PAPER
Theoretical Results about MIMO Minimal Distance Precoder and Performances Comparison
Baptiste VRIGNEAU†,Student Member, Jonathan LETESSIER†, Philippe ROSTAING†a), Ludovic COLLIN†, andGilles BUREL†,Members
SUMMARY This study deals with two linear precoders: the maximiza- tion of the minimum Euclidean distance between received symbol-vectors, called here max-dmin, and the maximization of the post-processing signal- to-noise ratio termed max-SNR or beamforming. Both have been designed for reliable MIMO transmissions operating over uncorrelated Rayleigh fad- ing channels. Here, we will explain why performances in terms of bit error rates show a significant enhancement of the max-dminover the max-SNR whenever the number of antennas is increased. Then, from theoretical developments, we will demonstrate that, like the max-SNR precoder, the max-dminprecoder achieves the maximum diversity order, which is warrant of reliable transmissions. The current theoretical knowledge will be applied to the case-study of a system with two transmit- or two receive-antennas to calculate the probability density functions of two channel parameters di- rectly linked to precoder performances for uncorrelated Rayleigh fading channels. At last, this calculation will allow us to quickly get the BER of the max-dminprecoder further to the derivation of a tight semi-theoretical approximation.
key words: MIMO,max-dminprecoder, beamforming, diversity order, 2D- channel parameters, BER approximation
1. Introduction
The multiple-input multiple-output (MIMO) systems under study consist of a linear precoder and a linear decoder under the full channel state information (CSI) assumption at both sides. The CSI is made available either through a feedback channel from the receiver to the transmitter, or whenever the transmitter acts also as a receiver in a time- or frequency- division duplex operation; one should note that the only jus- tification for transmit CSI is when the fading is sufficiently slow (quasi-static channel assumption).
The numbers of transmit- and receive-antennas arenT
and nR, respectively, i.e. (nT,nR) MIMO system. These precoded solutions permit reliable transmissions on condi- tion to use pertinent criteria like, for example, the min- imum mean-square error between the transmitted and re- ceived symbol vectors (MMSE) [1], the maximization of the minimum singular value of the global channel matrix (max- λmin) [2], the maximization of the post-processing signal-to- noise ratio (SNR) (max-SNR) [3], the minimization of the bit error rate (MBER) [4], or the maximization of the mini- mum Euclidean distance at the receiver side (max-dmin) [5]–
[7].
Manuscript received September 19, 2006.
Manuscript revised May 24, 2007.
†The authors are with the LEST UMR-CNRS 6165, 6 Av. Le Gorgeu, CS 93837, 29238 Brest Cedex 3, France.
a) E-mail: [email protected] DOI: 10.1093/ietcom/e91–b.3.821
Among them, only the max-dmin and max-SNR fixed- rate precoders exploit MIMO communication systems in or- der to get a full diversity gain and, therefore, a high link- reliability (see [3], [8] for the max-SNR precoder and this article for the max-dmin one). Another well-known, simple and powerful way of achieving the maximum diversity is the use of orthogonal space-time block codes (OSTBC) [9], [10]. These schemes require no transmit CSI, but the trans- mitted symbol rate is reduced; the best transmission rate is equal to 1 only for two transmit antennas. On the other hand, by using MIMO systems with precoding schemes the trans- mission rate is equal tob independent data substreams on condition thatb≤min(nT,nR) and, in the same time, a per- formance criterion (e.g. MMSE, max-dmin) can be applied to achieve reliable transmissions. For these reasons, the inves- tigations reported in this study were focused on the max-dmin and max-SNR fixed-rate precoders with a full perfect CSI.
In other respects, the optimization of thedmincriterion is a quite difficult issue because the solution depends on the constellation alphabet and on its size. A simple and ex- ploitable solution was proposed in [5] for the 4-QAM modu- lation with two data-streams (b=2) and an arbitrary number of transmit- and receive-antennas; the spectral efficiency of this scheme is, then, equal to 4 bit/s/Hz. In order to keep the same spectral efficiency for a relevant comparison of perfor- mances, the max-SNR precoder will use here a 16-QAM.
