Precoder-Blind Equalization Algorithm for MIMO-OFDM
3.4 Results and Discussions
3.4.3 BER performance
Figure 3.6 compares the BER performance of the proposed precoder with the JD equalizer with a similar system that uses the JD equalizer but without any precoder. A LS zero-forcing equalizer with CSI is used as benchmark. For the precoder-blind equalizer system, the equalizer uses two correlation matrices, Rˇzˇz(τ1) and Rˇzˇz(τ2), for joint diagonalization.
The 3-tap channel in (3.26) was used. Since the input signal s(i)m,ℓ is independently dis-tributed, the result reaffirms the idea that if the diagonal entries of the input signal power spectral density matrix are not distinct (no precoder is used), then it is not possible to identify and equalize FIR-MIMO channels using SOS of the received signal. This exhibits a verification of conditions stated in Section 3.1.2 which show that the SOS-based blind equalization can be performed if and only if the system input is colored.
Figure 3.7 shows the BER results of the proposed precoder-equalizer system as the number of OFDM symbols varies for different SNR values. Similar to previous simulations,
3.4 Results and Discussions 45
0 5 10 15 20 25 10−5
10−4 10−3 10−2 10−1 100
SNR [dB]
BER
BER vs. SNR, N t=2, N
r=4, q=2
LS equalizer with perfect CSI JD equalizer with precoder JD equalizer without precoder
Figure 3.6: Comparison of BER performance between system without coloring and with coloring for 3-tap channel.
3.4 Results and Discussions 46
Chapter 3: Chapter
Rˇzˇz(τ1) and Rˇzˇz(τ2) are used for joint diagonalization. As the figure shows, the BER of the proposed precoder-equalizer system approaches that of the LS equalizer when the number of symbols increases. This is because as more symbols are used, more accurate estimation of the correlation matrix can be obtained. Furthermore, at SNR = 18 dB, the proposed algorithm is able to equalize the channel using only 350 symbols with a BER of about 10−3. Compare to higher-order statistics techniques such as [10], when the number of symbols needed for equalization is in the order of 103, only a small amount of latency is incurred in the proposed SOS-based technique in order to equalize FIR-MIMO channels.
100 200 300 400 500 600 700 800 900 1000
10−5 10−4 10−3 10−2 10−1 100
Number of OFDM symbols
BER
BER vs. Number of OFDM symbols, N=2, M=4, q=2
LS equalizer with perfect CSI at SNR = 18dB JD equalizer with proposed precoder at SNR = 18dB JD equalizer with proposed precoder at SNR = 14dB JD equalizer with proposed precoder at SNR = 10dB
Figure 3.7: BER vs. different number of received OFDM symbols for 3-tap channel at SNR = 10, 14, 18 dB.
Figures 3.8 and 3.9 compare the BER performance of the proposed precoder-blind equalizer system with the precoder-equalizer scheme in [22] and the periodic precoding scheme reviewed in Section 3.2.1 [21]. Results using a LS equalizer and an identical
3.4 Results and Discussions 47
system that uses no equalization are also shown as benchmarks. As seen in the figures, the performance gap between the LS equalizer and the proposed one remains virtually unchanged as the channel spectrum changes. This shows that the performance of the proposed scheme is insensitive to various channel responses. This can be explained by observing the equation of the precoder in (3.23). Since the precoder is composed of cosine functions, the spectrum of the precoder will fluctuate periodically in the frequency domain.
Since the amplitude of the cosine, Ci,τp, is set to a small value, even if the minimum value of the cosine term coincides with the spectral null of the channel, this will not greatly impact the BER.
Compared with the precoders in [22], the proposed precoders perform better by at least 1 dB in all of the simulated channel conditions. Compared with the periodic precoding scheme in [21], the presented system has a comparable BER performance with the largely diminished latency. Recall the algorithm reviewed in Section 3.2.1 [21], a huge range of lag of cyclic correlation matrix is incurred. Besides, SOS knowledge of the transmitted signal and the received signal are both needed. For eRuu[k, τ ], a latency of M is incurred since eRuu[k, M] and eRuu[k, −M] are required. For eRxx[k, τ ], eRxx[k, −2K − M − q + 3]
and eRxx[k, 2K + M + q − 3] induce a latency of 2K + M + q − 3. To note that the latency of the periodic precoding scheme will increase if the size of IFFT/FFT or channel order increases. Under the conditions of simulation where M = 64 and K = 80, the latency of the periodic precoding scheme is q + 221 which is a relatively huge number while the proposed precoder-equalizer system has a small latency from τ2 = 2. The latency of the proposed precoder-blind system is dictated by the precoder design and independent of IFFT/FFT size. A comparison to both [22] and [21] is summarized in Table 3.4. Recall that M is the IFFT size, v is the length of CP, q is the channel order, and N is the size of correlation matrices of the whitened signal.
