Ordered Statistics Based Rate Allocation Scheme for Closed Loop MIMO-OFDM System ∗

PAPER

Ordered Statistics Based Rate Allocation Scheme for Closed Loop MIMO-OFDM System ^∗

Le ZHANG

^†a)

, Student Member, Xiang HE

^†

, Hanwen LUO

^†

, Nonmembers, and Xiaoying GAN

^†

, Student Member

SUMMARY A new approach to low-complexity rate allocation scheme in closed loop MIMO-OFDM system is proposed. The new scheme utilizes ordered statistics of channel matrix’s singular values to simplify the ideal scheme which uses water filling in both frequency and space domain. Un- like the conventional simplified algorithm called FFC [1] (“frequency flat constraint”), the proposed scheme has no restrictions of the numbers of antennas. The improvement of SNR gain with this new scheme is about 0.8 dB in 2-antenna systems with a little more complexity than FFC.

key words: Wishart, MIMO-OFDM, rate allocation

1. Introduction

With wireless services increasingly in demand, there is need for more throughput per bandwidth to accommodate more users with higher data rates while maintaining a guaranteed quality of service [2]. 3G and Beyond 3G should deliver higher data transmission rate and more diverse services [3].

At the physical layer, systems with multiple input multiple output (MIMO) technologies and orthogonal frequency di- vision multiplexing (OFDM) have received increasing atten- tion in the recent years. On MIMO-OFDM fading channels, the capacity-achieving rate allocation is based on water- pouring over space, frequency, and time [4].

Various resource allocation schemes have been pro- posed for such systems. One kind of rate allocation strategy for MIMO-OFDM systems is based on SVD (singular value decomposition) and receiver-to-transmitter feedback. In this paper, we study the bit allocation scheme for this MIMO- OFDM closed loop system using a statistical property of the channel matrix’s singular values.

The rest of this paper is organized as follows. Section 2 describes the MIMO-OFDM channel model and the prob- lem statement. Section 3 derives the ordered marginal P.D.F (probability density function) of Gaussian random matrix’s singular values. We demonstrate that they have a central- ized property with increasing antennas. We make use of this result in Sect. 4 to make a “worst case” analysis of FFC scheme under continuous water filling. In Sect. 5, we

Manuscript received December 19, 2005.

Manuscript revised April 10, 2006.

†

The authors are with the Department of Electronics Engineer- ing, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China.

∗

This work is sponsored by National Hi-Tech Research and Development Program of China (National 863 Program) under Grant No. 2003AA123310, and NSFC under Grant No. 60332030 and No. 60572157.

a) E-mail: [email protected] DOI: 10.1093/ietcom/e89–b.12.3274

explain the ordered statistics based rate allocation scheme.

Section 6 compares the new algorithm with FFC scheme and shows that it yields satisfactory performance. Section 7 con- cludes the paper with a brief summary.

2. System Model and Bit Allocation Problem

Consider a MIMO-OFDM system with t transmit antennas and r receive antennas given by:

y = H

n

x +
n (1)

where
y is the received vector,
x is the transmitted vector,

n is usually modeled as a zero mean i.i.d (independent and identically distributed) Gaussian vector with covariance σ

²

, H

n

is the channel matrix, which is modeled as an i.i.d Gaus- sian matrix. Channel state information (CSI) is only known at the receiver side. The receiver makes an SVD of H

n

as in (2) and feeds the singular vector matrix D

n

back to the transmitter. The transceiver then makes substitution shown in (3) to obtain a set of AWGN subchannels.

y = V

n

D

_n

U

_n

x +
n (2)

y

^

= D

n

x

^

+
n

^

(3)

where
y

^

= V

^H_n

y,
x

^

= U

n

x,
n

^

= V

_n^H

n, D

n

= diag

s

_1,n

, s

_2,n

, · · · s

t,n



, s

i,n

is the ith singular value of H

_n

, U

_n

, V

n

are unitary matrices. Note that the amount of feedback data increases with the antenna number.