The interest of the max-SNR, i.e. the significant im- provement of performances induced by addition of only two antennas respectively dedicated to transmission and recep- tion was evidenced in [11], [12] by deriving the symbol er- ror probability (SEP). In a same way, the present article reports on the significant improvement of the max-dmin in- duced by increasing the number of antennas in operation and evidenced from bit error rate (BER) simulations. It will also demonstrate that, like the max-SNR, the max-dminprecoder achieves the maximum diversity ordernT×nR. A thorough analysis of this enhancement will highlight the benefits of using this recently-designed precoder against the max-SNR precoder.
We will also compare these two precoders in the special case-study of min(nT,nR)=2, i.e. (2,nR) or (nT,2) MIMO systems. This configuration corresponds to a communica- tion scenario where a base station with several antennas is linked to a mobile transmitter/receiver equipped with only two antennas. The study of these two-dimensional systems will permit us to define new random variables (RV) based Copyright c2008 The Institute of Electronics, Information and Communication Engineers
on a cartesian-to-polar transformation of the two channel matrix singular values. The determination of the probabil- ity density functions (pdfs) for a Rayleigh uncorrelated en- vironment will show the independence of the transformed RV. Then, this calculation will be used to design and de- rive a well-suited numerical tight approximation of the BER in order to quickly compare the respective performances of max-dminand max-SNR precoders.
The paper is organized as follows: Sect. 2 will intro- duce the channel model and the simplifications made about, in particular, the eigen-mode representation. Section 3 will briefly describe the principle and characteristics of the two schemes under study. Section 4 will report on our BER simulations of max-dminand max-SNR precoders before ex- plaining qualitatively the main difference between them.
Section 5 will deal with the theoretical characteristics of the max-dmin precoder and of the semi-theoretical approx- imation of BER. At last, our conclusions will be drawn in Sect. 6.
2. Channel Model and Eigen-Mode Representation Let us consider a narrowband spatial multiplexing MIMO system withnR receive andnT transmit antennas. The sys- tem transmitsb≤min(nT,nR) independent symbol streams through a precoder matrix,F, and a decoder matrix,G, de- signed on assuming a perfect channel knowledge at both sides. The input-output relation is:
y=GHFs+Gn (1)
whereyis theb×1 received signal vector,His thenR×nT
Rayleigh fading channel matrix with† Nc(0,1) i.i.d. ele- ments,Fis thenT × bprecoder matrix,Gis theb × nR
decoder matrix,sis theb × 1 transmitted symbol vector, andnis thenR×1 additive white Gaussian noise (AWGN) vector. Let us assume thatb≤rank(H)≤min(nT,nR) and:
E[ss∗]=Ib, Rn=E[nn∗]=σ2nInR, E[sn∗]=0. (2) By using the following decompositionsF=FvFd andG= GdGv, the input-output relation (1) can be re-written as:
y=GdHvFds+Gdnv (3) where Hv = GvHFv is the eigen-mode channel matrix, nv =Gvnis the additive noise vector on the channel eigen- mode with the covariance matrixRnv =E[nvn∗v]=σ2nIb; in addition, the unitary matricesGvandFvare chosen in order to diagonalize the channel and reduce its dimension tob.
This procedure, based on the singular value decomposition (SVD) ofH, is frequently used for MIMO systems. The matrixFd results from optimization under a specific crite- rion described in the next section. In addition, the SNR is defined as:
Φ = ET
σ2n
(4) whereET is the total average transmitted power; the pre- coder must verifyF2F =Fd2F = ET. On the other hand,
the symbol decision is the maximum likelihood rule (ML) and is unaffected byGd; thus, this matrix can be equal to the identity matrix with no loss of generality. As a result, the matrixHvcorresponds tobindependent subchannels and is equal to diag(√
λ1, . . . ,√
λb) whereλ1 ≥ · · · ≥λb are theb highest eigenvalues ofHH∗.
