document5

1188 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013

A New BER Upper Bound forMultiuser Downlink Systems

Dianjun Chen and Takeshi Hashimoto, Member, IEEE

Abstract—We consider improvement of bit error rate (BER)upper bounds, based on the notion of decomposable error vectors(DEVs) introduced by S. Verdu, for downlink multiuser space-time systems over Rayleigh fading channels. Although omission ofDEVs and limiting before averaging (LBA) technique are knownto improve the standard union bound (UB) considerably, it ismathematically intractable to obtain a bound in a closed-form.In this paper, assuming maximum-likelihood (ML) detectionfor downlink multiuser space-time spreading (STS) systems, weintroduce a new modification of the decomposability of errorvectors and propose a new BER upper bound. We prove thatthe proposed bound is tighter than the standard union bound andthe bound based on the overall decomposability. The numericalresults show that the proposed bound is also tighter thanthe bound based on positive-sum-decomposability (PSD) in thescenario of downlink multiuser transmission.

Index Terms—BER upper bound, maximum-likelihood decod-ing, improved union bound, MIMO fading channel, indecompos-able error vector, space-time spreading (STS), CDMA.

I. INTRODUCTION

PERFORMANCE analysis of maximum likelihood (ML)detection has fundamental importance in designing a

transmission system over channels with interference and fad-ing. Calculating the bit error rate (BER) of a non-orthogonalsystem over fading channels is difficult in general and theunion bound (UB), the sum of all the pairwise error probabil-ities (PEPs), is a standard tool for performance analysis [1].When the system is operative in a severe fading environment,the UB becomes loose and sometimes is not even convergentat low signal-to-noise power ratios (SNRs). Thus, there havebeen many efforts to tighten the UB [2]-[16].

The progressive union bound (PUB) in [2] is an enhance-ment of the UB, where each candidate error vector is comparedwith multiple adversary error vectors contrary to the UBwhere each candidate error vector is compared with the singleadversary vector, the correct vector. Although the enhancementgives an improved bound, each term in the bound includes amultiple integral and hence a closed form expression for thebound is difficult to obtain. In [3], an approximate expressionfor the PUB is obtained first by replacing each multiple

Manuscript received December 30, 2011; revised August 4, 2012. Theassociate editor coordinating the review of this paper and approving it forpublication was G. Bauch.

Some materials of this paper were presented at IEICE Conference onInformation Theory, Nagoya, March 2006, and PIMRC 2009, Tokyo, Sept.2009.

D. Chen is with Beijing University of Posts and Telecommunications (e-mail: dianjun [email protected]).

T. Hashimoto is with The University of Electro-Communications, Tokyo(e-mail: [email protected]).

Digital Object Identifier 10.1109/TCOMM.2013.012313.110850

integral with a saddle point (SP) approximation and then bytruncating the Taylor series expansion of the SP approximationat the second order term. However, the resulting expression isonly an approximation of the PUB and may not even be anupper bound. In [4], from the observation that any probabilitynever exceeds one, a technique called limiting before averag-ing (LBA) is proposed, where the sum of conditional PEPs,conditional on the channel state information (CSI), is replacedwith one whenever the sum exceeds one in the averagingprocess with respect to CSI. The technique improves the UBat low SNRs, but it does not allow analytical expressionsand, moreover, time-consuming averaging is required. As aneffort to obtain bounds with explicit expressions, in [5]-[7], the Gallager bound that was originally used for upperbounding a key thresholding function in information theory[8]is employed to obtain bounds which are looser but moremanageable than the LBA bound. A specialized form of theGallager bound is also proposed in [9] and is applied to codedsystems in [9] and [10]. An alternative approach is to explicitlyexpurgate unnecessary terms from the union bound.

The notion of indecomposable error vectors (IEVs) wasintroduced by S. Verdu to reshape the union bound forthe multiuser detection (MUD) problem over the AWGNchannel [11]. The heart of the idea is that error vectorssatisfying certain conditions are decomposable to error vectorsof smaller weights and that only those error vectors thatare indecomposable in this sense are sufficient for upperbound. The notion has been specialized to fading channels andapplied to performance evaluation for code-division multipleaccess (CDMA) [12][13], for bit-interleaved coded modulation(BICM) [14], and for space-time coding [3][15][16]. In mostof these proposals, that is, in [3], [12], and [14]-[16], anerror vector is considered to be removable only when it isdecomposable for all the CSI, the overall decomposability,and hence the improvement over the UB is limited. Mohassebet al. [13], on the other hand, proposed a new modification tothe decomposability, called the positive-sum decomposability(PSD), and derived a new BER upper bound for the CDMAuplink channel. The new definition reflects the CSI on thedecomposability of the respective error vectors and yet issimple enough to allow numerical evaluation.

In this paper, for a downlink multiple input-multiple output(MIMO) system over a flat fading channel, we propose anew upper bound on the BER of ML detection based ona new sufficient condition for the decomposability of errorvectors. To this end, we revisit the upper bound based onVerdu’s decomposability [11] and the LBA argument [4].These two bounds are actually difficult to evaluate for fading

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CHEN and HASHIMOTO: A NEW BER UPPER BOUND FOR MULTIUSER DOWNLINK SYSTEMS 1189

channels and we can only recourse to Monte-Carlo method toevaluate them. We then proceed to propose a simple methodto reduce the difficulty in incorporating the decomposabilityinto the BER upper bound. The new upper bound is tighterthan the bound based on the overall decomposability [12][14]-[16] as well as the bound based on the PSD [13] specializedto downlink channel. We give some numerical results tocompare the proposed bound and other bounds, some of whichare evaluated numerically and the rest are evaluated throughMonte-Carlo simulation.

The rest of this paper is organized as follows. The systemmodel and maximum likelihood detector are described inSection II. Upper bounds of bit error probability conditionedon given path coefficients are discussed in Section III. The pro-posed average BER upper bound is explained in Section IV.Some numerical results are shown in Section V. A conclusionis given in the last section.

II. SYSTEM DESCRIPTION

We consider a single-cell synchronous downlink MIMOCDMA system based on space-time spreading (STS). Thebase station transmits signals with L antennas and each ofK users (mobile terminals) receives the CDMA signals withM antennas. A signature waveform s

(l)k (t) of length T is used

for the transmission of the information bit of user k from thelth antenna and is given as

s(l)k (t) =

N−1∑n=0

s(l)k,nψ(t− nTc),

where Tc = T/N is the chip length, ψ(t) is a chip waveformwhich is confined in [0, Tc] and normalized as

∫ Tc

0 ψ2(t)dt =

1, and s(l)k = [s

(l)k,0, s

(l)k,1, · · · , s(l)k,N−1]

T is the associatedsignature sequence of length N , the processing gain of thesystem. The signature sequences are normalized as

L∑l=1

N−1∑n=0

|s(l)k,n|2 = 1, k = 1, · · · ,K. (1)

We assume BPSK modulation. The information bit bk ∈{−1, 1} of user k is multiplied with s

(l)k (t), � = 1, · · · , L,

and transmitted from the respective antennas synchronouslywith signals from other users. Throughout this paper, user 1is the desired user and only the receiver of the desired user isconsidered.

