MIMO Single Carrier Zero Padding Block Transmission Systems
3.3 Channel Equalization
Once the received data ¯x(i) = He¯s(i)+ ¯w(i) is available and the channel is identified, the minimum mean square error (MMSE) or zero forcing (ZF) equalization methods [13, 14] can be used to recover the modulated sources sk(n). For example, with an MMSE equalizer, Ge, we estimate ¯s(i) by ¯s(i) = Gex(i). Since the precoding scheme is applied at the transmitter,¯ we need to multiply the estimated ¯s(i) by P−1 to obtain an estimate of ¯v(i), where ¯v(i) = [v(iN )T v(iN + 1)T · · · v(iN + M − 1)T]T, and P = I M
L+1 ⊗ (diag[p(0), · · · , p(L)] ⊗ IK).
In other words, the estimated ¯v(i) can be obtained by
¯v(i) = P−1Gex(i).¯ (3.32)
From (3.32), we know the equalization performance is related to P−1 and Ge. Because Ge is formed from the estimated channel coefficients, we expect good channel identification to bring an accurate Ge and thus improves the equalization performance. Also we know using the optimal precoding sequence in (3.25), the identification performance improves as τ decreases. Hence using a small τ brings good channel estimation and improves the accuracy of Ge, which is expected to improves the equalization performance. However, using a small τ would make the diagonal gain p(k)−1 = √1
τ in P−1, k = 1, 2,· · · , L, becomes large, which results in large noise amplification at the receiver and hence is more likely to cause decision error. Therefore using a small τ would amplify the noise and the equalization performance deteriorates as τ decreases.
In summary, although decreasing τ improves the accuracy of Ge, it would cause an increased amplification of noise, and vice versa. Hence there is a trade-off on the selection of τ when channel equalization is performed. In the work of [15, 16, 28], this trade-off is also observed. We will give a simulation example to demonstrate this trade-off in the next section.
3.4 Simulation Results
In this section, we use several examples to demonstrate the performance of the proposed method. The channel NRMSE, SNR, and the number of Monte Carlo runs are the same as those given in Section 2.5. The source symbols are i.i.d. QPSK signals. The channel noise is zero mean, temporally and spatially white Gaussian.
1) Simulation 1 – optimal selection of precoding sequences
In this simulation, we use the model (2.33) to demonstrate the performance of the proposed method. The length of symbol blocks is M = 27, which is zero padded to blocks of length M + P = 30. It means P = 3(= L + 1) and transmission efficiency is 90%. In experiment 1, we use 5 precoding sequences which all satisfy (3.21) and (3.22) to illustrate the effect of the precoding sequences on the identification performance. The first sequence S0 are chosen based on (3.25) for τ = 0.6, i.e., S0 is chosen as {√
1.8√ 0.6√
0.6}.
The sequences S1, S2, SA, and SB are chosen as {√ 0.6√
1.8√
0.6}, {√ 0.6√
0.6√ 1.8}, {√
0.6√ 1.0√
1.4}, and {1 1 1} (i.e., no precoding), respectively. Figure 3.2 shows that for SNR=10 dB, the NRMSE decreases as the number of symbol blocks increases for every precoding sequence. As expected, the optimal precoding sequence S0 yields the smallest NRMSE.
In experiment 2, we use the precoding sequences that satisfy (3.25), but with different τ to test the effect of τ on the identification performance. Figure 3.3 shows that when the number of symbol blocks = 100, the NRMSE decreases as SNR increases and is roughly constant for SNR ≥ 20 dB for different τ. Figures 3.3 also shows that the identification performs better for smaller τ .
2) Simulation 2 – channel order overestimation
In this simulation, we use the channel model (2.33) with SNR = 10 dB, fix the number of symbol blocks at 300, and use the precoding sequence that satisfies (3.25) with τ = 0.6.
For each upper bound ˆL, 0≤ (ˆL−L) ≤ 6, we choose P = ˆL+1 and M = 9P for simulation such that the transmission efficiency is maintained at 90%. Figure 3.4 shows the NRMSE increases with increasing channel order overestimation for each τ . We see that periodic precoding improves robustness to channel order overestimation. For example, without precoding (τ = 1), the NRMSE increases about 6 dB for ( ˆL− L) = 3. With precoding, (τ = 0.4), the corresponding increase in NRMSE is about 1.5 dB.