3. Description of the Precoders
3.1 Maximumdmin(max-dmin) Precoder
The minimum Euclidean distance between received vectors affects the system performances, especially with the ML de- tector [13]. The minimum Euclidean distance is defined by:
dmin(Fd)= min
(sk,sl)∈ C,sksl
HvFd(sk−sl) (5) whereskandslbelong toC, the set of all possible transmit- ted symbol vectors. The max-dminprecoder is given by:
Fddmin =arg max
Fd
dmin(Fd) (6)
under the power constraint,Fd2F =ET. Computing Eq. (6) is difficult because the expression ofdmin under considera- tion is the exact one and depends on both the constellation size and subchannels. One should note that the authors in [2]
used approximations of dmin. A very exploitable solution was given in [5] for two independent data streams, b = 2 and a 4-QAM. This precoder design is based on a change of variable:
λ1=(ρcosγ)2 λ2=(ρsinγ)2 ⇔⎧⎪⎪⎪⎨
⎪⎪⎪⎩ γ=arctan λ2
λ1
ρ= √ λ1+λ2
(7) whereρis a positive real parameter related to the channel gain, andγis an angle linked to the eigenvalues ratio meet- ing the conditionλ1≥λ2>0, i.e.π/4≥γ>0. It is worth noting that Hv is totally defined by ρand γ. Moreover, a small γmeans that the first subchannel is privileged (λ1 λ2), whereas a value close toπ/4indicates two close subchannels (λ1 λ2). Then, the solution given in [5] simply depends on the value ofγaccording to a constant thresholdγ0:
- if 0≤γ≤γ0, Fddmin=Fr1=
ET
⎛⎜⎜⎜⎜⎜
⎝
3+√ 3 6
3−√ 3 6 ei12π
0 0
⎞⎟⎟⎟⎟⎟
⎠ (8)
- ifγ0≤γ≤π/4, Fddmin=Focta=
ET
2
cosψ 0 0 sinψ
1 eiπ4
−1 eiπ4
(9)
where⎧⎪⎪⎪⎨
⎪⎪⎪⎩ψ=arctan √cos2−γ1 γ0=arctan
3√ 3−2√
6+2√ 2−3 3√
3−2√
6+1 17.28◦ . (10)
†Nc(0,1) is the zero-mean and unit-variance complex normal distribution,Inis the identity matrixn×n, (.)∗is the transpose con- jugate,.Fis the Frobenius norm, diag(.) is a diagonal matrix, and E[.] is the mathematical expectation.
The term,ψ, is related to the eigen-mode power allocation, and the constant threshold,γ0, permits the precoder to use either one or two subchannels, i.e. Eqs. (8) and (9) respec- tively. The value ofγ0 is obtained by considering thatFr1
andFoctaprovide the samedmin. Equations (8), (9) and (10) can be directly computed to design the max-dmin precoder for a given channel matrix or, more precisely, for a given value ofγ.
In order to further study the theoretical behavior of the max-dmin precoder, let us define the optimized normalized minimum Euclidean distance asdmin(γ)=dmin(Fd)/(
ETρ).