We assume that each pair of one transmit antenna, say thelth transmitting antenna, of the base station and one receiveantenna, say the mth antenna, of the desired user constitutesa frequency flat quasi-static Rayleigh fading channel withchannel coefficient ζl,m. The channel coefficients are mutuallyindependent, circularly symmetric complex Gaussian randomvariables with mean zero and variance 1/2 per dimensionand are fixed during a bit interval. We also assume that thereceiver completely knows the channel state information (CSI)ζl,m, l = 1, 2, · · · , L and m = 1, · · · ,M , but the transmitter(base station) does not.

The signal received by the mth antenna is

rm(t) =

K∑k=1

L∑l=1

Ak√Mbkζl,ms

(l)k (t) + wm(t),

where Ak is the amplitude assigned to user k and wm(t)is a zero-mean white Gaussian noise (WGN) with one-sidedpower spectral density per dimension N0. We assume thatthe noise signals received by different antennas are mutuallyindependent. For the information vector b = [b1, · · · , bK ]T ,the received signal is expressed as

rm(t) =1√MψT (t)SZmAb+ wm(t),

where the superscript T denotes transpose and we let

ψ(t) =[ψ(t), · · · , ψ(t− (N − 1)Tc)

]TS =

[S1, · · · , SK

]N×KL

Sk =[s(1)k , · · · , s

(L)k

]N×L

Zm = diag(ζm, · · · , ζm

)KL×K

ζm =[ζ1,m, · · · , ζL,m

]TA = diag

(A1, · · · , AK

)K×K

.

In the above expressions, diag(a1, a2, · · · , a�) denotes an� × � diagonal matrix with the specified diagonal elements ifai are scalars and denotes an � × � block diagonal matrix ifai are vectors or matrices. The cross-correlation matrix of allthe spreading sequences is given by

R = SHS =

⎡⎢⎣

R11, · · · , R1K

.... . .

...RK1, · · · , RKK

⎤⎥⎦KL×KL

with L×L matrices Rkk′ = SHk Sk′ where the superscript H

denotes conjugate transpose.According to the previous assumption, moreover, the chan-

nel coefficient vector ζm consisting of L independent circu-larly symmetric random variables ζ�,m with variance one hasa probability density function (pdf) [17]

p(ζm) = π−L exp(−‖ζm‖2) . (2)

With ML multiuser detection (MUD) under perfect CSI,we select an information vector b that maximizes the log-likelihood function Ω(b) given by

Ω(b) =

M∑m=1

{2√M

�[(ZmAb)Hym

]

− 1

M(ZmAb)HRZmAb

}. (3)

In this expression ym = [yTm,1, · · · ,yT

m,K ]T with ym,k =

[y(1)m,k, · · · , y(L)

m,k]T is the output of the series of the chip-

matched filter and code-matched filter bank located at theoutput of the mth receive antenna and is given by

ym =1√M

RZmAb+ nm, (4)

where nm =∫ T

0 SHψ(t)wm(t)dt.

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III. CONDITIONAL BER UPPER BOUNDS

A. Conditional union bounds

Given information vectors b and b′, the normalized differ-ence e = [e1, e2, · · · , eK ]T = (b− b′)/2 is called an errorvector [18]. The set of error vectors which affect user k is Ek

={e ∈ {−1, 0, 1}K ; ek �= 0

}and the set of all error vectors

is E =⋃K

k=1 Ek. For each e ∈ E, let W (e) =∑K

k=1 |ek|.The set of admissible error vectors (for the desired user)

under the condition that b is transmitted is given by A1(b)= {e ∈ E1; e = (b − b′)/2 for some b′}. Given CSI ζ ={ζ1, · · · , ζM}, the conditional error probability is1

Pe,1(ζ) = Pr[Ω(b−2e)≥ Ω(b) for an e ∈ A1(b)|ζ].Since transmitted bits are equiprobable, the probability that agiven e is admissible is Pr [e ∈ A1(b)] = 2−W (e). We cansee that the event Ω(b − 2e) ≥ Ω(b) is independent of thetransmitted b since (3) and (4) show that the left-hand sidedoes not include b as

Ω(b− 2e)− Ω(b) =

M∑m=1

{ 4√M

� [(ZmAe)Hnm

]− 4

M(ZmAe)HR(ZmAe)

}. (5)

Thus, we have a conditional union bound (UB) for the givenζ

Pe,1(ζ) ≤∑e∈E1

2−W (e) Pr[Ω(b−2e) ≥ Ω(b)|ζ]. (6)

Applying the LBA argument [4], we also have the conditionalLBA union bound (LBA-UB)

Pe,1(ζ)≤min

{1,∑e∈E1

2−W(e)Pr[Ω(b−2e)≥Ω(b)|ζ]}. (7)

B. Upper bounds based on decomposability

It is shown in [11] (also see [18]) that some error vectorscan be omitted from the summation in (6).

We say that a pair of error vectors, {e′, e′′}, is an E1-partition of e if

1) e = e′ + e′′

2) e′ ∈ E1 and e′′ ∈ E3) e′i = e′′i = 0 whenever ei = 0, i = 1, · · · ,K

and introduce the set D1(e) of all the E1-partitions of e. Forour system, the definition of decomposable error vectors isstated as follows[11][18].

Definition 1: Given ζ, an error vector e ∈ E1 is saiddecomposable to {e′, e′′} ∈ D1(e) or, simply, decomposableif the E1-partition of e satisfies η(e′, e′′) ≥ 0, where

η(e′, e′′) =1

M

M∑m=1

ζHmG(e′, e′′)ζm

G(e′, e′′) =

K∑i=1

K∑j=1

AiAje′ie

′′j (Rij +Rji).

1The equality holds under the assumption that the tie Ω(b− 2e) = Ω(b)always leads to an error.

The e is said indecomposable if there is no such E1-partitions.It is not difficult to see that, if we let

S(e) =1

M

M∑m=1

(ZmAe)HR(ZmAe),

then the condition η(e′, e′′) ≥ 0 is equivalent to S(e) ≥S(e′) + S(e′′). As discussed in [20], the decomposabilityof e is intimately connected to the minimality of S(e) as aquadratic form of ζm given by

S(e) =1

M

M∑m=1

ζHmH(e)ζm, (8)

where we let

H(e) =

K∑k=1

K∑k′=1

AkekAk′ek′Rkk′ . (9)

Let F ζ1 = {e ∈ E1 : η(e′, e′′) < 0 for all {e′, e′′} ∈ D1(e)}

be the set of all indecomposable error vectors for the givenζ. It is Verdu’s discovery that the decomposable error vectorscan be ignored in the union bound [11][18]. Thus, omittingall the decomposable error vectors gives a tighter upperbound, called the conditional Verdu bound (VB),

Pe,1(ζ) ≤∑e∈F ζ

1

2−W (e) · Pr[Ω(b− 2e) ≥ Ω(b)|ζ]. (10)

We note that F ζ1 includes all the error vectors with weight

one since they are always indecomposable.With the LBA argument, the bound is further tightened to

give an improvement, called the conditional LBA Verdu bound(LBA-VB).