In this paper, we study the bit allocation scheme for this MIMO-OFDM closed loop system. Assume N subcarriers are used and r=t=M. Then there are NM subchannels alto- gether, indexed by {n, m} , n ∈ {1, · · · , N} , m ∈ {1, · · · , M}.

Let the rate allocated to channel (n, m) be r

m,n

, Then bit al- location problem is to choose the set of {r

m,n

} to minimize the instantaneous SNR requirement



N n=1



M m=1

2

^rm,n

− 1

s

^_m,n²

(4a)

subject to



N n=1



M m=1

r

m,n

= R, r

m,n

≥ 0 (4b)

where R is the overall transmission rate. s

^m,n2

= s

²m,n

/σ

²

.

Copyright c  2006 The Institute of Electronics, Information and Communication Engineers

Optimization of the problem can be achieved by wa- ter filling in both frequency and space domain and its com- plexity is proportional to the number of unknown variables r

m,n

. Thus in practice, this ideal algorithm is virtually in- feasible due to the huge number of subcarriers (e.g. 1024).

A simplified algorithm [1] uses FFC (“frequency flat con- straint”), which equally allocates R among all subcarriers and performs water filling only within subchannels of the same subcarrier. The bit allocation problem is to find the set of {r

m,n

} to minimize the instantaneous SNR requirement



M m=1

2

^rm,n

− 1

s

^m,n2

(5a)

subject to



M m=1

r

m,n

= R, r

m,n

≥ 0 (5b)

The average performance loss of FFC schemes is neg- ligible only when antenna number is larger than 4, and to ensure better worst-case performance even more antennas must be used. Thus, FFC simplifies water filling procedure at the cost of both more antenna number and more feedback data.

In this paper we aim to provide an alternative bit al- location scheme by studying the ordered statistics of H

n

’s singular values. The new scheme’s performance loss is sig- nificantly smaller compared to FFC in the case of systems with smaller number of antennas.

3. Marginal P.D.F of Gaussian Random Matrix’s Or- dered Singular Values

Joint P.D.F. of i.i.d Gaussian random matrix’s unordered sin- gular values was a classical result given by Wishart in 1920s [5]. The marginal P.D.F of the smallest singular value is later given in [6], which is used to decide the number of the receive antennas. We find that, with similar method, the marginal P.D.F. of all ordered singular values can be derived.

We use the same notation and assumption as those in equa- tion (2) and (3). H

^H_n

H

n

is conventionally called a Wishart matrix. Let its unordered eigenvalue be λ

j

and ordered one be λ

( j)

, then

P 

λ

(k)

≥ a 

= P 

λ

(k−1)

≥ a 

+ 

S

P 

∀i ∈ S, λ

i

< a, ∀ j ∈ S

^C

, λ

j

≥ a  (6)

where S ⊂ {1, 2, · · · , t} , card(S ) = k − 1 is the cardinality of S,k ≥ 2, S

^C

is the complementary set of S. The 2nd term on the right side of (6) can be expressed as (7). Here p(λ

1

· · · λ

k

) is the joint P.D.F of the eigenvalues.



S

P 

∀i ∈ S, λ

i

< a, ∀ j ∈ S

^C

, λ

j

≥ a 

= 

S



_∞

a

· · ·



_∞












a

t−(k−1)



a 0

· · ·



a










0 k−1

p(λ

1

· · · λ

t

)dλ

i1

· · ·

dλ

i_k−1

dλ

j₁

· · · λ

j_t−(k−1)

(7)

The joint P.D.F is given by (8) according to [5].

p (λ

1

, · · · λ

t

) = c

⎛ ⎜⎜⎜⎜⎜

⎝



t i=1

λ

^si

exp (−λ

i

)

⎞ ⎟⎟⎟⎟⎟

⎠ ·



1≤i< j≤t

 λ

i

− λ

j



2

(8) where c is the normalization constant.

