New Closed-Form of the Largest Eigenvalue PDF for Max-SNR MIMO System Performances

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PAPER

New Closed-Form of the Largest Eigenvalue PDF for Max-SNR MIMO System Performances

Jonathan LETESSIER^†,Member, Baptiste VRIGNEAU^†,Student Member, Philippe ROSTAING^†a), andGilles BUREL^†,Members

SUMMARY Multiple-input multiple-output (MIMO) maximum-SNR (max-SNR) system employs the maximum ratio combiner (MRC) at the receiver side and the maximum ratio transmitter (MRT) at the transmitter side. Its performances highly depend on MIMO channel characteristics, which vary according to both the number of antennas and their distribution between the transmitter and receiver sides. By using the decomposition of the ordered Wishart distribution in the uncorrelated Rayleigh case, we derived a closed-form expression of the largest eigenvalue probability density function (PDF). The final result yields to an expression form of the PDF where polynomials are multiplied by exponentials; it is worth underlining that, though this form had been previously observed for given couples of antennas, to date no formally-written closed-form was available in the lit- erature for an arbitrary couple. Then, this new expression permits one to quickly and easily get the well known largest eigenvalue PDF and use it to determine the binary error probability (BEP) of the max-SNR.

key words:Wishart matrix, PDF, largest eigenvalue, Rayleigh fading chan- nel, MIMO, maximum SNR, MRT/MRC

1. Introduction

In the case of the full channel state information (CSI) available at the transmitter, the reliability of the transmission of multiple-input multiple-output (MIMO) systems can be improved by choosing a communication strategy, sometimes referred as max-SNR or MIMO maximal ratio transmitter (MRT) and maximal ratio combiner (MRC) [1], [2]. In this solution the signals are transmitted along the strongest direction of the channel, i.e. the direction of the eigenvec- tor corresponding to the largest eigenvalue of^∗HH^∗ where H = [hi j] is the MIMO channel matrix, hi j is the channel gain from the jth transmit antenna toith receive antenna and j =1. . .nT,i =1. . .nRwithnT andnR the respective numbers of transmit and receive antennas. The input-output relation is then:

y=

P₀w^∗RHwTs+w^∗Rn, (1) wherewT andwR are the principal right and left singular vectors ofH, respectively [3],sis the transmit symbol with E[|s|²]=1,P₀is the total transmit power andnis the complex circular Gaussian noise vector with the covariance ma- trixRn = E[nn^∗] = σ²InR. From (1), the channel matrix can be seen as only the largest singular valueσmax= √

λmax

Manuscript received September 15, 2006.

Manuscript revised August 22, 2007.

†The authors are with the University of Brest, Lab-STICC Lab- oratory (UMR CNRS 3192), 6 av. Le Gorgeu, CS 93837, 29238 Brest Cedex 3, France.

a) E-mail: philippe.rostaing@univ-brest.fr DOI: 10.1093/ietfec/e91–a.7.1791

ofH: the receiver SNR, given byP₀λmax/σ², is thus maxi- mized [4]. The equivalent input-output relation becomes:

y=

P0λmaxs+n (2)

wheren=w^∗_Rnis a complex Gaussian random variable with E[|n|²]=σ².

In many applications, the eigenvalue PDF is needed to theoretically evaluate the system performances. Ratnarajah et al. [5] expressed the marginal unordered eigenvalue PDF so as to get an expression of the capacity for a MIMO system in a Rayleigh channel. In some applications, the largest eigenvalue PDF, p_λ_max(u), is required to evaluate, for example, the symbol error probability (SEP) of a max-SNR system or the outage capacity [6]. The traditional way to find the PDF of the largest eigenvalue is to integrate the joint PDF from the lowest eigenvalue up to the largest one. But, reaching easily and quicklyp_λ_max(u) by this method is tricky, in particular in the general case with (nT,nR). So, quite of- ten, the PDF is either bounded [7] or not expressed in a general form [8]. Kang et al. [9] used the classical Jacobi’s formula about the derivative of a determinant to find p_λ_max(u), i.e. _du^d det (A(u)) = det (A(u)) trace

A⁻¹(u)_dt^dA(u) , where det (A(u)) is the cumulative density function of λmax. In [10], we used this classical formula to extend the max-SNR MIMO performances in terms of symbol error probability (SEP) to the M-PSK and M-QAM modulations. To com- plete our study on the p_λ_max(u) function, we present here a new method based on the determinant definition of Vander- monde’s matrices applied to the Wishart matrices (HH^∗) and permutation simplifications to calculate theλmaxPDF. This method permits one to obtain directly p_λ_max(u) where polynomials are multiplied by exponentials.