Pe,1(ζ)≤min

⎧⎨⎩1,

∑e∈F ζ

1

2−W(e)Pr[Ω(b−2e)≥Ω(b)|ζ]⎫⎬⎭.

(11)The conditional LBA-VB is tighter than the conditional VB

(10) and conditional LBA-UB (7). It is, however, difficult toobtain closed form expressions for the unconditional LBA-VBand LBA-UB since averaging the conditional bounds (7) and(11) with respect to the probability of ζ requires complicatedmulti-dimensional integrals, and we can only recourse to theMonte-Carlo method to evaluate these bounds at the cost oflarge computational complexity. Even for the VB, which is theaverage of the conditional VB (10), no analytical expressionhas not been reported yet and hence the Monte-Carlo methodis also used.

C. Modifications to the decomposability

To reduce the difficulty in averaging the conditional VBover fading channels, we may modify the definition of thedecomposability. The modification that is most convenientfor analysis is to replace the non-negativity condition onη(e′, e′′) with the non-negative definiteness of G(e′, e′′),which leads to the definition of the “overall decomposability”since it then becomes decomposable in the sense of Verdufor all ζ [3][12][14]-[16]. We call the resulting bound theoverall decomposability bound (ODB). The ODB is, though

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computable, much looser than the VB in general and hence isnot considered below.

In [13], Mohasseb et al. introduced a better modification.Definition 2: Given ζ, an error vector e is said positive-

sum-decomposable (PSD) if it satisfies η(e) ≥ 0, where

η(e) =1

M

M∑m=1

ζHmH(e)ζm

H(e) =K∑i=1

K∑j=1,j �=i

AiAjeiej(Rij +Rji).

The following lemma is proved in [13].Lemma 1: An error vector e ∈ E1 is decomposable if it is

PSD.In [13], the average of the conditional bound with PSD

error vectors omitted is derived. We call the resultant boundthe PSD bound (PSDB) hereafter.

IV. NEW UPPER BOUND

To introduce a new bound, we first revisit some expressionsrelated to the VB.

A. The standard bound

Given ζ, the term 1√M

∑Mm=1 �[(ZAe)Hnm] in (5) is

a Gaussian random variable with mean zero and varianceN0

2 S(e). Thus, we have an expression

Pr [Ω (b− 2e) ≥ Ω (b)| ζ] = Q(√ 2

N0‖S (e)‖

)using the Gaussian Q-function Q(x) = 1√

2π

∫∞xe−

12 t

2

dt.We note, from (2), that the pdf of ζ is given by p(ζ) =∏M

m=1 p(ζm) = π−LM exp(−∑M

m=1 ‖ζm‖2). Then, aver-aging both sides with respect to the probability of ζ andemploying Craig’s representation of the Gaussian Q-function[19] in the same manner as in [16], we obtain the union bound(UB)

Pe,1 ≤∑e∈E1

2−W (e)PWhl(e),

where, for H(e) given by (9), PWhl(e) is given by

PWhl(e) =

∫all ζ

Q

⎛⎝√

2S(e)

N0

⎞⎠ p(ζ)dζ

=1

π

∫ π2

0

det

[IL +

H(e)

N0M sin2 θ

]−M

dθ. (12)

The subscript of PWhl(e) denotes the integration over thewhole region.

B. Derivation of the new bound

The VB is given as

Pe,1 ≤ E

⎡⎣∑e∈F ζ

1

2−W (e) Pr[Ω(b− 2e) ≥ Ω(b)|ζ]⎤⎦

=∑e∈E1

2−W (e)PVD(e), (13)

where

PVD(e) =

∫ζ:e∈F ζ

1

Q

⎛⎝√

2S(e)

N0

⎞⎠ p(ζ)dζ. (14)

and the subscript “VD” denotes the average under Verdu’sdecomposability condition.

We note that, as discussed below (10), an error vector ewith W (e) = 1 is indecomposable irrespective of ζ and hencesatisfies PVD(e) = PWhl(e).

For an event E , let χ[E ] be the selection function of E thatassumes one if E is true and zero otherwise. Then, for an errorvector e ∈ E1 with W (e) > 1, we can modify PVD(e), byfirst making the indecomposability condition explicit and thenassuming the Craig’s representation, as follows.

PVD(e) =

∫χ [η(e′, e′′) < 0, all {e′, e′′} ∈ D1(e)]

×Q⎛⎝√

2S(e)

N0

⎞⎠ p(ζ)dζ

=1

π

∫ π2

0

{∫χ [η(e′, e′′) < 0, all {e′, e′′} ∈ D1(e)]

× exp

(− S(e)

N0M sin2 θ

)p(ζ)dζ

}dθ.

We note that, for each (e′, e′′)∈D1(e), the following inequal-ity holds.∫

χ [η(e′, e′′) < 0, all {e′, e′′} ∈ D1(e)]

× exp(− S(e)

N0M sin2 θ

)p(ζ)dζ

≤∫

η(e′,e′′)<0

exp(− S(e)

N0 sin2 θ

)p(ζ)dζ

Thus, we have the desired upper bound PMP(e) of PVD(e)

PMP(e)

=1

π

∫ π2

0

⎧⎪⎨⎪⎩ min

{e′,e′′}∈D1(e)

∫η(e′,e′′)<0

exp

(− S(e)

N0 sin2 θ

)p(ζ)dζ

⎫⎪⎬⎪⎭dθ

=1

π

∫ π2

0

min{e′,e′′}∈D1(e)

α(e′, e′′, θ)dθ, (15)

where we let

α(e′, e′′, θ) =∫

η(e′,e′′)<0

exp

(− S(e)

N0 sin2 θ

)p(ζ)dζ. (16)

The subscript “MP” denotes the minimal partition (MP)bound.

The following theorem is proved in Appendix A.Theorem 1: The function α(e′, e′′, θ) is expressed as

α(e′, e′′, θ) = γ(e′, e′′, θ) det[IL +

H(e)

N0 sin2 θ

]−M

with a function γ(e′, e′′, θ) satisfying

γ(e′, e′′, θ) ≤ 1.

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Let λ1, · · · , λP be the distinct nonzero eigenvalues ofG(e′, e′′) relative2 to IL + H(e)

N0M sin2 θand let n1, · · · , nP be

the associated multiplicities. Then, the function γ(e′, e′′, θ) isexpressed as

γ(e′, e′′, θ) = −∑

p:λp<0

βp

where, for μp = Mnp,

βp =1

(μp−1)!

dμp−1

dsμp−1

⎡⎣

∏q

(−M

λq

)μq

s∏

q:q �=p

(s−M

λq

)μq

⎤⎦s= M

λp

with the convention∏

q =∏P

q=1.Substituting the above α(e′, e′′, θ) into (15) and letting

PMP(e) = PWhl(e) for e with W (e) = 1, we obtain a newupper bound

Pe,1 ≤∑e∈E1

2−W (e)PMP(e). (17)

Comparison of (12) and α(e′, e′′, θ) in the theorem revealsthat the new upper bound is no larger than the UB.