This definition slightly differs from [5], and the result de- pends on onlyγ:
dmin(γ)=
⎧⎪⎪⎪⎪⎪⎪⎪
⎨⎪⎪⎪⎪⎪
⎪⎪⎩
1− 1
√3cosγ if 0<γ≤γ0
(4−2√
2) cos2γsin2γ 1+(2−2√
2) cos2γ otherwise
. (11)
A prerequisite to the comparison of thisdmin-based pre- coder with the max-SNR maximizing the post processing SNR is the calculation of the normalized dmin and post- processing SNR criteria in both cases. For an AWGN, the post-processing SNR,Γ, was defined in [2] as:
Γ =trace
GHF(GHF)∗(GRnG∗)−1
. (12)
Equations (2) and (3) permit a simplification of (12) as fol- lows:
Γ = ||HvFd||2F σ2n
. (13)
Let us now determine the post-processing SNR for the opti- mizeddminscheme:
Γ=⎧⎪⎪⎨
⎪⎪⎩ Φρ2cos2γ if 0<γ≤γ0
Φρ2 (3+21+(3√2) cos+2√2) cosγ4+sinγ22γ otherwise . (14) 3.2 Maximum SNR (max-SNR) Precoder
In a similar way, the precoder Fd optimizes the post- processing SNR (13):
FΓd=arg max
Fd
Γ. (15)
One should note thatλ1 ≥λ2 and that the upperbound for the post-processing SNR is:
Γ≤ ETλ1
σ2n
= Φλ1. (16)
The straightforward solution achieving the upperbound and consequently the optimization is a family of matrices de- fined by:
FΓd= ET
u v 0 0
(17)
whereuandvare two complex parameters meeting the con- straint |u|2 +|v|2 = 1. The precoder Fr1 is a solution of Eq. (15), so the max-dminsolution for 0≤γ ≤γ0gives the optimized post-processing SNR. The well-known max-SNR precoder [3], [11] is a particular solution of Eq. (17) corre- sponding to a single transmit symbol:u=1,v=0 andb=1.
This solution considers only the subchannel associated to λ1, i.e. Hv = √
λ1 and Fd = √
ET. The equivalent input- output relation is then:
y=
ETλ1s+n=
ETρcosγs+n (18) wheresis the single transmit symbol, andn is an AWGN with varianceσ2n. The optimized post-processing SNR is given by:
Γ = Φρ2cos2γ. (19)
In addition, for further comparisons the normalized mini- mum Euclidean distance for a 16-QAM is:
dmin(γ)=
√10
5 cosγ. (20)
The next section will evidence results from BER simulations showing different behaviors for the two precoders. A first study will permit us to qualitatively explain these observa- tions in order to further focus on theoretical developments.
4. Comparison of Precoder BERs after Increase in the Number of Antennas
4.1 Evidence of max-dminInterest from Simulations The effect of channel estimation errors was investigated in [5] where it was found to have a similar impact on both pre- coders. Consequently, all of the BERs were simulated with 104 known Rayleigh matrices (perfect CSI) under the fol- lowing conditions: b =1 and 16-QAM for the max-SNR andb =2 and 4-QAM for the max-dmin. Figure 1 shows
Fig. 1 BER simulations of max-dminand max-SNR for symmetric (2,2), (3,3) and (4,4) systems with an uncorrelated Rayleigh fading channel.
Fig. 2 max-dminto max-SNR ratios ofdmin(Rdmin) and post-processing SNR (RΓ) as a function ofγand pdf ofγfor (2,2) and (2,10) systems (nS ={0,8}).
the BER plots as a function of the SNRΦfor the max-dmin
and max-SNR precoders. It highlights that the gap between both precoders is increasing with the number of antennas;
for the (2,2) system, the max-SNR is slightly better than the max-dmin at low SNR, and vice versa at high SNR. The SNR difference between the two precoders is about 1.8 dB for the (3,3) system and reaches nearly 2.5 dB for the (4,4) one. The max-dmin precoder takes, therefore, more bene- fit from the spatial diversity than the max-SNR due to the significant enhancement of performances induced by using more antennas.
4.2 How DoesdminPrecoder Structure Better Exploit Ad- ditional Antennas?
The expressions of the normalized minimum Euclidean dis- tances (11) and (20) show thatρis a scale factor for both precoders. Onlyγhas an impact on the behaviors of both precoders, and the results of the performances can be qual- itatively explained by the study ofγ (Fig. 2). In order to compare the two precoders, let us now focus on the evolu- tion of both criteria as a function ofγ. Figure 2 depicts the evolution ofdmin and post-processing SNR ratios versusγ defined as:
Rdmin=dmin(Fddmin)
dmin(FΓd) >1 and RΓ=Γ(Fddmin)
Γ(FΓd) ≤1. (21) Let us denote nS=|nT−nR| to represent the configuration asymmetry. By using the theoretical result explained in Sect. 5.2 for min(nT,nR)=2, the pdfs ofγare superimposed on the same figure whennS=0 ornS=8 for (2,2) or (2,10) systems, respectively. Depending on whetherγis lower or greater thanγ0(10), the max-dminprecoder solution is either (8) or (9), which leads to different behaviors. Figure 2 shows clearly that RΓ=1 whenγ≤γ0(both precoders are solutions of Eq. (15)); on the other hand, whenγ > γ0 the max-SNR has a slightly better post-SNR (RΓ<1), but Rdmin is signifi-
Table 1 Probability ofFoctaprecoder choice according tonS.