This paper deals with the general case (nT,nR) of the largest eigenvalue PDF worth being used in many applications [6], [9]. Section 2 presents the method to obtain the closed-form expression of p_λ_max(u) with an explicit expression of the largest eigenvalue PDF. In Sect. 3, a bit error probability (BEP) for M-PSK and M-QAM is recalled for comparisons ofp_λ_max(u) behaviors and BEP gains for diﬀer- ent numbers of transmit and receive antennas, and the conclusion is drawn in Sect. 4.

∗The superscript * stands for the complex conjugate transpose.

Copyright c2008 The Institute of Electronics, Information and Communication Engineers

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2. Probability Density Function of the Largest Eigen- value of the Wishart Matrix, W

In order to determine the PDF of the largest eigenvalue to the Wishart distribution matrix ofW =HH^∗in a closed-form for a given and arbitrary couple (nT,nR) let us assume that Hhas independent and identically distributed (iid) entries according toCN(0,1) and denoteCN(0, σ²) the complex Gaussian distribution with independent real and imaginary parts distributed according to zero-mean ^σ₂²-variance Gaus- sian distribution.

The PDF of the unordered eigenvalues of the matrixW is given by [11]:

p(λ1, ..., λm)=α m

i=1

λⁿ_i^Se^−λⁱ

1≤i<j≤m

(λi−λj)², (3)

wherens = max(nT,nR)−m, m = min(nT,nR) and α = 1/m

i=1(nT−i)!(nR−i)! is the normalization constant.

Expressing p_λ_max(u) from (3) requires some rewriting.

Let us, at first, use the determinant of Vandermonde’s matrix, Θ. The determinant of Θ can be expressed as [9, Eq. (3)]:

|detΘ|=

1≤i<j≤m

(λi−λj) (4)

with Θ =

⎛⎜⎜⎜⎜⎜

⎜⎜⎜⎜⎜

⎜⎝

1 1 . . . 1

λ1 λ2 . . . λm

... ... ...

λ^m₁⁻¹λ^m₂⁻¹ . . . λ^mm⁻¹

⎞⎟⎟⎟⎟⎟

⎟⎟⎟⎟⎟

⎟⎠. (5)

Then, by replacing (4) in (3), the PDF of the unordered eigenvalues can be rewritten as follows [12]:

p(λ1, ..., λm)=α

⎛⎜⎜⎜⎜⎜

⎝ m

i=1

λⁿ_i^Se^−λⁱ

⎞⎟⎟⎟⎟⎟

⎠(detΘ)². (6) Equation (6) permits one to use the explicit calculation of determinant of Vandermonde’s matrix:

detΘ = m!

l=1

ε(ρl) m

i=1

λ^ki^l,i, (7)

whereρl is thelth permutation in the set of permutations {ρ1, ρ2, . . . , ρm!}. Such a permutation is a rearrangement of the set {1,2,3, . . . ,m}. Let us denote kl the vectork₁ = [0,1, . . . ,m− 1] permuted by ρl and ◦, the permutation operator and thus kl = ρl ◦ k₁. Moreover, let us define kl,i = [ρl◦k1]i = [kl]i the ith element of kl andε(ρl) the permutation sign, which depends on the number of transpositions. If the number of transpositions is even, then ε(ρl)= +1; on the other hand, if it is odd, thenε(ρl)=−1.

The use of (7) in (6) leads to:

p(λ1, ..., λm)=α m!

l=1

ε(ρl) m!

p=1

ε(ρp) m

i=1

λⁿ_i^s⁺^k^l^,ⁱ⁺^k^p^,ⁱe^−λⁱ. (8) One should note that the expression of (8) is useful for the derivation of the PDF ofλmax.

2.1 Determination ofp_λ_max(u)

The expression of p_λ_max(u) is obtained by determining, first, the cumulative density function (CDF) ofλmax, and then, its derivative. The cumulative density function ofλmax is calculated by integrating the PDF of the unordered eigenvalues mtimes overλi(i=1. . .m) [13]:

P(λmax<u)= u

0

. . . u

0

p(λ1, ..., λm)dλ1...dλm (9) By using (8) and (9) the CDF can be expressed as follows:

P(λmax<u)

=α m!

l,p=1

ε(ρl)ε(ρp) m

i=1

u 0

λⁿ_i^s⁺^k^l,i⁺^k^p,ie^−λⁱdλi

=α m!

l,p=1

ε(ρl)ε(ρp) m

i=1

Γˆu

ns+kl,i+kp,i+1

(10) where ˆΓu(p) = Γu(p)Γ(p) with Γ(p) being the complete Gamma function (Γ(p)=(p−1)! for pa positive integer), andΓu(p) the incomplete Gamma function is expressed as:

Γu(p)=1/Γ(p)×

u 0

t^p⁻¹e⁻^tdt=1−e⁻^u

p−1

k=0

u^k/k! (11) For a fixed permutation couple (ρp, ρl), there is a unique permutation ρr such that ρp = ρl◦ ρr. Then (10) can be rewritten as:

P(λmax<u)

=α m!

r=1

ε(ρr) m!

l=1

m i=1

Γˆu

ns+

ρl◦ {k1+kr}

i+1 . (12) In (12), the permutation signε(ρr) is obtained by:

ε(ρl)ε(ρp)=ε(ρl)ε(ρl◦ρr)

=ε(ρl)ε(ρl)ε(ρr)

=ε(ρr), (13)

and

ρl◦ {k1+kr}

iis calculated by:

kl,i+kp,i=[ρl◦k1]i+[ρp◦k1]i

=[ρl◦k1]i+[ρl◦ρr◦k1]i

=ρl◦ {k1+ρr◦k1}

i. (14)

Since multiplication is commutative, the permutation

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ρlhas no eﬀect on the result of the product in (12), which is found to be equal to a constant,κr, for a fixed permutation ρr:

κr= m

i=1

Γˆu(ns+ρl◦ {k₁+kr}i+1)

= m

i=1

Γˆu(ns+{k₁+kr}i+1). (15) Then, for a given ρr, the sameκr is added m! times.

Moreover, ask₁ = [0,1, . . . ,m−1], thenk_1,i+1 = i for i=1, . . . ,m. Finally, (12) becomes:

P(λmax<u)=αm!

m!

r=1

ε(ρr) m

i=1

Γˆu

ns+kr,i+i. (16) Thanks to (16), the calculation ofP(λmax <u) deriva- tive overugives the PDF ofλmax, and then leads to the fol- lowing expression:

p_λ_max(u)=αm!e⁻^u m!

r=1

ε(ρr) m

j=1

uⁿ^s⁺^k^r,j⁺^j⁻¹ m

i=1,ij

Γˆu(ns+kr,i+i), (17)

where the derivative overuof the product in (16) is equal to:

d du

⎧⎪⎪⎨

⎪⎪⎩

m i=1

Γˆu(ns+kr,i+i)⎫⎪⎪⎬

⎪⎪⎭

=e⁻^u m

j=1

uⁿ^s⁺^k^r,j⁺^j⁻¹ m i=1,ij

Γˆu(ns+kr,i+1). (18)

Finally, by using (11) and (17) p_λ_max(u) is equal to:

p_λ_max(u)=αm!e⁻^u m!

r=1

ε(ρr) m

j=1

uⁿ^s⁺^k^r,j⁺^j⁻¹ m

i=1,ij

(ns+kr,i+i−1)!

⎛⎜⎜⎜⎜⎜

⎜⎝1−e⁻^u

n_s+k_r,i+i−1

a=0

u^a a!

⎞⎟⎟⎟⎟⎟

⎟⎠. (19) The general analytic expression (19) gives the PDF for an arbitrary (nT,nR) couple. For example, restricting to (2,nR) and (nT,2) systems and using (19) withm = 2, k1=[0,1],ε(ρ1)=1 andε(ρ2)=−1 lead to the same result as the one reported in [2]:

p_λ_max(u)= (ns+2)e⁻^uuⁿ^s ns!

⎛⎜⎜⎜⎜⎜

⎜⎝1−e⁻^u

n_s+2 k=0

u^k k!

⎞⎟⎟⎟⎟⎟

⎟⎠

+e⁻^uuⁿ^s⁺² (ns+1)!

⎛⎜⎜⎜⎜⎜

⎝1−e⁻^u

ns

k=0

u^k k!

⎞⎟⎟⎟⎟⎟

⎠

−2e⁻^uuⁿ^s⁺¹ ns!

⎛⎜⎜⎜⎜⎜

⎜⎝1−e⁻^u

ns+1 k=0

u^k k!

⎞⎟⎟⎟⎟⎟

⎟⎠. (20)

For the particular case wherens =0, each term of (20) can be calculated to lead to:

p_λ_max(u)=2e⁻^u−2e^−2u−2e^−2u−u²e^−2u +u²e⁻^u−u²e^−2u

−2ue⁻^u+2ue^−2u+2u²e^−2u. (21) After simplification and factorization by e⁻^u and e^−2u, p_λ_max(u) for the (2,2) MIMO system can be finally expressed as:

p_λ_max(u)=e⁻^u

2−2u+u²

−2e⁻^2u. (22) This example can be generalized and the major interest of (19) is that it can be rearranged as:

p_λ_max(u)= m n=1

φn(u)e⁻^nu (23)

Table 1 c_n,icoeﬃcients fromφn(x).