If G(e′, e′′) is non-negative definite in the theorem, then allλl are non-negative and hence we have γ(e′, e′′, θ) = 0. Thus,we immediately have the following corollary, which impliesthat the terms omitted in the ODB are also omitted from ourbound and hence that our bound improves the ODB.

Corollary 1: If e has an E1-partition (e′, e′′) which makesG(e′, e′′) non-negative definite, then PMP(e) = 0.

Now, we make some comparison between the VB, PSDB,and new bound. A formal expression for PSDB is obtainedfrom (13) and (14) with the replacement of the condition e ∈F ζ1 by the non PSD condition η(e) < 0 as

Pe,1 ≤∑e∈E1

2−W (e)PPSD(e),

where

PPSD(e) =

∫ζ:η(e)<0

Q

⎛⎝√

2S(e)

N0

⎞⎠ p(ζ)dζ.

The following theorem is easy to show.Theorem 2: When K = 2, the VB, PSDB, and new bound

are identical.Proof: If K = 2, there exist only two types of error vectors

with weights equal to one and two, respectively. Since theerror vectors e of weight one are always indecomposable, theVB, PSDB and new bound include the corresponding PEPPWhl(e). For the weight two error vectors e = [e1, e2]

T , ei �=0, there exists just one way to decompose it into e = [e1, 0]

T+[0, e2]

T , i.e., there is only one element in set D1(e). Thus, theequality PVD(e) = PMP(E) holds. Moreover, it is easy to seeη([e1, 0]

T , [0, e2]T ) = η(e). These results conclude that the

VB, PSDB, and new bound are identical.The following theorem, which is proved in Appendix B,

shows that every error vector omitted from the PSDB is also

2The λ satisfying det[A−λB] = 0 is called the eigenvalues of A relativeto B, If B is a positive definite Hermitian matrix, then λ is the eigenvalueof B− 1

2 AB− 12 .

omitted from our bound and hence that the number of termsin the proposed bound (17) is less than or equal to the numberof terms in the PSDB.

Theorem 3: PPSD(e) = 0 implies PMP(e) = 0.Unfortunately, the theorem does not necessarily assure

PMP(e) ≤ PPSD(e). Some discussions on this point will begiven later.

We notice that our integral form is simpler than the integralform (43) in [13] since the latter requires multi-dimensionalintegration. Although the PSDB was originally proposed foruplink channels, the computational requirement when used fordownlink channels does not change considerably as seen inAppendix C.

C. Looser upper bounds with low computational complexity

The bound (17) needs search for a pair (e′, e′′) ∈ D(e)minimizing α(e′, e′′, θ) for each θ, which leads to a largecomputational complexity when K is large. A computationallyless demanding form

PMP(e) ≤ min(e′,e′′)∈Di(e)

1

π

∫ π2

0

α(e′, e′′, θ)dθ

reduces the computation complexity at the cost of negligibleperformance loss. Since error vectors with a large weightin (17) are not influential compared to error vectors with smallweights, we may also consider the following hybrid bound toreduce the complexity for an appropriate threshold W < K .

Pe,1 ≤∑e∈E1

W (e)≤W

2−W (e)PMP(e) +∑e∈E1

W (e)>W

2−W (e)PWhl(e)

V. NUMERICAL RESULTS

We compare the new analytical bound proposed in thispaper with known analytical bounds, UB and PSDB, andwith Monte-Carlo-based non-analytical bounds,3 MC LBA,MC VB, and MC LBA-VB. From the discussions in the lasttwo sections, we know the superiority relationships betweenbounds: MC LBA-VB ≤ MC VB ≤ Proposed bound ≤ UB,and MC LBA-VB ≤ MC LBA ≤ UB. We can confirm theserelationships through the following numerical results. Besidesthese, we are interested in the relationships between theproposed bound, PSDB, and MC-LBA. The theoretical resultsup to here could not reveal the relationships, although ourbound has a great chance to surpass PSDB as the discussionin Section IV-B. The last subsection will be devoted to somepartial discussions on the relationship.

MIMO channels have two aspects of advantage, increasingcommunication reliability or/and transmission information rate[22]. For the numerical comparison between bounds, weconsider two STS schemes in the following subsection. Thefirst one is a diversity gain oriented STS scheme, while thesecond one is an information rate oriented STS scheme. Innumerical comparison, we also consider an uplink systemsince the PSDB was originally proposed for uplink systems.

3“MC” stands for Monte-Carlo simulation.

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0.001

0.01

0.1

1

0 2 4 6 8 10 12 14

BE

R

SNR(dB)

UBPSDB

Proposed boundMC LBA-UB

MC VBMC LBA-VB

Simulation

Fig. 1. Bounds for the first STS scheme with L = 2 and K = 8, N = 15,and M = 1 (equal power case).

A. Equal power scenarios

We first consider an equal-power scenario, that is, Ak = Afor all k. Because of the normalization (1), this assures thatthe signals of all users are received with equal power at thereceiver.

The first system used for evaluation is derived from the stan-dard STS system [23] where Hurwitz-Radon matrices are usedto assign signature sequences to the respective transmittingantennas. In this simulation, we employ M-sequences, insteadof orthogonal sequences, to introduce certain interference inthe system. In Fig. 1 and Fig. 2, we show the results for L = 2and L = 4, respectively. The other parameters are N = 15,K = 8, and M = 1. In these figures, the UB always gives theworst results and, among the analytical bounds, the proposedbound gives the best results. Among the Monte-Carlo-basedbounds, the MC LBA-UB always gives the worst results andthe MC LBA-VB gives the best ones. The MC VB givesalmost the same result as the proposed bound and both boundscan not be distinguished in Fig. 1, while the MC VB gives aresult rather close to the MC LBA-VB in Fig. 2.

The second STS system used for evaluation is the oneproposed in [24], where each user, say user k, is assigned witha PN sequence of length LN and L segments s(�)k , � = 1, · · · ,L, of the PN sequence are assigned to L antennas of the user.In Fig. 3, we give results for N = 8, K = 8, L = 2, andM = 2. The results lay somewhat between those of Fig. 1and Fig. 2, and again we notice that the proposed bound givesthe best result among the three analytical bounds.

An uplink system can be emulated with the first systemif there are more transmit antennas than users. That is, ifL ≥ K in the first system, then we can assign differentantennas to different users and can emulate a synchronousuplink transmission since each pair of a transmit antenna anda receive antenna constitutes a link which are independent ofother such links. We note, however, that L in the normalization(1) must be replaced with L, the number of antennas assigned

0.001

0.01

0.1

1

0 2 4 6 8 10

BE

R

SNR(dB)

UBPSDB


MC VBMC LBA-VB

Simulation

Fig. 2. Bounds for the first STS scheme with L = 4 and K = 8, N = 15,and M = 1 (equal power case).