3) Simulation 3 – a 3-input 2-output channel
In this simulation, we use the 3-input 2-output model (2.35) to illustrate the perfor-mance of the proposed method for channel with more transmitters than receivers. We use M = 27 and P = 3. In experiment 1, we use the same precoding sequences S0, S1, and S2 which are used in simulation 1. Figure 3.5 shows that for SNR=10 dB, the NRMSE de-creases as the number of symbol blocks inde-creases for each precoding sequence. The optimal precoding sequence S0 yields the smallest NRMSE.
In experiment 2, we use the precoding sequences that satisfy (3.25), but with different τ to test the effect of τ on the identification performance. Figure 3.6 shows that the channel NRMSE decreases as SNR increases for each τ and that the identification method performs better for smaller τ .
4) Simulation 4 – trade-off in selecting τ
In this simulation, we discuss the trade-off in selecting τ when channel equalization is performed. We use the MMSE equalizer [13, 14]. We generate 150 2-input 2-output complex random channels based on the IEEE 802.11a standard [37, p. 336]. The sampling frequency is 20 MHZ and the the delay spread is 35 nsec (for home environment). Thus the orders of the channels are L = 7. We use M =56 and P = L + 1 = 8 such that N = M + P = 64. The number of symbol blocks is 250. We use the optimal precoding sequences which satisfy (3.25) with various τ .
Figure 3.7 shows that the identification performs better for smaller τ . Figure 3.8(a) shows that for τ ∈ [0.1, 0.8], the bit error rate (BER) performance deteriorates as τ de-creases and the BER for τ = 0.7 and τ = 0.8 are very close. Figure 3.8(b) shows that for large τ , τ ≥ 0.8, the BER performance improves as τ decreases. Figure 3.8 shows that there is a trade-off between identification accuracy and noise amplification: a small τ means large noise amplification and an accurate channel estimate, and vice versa. For this example, it seems a τ between 0.7 and 0.8 is a good choice for BER performance.
5) Simulation 5 – comparison with the subspace method
In this simulation, we again generate 300 2-input 2-output channels based on IEEE 802.11a standard. We use the precoding sequences that satisfy (3.25) with τ = 0.8. We use Gray-coded QPSK and 16-QAM input symbols for simulation. We compare the iden-tification and MMSE equalization performances of the proposed method with those of the subspace method [27] for MIMO SC-ZP systems.
Figure 3.9(a) shows that when the number of symbol blocks is 200, the identification performance of the proposed method is better than that of the subspace method except SNR > 16 dB. The proposed method yields almost the same identification performance for QPSK and 16-QAM input symbols. Figure 3.9(b) shows that the equalization performance of the proposed method is better than that of the subspace method except SNR > 16 dB. Figure 3.9 shows that the identification and equalization performance of the proposed method is better than those of the subspace method for low to medium SNR. The subspace method gives smaller BER than the proposed method for SNR> 16 dB.
6) Simulation 6 – identification using different sizes of covariance matrices
In this simulation, we use the channel model (2.33) to compare the channel NRMSE when we use the first 3, 15, and 30 block rows of ¯x(i) to form the covariance matrices for identification. We use the precoding sequences that satisfy (3.25) with τ = 0.6 and fix the number of symbol blocks at 100. Figure 3.10 shows that when we use the more information of the received signal ¯x(i), the identification performance improves. However, as we indicate at the end in Section 3.2.1, the computational load of solving vec(HH∗)
increases as the used information increases. If we define a “flop” to be a single complex multiplication or addition [35], then due to the sparse and lower-triangular structure of G−1, there requires about 4.3× 103 flops to solve vec(HH∗) for the first 3 block rows of
¯
x(i) (see (3.14)); while for the first 15 and 30 block rows of ¯x(i), the solution of vec(HH∗) is obtained via the least square approach, which is solved by the QR factorization [35, p.240], and the flop counts are roughly 8.4× 105 and 3.44× 106 flops, respectively .
