Transmission Systems
4.2.3 Extension to MIMO Zero-Padding OFDM Systems
The proposed method can be extended to the MIMO ZP-OFDM systems. In this case, at the transmitter, each ¯sk(i) is multiplied by the IFFT matrix F−1 = F∗ before entering the zero padding block, F1 [27] (see Figure 4.2). Here F∈ CM×M is an FFT matix. Thus we know the input to F1 is F∗¯sk(i) for OFDM case. Since F is a unitary matix [26], ¯sk(i) and F∗¯sk(i) are both zero mean and have the same second-order statistics. Hence
E[F∗¯sk(m)] = 0, E[(F∗¯sk(m))(F∗¯sk(n))∗] = δ(m− n)IM.
Hence the first and second-order statistics of u(n) for OFDM case are the same as those for single carrier case. Therefore, following the same method given in Section 4.2.1, we can identify the channel impulse response matrix for ZP-OFDM systems.
- S/P ===⇒ IFFT ====⇒ ZP ====⇒P/S
Figure 4.2. An MIMO ZP-OFDM baseband model
4.3 Simulation Results
In this section, we use several examples to demonstrate the performance of the proposed method. The channel NRMSE and the number of Monte Carlo runs are the same as those given in Section 2.5. The input source symbols are i.i.d. QPSK signals. The signal-to-noise ratio (SNR) at the output is defined as SNR = EE[w(n)[t(n)222]
2], where t(n) = [t1(n) · · · tJ(n)]T is the signal component of the received signal (see Figure 4.1 and 4.2). Except Simulation 1, the channel noise is zero mean, temporally and spatially white Gaussian.
1) Simulation 1 – color noise case
In this simulation, we use the channel model (2.33) to demonstrate the performance of the proposed method when the channel noise is colored. 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%. The additive color noise w(n) is generated by passing a zero mean, unit variance, temporally and spatially white Gaussian vector sequence wv(n)∈ R2 through an FIR filter C(z) = C(0)+C(1)z−1+C(2)z−2whose output is w(n) = C(z)wv(n), where
C(0) =
⎡
⎣ 0.283 + 0.181i 0.185 + 0.115i
−0.135 + 0.192i 0.136 + 0.235i
⎤
⎦ , C(1) =
⎡
⎣ 0.185 + 0.126i 0.165 + 0.235i
−0.154 + 0.102i 0.108 + 0.338i
⎤
⎦ ,
C(2) =
⎡
⎣ 0.089 + 0.181i 0.089 + 0.235i 0.089 + 0.126i 0.108 + 0.159i
⎤
⎦
In this case, Kw(0), Kw(1), and Kw(2) defined in assumption (C1) are shown as follows:
Kw(0) =
⎡
⎣ 0.397 0.208− 0.159i 0.208 + 0.159i 0.350
⎤
⎦ , Kw(1) =
⎡
⎣ 0.242− 0.067i 0.121 − 0.120i 0.171 + 0.101i 0.199 + 0.011i
⎤
⎦ ,
Kw(2) =
⎡
⎣ 0.101− 0.068i 0.086 − 0.037i 0.090 + 0.031i 0.064 + 0.038i
⎤
⎦
Figure 4.3(a) shows the NRMSE decreases as the number of symbol blocks increases. Figure 4.3(b) shows that the noise NRMSE also decreases as the number of symbol blocks increases, where the noise NRMSE is similarly defined as in (2.32) except H is replaced by K = [Kw(0)T Kw(1)T Kw(2)T]T and H(i) is replaced by K(i) = [ K(i)w(0)T K(i)w(1)T K(i)w(2)T]T.
2) Simulation 2 – random channels case
In this simulation, we generate 500 2-input 2-output random channels with order L = 2 to demonstrate the performance of the proposed method. Each element in the channel impulse response matrix is complex Gaussian distribution with zero mean and unit variance.
We use M = 18 and P = 2(= L) (transmission efficiency is 90%). Figure 4.4 shows for different number of symbol blocks, the NRMSE decreases as SNR increases and is roughly constant for SNR≥ 20 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 = 18 and P = 2. Figure 4.5 shows for different number of symbol blocks, the NRMSE decreases as SNR increases and is roughly constant for SNR ≥ 20 dB.
4) Simulation 4 – channel order overestimation
In this simulation, we use the channel model (2.34) to demonstrate the performance of the proposed method by comparing with the subspace method [27], which is also for MIMO zero padding block transmission systems. For each upper bound ˆL, 0 ≤ (ˆL − L) ≤ 6, we choose P = ˆL and M = 9P for simulation such that the transmission efficiency is
maintained at 90%. Figure 4.6 shows when the number of symbol blocks is fixed at 500, the NRMSE increases with increasing channel order overestimation for different SNR. When SNR=0 and 5 dB, the proposed method performs better than the subspace method. When SNR=10 dB, the subspace method performs better than the proposed method. Figures 4.6 shows that the proposed method is more robust to channel order overestimation than the subspace method when SNR is low.