0.001

0.01

0.1

1

0 2 4 6 8 10

BE

R

SNR(dB)

UBPSDB


MC VBMC LBA-VB

Simulation

Fig. 3. Bounds for the second STS scheme with K = 8, L = 2, N = 8,and M = 2 (equal power case).

to each user. In Fig. 4, we give numerical results for theemulated uplink system, where the signal of each of 8 usersis spread by an M-sequence and is transmitted from one of8 transmit antennas. This system, with transmitter parametersK = 8, L = 8, and N = 15 and with receiver parameterM = 1, emulates a synchronous uplink system consisting ofeight single-antenna mobile users and a single-antenna basestation as discussed in [12] and [13]. In this emulated uplinksystem, the proposed bound and PSDB give almost the sameresults, although the PSDB gives a little bit better result at lowSNRs. The MC LBA-UB does not give a good result again.Since Theorem 3 claims that the proposed bound includes nomore terms than the PSDB, Fig. 4 seems to suggest that thePSD criterion gives a smaller integral region for each errorpattern.

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0.01

0.1

1

0 2 4 6 8 10 12 14

BE

R

SNR(dB)

UBPSDB


MC VBMC LBA-VB

Simulation

Fig. 4. Bounds for the emulated uplink system with K=8, L=8, N=15, andM=1 (equal power case).

We can summarize the results for the equal power sce-nario as follows. For downlink transmission, we observed thesuperiority ordering “MC LBA-VB < MC VB < Proposedbound < {MC LBA-UB, PSDB} < UB” at a relatively largeSNR, while LBA-type bounds show better performance ata small SNR. The superiority between MC LBA-UB andPSDB is not fixed. For uplink transmission, the superiorityordering between bounds is basically the same as downlinktransmission with the exception that the performance of thePSDB is almost the same as or slightly better than theproposed bound.

B. Unequal power scenarios

BER characteristics become quite different when trans-mit power can be different for each user. To simplify thesituation, we control only the transmit power of the unde-sired users and consider two unequal-power scenarios: thedouble-amplitude scenario and the half-amplitude scenario.The double-amplitude scenario is that the desired user hasan amplitude double the amplitude of the rest of users, i.e.,A1 = A and Ak = 1

2A for k = 2, · · · ,K . The half-amplitudescenario is that the desired user has an amplitude half theamplitude of each of the other users, i.e., A1 = A andAk = 2A for k = 2, · · · ,K . We compute the bounds forthe downlink/uplink STS systems as in Fig. 1, Fig. 2, Fig. 3,and Fig. 4.

In Fig. 5 and Fig. 6, we give the bounds and simulationresults for the system considered in, respectively, Fig. 1 andFig. 2.

We can immediately notice that the simulation results forboth scenarios are almost the same. The reason may be thatML multiuser detection, together with the low spectrum effi-ciency of the system, could protect the desired user sufficientlyagainst multiuser interference. The figures also reveal that allthe bounds give far looser results for the double-amplitudescenario than for the half-amplitude scenario. Among the

0.01

0.1

1

0 5 10 15 20

BE

R

SNR(dB)

double-amplitude

half-amplitude

UBPSDB


MC VBMC LBA-VB

Simulation

Fig. 5. Bounds for the first STS scheme with L = 2 and K = 8, N = 15,and M = 1 (unequal power case).

0.01

0.1

1

0 2 4 6 8 10 12

BE

R

SNR(dB)

double-amplitude

half-amplitude

UBPSDB


MC VBMC LBA-VB

Simulation

Fig. 6. Bounds for the first STS scheme with L = 4 and K = 8, N = 15,and M = 1 (unequal power case).

analytical bounds, however, the proposed bound give the bestresults.

In Fig. 7, we give the bounds and simulation results for thesystems considered in Fig. 3. We can observe the similar phe-nomena as observed in Fig. 5 except that the simulation resultfor the double-amplitude scenario becomes clearly worse thanfor the half-amplitude scenario.

The reason for the respective behaviors of the bounds in thetwo scenarios may be considered as follows. When an errorevent of weight one is considered, its PEP does not depend onthe transmit power of the other users since no other users areinvolved in the error. When we consider an error event e ofweight more than one, the matrix H(e) given in (9) includescontributions from other users now and has larger diagonalelements for the half-amplitude scenario than for the double-amplitude scenario for the same the transmit power of the

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0.001

0.01

0.1

1

0 2 4 6 8 10 12

BE

R

SNR(dB)

double-amplitude

half-amplitude

UBPSDB


MC VBMC LBA-VB

Simulation

Fig. 7. Bounds for the second STS scheme with K = 8, L = 2, N = 8,and M = 2 (unequal power case).

0.01

0.1

1

0 2 4 6 8 10 12

BE

R

SNR(dB)

double-amplitude

half-amplitude

UBPSDB


MC VBMC LBA-VB

Simulation

Fig. 8. Bounds for the emulated uplink system with K=8, L=8, N=15, andM=1 (unequal power case).

desired user. If the interference between users are not quitestrong, then the quadratic form (8) gives a larger value forthe half-amplitude scenario and hence a smaller PEP for thatscenario.

In Fig. 8, we also show the results of the emulated uplinktransmission considered in Fig. 4. For the half-amplitude case,the MC VB and MC LBA-VB can not be distinguished. Again,we can deduce the same conclusion between the PSDB andproposed bound as for Fig. 4.

We can summarize the results for the unequal power sce-nario in the same manner as for the equal power scenario. Theonly difference seems to be that all the bounds become quiteloose for the double-amplitude setting although simulationresults do not differ much.

C. Discussions on the proposed bound and PSDB

We have seen that the proposed bound gives performancessuperior to the PSDB in the downlink simulation while itsperformances are approximately equal to or slightly worsethan the PSDB in the uplink simulation. Although theoreticalcomparison of the proposed bound and PSDB is difficult,we can see some evidences that may support the aboveobservations.

For the following discussion, we introduce some notations.An error vector p = [p1, · · · , pK ]T ∈ E is an error profile ifpi is either 0 or 1 for all i and let EP1 is the set of all errorprofiles p such that p1 = 1. An error vector e ∈ E is saidto have an error profile p if |ei| = pi for all i. For an errorprofile p ∈ EP1, let E1(p) be the set of all e ∈ E1 with errorprofile p. Especially, let E+

1 (p) be the set of all e ∈ E1(p)with e1 = 1.

The following lemma is due to [20].Lemma 2: For each p ∈ EP1, an error vector e ∈ E+

1 (p)is indecomposable if and only if

S(e) < mine∈E+

1 (p), e �=eS(e).

