. -+-No Preceding (2x2 ML) . -+-SVD-based __ Capacity-based --e-ORO-based - Enhanced ORO-based .. 1O-6�==C==C:::==C"--__ ,--__ -,-__ ---,,--__ -,-__ �L..J 2 4 6 8 10 12 Average SNR(d8) 14 16 18

Fig. 6. BER performance comparison for precoder selection with Nt =

4, Nr = 2, M = 2, and B = 64.

Fig. 6 compares the BER performance of precoding for the case with

Nt

=

4, Nr

=

M

=

2,

and

B

=

64.

Similarly, the

proposed selection methods outperform the SVD-based and capacity-based methods in high SNR regimes. In this set of simulations, the gap between the QRD-based and the enhanced QRD-based method is not obvious since we only have two permutation patterns for

M

=

2.

Fig. 7 shows the performance improvement for joint pre­ coding and receive antenna selection. In this case, we let

Nt

=

4, Nr

= 3,

M

=

2,

and

B

=

16,

which means we

have 3 receive antenna subsets to choose. As we can see, the method is very effective for performance improvement. The proposed joint selection methods outperform other selection methods. Compared to the result in Fig. 6, the capacity-based method exhibits some performance loss for high SNR. This

ffi 10-' .••. . '" . --+-No Joint Selection (2x2 ML) . -+-SVO-based __ Capacity-based -e-ORO-based

--Enhanced ORO-based

1O-6L::::====:I::::====:::::!.. __ '--__ --'-__ ----L��___.J

4 6 8 10

Average SNR(dB) 12 14 16

Fig. 7. BER perfonnance comparison for joint precoder and receive antenna selection with Nt = 4, Nr = 3, M = 2, and B = 16.

0: LU '" --+-Precoder Selection, 8=4 -+-Precoder Selection, 8=8 __ Precoder Selection, 8=16 -e-Precoder Selection, 8=64

--Joint Selection, 8=4

1O-6l':===:::i::::==S===�_'____ __ ___'__ ___ '__ __ ___'___�

4 6 8 10 12

Average SNR(dB) 14 16 Fig. 8. BER perfonnance comparison between joint selection (Nt = 4, Nr = 3, and M = 2) and precoder selection (Nt = 4, Nr = 2, and

M= 2).

can be explained by the fact that it maximizes the channel capacity, not the free distance. Thus, its performance may degrade for some channel conditions, Besides, we observe that the gap between the QRD-based and SVO-based method is reduced somewhat since we only have

(�)

24 =

48

candidate

matrices in this case. Fig. 8 shows the reduction of the required feedback bits when the joint selection scheme is considered, Here, all results are obtained with the enhanced QRD-based method, We observe that at least log

�

4 - log

�

=

4

bits can be

saved when an extra antenna is used at the receiver. Note that increasing the receive antenna may not be always possible for some applications due to the size constraint at the receiver. Thus, the joint selection method can be viewed as a tradeoff between the feedback bits and the number of receive antennas.

VI. CONCLUSIONS

In this paper, we propose a QRD-based precoder selection method for ML receivers. Theoretical and simulation results indicate that the QRD-based method is not only better than the conventional SVO-based method, but also has lower computa­ tional complexity. To further improve the performance, we also propose the enhanced QRD-based method that can provide a more accurate estimate of the free distance. Using Givens rotations, the computational complexity of the enhanced QRD­ based method can be reduced effectively. Besides, we combine the precoding with receive antenna selection, and solve the selection problem using the proposed methods. Simulations show that the proposed approaches can provide the significant performance improvement. Moreover, the proposed QRD­ based approaches will exhibit a significant advantage when sphere-decoding (SO) [15], an efficient algorithm for the ML detection, is used at the receiver. Note that the QRD is also required in the SO algorithm, which implies that the same QRD unit can be shared by proposed selection methods and the SO algorithm. Based on the above reasons, we conclude that the QRD-based selection algorithms will be much more efficient in real-world applications.

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