Using similar procedure described in [6], the integral of the joint P.D.F has the form in (9).



a 0

· · ·



a










0 t

p( λ

1

· · · λ

t

)d λ

i1

· · · dλ

ik−1

d λ

j1

· · ·

dλ

j_t−(k−1)

= t!c det  A

r,t,a



(9)

where A

r,t,a

=

⎛ ⎜⎜⎜⎜⎜

⎜⎜⎜⎜⎜

⎜⎜⎜⎜⎜

⎝

Γ ˜

a

(s + 1) · · · Γ ˜

a

(r) Γ ˜

a

(s + 2) · · · Γ ˜

a

(r + 1)

... ...

Γ ˜

a

(r) · · · ˜Γ

a

(r + t − 1)

⎞ ⎟⎟⎟⎟⎟

⎟⎟⎟⎟⎟

⎟⎟⎟⎟⎟

⎠

, in which

Γ ˜

a

(p) = 

_∞

a

t

^p−1

e

^−t

dt and s = r − t.

By using the same method, we find that (10) also holds.



S

P 

∀i ∈ S, λ

i

< a, ∀ j ∈ S

^C

, λ

j

≥ a 

= t!c det 

B

r,t,a,s

 (10)

where B

r,t,a,s

is modified from A

r,t,a

by changing k − 1 columns indexed by S as shown in (11)

B

r,t,a,s

(i, j) =

 1 − A

r,t,a

(i, j) , ∀ j ∈ S

A

r,t,a

(i , j) , ∀ j ∈ S

^C

(11) We see that (9) is a special case of (11). By substituting (10) into (6) and (7), we arrive at (12), which is our major result.

P  λ

(k)

≥ a  = P  λ

(k−1)

≥ a  + t!c 

S

det  B

r,t,a,s



(12) Marginal cumulative probability density function (C.D.F) of H

n

’s ordered singular value s

(k)

can be calculated with (13).

P 

s

(k)

≤ a  = P 

λ

(k)

≤ a

²



= 1 − P 

λ

(k)

> a

²



(13)

With increasing antenna number, s

(k)

for a fixed k be-

comes more centralized around its mean. This is verified by

Table 1 in which we estimate the average variance ¯ σ

²

of s

(k)

,

which decreases by almost 45% when r increases from 2 to

6.

Table 1 Average variance of singular values.

t,r 2 3 4 5 6

σ¯² 0.165 0.137 0.119 0.104 0.095

4. Proof of Convergence for Frequency Flat Constraint Scheme

The P.D.F derived above encourages us to explore the rea- son for FFC scheme to approach ideal performance when r and t increase. We do so by comparing the ideal water- filling solution to (4) with the FFC water-filling to (5). As- sume R in (4) and (5) is sufficiently large so both degenerate to the Lagrange problems. Then their solutions can be ex- pressed analytically as (14a) and (14b) for ideal waterfilling and frequency-flat-constraint respectively, whose derivation is present in the following Appendix.

r

m,n

= 1 MN

⎧⎪⎪ ⎨

⎪⎪⎩− 

m



n

log

₂

s

^_m,n²

ln 2

⎫⎪⎪ ⎬

⎪⎪⎭

+ log

₂

s

^_m,n²

ln 2 + R

MN

(14a)

r

m,n

= 1 M

⎧⎪⎪ ⎨

⎪⎪⎩− 

m

log

₂

s

^_m,n²

ln 2

⎫⎪⎪ ⎬

⎪⎪⎭

+ log

2

s

^_m,n²

ln 2 + R

MN

(14b)

The average SNR per antenna per tone S NR

av

is de- fined in (15a) for ideal water filling and in (15b) for FFC scheme

S NR

av

=



N n=1



M m=1

2

^rm,n

− 1

s

^m,n2

(15a)

S NR

av

=



M m=1

2

^rm,n

− 1

s

^m,n2

(15b)

Then the average SNR per antenna per tone S NR

⁽¹⁾av

is (16b) for ideal water-filling and S NR

⁽²⁾av

is (16c) for FFC scheme and their derivations can also be found in the ap- pendix. We note that the 1st term grows exponentially with R, while the 2nd term remains unchangeable. Thus