3.5 Summary
In this chapter, we propose a blind identification method for MIMO FIR channels in the SC-ZP block transmission systems using periodic precoding. The identifiability condition requires that the channel impulse response matrix is full column rank. The channel can have more transmitters or more receivers. The performance of identification algorithm depends on the choice of the precoding sequence. We propose a two-level optimal precoding scheme that minimizes the noise effect in the estimation of the covariance matrix Rf. The effect of the optimal precoding sequence on channel equalization is also discussed.
Compared with the subspace method [27], the proposed method is shown to have better performance from low to medium SNR. Besides, the computations involved in the algorithm are relatively simple: only covariance matrix estimation, a multiplication of vec(Rf) by a lower triangular matrix to obtain vec(HH∗), and an eigen-decomposition of a J (L + 1)× J (L + 1) matrix, the main computational load; whereas, the computations of the subspace method requires a covariance matrix estimation, and two main computational loads: an eigen-decomposition of a J (L + M )× J(L + M) matrix and a singular value decomposition of a (J N− KM)N × J(L + 1) matrix. Since N = M + P and M > L, hence the subspace method requires substantially more computations than the proposed method.
10 20 30 40 50 60 70 80 90 100
−18
−16
−14
−12
−10
−8
−6
−4
−2 0
Number of symbol blocks
Channel NRMSE(dB)
sequence S0 sequence S1 sequence S2 sequence SA sequence SB
Figure 3.2. Channel NRMSE versus number of symbol blocks
0 5 10 15 20 25 30 35 40
−30
−25
−20
−15
−10
−5
SNR
Channel NRMSE(dB)
τ=0.2 τ=0.4 τ=0.6 τ=0.8 τ=1.0
Figure 3.3. Channel NRMSE versus output SNR
0 1 2 3 4 5 6
−30
−25
−20
−15
−10
−5
Overestimated Channel Order
Channel NRMSE(dB)
τ=0.2 τ=0.4 τ=0.6 τ=0.8 τ=1.0
Figure 3.4. Channel NRMSE versus ( ˆL− L)
20 40 60 80 100 120 140 160 180 200
−22
−20
−18
−16
−14
−12
−10
−8
−6
−4
Number of symbol blocks
Channel NRMSE(dB)
sequence S0 sequence S1 sequence S2
Figure 3.5. Channel NRMSE versus number of symbol blocks
0 5 10 15 20 25 30 35 40
−28
−26
−24
−22
−20
−18
−16
−14
−12
−10
SNR(dB)
Channel NRMSE(dB)
τ=0.2 τ=0.4 τ=0.6 τ=0.8 τ=1.0
Figure 3.6. Channel NRMSE versus output SNR
0 5 10 15 20
−35
−30
−25
−20
−15
−10
−5
SNR
Channel NRMSE(dB)
150 random channels with L=7, number of symbol blocks = 250
τ=0.1 τ=0.3 τ=0.5 τ=0.7 τ=0.8 τ=0.9 τ=0.93 τ=0.96 τ=0.99
Figure 3.7. Channel NRMSE versus output SNR
0 5 10 15 20
150 random channels with L=7, number of symbol blocks = 250 τ=0.1
150 random channels with L=7, number of symbol blocks = 250 τ=0.7
Figure 3.8. BER versus output SNR
0 5 10 15 20
300 random channels with L=7, number of symbol blocks = 200 QPSK, proposed method 16 QAM, proposed method QPSK, subspace method 16 QAM, subspace method
(a) Channel NRMSE versus output SNR
0 5 10 15 20
300 random channels with L=7, number of symbol blocks = 200
QPSK, proposed method 16 QAM, proposed method QPSK, subspace method 16 QAM, subspace method
(b) Bit error rate versus output SNR
Figure 3.9. Comparison with the subspace method
0 5 10 15 20 25 30 35 40
−28
−26
−24
−22
−20
−18
−16
−14
−12
−10
−8
SNR
Channel NRMSE(dB)
the first 3 block rows of the received data the first 15 block rows of the received data the first 30 block rows of the received data
Figure 3.10. Channel NRMSE versus number of symbol blocks