Optimal diversity vs multiplexing tradeoff for frequency selective mimo channels

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OPTIMAL DIVERSITY VS MULTIPLEXING TRADEOFF FOR FREQUENCY SELECTIVE MIMO CHANNELS

Abdelkader Medles Bell Labs, Lucent Technologies

The Quadrant, Stonehill Green Swindon, SN5 7DJ, UK Email: medles@lucent.com

Dirk T.M. Slock Eurecom Institute 2229 route des Crˆetes, B.P. 193 06904 Sophia Antipolis Cedex, FRANCE

Email: slock@eurecom.fr

ABSTRACT

In this paper we derive the optimal diversity versus multiplexing tradeoff for a frequency selective i.i.d. Rayleigh MIMO channel.

This tradeoff is shown to be better than the one of the frequency flat MIMO channel. Traditional approaches for frequency selective channels use OFDM techniques in order to exploit the diversity gain due to frequency selectivity. We show that although coding in OFDM over a sufficient subset of subcarriers allows to exploit full diversity as such (at fixed rate), such an approach leads to a suboptimal diversity vs multiplexing tradeoff.

1. INTRODUCTION

Frequency selective channels, also known as inter-symbol interfer- ence (ISI) channels, are regarded in general as troublesome channels from a performance point of view and practical systems assume the use of the OFDM approach to avoid this type of channels.

However, recent studies [1, 2] show that SISO/SIMO frequency selective channels improve performance and allow to achieve better diversity versus multiplexing gain compared to frequency-flat channels. In this paper, we generalize the optimal diversity versus multiplexing tradeoff (introduced first for flat MIMO channels in [3]) to frequency selective MIMO channels. The tradeoff obtained provides at any rate more diversity gain than in the case of flat channels. Furthermore, this tradeoff is better than the one obtained via coding over a set of linearly independent or even un- correlated subcarriers of an OFDM system. Such a result is very surprising and suggests to stay in the time domain or code across more subcarriers for more efficient diversity exploitation.

Consider linear digital modulation over a linear channel with additive white circular complex Gaussian noise with varianceσ²_v. The number of antennas isN_tat the transmitter side andN_r at the receiver side. The channel is frequency selective, with a delay spreadL. The complex received signal at antennaris

y^r_k=

L−1 l=0

Nt

t=1

H^rt_l x^t_k−l+v^r_k (1)

wherex^t_kis the transmitted signal at antennatat timek,H^rt_l , l= 0, . . . , L−1are the complex coefficients of the channel impulse response from antennatto antennar, andv^r_kis the noise at antenna r. Collecting the received signals in one vector symbol

The Eurecom Institute’s research is partially supported by its industrial members: Bouygues Télécom, Fondation d’entreprise Groupe Cegetel, Fondation Hasler, France Télécom , Hitachi, ST Microelectronics, Swiss- com, Texas Instruments, Thales.

y_k= [y¹_k, . . . ,y^N_k^r]^T, we can then write y_k=

L−1 l=0

H_lx_k−l+v_k (2)

wherex_k = [x¹_k, . . . ,x^N_k^t]^T,H_l : N_r×N_t, l = 0, . . . , L−1 are the matricial coefficients of the channel impulse response, and v_k = [v¹_k, . . . ,v^N_k^r]^T,v_k ∼ CN(0, σ²_vI_N_r). Superscripts ^T, ^H denote transpose and Hermitian transpose respectively.

1.1. Block (Frame) Transmission

We consider a transmission over a large block of lengthT >> L, the channel is assumed to be constant over the block, and varying independently from one block to another. The received signal can be written as

Y=AT(H)X+V, (3) whereY= [y^T₁,y^T₂, . . . ,y^T_T+L−1]^T,X= [x^T₁,x^T₂, . . . ,x^T_T]^Tand V= [v^T₁,v^T₂, . . . ,v^T_T+L−1]^T. AT(H)is the followingN_r(T + L−1)×N_tT block Toeplitz matrix

AT(H) =

H₀ 0 . . . 0

H₁ H₀ . .. ... ... H₁ . .. 0 H_L−1 ... . .. H₀

0 H_L−1 ... H₁ ... . .. . .. ...

0 . . . 0 H_L−1

. (4)

For a general input signal the mutual information per symbol period, is

I_T(H) = 1

T+L−1I(X;Y|H). (5) LetRXX = E XX^H be the input correlation. The power constraint is given by trRXX ≤ T P, whereP is the maximum transmit power per symbol period. For a given color of the transmitted signal (RXX), the mutual information per symbol period is maximized for a Gaussian centered input, for which it is given by I_T(H) = 1