R

av

=

∆

R

MN , x

m,n

= 1/s

∆ ^m,n2

(16a)

S NR

⁽¹⁾_av

=2

^Rav

⎛ ⎜⎜⎜⎜⎜

⎜⎜⎝



N n=1

⎛ ⎜⎜⎜⎜⎜

⎝



M m=1

x

m,n

⎞ ⎟⎟⎟⎟⎟

⎠

1/M

⎞

⎟⎟⎟⎟⎟

⎟⎟⎠

1/N

−



M m=1



N n=1

x

m,n

MN

(16b)

S NR

⁽²⁾_av

=2

^Rav



N n=1

⎛ ⎜⎜⎜⎜⎜

⎝



M m=1

x

m,n

⎞ ⎟⎟⎟⎟⎟

⎠

1/M

/N

−



M m=1



N n=1

x

m,n

MN

(16c)

S NR

⁽¹⁾av

S NR

⁽²⁾av

≈

⎛ ⎜⎜⎜⎜⎜

⎝



N n=1

 

_M

m=1

x

m,n 1/M

⎞

⎟⎟⎟⎟⎟

⎠

1/N



N n=1



_M



m=1

x

m,n 1/M

/N

=



_N



n=1

y

n 1/N



N n=1

y

n

/N y

n

=

⎛ ⎜⎜⎜⎜⎜

⎝



M m=1

x

m,n

⎞ ⎟⎟⎟⎟⎟

⎠

1/M

(16d)

P

⎛ ⎜⎜⎜⎜⎜

⎝



_M



m=1

U

_m⁻²

1/M

≤ y

n

≤



_M



m=1

L

⁻²_m

1/M

⎞

⎟⎟⎟⎟⎟

⎠ ≈ 1 L

m

= sup !

l """P(s

^m,n

≤ l) < 0.01 # U

m

= inf !

u """P(s

^m,n

≥ u) > 0.99 #

(16e)

where sup {x} is the supremum of x and inf{x} is the infimum of x. Worst case is when y

n

that minimizes (16d). It is easily verified that the worst case must be achieved on the boundary. That is, y

n

takes either the minimal value or the maximal value in its possible value range. FFC works best when s

m,n

are similar for different n and fixed m, so the worst case appears when they are as different as possible. Thus minimization of (16d) is simplified into a single parameter function of n

L

, which is the number of y

n

that takes minimal value. n

L

is given in (17a).

n

L

N ≈ (a − b) − b (ln a − ln b)

(a − b) (ln a − ln b) = p

^∆

(17a)

a =

^∆

⎛ ⎜⎜⎜⎜⎜

⎝



M m=1

U

m⁻²

⎞ ⎟⎟⎟⎟⎟

⎠

1/M

, b =

^∆

⎛ ⎜⎜⎜⎜⎜

⎝



M m=1

L

⁻²m

⎞ ⎟⎟⎟⎟⎟

⎠

1/M

(17b) p in (17a) for 2-antenna system is computed to be 22%

and the worst case performance loss per antenna per tone compared to ideal water-filling shifts from 4 to 8 dB with rate/bits/sec/tone increasing from 5 to 30. Within the same range of R, loss of system with 4 antennas increases from 2 to 4 dB, where p is around 29%.

5. Ordered Statistics Based Rate Allocation Scheme

One way of utilizing ordered statistics in rate allocation is

to group subchannels with similar gain and allocate same

rate to the subchannels within the same group if the channel

gain’s variance is small within every group [7]. The opti-

mal algorithm thus requires sorting all subchannels based

on their gains, which is still too complex due to the huge

number. Here we propose a suboptimal algorithm based on ordered statistics of singular values.