Since S(e) is a quadratic function of ζ, we may confine ourdiscussion on the unit sphere defined by ‖ζ‖2 = 1. On the unitsphere, let us call the set of those ζ that make S(e) minimumin the sense of the lemma the region of e. We note that thoseregions constitute a partition of the unit sphere. An additionale ∈ E+

1 (p) uniquely determines a partition {e′, e′′} in sucha manner that e − 2e′′ = e and the condition η(e′, e′′) = 0gives a potential boundary between the regions of e and of e.It is a potential boundary since the region of e is the unionof the subsets confined by these potential boundaries.

1) Downlink case: Let us consider the first downlinksystem with Ak = 1 for all k, L = 2, K = 8, N = 15,and M = 1, the system in Fig. 1, and let us consider anerror vector e = [+1, +1, +1, 0, · · · , 0]T with an errorprofile p = [1, 1, 1, 0, · · · , 0]T , which involves only thefirst three users. To simplify the notation, we only considerthe first three users and let the related error vectors be e(0) =[+1,+1,+1]T , e(1) = [+1,−1,−1]T , e(2) = [+1,−1,+1]T ,and e(3) = [+1,+1,−1]T . Spreading sequences are assignedas s1 = [m1,−m2], s2 = [m2,m1], s3 = [m3,−m4] wheremi, i = 1, 2, 3, are M-sequences. For H(e) given in (9), S(e)= ζHH(e)ζ and, for the above error vectors, we have

H(e(0)) =3

2

[1− 2

N1N

1N 1 + 2

3N

],

H(e(2)) =3

2

[1 + 2

3N1N

1N 1− 2

N

],

and

H(e(1)) = H(e(3)) =3

2

[1 + 2

3N − 13N− 1

3N 1 + 23N

].

Since L = 2, the unit sphere is now a unit circle andeach point ζ on the unit circle is uniquely specified byan angle φ as ζ1 = cosφ and ζ2 = sinφ. Then, fromthe symmetry of H(e(0)) and H(e(2)), we can see that thecondition η([1, 0, 1]T , [0, 1, 0]T ) < 0, or equivalently thecondition S(e(0)) < S(e(2)), specifies a subset of the unit

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circle consisting of −π4 < φ < π

4 and 3π4 < φ < 5π

4 . On theother hand, the inequalities 1

2

[S(e(0)) + S(e(2))

] ≤ S(e(i)),i = 1, 3, imply that e(1) and e(3) are always decomposable.Since there are only two regions and the boundary betweenthem is given by a single condition η([1, 0, 1]T , [0, 1, 0]T ) =0, we must have PVD(e) = PMP(e), which partially explainthe evidence that the proposed bound and VB almost coincidein Fig. 1.

For the PSDB, on the other hand, we have η([1, 1, 1]T ) =ζHH([1, 1, 1]T )ζ for

H([1, 1, 1]T ) =

[ − 3N

1N

1N

1N

]

A simple geometric considerations reveals that the subsetdetermined by the condition η([1, 1, 1]T ) < 0 consists of− tan−1 3 < φ < π

4 and π − tan−1 3 < φ < π + π4 . This

range is wider than the region for e(0) and hence PPSD(e) >PMP (e).

A similar situation is expected whenever L is not largecompared to K .

2) Uplink case: In the emulated uplink transmission withK = L and M = 1, every channel coefficient is uniquely asso-ciated to an user, and we have S(e) = ζHdiag(e)F diag(e)ζ,where F is a K×K matrix with its (i, j) element ri,j = sHi sjand, when the M-sequences are employed, is given by

F =N + 1

NIK − 1

N1K1T

K ,

where IK is the K×K identity matrix and 1K is the all-onevector of length K . It is not difficult to see that the matrix Fhas the minimum eigenvalue λ1 = N+1−K

N for the associatedeigenvector v1 = 1K and K − 1 duplicated eigenvalues λk =N+1N for the associated eigenvectors vk perpendicular to 1K .In the following discussions, we consider an all-one error

profile p with pi = 1 for all i and consider an e ∈ E+1 (p).

For other case, we can safely discard correctly detected usersfrom our consideration due to the uniform cross-correlationproperty of the M-sequences.

From the property of F, we can see that S(e) takes itsminimum N+1−K

N for ζ parallel to e on the unit sphere‖ζ‖2 = 1. If e1 ∈ E+

1 (p) is different from e ∈ E+1 (p)

only at one position, say at the last position, then the partition{e′, e′′} of e associated with e1 is given by e′′ which is allzero except at the last position. It is not difficult to see, onthe unit sphere, that η(e′, e′′) = 0 occurs at the midpointbetween ζ that minimize S(e) and ζ1 that minimize S(e1)and that S(e) assumes N+2−K

N at the midpoint. The errorvector e has at least K such neighbouring error vectors ase1 and S(e) assumes the same value at the midpoints onthe respective boundaries. Thus, any particular choice of acondition η(e′, e′′) < 0 may not uniquely minimize PMP(e).

On the other hand, the PSD condition η(e) < 0 is equivalentto S(e) < 1 and the associated integral region includes all theK midpoints on the boundary discussed above when K > 2.When K is small compared to N , however, the values ofS(e) at those midpoints are not quite different from one. Thenthe PSDB has more chances to be better than the proposedbound since the area of the integral region is smaller forthe PSDB than for our bound. This conclusion seems to

hold whenever the signature sequences have a uniform cross-correlation property like M-sequences and L is large comparedto K .

The above conclusion is based on the assumption that M-sequences are used and may not hold if other sequencesare used. In fact, although the result is not shown here, thesmall superiority of PSDB completely disappear when randomsequences are employed.

VI. CONCLUSION

After reviewing the definition of IEVs and its extensions,we proposed a new closed-form BER upper bound for the MLreceiver over downlink MIMO fading channels. The proposedupper bound was shown to include less terms than the positive-sum-decomposability bound (PSDB), which was originallyproposed for uplink channels, and shown to be better thanknown analytical bounds including the PSDB by simulationover downlink MIMO channels. We expect that the proposedbound is superior to PSDB whenever L is small comparedto K . An interesting observation is that the proposed boundmay be also useful for uplink channels and give results quiteclose to, though a little bit worse than in low SNR regions,the PSDB, which was originally proposed for uplink channels.Thus, the proposed bound seems to give almost the best resultsfor both uplink and downlink channels.

APPENDIX APROOF OF THEOREM 1

From the definition (16), we have

α(e′, e′′, θ) =

∫· · ·

∫η(e′,e′′)<0

exp

(−∑M

m=1 ζHmH(e)ζm

N0M sin2 θ

)

× 1

πMLexp

(−

M∑m=1

ζHmζm

)dζ1 · · · dζM

=

∫· · ·

∫η(e′,e′′)<0

1

πMLexp

[−

M∑m=1

ζHm

(I+

H(e)

N0M sin2 θ

)ζm

]

×dζ1 · · · dζMWe make some transformations to simplify the integration.First we consider the eigenvalue decomposition

I+H(e)

N0M sin2 θ= UH

ihΛihUih

where Uih is a unitary matrix, Λih is a diagonal matrixconsisting of eigenvalues of I+H(e)/N0M sin2 θ. Since Λih

is positive-definite, we let

ζm = [ζm,1, · · · , ζm,L]T = Λ

12

ihUihζm,m = 1, · · · ,M.