nS 0 1 4 8
P[chooseFocta]=P[γ > γ0](%) 56 82.8 99.2 99.99
cantly elevated: According to BER simulations, thedminen- hancement is more important than the post-processing SNR loss. Additionally, it is worth noting that thedmin ratio is constant forFr1with a little gain (1.03) due to the rotation π/12(8) in the received constellation. Thus, the difference be- tween both precoders is large when the max-dmin precoder (Focta) uses the two subchannels, whereas the max-SNR pre- coder always suppressesλ2. One should note that whennS
is increasing, the plots ofγpdfs show a shift on the right side corresponding to higher values ofγ. The second gain,λ2, is then closer toλ1, but this subchannel is still neglected by the max-SNR. Moreover, Table 1 shows that, statistically, the max-dmin quickly uses only the Focta precoder whennS is increasing. These statistical observations together with the previous comparison of behaviors explain why the max-dmin
precoder provides a better BER than the max-SNR.
5. Theoretical Results for max-dmin
5.1 Proof of Full Diversity of max-dmin
The maximum order diversity of the max-dminprecoder can be shown by a development alike the one reported in [14, pp.99–100] for the max-SNR. So, let us consider the nearest neighbor union bound to the symbol (vector) error probabil- ity (SEP) when the ML detection rule is applied. It gives:
SEP≤ Ne
2 erfc
⎛⎜⎜⎜⎜⎜
⎜⎜⎜⎝
d2min(Fddmin) 4σ2n
⎞⎟⎟⎟⎟⎟
⎟⎟⎟⎠ (22)
where Ne is the average number of the nearest neighbors per symbol vector [15], and erfc() is the complementary er- ror function. One should note that the max-dmin is a par- ticular case where the nearest neighbors are all located at the same optimized distance, dmin(Fdmin). Thus, the con- stant, Ne, corresponds exactly to the average number of symbol vectors at this minimum distance, and Eq. (22) pro- vides a close approximation to the actual probability error [15]. Equation (11) can be rewritten to expressdmin2 versus λ1=ρ2cos2γ:
d2min(Fdmin)=
ETλ1ξ if 0< γ≤γ0
ETλ1χ(γ) ifγ0< γ≤π/4 (23) whereξ=1−√13andχ(γ)= 1+(4−2(2−2√√2) sin2) cos2γ2γ. This increasing function verifiesξ≤χ(γ)≤1 forγ∈[γ0,π/4] and, thus, the distance can be upper- and lower-bounded as follows:
ETξλ1≤d2min(Fdmin)≤ETλ1. (24) Moreover, using the inequality||H||2F/min(nT,nR) ≤ λ1 ≤
||H||2F leads to:
ETξH2F
min(nT,nR)≤d2min(Fdmin)≤ETH2F. (25) By using the Chernoffbound (i.e. erfc(x)e−x2forx1) and the resultE[exp(−xH2)]=(1+x)−nTnR (entries ofH are assumed to beNc(0,1) i.i.d.), at high SNR regime, the upper- and lower-bounds of the average SEP are as follows:
Ne
2 Φ
4 −nTnR
≤SEP≤Ne
2
Φξ
4min(nT,nR) −nTnR
(26) The diversity order of the max-dminis, thus, maximum and equal tonT×nR.