The algorithm first performs local sorting within chan- nels of the same subcarrier, based on singular values of H

n

. Let s

_(i),n

be its ith singular value. If the system has M antennas and the ordered singular values are divided into C categories, then each category is indexed by {i, q} , i ∈ {1, · · · M} , q ∈ {1, · · · C}. w

i,q

is the number of subcarriers belonging to this category. r

i,q

is its rate. P

i,q

is its probabil- ity. For different n, we classify s

_(i),n

into different categories based on predefined “quantization threshold.” For arbitrary numbers of categories, the thresholds can be computed by applying marginal P.D.F we derived in section II to Lloyd quantization design algorithm [8] or by making P

i,q

equal to each other. That is, P

i,q

= 1/C. The rate allocation problem is to compute the set of {r

m,n

} so as to minimize the instan- taneous SNR requirement



i



q

⎛ ⎜⎜⎜⎜⎜

⎜⎝(2

^ri,q

− 1) 

n∈q

1 s

^_(i),n²

⎞ ⎟⎟⎟⎟⎟

⎟⎠ (18a)

subject to



i



q

w

i,q

r

i,q

= R, r

i,q

≥ 0 (18b)

whose water-filling solution is r

i,q

= max ⎧⎪⎪⎨

⎪⎪⎩0, log

²

λ − log

₂

ln 2 w

i,q



n∈q

1 s

^_(i),n²

⎫⎪⎪⎬ ⎪⎪⎭ (19) It is slightly different from conventional water-filling solu- tion in space and time in that the bottom area of every “bin”

changes from 1 to w

i,q

. In practice, since only integer val- ues are available for r

i,q

, (18) can be solved by the greedy algorithm [9].

6. Experimental Results

We compare the average performance of proposed algorithm with conventional water-filling scheme and FFC scheme un- der 100 randomly generated MIMO-OFDM channels. Bit allocation is discrete and uses greedy algorithm. We con- sider the case of 64 subcarriers. Figures 1 and 2 compare the efficiency of the algorithm in terms of the SNR and the ordered singular values are divided into 4 categories.

Figure 1 shows the SNR loss of the new scheme com- pared with “ideal” algorithm, which uses greedy algorithm in both space and frequency domain. When 2 antennas are used, the new scheme introduces 0.7 dB penalty at most.

The performance loss decreases with increasing antenna number, indicating that FFC is not the only algorithm that exhibits this property. Figure 2 shows the SNR gain of the new scheme compared with FFC scheme. We observe a 0.8 dB gain when 2 antennas are used, which means that it is more suitable for small antenna array. And for large antenna array, its performance is similar to FFC scheme.

We notice in these 2 figures that there are fluctuations

in the SNR losses and gains. The major reason is due to the granularity of the water-filling algorithm. The coefficient

Fig. 1 SNR loss per tone per antenna compared with ideal water-filling.

Fig. 2 SNR gain per tone per antenna compared with FFC scheme.

Fig. 3 SNR gain per tone per antenna compared with FFC scheme with 6 categories and 2 antennas.

Table 2 Complexity comparison of the three schemes.

Algorithm Number of water-filling

Conventional MN

FFC M

Ordered-statistics MC

before r

i,q

is 1 for conventional and FFC algorithm, but is w

i,q

for the proposed algorithm. That means our algorithm has a larger granularity. So in some cases, w

i,q

bits are allo- cated, and sometimes due to some constraints, they cannot be allocated, then the performance suffers. If the number of categories is large enough so that w

i,q

approaches 1, the fluctuations will be smaller. In Fig. 3 the SNR gain of the proposed scheme with 6 categories for two antennas com- pared with FFC scheme is shown. The fluctuation is not as wide as Fig. 2 and does not exhibit a wave-like shape. We also compare the complexity of the proposed scheme and the FFC scheme in terms of the number of water-filling as listed in Table 2. The complexity of the proposed scheme is sig- nificantly lower than the conventional water filling scheme and a little more complex than the FFC scheme depending on the number of category.