Figure 3.10 shows the BER result when different lags are chosen for joint
diagonaliza-3.4 Results and Discussions 48
Chapter 3: Chapter
Table 3.4: Performance comparison
Performance Indices
Proposed Precoder-Blind System
Periodic Precoder System [21]
Using Precoder in [22]
SNR at BER = 10−4 for 3-tap channel
19.5 dB 19 dB 21 dB
Number of symbols for SOS at SNR=18 dB
350 500 350
Latency from corre-lation matrix
Controlled by precoders
3M + 2v + q − 3 10 ∼ 20
Computational Complexity
O(N3) O(M3) O(N5)
3.4 Results and Discussions 49
0 5 10 15 20 25
LS equalizer with perfect CSI JD equalizer with proposed precoder JD equalizer with precoder in [22]
Equalizer with precoding in [21]
No equalization
Figure 3.8: Comparison of BER vs. SNR with different algorithms for 3-tap channel.
0 5 10 15 20 25 30 35 40 45 50
LS equalizer with perfect CSI JD equalizer with proposed precoder JD equalizer with precoder in [22]
Equalizer with precoding in [21]
No equalization
Figure 3.9: Comparison of BER vs. SNR with different algorithms for 7-tap channel.
3.4 Results and Discussions 50
Chapter 3: Chapter
tion at the receiver. As explained in Section 3.2.2, if τ1 = 1 and τ2 = 2, then Rˇzˇz(1) and Rˇzˇz(2) become the only correlation matrices that are needed for joint diagonalization.
This is reaffirmed by the simulation results in Figure 3.10 when the best BER perfor-mance is attained when only Rˇzˇz(1) and Rˇzˇz(2) are used for equalization. When other correlation matrices are used, the BER curves saturate to a noise floor. Since the choice for τp is chosen at the transmitter and it determines exactly which, as well as how many, correlation matrices should be used at the receiver for equalization, the transmitter has complete control on the computational complexity of the equalizer.
0 5 10 15 20 25
10−5 10−4 10−3 10−2 10−1 100
SNR [dB]
BER
BER vs. SNR, Nt=2, Nr=4, q=2
LS equalizer with perfect CSI Proposed precoder with τ range:1~2 Proposed precoder with τ range:3~4 Proposed precoder with τ range:5~6 Proposed precoder with τ range:7~8 Proposed precoder with τ range:9~10
Figure 3.10: BER vs. SNR with different lag for 3-tap channel.
As discussed earlier, the design parameter P cannot be increased indefinitely in order to enhance equalization performance at the expense of increase computational complexity.
Likewise, P also cannot be made too small since it will adversely affect the equalization performance. Figure 3.11 shows the result of the proposed algorithm when P is allowed to vary from 1 to 5. As seen from the figure, the BER is smallest when P = 2. Therefore,
3.4 Results and Discussions 51
P cannot be made arbitrary small in order to minimize computational complexity at the receiver, but it also cannot be made arbitrarily big since it will adversely impact the BER performance since this will induce too much amplitude variation into the transmitted bitstream.
Amplitude variation is dominated by not only P but also coefficients of the precoder.
Obviously, larger coefficients lead to more amplitude variation of the spectrum. To discuss the effects the coefficients of the precoder have on the BER performance, we define a ratio γ = κ/µ, where κ is the absolute value of the maximum coefficient and µ is the absolute value of the minimum coefficient. When P is fixed, a large γ induces more amplitude variation. Figure 3.12 shows the BER performance under distinct values of γ. This is obtained by varying κ while µ is held constant. It can be seen that the best BER occurs when γ = 10, and the BER will become worse no matter if γ is larger or smaller than 10. Among all the values that have been tested, γ = 16 gives the worst BER because it induces too much amplitude variation into the transmitted signal. This implies that there exists an optimal choice for the precoder coefficients. If the magnitude of the coefficients is too small, then the accuracy of the estimates of U is not sufficient. If the magnitude is too large, then too much amplitude variation is introduced in the spectrum such that the BER will degrade. Hence, the effect of γ is similar to that of P .
3.4 Results and Discussions 52
Chapter 3: Chapter
LS equalizer with perfect CSI Proposed precoder, P=1 Proposed precoder, P=2 Proposed precoder, P=3 Proposed precoder, P=4 Proposed precoder, P=5
Figure 3.11: BER vs. SNR with different P for 3-tap channel.
0 5 10 15 20 25
LS equalizer with perfect CSI JD equalizer with γ = 4 JD equalizer with γ = 7 JD equalizer with γ = 10 JD equalizer with γ = 13 JD equalizer with γ = 16
Figure 3.12: BER vs. SNR with different γ for 3-tap channel.
3.4 Results and Discussions 53