5.2 Rayleigh Fading Channel Study withγandρParame- ters
Let us now setnT or nR equal to two i.e. (2,nR) or (nT,2) systems. This asymmetric configuration corresponds to a base station with several antennas and a mobile transmit- ter/receiver restricted to two antennas. Moreover, Fig. 1 shows that the max-dmin exploits the spatial diversity bet- ter than the max-SNR, and the case (2,2) is the most un- favorable since the BERs of the two precoders are close.
These considerations led us to theoretically compare the two precoders for (2,2+ns) or (2+ns,2) MIMO systems with ns={0,1,2, . . .}in the two next paragraphs.
It is worth noting that, usually, performances are stud- ied from the statistics ofλi [16], [17]. On the other hand, the present study uses theoretical results aboutρandγ. The joint density probability of the two eigenvaluesλ1 andλ2
was given in [18]:
fλ1,λ2(λ1, λ2)= (λ1λ2)ns
nS!(nS +1)!e−(λ1+λ2)(λ1−λ2)2. (27) Then, the joint pdf ofρandγis obtained by using the change of variables defined in Eq. (7):
fρ,γ(ρ, γ)= fλ1,λ2(ρ2cos2γ, ρ2sin2γ)|J| (28) where |J| is the Jacobian determinant and is equal to 2ρ3sin 2γ. Finally, the result is:
fρ,γ(ρ, γ)
= nS2!(n−2nSS++11)!cos22γ(sin 2γ)2nS+1ρ7+4nSe−ρ2. (29) The two marginal pdfs can be calculated by integrating this equation according toρorγ. The pdf ofγis:
fγ(γ) = +∞
0
fρ,γ(ρ, γ)dρ
=2−2nS(2nS +3)!
nS!(nS +1)! cos22γ(sin 2γ)2nS+1 (30) and the pdf ofρis:
fρ(ρ)= π/4
0
fρ,γ(ρ, γ)dγ= 2
(2nS +3)!ρ7+4nSe−ρ2. (31)
Table 2 Average number of nearest neighborsNeand false bitsNb. FΓd Fr1 Focta
Ne 3 7/2 7
Nb 1 1.471 1.488
Thus, we can easily verify that the two random variables are independent:
fρ,γ(ρ, γ)= fρ(ρ)fγ(γ). (32) This interesting result allows us to develop a semi- theoretical approximation of the BER.
5.3 Semi-Theoretical Approximation of the BER
The theoretical determination of BER is not trivial [12], and finding the exact BER of max-dminwould be very hard.
These considerations led us to propose an efficient average SEP approximation of the max-dmin or max-SNR precoder.
The expectation over the random variables ρandγof the tight error probability (22) can be expressed as:
SEP=
π/4 0
∞
0
Ne
2 erfc ρ2η(γ)
fρ,γ(ρ, γ)dγdρ (33) whereη(γ) = d2min(γ)Φ/4. The termd2min(γ) is either the normalized minimum distance of max-dmin (11) or the nor- malized minimum distance of max-SNR (20). The num- ber of the nearest neighbors depends on the chosen precoder and can be counted once the received constellation has been plotted. In the same way, the corresponding average number of false bits per 4 bits symbol vector error, denotedNb, can be counted, and the non-averaged BER-SEP relation is then given by:
BER= Nb
4 SEP. (34)
The max-dminprecoder is unable to minimizeNbbecause of its structure and of the large number of neighbors (Table 2) conversely to the 16-QAM used by the max-SNR, which, thanks to the Gray coding, achieves Nb =1. Indeed, the Gray coding requires a number of nearest neighbors lower than the number of bits corresponding to a transmitted vec- tor (4 bits here). The values for the Fr1 and the Focta are given in Table 2. On the other hand, the erfc function has a tight upper bound [19]:
erfc(x) N
i=1
αie−βix2 (35)
whereNis the development order, andαiandβiare coeffi- cients. The use of Eqs. (32) to (35) leads to:
BER N
i=1
αi
π/4 0
τγ 2 fγ(γ)
+∞
0
e−βiη(γ)ρ2fρ(ρ)dρdγ (36)
