In this section, we use several examples to demonstrate the performance of the proposed method. The channel normalized root-mean-square error (NRMSE) is defined as
NRMSE = 1 estimate of channel impulse response matrix H after removing the unitary matrix ambiguity by the least squares method [20, 21], and I = 500 is the number of Monte Carlo runs. The input source symbols are independent and identically distributed (i.i.d.) QPSK signals.
The channel noise is temporally and spatially white Gaussian. The signal-to-noise ratio (SNR) at the output is defined as SNR = P1
P −1
n=0E[t(n)22]
E[w(n)22] , where t(n) = [t1(n) · · · tJ(n)]T is the signal component of the received signal (see Figure 2.1).
1) Simulation 1 – optimal selection of precoding sequences In this simulation, we use the following model
H(z) =
to demonstrate the effect of different precoding sequences on the performance of the pro-posed method. In experiment 1, the first sequence is chosen as {0.767 1.07 1.07 1.07},
which satisfies (2.28) and (2.29). The second and third sequences are chosen based on (2.30) for P = 4 and τ = 0.5878 with the two possible peak positions: m = 0 and m = 3.
By computation, the corresponding µ for the three cases are 40.0, 4.66 and 22.1, respec-tively. Thus m = 0 is the optimal selection. Figure 2.2 shows that for SNR=10 dB, there are about 5∼7 dB and 5∼9 dB difference in NRMSE between the optimal one and two others.
In experiment 2, we use the precoding sequences that satisfy (2.30) with m = 0, but with different τ to test the effect of τ on the identification performance. Figure 2.3 shows that for each sequence, when the number of samples (for each transmitter) is fixed at 1000, the NRMSE decreases as SNR increases and is roughly constant for SNR ≥ 20 dB. Figure 2.3 also shows that the identification performs better for smaller τ , which is consistent with the conclusion at the end of Section 2.3.1.
2) Simulation 2 – channel order overestimation
In this simulation, we use the following channel model
H(z) = dB, and 1000 samples (for each transmitter) for simulation. The precoding sequences are chosen as (2.30) with m = 0 and τ = 0.2, 0.4, 0.6, and 0.8. Figure 2.4 shows the NRMSE increases with increasing channel order overestimation. We see the proposed method is quite robust to channel order overestimation when τ is small. For example, with τ = 0.4, when ( ˆL− L) increases from 0 to 3, the NRMSE increases from -25.5dB to -21dB, which is still a low value.
3) Simulation 3 – a 3-input 2-output channel
In this simulation, we use the 3-input 2-output model to illustrate the performance of the proposed method for channel with more transmitters than receivers. Note that H is full column rank, but the channel is not irreducible [21]
because H(0) is not full rank, and it is not column reduced [21] either because H(2) is not full rank. In experiment 1, the precoding sequences (P = 4, τ = 0.5878) are given as in (2.30) with m = 0 and m = 3, respectively. Figure 2.5 shows that the NRMSE decreases as the number of data samples increases for SNR=10 dB. As expected, m = 0 case (the optimal selection) is better than m = 3 case.
In experiment 2, we use the precoding sequences that satisfy (2.30) with m = 0, but with different τ to test the effect of τ on the identification performance. Figure 2.6 shows that for each sequence, when the number of samples (for each transmitter) is fixed at 1000, the NRMSE decreases as SNR increases and is roughly constant for SNR ≥ 25 dB. Figure 2.6 also shows the identification performs better for smaller τ .
4) Simulation 4 – channel equalization performance
In this simulation, we use the channel model given in (2.34) to demonstrate the perfor-mance of the proposed method for channel equalization. We use the precoding sequences that satisfy (2.30) with m = 0, but with different τ to test the effect of τ on the equalization performance. For simplicity, we use the minimum mean square error (MMSE) equalizer.