T+L−1ln det(I_N_r_(T+L−1)+ 1

σ_v²AT(H)RXXA^H_T(H)). (6) Stationary Input

For large block lengthT >> L, we consider the case of stationary input. We can then define the second order statistics of the input

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S_i= Ex_kx^H_k−i. The power spectral density function of the input is thenS(z) = _iS_iz⁻ⁱ(S(z)is the z-transform ofS_i).RXXis now a block Toeplitz matrix ofS_i:

RXX=

S₀ S₁ . . . S_L−2 S_L−1 S^H₁ S₀ . .. . .. S_L−2

... . .. . .. . .. ... S^H_L−2 . .. . .. S₀ S₁ S^H_L−1 S^H_L−2 . . . S^H₁ S₀

. (7)

Frequency Selective Block Fading Rayleigh Model

We construct this model as a generalization of the flat Rayleigh MIMO channel model.H_l, l= 0, . . . , L−1are i.i.d, and eachH_l has i.i.d. Gaussian componentsH^rt_l ∼ CN(0,_L¹)for1≤r≤N_r and1≤t≤N_t.

1.2. Continuous transmission (T→ ∞)

The mutual informationI(H)in this case corresponds to the limit ofI_T(H), and is given in the following lemma.

Lemma 1:The limit ofI_T(H),I(H) = lim

T→+∞I_T(H), is I(H) = 1

2πj

dz

z ln det(I+ 1

σ²_vH(z)S(z)H^†(z)). (8) Proof:This is a generalization of the SISO case shown in [4], the proof is similar.

1.3. Bounds on the mutual information (MI) for white input LetJ_T(H)denote the MI for spatiotemporally white input,

J_T(H) = 1

T+L−1ln det(I_N_r_(T+L−1)+ρAT(H)A^H_T(H)). (9) whereρ = σ²_x

σ²_v = ^P

Ntσ_v². For large block lengthT,J_T(H)converges to

J(H) = 1 2πj

dz

z ln det(I_N_r+ρH(z)H^†(z)). (10) For finite block length, the following lemma gives useful bounds onJ_T(H)in terms of the infinite block length mutual information (J(H)).

Lemma 2 :For white input, the mutual information of a block of lengthT is bounded by

(1−_T+L−1^L−1 )J(H)≤J_T(H)≤J(H). (11) Proof :The proof is omitted for lack of space.

2. DIVERSITY AND MULTIPLEXING FOR FREQUENCY SELECTIVE MIMO CHANNELS

In [3], Zheng and Tse give a new definition of the diversity and spatial multiplexing that considers adaptive SNR schemes. For the generalization of the tradeoff to frequency selective MIMO channel we consider the same definitions. SchemeC(ρ)is a family of codes of block lengthT(one for each SNR level), that supports a bit rateR(ρ). This scheme is to achievespatial multiplexingrand diversity gaindif the data rate satisfies

ρ→∞lim R(ρ)

ln(ρ) =r , (12)

and the average error probability satisfies

ρ→∞lim lnP_e(ρ)

ln(ρ) =−d . (13)

For eachr,d^∗(r)is defined to be the supremum of the diversity advantage achieved over schemes. The maximal diversity gain is defined byd^∗_max =d^∗(0)and the maximal spatial multiplexing gain isr^∗_max=sup{r:d^∗(r)>0}.

We will use the special symbol .

=to denote the exponential equality,i.e., we writef(ρ) .

=ρ^bto denote

ρ→∞lim lnf(ρ)

ln(ρ) =b . (14)

Zheng and Tse considered aflat Rayleigh MIMO channel, and using properties of the distribution of eigenvalues they showed the following results.

Optimal Tradeoff Curve for a flat MIMO channel : Assume T ≥ N_r +N_t−1. The optimal tradeoff curved^∗(r)is given by the piecewise-linear function connecting the points(k, d^∗(k)), k= 0,1, . . . , p, where

d^∗(k) = (p−k)(q−k). (15) withp= min{N_r, N_t}, q= max{N_r, N_t}. In particulard^∗_max= N_tN_randr_max^∗ = min(N_r, N_t).

2.1. Behavior of the diversity vs multiplexing optimal tradeoff for a frequency selective Rayleigh MIMO channel

In the case of a frequency selective channel the optimal tradeoff curved^∗(r)has a behavior characterized by the following theorem.