(nT,nR) n x⁰ x¹ x² x³ x⁴ x⁵ x⁶ x⁷ x⁸ x⁹ x¹⁰ x¹¹ x¹²

(2,2) 1 2 −2 1 2 −2

(2,3) 1 3 −2 1/2

2 −3 −1

(2,4) 1 2 −1 1/6

2 −2 −1 −1/6

(2,6) 1 1/4 −1/12 1/120

2 −1/4 −1/6 −1/20 −1/120 −1/1440

(2,8) 1 1/90 −1/360 1/5040

2 −1/90 −1/120 −1/336 −1/1512 −1/10080 −1/100800 −1/1814400

(3,3) 1 3 −6 6 −2 1/4

2 −6 6 −3 −1 −1/2

3 3

(4,4) 1 4 −12 18 −34/3 7/2 −1/2 1/36

2 −12 24 −24 8/3 −4/3 4/3 −4/9 1/18 −1/72 3 12 −12 6 14/3 23/6 4/6 1/12

4 −4

(5,5) 1 5 −20 40 −110/3 215/12 −29/6 13/18 −1/18 1/576

2 −20 60 −90 40 −45/4 33/4 −139/24 17/8 −59/96 115/864 −11/576 1/576 −1/10368 3 30 −60 60 10 55/4 −7 2/3 1/6 9/64 35/864 11/864 1/864 1/6912 4 −20 20 −10 −40/3 −185/12 −77/12 −103/72 −11/72 −1/144

5 5

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whereφn(u) is thenth polynomial multiplied by the expo- nentiale⁻^nu. The polynomial is defined byφn(u)=cn,0u⁰+ cn,1u¹+. . .+cn,Dnu^Dⁿ whereDn is the maximum degree.

Then, (19) shows that the PDF ofλmaxfor an arbitrary number ofnT transmit andnRreceive antennas can be written in a polynomial form multiplied by an exponential. Thus, application of the method proposed here allows one to getcn,i

simply by calculating, at first, Eq. (19) and then by summing its coeﬃcients, which can be done with a symbolic program.

One should note that to getφn(u), its calculation requires a two-step process so as to:

• at first, calculate all the sums and products in (19),

• and secondly, simplify thee⁻^nu-factorized polynomials.

The coeﬃcients of φn(u) are given in Table 1 for diﬀerent (nT,nR) MIMO systems.

Equation (23) was previously described by Wennstr¨om et al. [13]. It is worth noting that the lack of explicit expression forcn,idrove Dighe et al. [14] to use a curve fitting approach.

The PDF p_λ_max(u) will be validated in Sect. 3.2 by sim- ulations.

3. Application to the Max-SNR Error Probability

3.1 Expression of Error Probability

Though the polynomial form (23) was already used to successfully provide the max-SNR BEP [14] or the outage capacity [6], it sounded to us worth extending, hereafter, the max-SNR BPSK case [14, Eq. (29)] toM-QAM andM-PSK in a Rayleigh fading channel by using two modulation co- efficients,αM andβM. The coefficient,αM, corresponds to the average number of neighbors separated byd_minand di- vided by log₂(M);dminis the minimum distance of the transmit constellation. The coefficient,βM, is linked tod_minwith d²_min/4=βMP0. Moreover, in Sect. 3.2, we will show from case-studies how PDF and, thus, performances are both af- fected by any change in the total number of antennas and in their distribution between the transmitter and the receiver sides.

The max-SNR average BEP is given by:

P¯e= _∞

0

αMerfc

⎛⎜⎜⎜⎜⎜

⎝

βM

uP0

σ²

⎞⎟⎟⎟⎟⎟

⎠p_λ_max(u)du (24) where αM = 1/2, βM = 1 for BPSK and αM = 2/log₂(M)

1− ^√¹_M

,βM=_2(M³₋₁₎forM-ary squared QAM andαM = 1/(log₂(M)),βM = sin²(π/M) forM-PSK. Ex- cept for the BPSK case, (24) is an approximation of the ex- act average BEP. However, it is worth recalling that (24) is a commonly used approximation and gives a very tight upper bound at high SNR [15, Sect. 8.1.1].

The average BEP expressed with (23) and (24) is directly obtained by extension of the MRC case [16, chap.7].

The final analytical expression is:

Fig. 1 p_λ_max(u) for constant total number of antennas 6, 8 and 10.