Then, we have a transformation ζm = UHihΛ

− 12

ih ζm and

η(e′, e′′) =1

M

M∑m=1

ζH

mFζm

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where F = Λ− 1

2

ih UihG(e′, e′′)UHihΛ

− 12

ih . Moreover, the abovetransformation gives

α(e′, e′′, θ) =

∫· · ·

∫η(e′,e′′)<0

1

πMLexp

(−

M∑m=1

ζH

mζm

)

×|J|dζ1 · · ·dζMwhere J is Jacobian matrix and

|J| = det

[I+

H(e)

N0M sin2 θ

]−M

.

Since F is also a Hermitian matrix, we next consider anothereigenvalue decomposition F = UfΛfUf . Then, we have

η(e′, e′′)=1

M

M∑m=1

ζH

mFζm =1

M

M∑m=1

(Uf ζm)HΛf (Uf ζm)

=1

M

M∑m=1

ζH

mΛf ζm

where we let ζm = [ζm,1, · · · , ζm,L] = Uf ζm and ζ = UHf ζ

for m = 1, · · · ,M . Thus,

α(e′, e′′, θ)= |J|∫

· · ·∫

η(e′,e′′)<0

1

πMLexp

(−

M∑m=1

ζH

mζm

)dζ1 · · · dζM .

We note that ζm,l, l = 1, · · · , L,m = 1, · · · ,M maybeconsidered i.i.d. complex-valued Gaussian variables with zeromean and variance 1. Thus, we have

α(e′, e′′, θ) = |J|Pr[ 1

M

M∑m=1

ζH

mΛf ζm < 0].

We recall η(e′, e′′) that, for Λf = diag(λ1, · · · , λL),

η(e′, e′′) =1

M

M∑m=0

ζH

mΛf ζm =

L∑l=1

λl

( 1

M

M∑m=1

|ζl,m|2)

The probability density function (PDF) of |ζl,m|2 is an expo-nential function as

f|ζl,m|2(x) ={e−x, x ≥ 00, x < 0

.

Then the characteristic function of |ζl,m|2 is

1

1− jν.

For λl �= 0, we attain the characteristic function of η(e′, e′′)as

ϕη(jν) =

L∏l=1λl �=0

1

(1− jν λl

M )M. (18)

Let λp, p = 1, · · · , P denote the distinct nonzero eigenvaluesof F with multiplicities n1, · · · , nP , respectively. We rewrite(18) as

ϕη(jν) =P∏

p=1

1

(1 − jνλp

M )Mnp

From Levy formula [25]

Pr[η(e′, e′′) < x2]− Pr[η(e′, e′′) < x1]

=1

2πlimr→∞

∫ r

−r

e−jνx1 − e−jνx2

jνϕη(jν)dν,

with x2 = ∞ and x1 = 0, we have

Pr[η(e′, e′′) ≥ 0] =1

2π

∫ ∞−jε

−∞−jε

1

jνϕη(jν)dν

where a small positive number ε has been inserted in orderto move the path of integration away from the singularity atν = 0 and it must be positive in order to satisfy Pr[η(e′, e′′) ≥0] = 1 when λp ≥ 0 for all p.

Let s = jν, we have

Pr[η(e′, e′′) ≥ 0] =1

2πj

∫ ε+j∞

ε−j∞

1

s

P∏p=1

(−M

λp

)Mnp

(s− M

λp

)Mnpds.

Let us consider the integral path Γ made up of the linesegment, called line A, joining the points ε − j∞ , ε + j∞,and a half circle with radius r, called line B, beginning at thesecond and ending at the first of these two points and lyingin the left-half plane. For line B,

∣∣∣∣∣∣∣1

2πj

∫line B

1

s

P∏p=1

(−M

λp

)Mnp

(s− M

λp

)Mnpds

∣∣∣∣∣∣∣≤ 1

2π

∫line B

P∏p=1

∣∣∣∣∣−M

λp

s− Mλp

∣∣∣∣∣Mnp

1

|s|ds → 0 as r → ∞.

Then, we have the following result

Pr[η(e′, e′′) ≥ 0] = limr→∞

1

2πj

∫Γ

1

s

P∏p=1

(−M

λp

)Mnp

(s− M

λp

)Mnpds

= 1+∑

p′:λp′<0

βp′

where

βp′ =

1

(Mnp′ − 1)!

dMnp′−1

dsMnp′−1

⎡⎢⎢⎢⎢⎢⎣1

s

P∏p=1

(−M

λp

)Mnp

P∏p=1

λp �=λp′

(s− M

λp

)Mnp

⎤⎥⎥⎥⎥⎥⎦

∣∣∣∣∣∣∣∣∣∣∣s= M

λp′

.

We conclude that

α(e′, e′′, θ) = −det

[I+

H(e)

N0M sin2 θ

]−M∑p′:λp′<0

βp′ .

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APPENDIX BPROOF OF THEOREM 3

For a given ζ, let

ai,j =AiAjeiej

M

M∑m=1

ζHm(Rij +Rji)ζm

Then, for an error vector e, we have η(e) = 2∑

i,j;i<j ai,j . Inthe rest of the proof, we fix an e and ζ and let η = η(e). Weshow that η ≥ 0 ensures the existence of {e′, e′′} ∈ D1(e)such that η(e′, e′′) ≥ 0.

Let K = {1, 2, · · · , K}. Then, an E1 partition {e′, e′′} of eis characterized by a subset Ko ⊂ K satisfying 1 �∈ Ko and Ko

�= ∅ in such a manner that e′i = 0 for i ∈ Kco and e′′j = 0 for j

∈ Ko, and we have η(e′, e′′) =∑

i∈Kco

∑j∈Ko

ai,j , where Kco

is the complement in K. Now, let η(Ko) =∑

i∈Kco

∑j∈Ko

ai,jand let FK be the set of all such subsets Ko. We show thatthere exists at least one Ko ∈ FK satisfying η(Ko) ≥ 0whenever η ≥ 0.

According to combinatoric arguments, we can see that thereare

∑K−1k=1

(K − 1

k

)distinct subsets in FK . For a given Ko, we

say that a pair {i, j} is a feasible pair if i ∈ Kco and j ∈ Ko

or if i ∈ Kco and j ∈ Ko.

First, let us consider a pair {i, j} such that i, j �= 1 and i �=j. Then, among

(K − 1

k

)subsets Ko with |Ko| = k, 2

(K − 3k − 1

)of them make {i, j} feasible. From this, the number of subsetsin FK which make {i, j} feasible is 2

∑K−2m=1

(K − 3k − 1

)= 2K−2.