5) Simulation 5 – channel estimation and equalization of a 2-input 2-output ZP-OFDM system
In this simulation, we use a ZP-OFDM system with the same channel model (2.34), and M = 18, P = 2. We compare the performance of the proposed method with that of the subspace method [27]. Figure 4.7(a) shows when SNR = 0 and 5 dB, the performance of the proposed method is better than that of the subspace method. However, when SNR = 10 dB, the performance of the subspace method is better than that of the proposed method.
Figure 4.7(b) shows when the number of blocks is 100 (300), the proposed method performs better than the subspace method when SNR below about 8 dB (6 dB). Figures 4.7(a) and 4.7(b) show that the proposed method has better performance than the subspace method under low SNR.
Figure 4.8 shows the simulation results for the zero forcing equalization of the proposed method and the subspace method. The number of symbol blocks is 500 (where the number of symbols = 18× 2 × 500 = 18000). We first identify the channel using the first 25, 50, 250, and 500 symbol blocks, respectively, and then do equalization. In each sub-figure of Figure 4.8, we see the proposed method performs better than the subspace method under low SNR, whereas the subspace method performs better under high SNR. Besides, from Figure 4.8, we can also observe the tendency that when the number of symbol blocks used for identification increases, the equalization performance of the proposed method and the subspace method would tend to be identical. Simulation result in Figure 4.9 shows when the number of symbol blocks for identification and equalization is 5000, the performance of these two methods are almost identical.
4.4 Summary
In this chapter, we have proposed a blind identification method for the MIMO FIR channels in zero padding block transmission systems without using the periodic precoding.
The method exploits the structure provided by zero padding. The channel noise may be temporally and spatially colored, the channel identifiability condition requires the channel impulse response matrix is full column rank, and the channel can have more transmitters or more receivers.
Compared with the subspace method [27], the proposed method is shown to have better performance under low SNR. Besides, the computations involved in the algorithm are relatively simple: only covariance matrix estimation, matrix subtractions, 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 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.
50 100 150 200 250 300 350 400 450 500
(a) Channel NRMSE versus number of symbol blocks
(b) Noise NRMSE versus number of symbol blocks
Figure 4.3. Color noise case
0 5 10 15 20 25 30 35 40
Figure 4.4. Channel NRMSE versus output SNR
0 5 10 15 20 25 30 35 40
−18
−17
−16
−15
−14
−13
−12
−11
−10
SNR(dB)
Channel NRMSE(dB)
100 symbol blocks 300 symbol blocks 500 symbol blocks
Figure 4.5. Channel NRMSE versus output SNR
0 1 2 3 4 5 6
−30
−25
−20
−15
−10
−5 0
Overestimated Channel Order
Channel NRMSE(dB) SNR=0 dB: proposed method
SNR=0 dB: subspace method SNR=5 dB: proposed method SNR=5 dB: subspace method SNR=10 dB: proposed method SNR=10 dB: subspace method
Figure 4.6. Channel NRMSE versus ( ˆL− L)
50 100 150 200 250 300 350 400 450 500
(a) Channel NRMSE versus number of symbol blocks
100 blocks: the proposed method 100 blocks: subspace method 300 blocks: the proposed method 300 blocks: subspace method
(b) Channel NRMSE versus output SNR
Figure 4.7. An OFDM system case
0 5 10 15 20 25
(a) 25 symbol blocks for identification
0 5 10 15 20 25
(b) 50 symbol blocks for identification
0 5 10 15 20 25
(c) 250 symbol blocks for identification
0 5 10 15 20 25
(d) 500 symbol blocks for identification
Figure 4.8. An OFDM system: symbol error rate versus output SNR
0 5 10 15 20 25 10−5
10−4 10−3 10−2 10−1 100
SNR
Symbol Error Rate
proposed method subspace method
Figure 4.9. An OFDM system: symbol error rate versus output SNR
Chapter 5
Conclusions
We develop three blind identification algorithms for MIMO frequency selective wire-less communication channels. Instead of computing the channel impulse response matrix directly from the covariance matrix of the received data, as in subspace methods, the algo-rithms compute the channel product matrices first and then determine the channel impulse response matrix via an eigenvalue-eigenvector decomposition. The algorithms are simple, in terms of the amount of computations required, as compared with subspace methods;
they allow a more relaxed identifiability condition and are applicable to MIMO systems with more transmitters or more receivers. Simulation results show that they are reason-ably robust with respect to channel order overestimation and has an NRMSE performance comparable to subspace methods.
The algorithms differ in precoding complexity. The three precoding considered are:
(i) periodic precoding, (ii) periodic precoding plus zero padding, and (iii) zero padding alone. As a result, for each of the three cases, the computation required to determine the channel product matrices are also different. The computations required are respectively (i) to solve a decoupled group of overdetermined linear systems of equations, (ii) to solve a lower triangular linear system, and (iii) to carry out a number of simple subtractions.
Future research can be focused on (i) direct blind equalization without identifying the channel first, and (ii) blind identification for MIMO time varying channels.