whereτγ =NeNb/4 depends on the chosen precoder accord- ing to the value ofγ(Table 2). Let us calculate the integral overρby using Eq. (31):
∞
0
e−βiη(γ)ρ2fρ(ρ)dρ=(1+βiη(γ))−(2nS+4). (37) Then, the result is expressed by:
BER
N
i=1
αi
π/4 0
τγ 2
fγ(γ)
(1+βiη(γ))2nS+4dγ (38) where the whole set of elements is known, which permits an easy numerical integration for the max-dmin and max-SNR precoders. Moreover, the denominator in Eq. (38) can be approximated at high SNR, which allows the factorization of the SNR term:
BER≤Φ−(2nS+4) N
i=1
αi
π/4 0
τγ
2
fγ(γ)dγ βid2min(γ)
4
2nS+4. (39)
Let us denoteκ(nS) thenS-dependent parameter as:
κ(nS)−(2nS+4)= N
i=1
αi
π/4 0
τγ
2
fγ(γ) βid2min(γ)
4
2nS+4dγ, (40)
it is also defined as the coding gain [14, pp.89]. As a result, the average BER can be upper-bounded at high SNR by:
BER≤(κ(nS)Φ)−(2nS+4). (41) Equation (41) verifies that the diversity order of max-dmin precoder is equal to 2nS +4=nTnR(with min(nT,nR)=2).
5.4 Approximation Results
Figure 3 shows the simulated and approximated BERs for max-dmin and max-SNR precoders with nS = 8 and the
Fig. 3 Simulated and approximated BERs as a function of SNR for max-dminand max-SNR precoders fornS =8 and min(nT,nR)=2.
erfc approximation of order 2 Eq. (35) withαi andβi from [19] (erfc(x) 16e−x2+12e−43x2). The approximation of the max-SNR is useful for comparison of results. Indeed, the pdf ofλ1can be found in [20], and the performances of the max-SNR can be deduced from [12], [21]. The approxima- tion is a tight upper bound especially for middle and high SNRs (Φ>4 dB). This approximation allows one to quickly predict and compare the performances of both precoders.
At a fixed SNR, Fig. 4 shows the BERmax-SNR-to- BERmax-dmin ratio as a function ofnS. The BERs were sim- ulated or approximated by using (38). A first comparison of the simulated and approximated BER ratios atΦ =8 dB indicates that the approximated ratio, i.e. the BER ratio cal- culated with the approximation, is very close to the ratio issued from simulations. According to Fig. 3 the approxi- mation is correct at high SNR. These data led us to consider that the use of the approximation ratio was valid at SNRs greater than 8 dB, and in particular whenΦ =12 and 16 dB (Fig. 4). It is worth noting that the BER ratio is increas- ing linearly with nS in a semilogarithmic scale where the slope is SNR-dependent. Thus, the results issued from the max-dminare better than those from the max-SNR when the number of antennas is increased (min(nT,nR)=2).
We demonstrated that the max-dminand max-SNR pre- coders have the same diversity order or, equivalently, the same slope of the asymptotic BER at high SNR. Let us de- fine the asymptotic SNR gainGdBas:
GdB=κ(nS)max-ddB min−κ(nS)max-SNRdB . (42) The termGdBrepresents the SNR difference in dB between the two precoders at high SNR. In order to evaluate the in- fluence of the number of antennas, Fig. 5 depicts this SNR gain versusnS. It shows a strong enhancement of the gain whennS falls within 0 and 4, and a saturation effect whennS
is greater than 4. The max-dminprecoder takes more profit of additional antennas than the max-SNR.
Fig. 4 Simulated BER ratio forΦ=8 dB and semi-theoretical BER ratios as a function ofnS forΦ={8,12,16}dB and min(nT,nR)=2.
Fig. 5 Evolution of the SNR gainGdBbetween max-dminand max-SNR at high SNR as a function ofnS.GdBis obtained with the semi-theoretical approximation for min(nT,nR)=2.