(2,(LNr−2)(Nt−2)) (r,(LNr−r)(Nt−r))

(Nt,0) AsymptoticDiversityGain:dout(r)

Spatial Multiplexing Gain: r (1,(LNr−1)(Nt−1)) (0, LNrNt)

Fig. 1. Asymptotic diversity vs. multiplexing tradeoff for fre- quency selective channel andN_t≤N_r

Theorem 1:AssumeT ≥Lq+p−1. The optimal tradeoff curve is characterized by a diversity advantage functiond^∗(r)which is bounded as follows:

d_out((1 +L−1

T )r)≤d^∗(r)≤d_out(r). (16) d_out(r)is the asymptotic (inT) diversity advantage given by the piecewise-linear function connecting the points(k, d_out(k)),k=

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0,1, . . . , p, and

d_out(k) = (p−k)(Lq−k). (17) Independently of the block length the maximum achievable diversity advantage isd^∗_max =d^∗(0) =LN_rN_t. However, the maximal spatial multiplexing gainr_max^∗ depends on the block length T, and is bounded by(1−_T+L−1^L−1 )p≤r_max^∗ ≤p. Then asymp- toticallyr_max^∗ =p.

Remarks :

• This theorem defines a region in the plane of diversity vs multiplexing, whered^∗(r)is situated. However, for large block lengthT >> L, this region is restricted to the curve d_out(r),d^∗(r) =d_out(r).

• ForT >> Landr=k, k= 0,1, . . . , p,(p−k)(Lq−k)−

(p−k)(q−k) = (p−k)(L−1)qis the supplementary diversity gain provided by a frequency selective channel w.r.t.

to a flat one with the same number of antennas.

• OFDM systems assume in general coding of the transmitted signal overLindependent subcarriers. This is equivalent to coding overLindependent blocks of a MIMO flat block fading channel. The optimal diversity gain is then [3] the piecewise-linear function connecting the points(k, L(p− k)(q−k)),k= 0,1, . . . , p. The difference(p−k)(Lq− k)−L(p−k)(q−k) = (p−k)(L−1)k, is the supplementary diversity gain provided by the frequency selective channel, and can be attained by exploiting the diversity in the time domain (TDMA) or by coding across more subcarriers. The exact number of needed subcarriers is left here as an open problem . Note that coding acrossLindependent subcarriers in an OFDM system allows to attain the unre- ducedd^∗_max =LN_tN_r, which motivated the introduction of coding over such a reduced set of subcarriers.

In the following we present the proof of Theorem 1.

2.2. Preliminary Results

Bounds onI(H)for general input color

For a general input spectrum, under the power constraint ( ¹

2πj dz

z tr{S(z)} ≤P), the mutual informationI(H)for infinite block length is given in (8). The following lemma gives an upper bound onI(H).

Lemma 3: The mutual information for infinite block length is bounded as follows

I(H)≤ln det(I+ρ LN_tH¯H¯^H), (18)

whereH¯ =

H₀ ... H_L−1

= [H^T₀, . . . ,H^T_L−1]^T in the caseN_t ≤ N_r, andH¯= [H₀,H₁, . . . ,H_L−1]forN_t≥N_r(for both casesH¯ has rankp= min{N_r, N_t}).

Proof : The proof is omitted for lack of space.

We introduce now a paraunitary precoder for white input (S(z) = σ_x²I_N_t) that will allow us to characterizeJ(H)in (10).

Introduction of the STS precoder:

The STS precoderT(z)was introduced in [5].T(z)is aN_t×N_t MIMO filter

T(z) = D(z)Q

D(z) = diag{1, z^−L, . . . , z^−L(N^t⁻¹⁾}, (19)

whereQis a unitary matrix (Q^HQ=I)

Q^s=√1 N_t

1 θ₁ . . . θ₁^N^t⁻¹ 1 θ₂ . . . θ₂^N^t⁻¹

... ... ... 1 θ_N_t . . . θ_N_t^N^t⁻¹

, (20)

where theθ_iare the roots ofθ^N^t−j = 0, j =√

−1. T(z)is paraunitury (T(z)T^†(z) = I_N_t), and so isT(z^L), hence using it as a precoder does not modify the mutual information for white Gaussian input

J(H) = _2πj¹ ^dz_z ln det(I_N_tx+ρH^†(z)H(z))

= _2πj¹ ^dz_z ln det(I_N_tx+ρT^†(z)H^†(z)H(z)T(z))

