3.4 Non-Linear SDM Detection Methods
3.4.2 SIC Detection
10-4 10-3 10-2 10-1 100 0 5 10 15 20 BER SNR [dB] sdm-ofdm-mld : 28-Aug-2006 nr=4 mt=3 mt=4 mt=5 mt=6
Figure 3.5: Bit Error Rate performance exhibited by the SDM-QPSK-OFDM system employing an ML
SDM detector of Equation (3.24) and mt = 3, 4, 5 and6 transmit antennas, as well asnr = 4 receive
antennas. The abscissa represents the average SNR recorded at the receive antenna elements. The system parameters are summarized in Table 2.1.
we aim for exploring the performance of the ML SDM detector in the overloaded system scenario, where the number of transmit antenna elements exceeds that of the receiver elements and thus we havemt > nr. Figure 3.5 demonstrates the achievable BER performance of the SDM-OFDM system employing the ML SDM detector as a function of the average SNR recorded at the receive antenna elements. We can see that as opposed to the MMSE SDM detector discussed in Section 3.4.1, the ML SDM detector exhibits a good performance both when we havemt≤ nr, as well as in the overloaded system scenario, when the number of transmit antenna elements exceeds the number of the receive antenna elements, i. e. when we havemt >nr.
3.4.2 SIC Detection
The SIC-assisted SDM detector was proposed by Foschini et al. in [84] and it was discussed in further detail in [97, 98, 138–140].
In order to commence our discourse, let us recall the philosophy of the linear SDM detector discussed in Section 3.3, where the detection of the transmitted signal vector x[n,k] was performed using a linear transformation described by Equation (3.1), namely by
ˆ
x[n,k] =WH[n,k]y[n,k], (3.38)
3.4.2. SIC Detection 99
As it was further inferred in Section 3.3, the corresponding SINR at the output of the linear SDM detector may vary considerably across different elements of the transmitted signal vector x[n,k], as substantiated by Equation (3.5). Consequently, as suggested in [28], the overall MSE at the output of the linear SDM detector employed is dominated by the SINR associated with the transmitted signal component having the lowest signal power [28] determined by ∑j|Hij|2. This observation suggests that a considerably higher
performance can be achieved by employing successive interference cancellation.
Following the SIC paradigm, the detection of the transmitted signal vectorx[n,k]associated with thekth OFDM subcarrier of the nth OFDM symbol is performed in a successive manner, where at each detection iteration i we detect a single vector component xi[n,k] using the linear MMSE SDM detection method
discussed in Section 3.3.1. We then modify the received signal vectory[n,k]by removing the remodulated interfering signal components and repeat the aforementioned linear detection process in order to estimate the next transmitted signal component xji+1. The iterative process described above is then repeated until the transmitted signal components associated with all transmitter antenna elements are detected. In this section we will demonstrate that the successive structure of the detection process results in a substantially improved SIR for the weaker signal components. Note that in our forthcoming derivation we, once again, omit the OFDM symbol and subcarrier indicesnandk, which does not restrict the generality of the results obtained, since the space-devision detection process described is performed independently for each pair of time and frequency domain indices[n,k].
More specifically, we commence our SIC detection process with a linear detection of the transmitted signal component xj1, as suggested by Equation (3.1), where we have
ˆ
xj1 =w
H
1y1, (3.39)
and w1 = (W)j1 is the j1th column of the SDM MMSE detector’s weight matrix described by Equation (3.13), whiley1is assumed to be identical to the original received signal vectory.
In the next step, the interference imposed by the just detected and remodulated signal componentxj1 is subtracted from the received signaly1, yielding
y2= y1−(H)j1Q(xˆj1), (3.40)
where(H)j1 is thej1th column of the channel matrixH, whileQ(x)represents the slicing or hard-decision operation performed in the receiver in order to estimate the transmitted information-carrying QAM/PSK symbol. The resultant partially-decontaminated signaly2comprises the contributions of a reduced number of interferers. In order to detect our next desired transmitted signal componentxj2 we now have to calculate the updated linear SDM detector weight matrix W2, which may be readily achieved by substituting the effective channel matrixHj1, obtained by zeroing columnj1of the original channel matrixH, into Equation
3.4.2. SIC Detection 100
(3.13) yielding
W2 =Hj1(HHj1Hj1+mtσw2I)−1, (3.41)
where we follow the notation employed in [138] and correspondingly Hj
i denotes the matrix obtained by
zeroing columns j1,· · · ,ji of the original matrix H. By substituting the terms xˆj1,w1and y1of Equation (3.39) by the corresponding termsxˆj2,y2of Equation (3.40) andw2= (W)2of Equation (3.41), we arrive at the desired estimate of next transmitted signal component. Finally, the iterative detection process described above is repeated, until all desired transmitted signal components are successfully detected.
As it was argued in [138], the order in which the detection of the transmitted signal components
xj[n],j= 1,· · · ,mtis performed is important for the overall performance of the detection process. More- over, as it was demonstrated in [138], the optimal ordering arises if the “best first” successive detection strategy is applied, where the best possible performance is achieved, when at each iteration iof the SIC detection process the desired signal component is selected according to the selection criterion of
ji+1 =argmax
j k
(Hj
i)jk
2, (3.42)
implying that the least attenuated , i.e. the highest-power antenna’s signal is detected first.
The SDM SIC detection process employing the MMSE detection method of Section 3.3.1 is summarised in Algorithm 9.
3.4.2.1 Performance Analysis of the SIC SDM Detector
In this section we present our performance results for the SDM-OFDM system employing the SIC SDM detection scheme described in Section 3.4.2. The simulation setup is identical to that described in Section 3.3.1.2 and the corresponding simulation parameters are summarised in Table 2.1.
Figure 3.6 characterizes the ability of the SDM-OFDM system employing the SIC SDM detector of Algorithm 9 to exploit the available MIMO multiplexing gain in the fully loaded system configuration, when the number of the transmit antenna elements mt is equal to that of the receiver antenna elements
nr. More explicitly, Figure 3.6 depicts the achievable BER performance of the SDM-OFDM SIC system considered as a function of (a) the average SNR recorded at the receiver antenna elements, as well as (b) versus the corresponding Eb/N0 value for various numbers of mt = nr = 1,· · · , 6transmit and receive antenna elements.
More specifically, the results portrayed in Figure 3.6 illustrate on the SNR scale that the SDM-OFDM SIC system having mt = nr = 6transmit and receive antennas exhibits an SNR gain of about2dB at the target BER of10−3, when compared to the same system employing a single antenna element at both the transmitter and receiver.
3.4.3. Genetic Algorithm-Aided MMSE Detection 101