The equalizer is a 17-tap Wiener filter with 12-tap reconstruction delay whose jth output ˆ
uj(k) is an estimate of uj(k) for j = 1, 2,· · · , K. Since the precoding scheme is applied at the transmitter, we need to multiply ˆuj(k) by the corresponding p(k)−1 to obtain an estimate of sj(k) for j = 1, 2,· · · , K. The number of samples is 1200. We first identify the channel using the first 400 samples and then do equalization.
Figure 2.7 shows that under low SNR, the proposed method performs better when τ is large; however, under high SNR, the proposed method performs better when τ is low. A
possible explanation is as follows.
Channel estimates become more accurate as τ becomes smaller, but the gains p(k)−1 =
√1
τ, k = 1, 2,· · · , P −1 become larger and result in larger noise amplification at the receiver.
Both channel estimation error and channel noise contribute to the (maximum likelihood) detection performance, i.e., the symbol error rate. In the low SNR region, the detrimental effect of noise amplification outweighs the benefit of small estimation error; whereas in the high SNR region, accurate channel estimation weighs more than the noise amplification effect. Hence we choose a small τ when SNR is high and a large τ when SNR is low.
5) Simulation 5 – Comparisons with other methods
In this simulation, we generate 100 2-input 4-output random channels with order L = 2;
each element in the channel impulse response matrix is a complex circular Gaussian random variable with unit variance. We compare the proposed method with a generalized space time block codes (GSTBC)[23] based method. Both methods require periodic precoding sequences. For the proposed method, the precoding sequence is chosen as {1.500 0.767 0.767 0.767}; whereas the entries in the precoding sequence for the GSTBC method is chosen as random entries with modulus 1 for each random channel simulation [23]. The performance of the proposed method is also compared with a linear prediction (LP)[2, chap. 6] based method, and an outer product decomposition algorithm (OPDA)[20]. Both methods do not require a periodic precoder. MMSE equalizers are used for the proposed method, LP method, and OPDA method. For the GSTBC method, we use the customized equalizer proposed in [23]. Figure 2.8(a) shows that when the number of samples is 1200 (for each transmitter), the identification performance of the proposed method is better than those of the other three methods excepting the GSTBC method for SNR ≥ 13 dB.
However, Figure 2.8(b) shows the equalization performance of the proposed method is only better than those of the LP and OPDA methods and worse than the GSTBC method.
The inconsistency of the channel estimation and equalization performance of the proposed method and the GSTBC method for SNR≤ 13 dB may be due to the different precoding sequences and equalizers used. Figure 2.9 shows that when the number of samples is 200 (for each transmitter), the identification and equalization performance of the proposed method is better than that of the GSTBC method for SNR≤ 15 dB. Figure 2.9 shows that
when the number of samples is small, the proposed method has better performance than the GSTBC method under low SNR.
2.6 Summary
In this chapter, we have proposed a blind identification method for general MIMO FIR channels using periodic precoding. We optimally design the precoding sequence against the effects of channel noise and numerical error, so as to increase the accuracy of channel estimation. The channel identifiability only requires that channel impulse response matrix H is full column rank, which is more relaxed than irreducible or column reduced. In addition, the channel can have more receivers or more transmitters. This algorithm can be used in the case of channel order overestimation. Simulation results are used to demonstrate the performance of the proposed method.
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Figure 2.2. Channel NRMSE versus number of samples
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Figure 2.3. Channel NRMSE versus output SNR
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Figure 2.4. Channel NRMSE versus ( ˆL− L)
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Figure 2.5. 3-input 2-output model: channel NRMSE versus number of samples
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Figure 2.6. 3-input 2-output model: channel NRMSE versus output SNR
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Figure 2.7. Symbol error rate versus output SNR
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Figure 2.8. Comparison of NRMSE and symbol error rate, number of input samples = 1200
(a) Channel NRMSE versus output SNR
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(b) Symbol error rate versus output SNR
Figure 2.9. Comparison of NRMSE and symbol error rate, number of input samples = 200