= _2πj¹ ^dz_z ln det(ρ R(z))

= _2πj¹ ^dz_z ln det(ρL^†(z)ΣL(z))

= ln det(ρΣ) =

Ntx

n=1

J_n,

(21) whereJ_n= ln Σnnand in the fourth equality we replacedR(z) =

1

ρI_N_tx+T^†(z)H^†(z)H(z)T(z)by its UDL matrix spectral factorization.L(z)is lower triangular, with diagonal elements,L_ii(z), i= 1, . . . , N_t, causal, monic¹and minimum phase². The lower triangular elementsL_ij(z), i > j, are arbitrary (non causal) trans- fer functions. W.r.t. a classical causal MIMO spectral factor, the degrees of freedom of the strictly causal upper triangular elements have been transferred to the anticausal part of the lower triangular elements. In the fifth equality we exploit the fact that det(L(z)) =^N_n=1^t L_nn(z)is causal, monic and minimum phase.

Let us denoteV_n = [e_n,e_n+1, . . . ,e_N_t], wheree_nis theN_t×1 vector with 1 at then^thposition and all other entries are zeros.

By identification from the UDL spectral factorization of R(z), i.e. L(z)^†ΣL(z) = R(z), we show that([V^H_nR(z)V_n]⁻¹)11 =

1

ΣnnLnn(z)L^†_nn(z), where the diagonal coefficientL_nn(z)is causal, monic and minimum phase, hence it satisfies

1 2πj

dz

z ln(L_nn(z)L^†_nn(z)) = 0³. Finally J_n= lnρΣnn=− 1

2πj dz

z ln([V^H_nρR(z)Vn]⁻¹)11. (22) Bounds onJ(H):

SIMO case Lemma 4:

For this case the mutual informationJ(H)is bounded by ln(1+ρ||h||²

γ_L )≤ 1 2πj

dz

z ln(1+ρh^†(z)h(z))≤ln

1+ρ||h||² (23) whereh=

h₀ ... h_L−1

represents the vectorized channel impulse response andγ_L=

L−1

l=0

L−1 l

2

.

1A monic SISO filter has first coefficient equal to 1

2A rationalf(z)is said to be minimum phase if all of its poles and zeros are inside the unit circle

3 1 2πj dz

z ln(f(z)f(z)^†) = 0forf(z)causal, monic and minimum phase

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MIMO case Let us introduce first

c_n= (q−n) ln

γqL(q−n+1) q−n

,1≤n≤q−1

0 , n=q (24)

Lemma 5:

ForN_t≥N_r,J_nis bounded by

−ln(N_tLγ_N_t_L)−c_n≤J_n−ln(1 +ρL s_N_t_(L−1)+n)≤c_n, (25) wheres_n, n= 1, . . . , N_tL,are the eigenvalues ofH¯^HH¯sorted in a nondecreasing order.

Lemma 6:

Using the results of lemma 5, we conclude for anyN_t,N_r,

−qln(qLγ_qL)−c≤J(H)−ln det(I+ρ LH¯H¯^H)≤c (26) wherec=^q_n=1c_n.

Proof : ForN_t ≥ N_r, the derivation is straightforward from Lemma 5, in fact forN_t ≥ N_rthere are onlyp=N_r non-zero eigenvalues inH¯^HH:¯ s_N_t_L−N_r₊₁, . . . , s_N_t_L. This ensures that

Nt

n=1ln(1 +ρLs_N_t_(L−1)+n) = ln det(I+ρ LH¯H¯^H). For the caseN_t ≤N_rwe observe that the instantaneous mutual information satisfies

J(H) = _2πj¹

dz

z ln det(I+ρH(z)H^†(z))

= _2πj¹

dz

z ln det(I+ρH(z)H^H(z))

= _2πj¹

dz

z ln det((I+ρH(z)H^H(z))^T)

= _2πj¹

dz

z ln det(I+ρH^T(z)H^∗(z)).

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In the second equality we replaced^†by^Hbecause the integral is done on the unit circle, the two operators are then equivalent. We conclude that the capacity is the same as for a new virtual channel H^T(z). As a consequence the result of the first case,N_t ≥N_r, holds also here after interchangingN_randN_t, and replacingH_i byH^T_i fori= 0, . . . , L−1.

