Sergey Loyka School of Information Technology and
Engineering (SITE), University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario,
Canada, K1N 6N5
E-mail: sergey.loyka@ieee.org

Francois Gagnon Department of Electrical Engineering Ecole de Technologie Superieure 1100, Notre-Dame St. West, Montreal Quebec, H3C 1K3, Canada E-mail: francois.gagnon@etsmtl.ca

An analytical technique for the outage and BER analysis of the nx2 V-BLAST algorithm with the optimal ordering has been presented in [1], including closed-form exact expressions for average BER and outage probabilities, and simple high-SNR approximations. The analysis in [1] is based on the following essential approximations:

1. The SNR was defined in terms of total after-projection signal and noise powers, and the BER was analyzed based on their ratio. This corresponds to a non-coherent (power-wise) equal-gain combining of both the signal and the noise, and it is not optimum since it does not provide the maximum output SNR.
2. The definition of the total after-projection noise power at each step ignored the fact that the after- projection noise vector had correlated components.
3. The after-combining noises at different steps (and hence the errors) were implicitly assumed to be independent of each other. Under non-coherent equal-gain combining, that is not the case. It turns out that the results in [1] hold also true without these approximations, subject to minor modifications only. The purpose of this note is to show this and also to extend the average BER results in [1] to the case of BPSK-modulated V-BLAST with more than two Rx antennas (eq. 18-20). Additionally, we emphasize that the block error rate is dominated by the first step BER at the high-SNR mode (eq. 14 and 21). 15-May-06 TWireless - Amendment 2(7) To accomplish the task, we employ the maximum ratio combining (MRC) weights, which also incorporate the orthogonal projection to eliminate interference from yet-to-be-detected symbols (i.e. zero- forcing (ZF) MRC weights, which maximize the SNR under the ZF constraint), and the expressions for the corresponding after-processing SNR. For such weights, the after-combining noises at different steps (and hence errors) are indeed independent of each other. Based on this, we demonstrate that the results in [1] apply to the case of the ZF-MRC processing with minor modifications only: the exact closed-form expressions for the average BER hold true up to a constant factor in terms of the average SNR. The standard baseband system model is given by [1, eq. 1 and 2]. The received signal after the interference cancellation at the i-th step i′ r is given by [1, eq. 3]. The inter-stream interference nulling (from yet-to-be-detected symbols) can be expressed as i i i i i i i q ′′ ′ = =

r P r P h P v

(1)

where

iP is the projection matrix onto the sub-space orthogonal to

{

}

1

2

,

,…

i

i

m

span

+

+

h

h

h

1 ( ) i i i i i + − +

− P Ι H H H H , where 1 2 [ … ] i i i m + +

H h h h and “+” denotes conjugate transpose. The first term in (1) represents the signal and the second one is the noise contribution. The analysis in [1] was based on the after-processing signal and noise powers defined as 2 si i P ⊥ = h (assuming that 2 1 iq

), where i i i ⊥= h Ph , and 2 2 0 ( ) vi i P n m i

= − + σ v P v , where v is the expectation over the noise. Consequently, the output SNR was obtained as 2 2 0 ( ) i si i vi P P n m i ⊥ ′ γ =

− + σ h

(2) and the outage and BER analysis was carried out in terms of i′ γ [1]. This definition of the output SNR corresponds to power-wise (non-coherent) combining with unit weights (i.e. equal gain) in terms of i′′ r , 15-May-06 TWireless - Amendment 3(7) which is not optimum. Additionally, the definition above ignores the fact that the after-projection noise in (1), i i ′ = v P v , has correlated components [2]: 2 2 0 0 ( ) i i i i + ′ ′

= σ ≠σ v R v v P I . Optimum Weights: To remove these approximations, we employ the optimum combining weights that maximize the output after-projection SNR in (1) and maintain orthogonality to 1 2 [ … ] i i i m + +

H h h h

(ZF-MRC weights) [2,3],

i i i ⊥ ⊥ = h w h

(3) Similar approach has been used before in the context of multiuser CDMA receivers [4,5]. Using the ZF- MRC weights in (4), the output SNR becomes [2,3], 2 2 0 i i ⊥ γ = σ h

(4) Comparing (4) to (2), one concludes that the optimum SNR has the same distribution (up to a constant factor) as the “power-wise” one, ( ) i i n m i ′ γ = − + γ . Hence, all the results in [1] in terms of the SNR distribution can be applied to the ZF-MRC processing with a minor modification, i.e. the substitution i i ′ γ →γ . Independence of the After-Combining Noises: Using the following propert

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