In the MMSE detection algorithm, the expected value of the mean square error between the transmitted vectorx and a linear combination of the received vectorwHris minimized
minE{(x−wHr)2} (6.23)
wherewis annR×nT matrix of linear combination coefficients given by [8]
wH =
HHH+σ2InT
−1
HH (6.24)
σ2is the noise variance and InT is an nT ×nT identity matrix. The decision statistics for the symbol sent from antennai at timet is obtained as
yti =wHi r (6.25)
wherewHi is theith row ofwH consisting ofnR components. The estimate of the symbol sent by antennai, denoted byxˆti, is obtained by making a hard decision onyti
ˆ
xit =q(yit) (6.26)
In an algorithm with interference suppression only, the detector calculates the hard decisions estimates by using (6.25) and (6.26) for all transmit antennas.
In a combined interference suppression and interference cancellation, the receiver starts from antennanT and computes its signal estimate by using (6.25) and (6.26). The received signalrin this level is denoted byrnT. For calculation of the next antenna signal(nT−1), the interference contribution of the hard estimatexˆtnT is subtracted from the received signal rnT and this modified received signal denoted byrnT−1is used in computing the decision statistics for antenna(nT −1)in Eq. (6.25) and its hard estimate from (6.26). In the next level, corresponding to antenna (nT −2), the interference fromnT −1 is subtracted from the received signalrnT−1and this signal is used to calculate the decision statistics in (6.25) for antenna(nT −2). This process continues for all other levels up to the first antenna.
After detection of leveli, the hard estimate xˆti is subtracted from the received signal to remove its interference contribution, giving the received signal for leveli−1
ri−1=ri− ˆxtihi (6.27)
wherehi is theith column in the channel matrixH, corresponding to the path attenuations from antennai. The operationxˆtihi in (6.27) replicates the interference contribution caused byxˆit in the received vector.ri−1is the received vector free from interference coming from
ˆ
xtnT,xˆtnT−1, . . . ,xˆit. For estimation of the next antenna signalxti−1, this signalri−1is used in (6.25) instead ofr. Finally, a deflated version of the channel matrix is calculated, denoted byHid−1, by deleting column i from Hid. The deflated matrix Hid−1 at the(nT −i+1)th cancellation step is given by
Hid−1=
h1,1 h1,2 ã ã ã h1,i−1 h2,1 h2,2 ã ã ã h2,i−1
... ... ... ... hnR,1 hnR,2 ã ã ã hnR,i−1
(6.28)
This deflation is needed as the interference associated with the current symbol has been removed. This deflated matrixHid−1is used in (6.24) or computing the MMSE coefficients and the signal estimate from antennai−1. Once the symbols from each antenna have been estimated, the receiver repeats the process on the vectorrt+1received at time(t+1). The summary of this algorithm is given below.
Summary of Linear MMSE Suppression and Successive Cancellation
Set i=nT
and rnT =r.
while i≥ 1 {
wH=[HHH + σ2InT]−1HH yit = wHiri
^
xti = q(yti)
ri−1 = ri − ^xithi
Compute Hid−1 by deleting column i from Hid. H=Hid−1
i=i−1 }
The receiver can be implemented without the interference cancellation step (6.27). This will reduce system performance but some computational cost can be saved. Using cancellation requires that MMSE coefficients be recalculated at each iteration, as H is deflated. With no cancellation, the MMSE coefficients are only computed once, asHremains unchanged.
The most computationally intensive operation in the detection algorithm is the computation of the MMSE coefficients. A direct calculation of the MMSE coefficients based on (6.24), has a complexity polynomial in the number of transmit antennas. However, on slow fading channels, it is possible to implement adaptive MMSE receivers with the complexity being linear in the number of transmit antennas.
The described algorithm is for uncoded LST systems. The same detector can be applied to coded systems. The receiver consists of the described MMSE interference suppressor/
LST Receivers 195
canceller followed by the decoder. The decision statistics,yti, from (6.25), is passed to the decoder which makes the decision on the symbol estimatexˆti.
The performance of a QR decomposition receiver (QR), the linear MMSE (LMMSE) detector (LMMSE) and the performance of the last detected layer in an MMSE detector with successive interference cancellation (MMSE-IC) are shown for a VBLAST structure withnT =4,nR=4 and BPSK modulation on a slow Rayleigh fading channel in Fig. 6.4.
Figure 6.4 also shows the interference free (single layer) BER which is given by [3]
Pb= 1
2(1−à)
nRk=nR−1 k=0
1 2(1+à)
k
(6.29)
whereà= γb
nR
1+nRγb andγb=ENbo.
Figure 6.4 V-BLAST example, nT = 4, nR = 4, with QR decomposition, MMSE interference suppression and MMSE interference suppression/successive cancellation
One of the disadvantages of the MMSE scheme with successive interference cancellation is that the first desired detected signal to be processed sees all the interference from the remaining(nT−1)signals, whereas each antenna signal to be processed later sees less and less interference as the cancellation progresses. This problem can be alleviated either by ordering the layers to be processed in the decreasing signal power or by assigning power to the transmitted signals according to the processing order. Another disadvantage of the successive scheme is that a delay of nT computation stages is required to carry out the cancellation process.
The complexity of the LST receiver can be further reduced by replacing the MMSE interference suppressor by a matched filter, resulting in interference cancellation only.
A laboratory prototype of a VLST system was constructed in Bell Laboratories [43]. The prototype operates at a carrier frequency of 1.9 GHz, uncoded 16-QAM modulation and a symbol rate of 24.3 k symbols/sec, in a bandwidth of 30 kHz with 8 transmit and 12 receive antennas. The system achieves a frame error rate of 10−2 at an SNR of 25 dB. The frame length is 100 symbols, 20 of which are used to estimate the channel in each frame, so that the efficiency within a frame is 80%. The ideal spectral efficiency is 25.9 bits/s/Hz, but if the bandwidth loss due to transmission of training sequences is included, the reduced spectral efficiency is 20.7 bits/s/Hz. This is much higher than the achievable spectral efficiency in the second generation of cellular mobile systems with a single element transmit/receive antenna.