2.3. Outage Probability Behavior

An outage event occurs when the mutual information of the channel does not support a target data rate

{H:I_T(H) = 1

T+L−1I(X;Y|H)< R}. (28) The mutual information depends on the input distribution and the channel realization. Since the mutual information is maximized for Gaussian inputs, we can then define the outage probability as

P_out(R) = min

RXX^≥0,tr{RXX^{}≤T P}P(I_T(H)< R). (29) Any covariance matrixRXXin the set{RXX ≥0,tr{RXX} ≤ T P}satisfiesRXX≤T PI_{T N}_t, whereI_{T N}_tis the identity matrix of sizeT N_t×T N_t.

From (6) we can then write that I_T(H)≤ 1

T +L−1ln det(I+N_tT ρAT(H)A^H_T(H)). (30)

The outage probability then satisfies

P[_T+L−1¹ ln det(I+N_tT ρAT(H)A^H_T(H))< R]

≤P_out(R)≤P[_T+L−1¹ ln det(I+ρAT(H)A^H_T(H))< R], (31) where for the upper bound we have just chosenRXX=_N^P

tI_{T N}_t. As has been observed in [3], the two constant scalings (N_tT and 1) ofρlead to the same exponential equality, hence

P_out(R) .

=P[ 1

T+L−1ln det(I+ρAT(H)(AT(H))^H)< R]. (32) Now, using the results of Lemma 2, namely (11), we get

P[J(H)< R]≤^· P_out(R)≤^· P[(1− L−1

T +L−1)J(H)< R]. (33) Finally, by the exploitation of Lemma 6, or hence (26), we conclude

P[ln det(I+ρH¯H¯^H)< R]≤^· P_out(R)≤^·

P[(1−_T+L−1^L−1 ) ln det(I+ρH¯H¯^H)< R]. (34) Remarks :

• The scale factorN_tT in (30) is very loose for large block length (T >> L), and no more valid forT → ∞. How- ever, for such cases practical signals are stationary and ad- mit a spectrumS(z),I_T(H)then converges to the mutual information of infinite block lengthI(H)(8). We can then use the upper bound of (18) which leads directly to the lower bound in (34). By consequence the reasoning is still valid for infinite block lengthT, with the upper bound in (34) remaining the same.

• For large block length (T >> L), (34) becomesP_out(R) .

= P[ln det(I+ρH¯H¯^H)< R]. The frequency selective channel behaves as the flat MIMO channelH¯= [H^T₀, . . . ,H^T_L−1]^T in the caseN_t ≤ N_r and H¯ = [H₀,H₁, . . . ,H_L−1]for N_t≥N_r.

Let the data rateR=rlnρ. In [3] it was shown that for aN_r×N_t flat Rayleigh distributed MIMO channel

P[ln det(I+ρHH^H)< rlnρ] .

=ρ^−d^out^(r), (35) whered_out(r)is given by the piecewise-linear function connecting the points(k, d_out(k)),k= 0,1, . . . , p, with

d_out(k) = (p−k)(q−k). (36) Recall thatp= min{N_r, N_t}, q= max{N_r, N_t}.

In our case,H¯ has the same probability distribution as that of a flat Rayleigh distributed MIMO channel of sizeN_rL×N_t for N_t≤N_r, andN_r×N_tLforN_t ≥N_r. These two cases lead to the same result

P[ln det(I+ρH¯H¯^H)< rlnρ] .

=ρ^−d^out^(r), (37) where nowd_out(r)is given by the piecewise-linear function connecting the points(k, d_out(k)),k= 0,1, . . . , p, with

d_out(k) = (p−k)(Lq−k). (38) Using this results it is straightforward to derive the following theorem.

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Theorem 2: For a frequency selective and i.i.d. Rayleigh distributed MIMO channel, the outage probability satisfies

ρ^−d^out^(r)≤^· P_out(R)≤^· ρ^−d^out⁽⁽¹⁺^L−1^T ^)r), (39) whered_out(r)is given by the piecewise-linear function connecting the points(k, d_out(k)),k= 0,1, . . . , p, and

d_out(k) = (p−k)(Lq−k). (40) Remark :From (39) we can see that for large blockT >> L, the outage probability satisfiesP_out(R) .

=ρ^−d^out^(r). 2.4. Proof of Theorem 1

In order to prove Theorem 1, we characterize the error probability of any coding scheme (for a given rateR = rlnρ) by P_e(ρ) = ρ^−d(r). We recall that d^∗(r) is the supremum of the diversity advantaged(r)achieved over all these schemes.