7. Conclusion

In this paper we propose a new bit allocation scheme based on ordered statistics of i.i.d Gaussian matrix’s singular val- ues. The new algorithm does not require very large num- ber of antennas to work well, thus it does not suffer from high feedback overhead. It is simpler than ideal water-filling scheme with significant reduction in unknown variables, and has better performance and offers more flexibility in trade- off between performance and complexity compared to FFC scheme since di fferent numbers of “thresholds” can be used for different performance requirements. Also our method can be extended to general channel matrix other than Guas- sian case as long as its ordered marginal P.D.F of singular values is obtainable either analytically or numerically. In simulation we show that when 4 categories are used, the new algorithm outperforms FFC scheme by 0.8 dB for 2-antenna system with similar complexity. With the increase of cate- gories, the performance fluctuation is also narrower.

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Appendix: Derivation of (14a), (14b), (16b) and (16c) The bit allocation problem (4) can be solved using the stan- dard Lagrange multiplier technique. Define the Lagrangian as (A· 1)

f (r

m,n

, λ) =



N n=1



M m=1

2

^rm,n

− 1 s

^_m,n²

− λ

⎛ ⎜⎜⎜⎜⎜

⎝



N n=1



M m=1

r

m,n

− R

⎞ ⎟⎟⎟⎟⎟

⎠ (A· 1) where λ is a Lagrange multiplier, then the solution for r

m,n

can be obtained by solving (A · 2)

⎧⎪⎪ ⎪⎪⎪⎨

⎪⎪⎪⎪⎪

⎩ d f dr

m,n

= 0 d f dλ = 0

(A· 2)

Then we get the following (A · 3) r

m,n

= log

2

λ − log

2

(ln 2 /s

^m,n2

) (A · 3)

After the summation over M and N, we get

log

₂

λ =

⎛ ⎜⎜⎜⎜⎜

⎝R +



N n=1



M m=1

log

₂

(ln 2/s

^m,n2

)

⎞ ⎟⎟⎟⎟⎟

⎠

$

MN (A· 4)

By Substituting (A· 4) into (A· 3), we arrive at (14a).

For derivation of (14b), the bit allocation problem (5) can be solved using the standard Lagrange multiplier tech- nique. Define the Lagrangian as (A · 5)

f (r

m,n

, λ) =



M m=1

2

^rm,n

− 1 s

^_m,n²

− λ

⎛ ⎜⎜⎜⎜⎜

⎝



M m=1

r

m,n

− R/N

⎞ ⎟⎟⎟⎟⎟

⎠ (A· 5) Then we get the following (A· 6)

r

m,n

= log

₂

λ − log

₂

(ln 2/s

^m,n2

) (A· 6)

After the summation over M, we get

log

₂

λ = R/MN +



M m=1

log

₂

(ln 2 /s

^m,n2

) /M (A · 7)

By Substituting (A · 7) into (A· 6), we arrive at (14b).

According to the definition of S NR

av

in (15a), we sub- stitute (14a) into (15a). And we can get (16b). The proce- dure is similar for (16c).

Le Zhang was born in 1979. He received the B.E. degree in communication engineering from Shanghai Jiao Tong University, Shanghai, China in 2001. He is now pursuing the Ph.D.

degree in Shanghai Jiao Tong University, Shang- hai, China. His main research interests include wireless communications and error correcting codes.

Xiang He received the B.E. and M.W. degrees from the Department of Electronics Engineering, Shanghai Jiao Tong University, Shanghai, China in 2003 and 2006, respectively. His research interests include channel cod- ing and MIMO theory.

Hanwen Luo was born in 1950. Now he is a professor in the Electronic Engineering Depart- ment, Shanghai Jiaotong University, Shanghai, China. He is currently the vice-chairman of In- stitute of Wireless Communication in Shanghai Jiao Tong University. His research interests in- clude mobile and personal communication.

Xiaoying Gan received the B.E. and Ph.D. degrees from the Department of Electron- ics Engineering, Shanghai Jiao Tong Univer- sity, Shanghai, China, in 2000 and 2005, respec- tively. Her research interests include adaptive resource allocation, partial feedback theory and cellular structure in wireless mobile communi- cation.