P¯e=αM

⎡⎢⎢⎢⎢⎢

⎢⎢⎢⎣1− m n=1

"

# _β_M_P₀

σ² βMP₀

σ² +nϕn

βMP0

σ² ⎤

⎥⎥⎥⎥⎥

⎥⎥⎥⎦ (25) whereϕn(x) is a rational polynomial given by:

ϕn(x)=

Dn

i=0

bn,i

⎛⎜⎜⎜⎜⎜

⎜⎜⎜⎝1+ i k=1

(2k−1)!!

(2k)!!

1+^x_nk

⎞⎟⎟⎟⎟⎟

⎟⎟⎟⎠. (26)

with bn,i = ^c_n^n,ii+1^i!, (2k−1)!! = 1×3×. . .×(2k−1) = (2k)!/(k! 2^k) and (2k)!!=2×4×. . .×(2k)=k! 2^k.

Equation (25) forαM=1/2 andβM=1 is equivalent to the BPSK closed-form presented in [13], [14]; it is useful to compare the BEP performances in theM-QAM andM-PSK cases [10].

3.2 Results

In order to validate the theoretical expression of p_λ_max(u), several PDFs were estimated with 100,000 matricesHwith iid complex Gaussian entries (i.e. CN(0,1) entries). Fig- ure 1 shows that theoretical and simulated curves of PDF fit perfectly for (2,4) and (4,4) systems. The good adequation with (19) was confirmed by other and numerous comparisons, not plotted here for the sake of clarity.

In other respects, Fig. 1 compares the theoretical PDFs corresponding to balanced and unbalanced distribution of various set of antennas and shows, for balanced distributions of antennas, a shift on the right side corresponding to higher values of λmax, and thus to a higher receiver SNR.

It also highlights that, the gap between the curves obtained with (2,8) and (5,5) antennas (sets of ten antennas) is larger than the one between the curves corresponding to (2,4) and (3,3) antennas (sets of 6 antennas). In conclusion, the distribution of antennas between transmitter and receiver must be balanced for the max-SNR solution to obtain the best performance in term of SNR. Otherwise, the max-SNR benefits of the number of uncorrelated channel gains oﬀer by large

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Fig. 2 4-QAM max-SNR BEP comparisons for constant total number of antennas 6, 8 and 10.

MIMO systems.

The average BEP in (24) depends onp_λ_max(u). Figure 2 depicts the max-SNR BEP for the same antenna configura- tions as in Fig. 1. The curves are plotted by using (25) for a 4-QAM:αM = 1/2, βM = 1/2. This figure evidences that the gap between (2,4) and (3,3) or (2,8) and (5,5) of the PDF curves remains true for the BEP curves. As demon- strated above a balance distribution of antennas between the transmitter and receiver sides must be privileged for performance enhancement. Note that, with 16 antennas, the performance gain for the (8,8) MIMO system compared to the (5,5) MIMO system is about 4 dB at 10⁻⁵.

4. Conclusion

We investigated the largest eigenvalue PDF of the Wishart matrix in the case of independent and uncorrelated Rayleigh fading channel to propose a new method to express it in an analytic closed-form by using the determinant definition of Vandermonde’s matrices and its decomposition of sums and permutations. The max-SNR performances were ex- tended to M-QAM and M-PSK modulations by using the polynomial multiplied by an exponential representation of the largest eigenvalue PDF. The BEP of the max-SNR system was improved by setting as many as possible antennas while privileging a balanced distribution between the transmitter and receiver sides.

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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 Uni- versity of Brest (France). Since 2005 he has been Assistant Professor and works as member of the Lab-STICC Laboratory (UMR CNRS 3192). His research interests are focused on MIMO systems.

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Baptiste Vrigneau was born at Lannion (France) in 1979. He received the M.Sc. degrees from ´Ecole Normale Sup´erieure of Cachan, France, in 2001, and the Ph.D. degree from Uni- versity of Brest, France, in 2006. He is a member of the Lab-STICC Laboratory (UMR CNRS 3192), France, and his research interest are focused on MIMO systems.

Philippe Rostaing received the Ph.D. degree 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 Lab-STICC Laboratory (UMR CNRS 3192).

His present research interests are in signal processing for digital communications with empla- sis on MIMO systems.

Gilles Burel was born in 1964. He received the M.Sc. degree from Ecole Supérieure d’Electricité, 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 Director of the Doctoral School SICMA and Associate Director of the Lab-STICC Laboratory (UMR CNRS 3192).

He is the author of 19 patents, one book and more than one hundred sci- entific papers. His present research interests are in signal processing for digital communications with emphasis on MIMO systems and interception of communications.