6. Conclusion
These additional investigations on a MIMO minimal dis- tance precoder allowed us to demonstrate from theory that the diversity order of the max-dmin precoder was maxi- mum and equal tonT ×nR. Inter-comparison of simulated BER of the max-dmin and max-SNR precoders for sym- metric systems highlighted the benefits of the dmin-based precoder when antennas were added. Moreover, accord- ing to the Rayleigh fading for min(nT,nR) = 2 MIMO systems, we determined the pdfs of the channel param- eters γ and ρ defined by the transformation of the two- dimensional Cartesian eigenvalues system into a polar one and observed that the variablesγandρare independent. At last, we derived a tight semi-theoretical approximation of the max-dmin BER, which enabled us to quickly compare the two precoders for asymmetric systems (min(nT,nR)=2):
we found that the max-dminprecoder was always better than the max-SNR precoder whennS was increasing (except for nS=0). However, for a highnS, using a MIMO system with min(nT,nR)=3 could be a better solution since the SNR gain showed some saturation. For this reason, it would be worth extending these promising results to MIMO systems with min(nT,nR)>2 in further investigations.
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Baptiste Vrigneau was born at Lannion (France) in 1979. He received the M.Sc. degree from ´Ecole Normale Sup´erieure of Cachan, France, in 2001, and the Ph.D. degree from University of Brest, France, in 2006. He is a member of the Laboratory for Electronics and Telecommunication System (LEST-UMR CNRS 6165), France, and his research interest are focused on MIMO systems.
Jonathan Letessier was born at Foug`eres, France, in 1978. In 2001 and 2002, he succes- sively received a B.Sc. degree and a M.Sc. one in Telecommunication and Communication Pro- cessing before successfully defending, in 2005, his Ph.D. in Electrical engineering at the Univer- sity of Brest (France). Since 2005 he has been Assistant Professor and works as member of the Laboratory for Electronics and Telecommunica- tion Systems (LEST-UMR CNRS 6165). His re- search interests are focused on MIMO systems.
Philippe Rostaing received the Ph.D. de- gree in electrical engineering from the Univer- sity of Nice-Sophia Antipolis, France, in 1997.
From 1997 to 2000, he was Assistant Professor at the French Naval Academy, Lanveoc, France.
Since 2000, he has been Assistant Professor of Digital Communications and Signal Processing at the University of Brest. He is a member of the Laboratory LEST - UMR CNRS 6165. His present research interests are in signal process- ing for digital communications with emplasis on MIMO systems.
Ludovic Collin received the Ph.D. de- gree in electrical engineering from the Uni- versity of Bretagne Occidentale, Brest, France, in 2002. From 1989 to 1999 he was with ORCA Instrumentation, Brest, where he de- veloped oceanographical instrumentation and acoustic modems. From December 1999 to September 2002, he was a Research and Teach- ing Assistant at French Naval Academy, Lan- veoc, France. Since September 2002, he has been Assistant Professor at the Institut of Tech- nology of Lannion, and at the ENSIETA, Brest. He is a member of the Laboratory for Electronics and Telecommunication Systems (LEST-UMR CNRS 6165) and his research interests are in MIMO systems and intercep- tion of communications.
Gilles Burel was born in 1964. He re- ceived the M.Sc. degree from ´Ecole Sup´erieure d’Electricit´e, Gif Sur Yvette, France, in 1988 and the Ph.D. degree from University of Brest, France, 1991. From 1988 to 1997 he was a member of the technical staffof Thomson CSF, then Thomson Multimedia, Rennes, France, where he worked on image processing and pat- tern recognition applications as project manager.
Since 1997, he has been Professor of Digital Communications, Image and Signal Processing at the University of Brest. He is a member of the Laboratory for Electron- ics and Telecommunication Systems (LEST-UMR CNRS 6165) and, within this laboratory, he is Director of the Signal Processing Team. He is the au- thor of 19 patents, one book and more than one hundred scientific papers.
His present research interests are in signal processing for digital communi- cations with emphasis on MIMO systems, interception of communications and chaotic transmissions.