Next, let us consider a pair {i, j} such that i = 1 and i �=j. Then, among subsets Ko with |Ko| = k,

(K − 1k − 1

)of them

make {i, j} feasible. From this, the number of subsets in FK

which make {i, j} feasible is∑K−1

m=1

(K − 1k − 1

)= 2K−2. Thus,

we conclude that every pair {i, j}, i �= j, is made feasiblewith exactly 2K−2 subsets in FK , and we have∑

all Ko∈FK

η(Ko) = 2K−3η

This shows that, if the right-hand side is non-negative, thereexists at least one Ko which makes η(Ko) non-negative andcompletes the proof.

APPENDIX CDERIVATION OF PSDB FOR OUR DOWNLINK SYSTEMS

From the PSD definition, its pairwise error probability canbe attain as

PPSD(e) =

∫ζQ

⎛⎝√

2S(e)

N0

⎞⎠χ [η(e) < 0] p(ζ)dζ

=

∫ ∞

0

Q

(√2X

N0

)∫ 0

−∞pX,η(X, v)dvdx

where X = S(e) = ζHH(e)ζ and pX,η(X, v) is joint pdf bynoting η(e) < 0. Since X and η(e) are two quadratic formsof the same channel path coefficients, their joint characteristicfunction can be written as

ΦXη(ωXωη) =1

det[IL − jωXAH(e)− jωηAH(e)

] .

We simplify the determinant as

det[IL − jωXAH(e)− jωηAH(e)

]= det[UH

iη]det[Λiη − jωXUiηAH(e)UHiη ]det[Uiη ]

= det[Λiη]det[IL − jωXWX ]

where UHiηΛiηUiη is the eigenvalue decomposition of IL −

jωηAH(e) and WX = Λ−1iη UiηAH(e)UH

iη . Therefore,

PPSD(e)

=1

2πj

∫ ∞

0

∫ 0

−∞

∫ ∞

−∞

∫ ∞

−∞Q

(√2X

N0

)ΦXη(ωX , ωη)

× exp(−jωXx− jωηv)dωXdωηdvdx

=1

2πj

∫ 0

−∞

∫ ∞

−∞

[1

2πj

∫ ∞

0

∫ ∞

−∞

×Q(√

2XN0

)exp(−jωXx)

det[Λiη]det[IL−jωXWX ]dωXdx

⎤⎦exp(−jωηv) dωηdv

=1

2πj

∫ 0

−∞

∫ ∞

−∞

1

det[Λ]

⎡⎣ ∑λi(WX )≥0

Ξ(λ(WX ))

2

×⎛⎝1− 1√

1 + N0

λi(Wx)

⎞⎠⎤⎦ exp(−jωηv)dωηdv

where λi(WX) is the eigenvalue of WX and

Ξ(λ(WX)) =

L∏i=1,i�=j,λ(WX ) �=0

λi(WX)

λj(WX)− λi(WX).

We notice that a double integral is remained, thus solvingthe eigenvalues of WX for every ωη and v, sampled from−∞ to ∞ and from −∞ to 0 respectively, is necessary whilecalculating numerical results. Therefore, its computation islarge.

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[8] R. G. Gallager, Information Theory and Reliable Communication. JohnWiley & Sons, 1968.

[9] K. Engdahl and K. Sh. Zigangirov, “Tighter bounds on the errorprobability of fixed convolutional codes,” IEEE Trans. Inf. Theory,vol. 47, pp. 1625–1630, May 2001.

[10] T. M. Duman and M. Salehi, “New performance bounds for turbocodes,” IEEE Trans. Commun., vol. 46, pp. 717–723, June 1998.

[11] S. Verdu, “Minimum probability of error for asynchronous Gaussianmultiple-access channels,” IEEE Trans. Inf. Theory, vol. IT-32, pp. 85–96, Jan. 1986.

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[12] Z. Zvonar and D. Brady, “Multiuser detection in single-path fading chan-nels,” IEEE Trans. Commun., vol. 42, pp. 1729–1739, Feb./Mar./Apr.1994.

[13] Y. Mohasseb, M. P. Fitz, and U. Mitra, “An improved bound on theperformance of maximum-likelihood multiuser detection receivers inRayleigh fading,” IEEE Trans. Inf. Theory, vol. 52, pp. 1184–1196,Mar. 2006.

[14] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modula-tion,” IEEE Trans. Inf. Theory, vol. 44, pp. 927–946, May 1998.

[15] J. Geng, M. Vajapeyam, and U. Mitra, “Union bound of space-timeblock codes and decomposable error patterns,” in Proc. 2003 IEEE Int.Symp. Inf. Theory, pp. 62.

[16] D. Chen and T. Hashimoto, “Performance analysis of MISO downlinkCDMA systems over Rayleigh fading channels,” IEICE Trans. Funda-mentals, vol. E87-A, pp. 2440–2447, Sept. 2004.

[17] S. Kotz and S. Nadarajah, Multivariate t Distributions and TheirApplications. Cambridge University Press, 2004.

[18] S. Verdu, Multiuser Detection. Cambridge University Press, 1998.[19] J. W. Craig, “A new, simple and exact result for calculating the

probability of error for two-dimensional signal constellations,” in Proc.IEEE Conf. on Military Commun., vol. 2, pp. 571–575.

[20] W. Luo and A. Ephremides, “Indecomposable error sequences in mul-tiuser detection,” IEEE Trans. Inf. Theory, vol. 47, pp. 284–294, Jan.2001.

[21] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes forhigh data rate wireless communication: performance criterion and codeconstruction,” IEEE Trans. Inf. Theory, vol. 44, pp. 744–765, Mar. 1998.

[22] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamentaltradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49,pp. 1073–1096, May 2003.

[23] B. Hochwald, T. L. Marzetta, and C. B. Papadias, “A transmitterdiversity scheme for wideband CDMA systems based on space-timespreading,” IEEE J. Sel. Areas Commun., vol. 19, pp. 48–60, Jan. 2001.

[24] D. Chen, T. Hashimoto, Z. Zhou, F. Zhao, and L. Shi, “Spreadingmatrices from PN sequences,” in Proc. 2010 Int. Symp. Commun. Inf.Technol., pp. 372–377.

[25] Y. G. Sinai, Kurs Teorii Veroyatnostej. (Japanese translation) MGU,1985.

Dianjun Chen received the B.S. degree in electronicengineering from Fudan University, Shanghai, in1988, the M.E. degree in radio engineering fromBeijing University of Posts and Telecommunications(BUPT) in 1993, and the Ph.D. degree from theUniversity of Electro-Communications, Tokyo, in2006. He is an associate professor at BUPT. Hiscurrent research interests include multiple antennassystems and CDMA systems.

Takeshi Hashimoto was born in Ibaraki Pref.,Japan, in 1952. He received the B.Es., M.Es andD.E. from Osaka University, respectively, in 1975,1977, and 1981. He was a Research Assistant inOsaka University from 1979 to 1986, was an Associ-ated Professor in Tokyo Denki University from 1986to 1992, and is a Professor in University of Electro-Communications, Tokyo, Japan. His main interestsare on information theory and its application.

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