To show the upper bound in (16), we can use the same proof of lemma 5 in [3]. This will lead to the following result

P_e(ρ)≥^· ρ^−d^out^(r). (41) Or in other terms, for any schemed(r) ≤ d_out(r). Hence, the supremum verifies

d^∗(r)≤d_out(r). (42) Proof of the lower bound in Theorem 1 :We need now to prove thatd_out((1 +^L−1_T )r) ≤d^∗(r), forT ≥Lq+p−1. To show this result we provide an upper bound on the error probability by choosing the input to be a random code from an i.i.d. Gaussian ensemble.

The proof is very similar to the one provided in [3] for the proof of theorem 2 of that paper. As has been done in [3], we can show that

P_e(ρ)≤P_out(R) +P(error, no outage). (43) Now using (39), we get

P_e(ρ)≤^· ρ^−d^out⁽⁽¹⁺^L−1^T ^)r)+P(error, no outage). (44) In the same way as in [3], we can show that for a given channel, the error probability is upper bounded by

P(error|H)≤ρ^{T r}det(I+ρ

2AT(H)A^H_T(H))⁻¹. (45) Using (11) and results of Lemma 6, we then show that

P(error|H) ≤ ρ^{T r}e^{−ln det(}I_Nr(T_+L−1)+^ρ₂AT(H)A^H_T(H))

≤ ρ^{T r}e^−T^2πj¹ ^dz^z ^{ln det(}I_Nr+^ρ₂H(z)H^†(z))

≤ e^{T c}¹ρ^{T r}det(I+^ρ₂LH¯H¯^H)^−T

(46) wherec¹ = pln(qLγ_qL) +c. This bound depends onH¯ only through thepnon zero eigenvalues ofH¯H¯^H, λ_i, i = 1, . . . , p sorted in a nondecreasing order. With the following change of variablesλ_iρ^−αⁱ, we then get

P(error|α)≤^· ρ^−T[

p

i=1(1−α_i)⁺−r]. (47) From the upper bound in (11) we can conclude that theno outage event is included in the event(A)^c, complement of the event(A) :

{J(H)< R}={α₁ ≥. . .≥α_p≥0,and ^p

i=1(1−α_i)⁺ <

r},

P(error, no outage)≤^·

(A)^c

p(α)ρ^−T[

p

i=1(1−αi)⁺−r]. (48) The remainder of the proof is the same as the one in [3]. For T≥Lq+p−1it leads to

P(error, no outage)≤^· ρ^−d^out^(r). (49) Finally

P_e(ρ) ≤^· ρ^−d^out⁽⁽¹⁺^L−1^T ^)r)+ρ^−d^out^(r)

≤· ρ^−d^out⁽⁽¹⁺^L−1^T ^)r),

(50) andd^∗(r)≥d_out((1 + ^L−1_T )r). This completes the proof.

3. OPTIMALITY OF THE MMSE ZF DFE EQUALIZER FOR SIMO CHANNELS

For a SIMO channel (N_t = 1) the optimal tradeoff curve is the linear functiond^∗(r) =LN_r(1−r),0≤r≤1. In [1] we show that the SNR at the output of the Decision Feedback Equalizer with a MMSE Zero-Forcing design verifies SNR_{DF E} ≥ ^ρh²

γL , h² =

L−1

i=0

h_i²2. We assume the use of a uniform QAM con- stellation of sizeρ^r(corresponding to a raterlnρ), and minimum distanced²= ³σ²_x

2(ρ^r−1).

The transmitted frame is of sizeT and no space time coding is used. The transmitted symbols are detected sequentially at the receiver using the MMSE ZF DFE, which reproduces the same sit- uation as the detection over a SISO flat channel with a SNR = SNR_{DF E}, and this at each step. P_e can then be upper-bounded byP_e ≤T P(E), whereP(E)is the detection error probability over the equivalent SISO flat channel. A point in the QAM con- stellation has at most four nearest neighbors, theP(E)can then be upper-bounded by four times the Pairwise Error Probability with minimum distance. Finally using the PEP expression in[1] we get

P_e≤4T(1 + 3

8c_L(ρ^r−1)ρ)^−LN^r .

=ρ^−LN^r^(1−r)=ρ^−d^∗^(r). (51) This shows the optimality of the MMSE ZF DFE equalizer for SIMO channels.

4. REFERENCES

[1] A. Medles and D.T.M. Slock. Decision-Feedback Equaliza- tion Achieves Full Diversity for Finite Delay Spread Chan- nels. InIEEE ISIT 2004, Chicago, Illinois, Jun-Jul 2004.

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