Chapter 3 ICI Cancellation
3.2 The Per-Subcarrier Adaptive ICI Cancellation Framework 44
We focus on the linear equalizers in the frequency domain. As mentioned in Chapter 1, a common means to reduce the complexity of ICI equalization is to approximate
the non-diagonal CFR matrix with a banded matrix. For example, the approximation by banded CFR matrices can be utilized to construct the so-called block and serial ICI equalizers [16]. The CFR matrix Hf in (3.1) is first approximated by a banded matrix B in which all but few elements on the selected 2Q + 1 diagonals are set to zero. The transmitted signal s can be estimated by solving the resulted linear model, either by ZF or MMSE criteria. Figure 3.2 illustrates the difference between the block and serial approaches. In the block approach, the whole system of linear equations is solved altogether, e.g, the banded matrix B is directly inverted in the ZF approach to find the estimation of s. While in the serial approach, for each subcarrier a local (2Q + 1) × (2Q + 1) banded CFR matrix is considered, i.e., consider B(Q)k , [Hf]{<k−Q:k+Q>N,<k−Q:k+Q>N}, and a truncated (2Q + 1) × (2Q + 1) system of linear equations is solved for each subcarrier. Apparently, the serial approach enjoys an even lower complexity than the block approach does, but it also results in a larger performance degradation since the truncation in building smaller systems of linear equations introduces larger approximation errors.
The approach of using banded CFR matrices is, however, highly inefficient, espe-cially taking into account the fact that each subcarrier can experience vastly different ICI scenarios. Intuitively, if different counter-ICI measures are adopted for different subcarrier, possible benefits of reduced complexity and/or enhanced performance may follow. A closer look at the ICI model in (2.35) and (2.37) provides details about the different ICI scenarios faced by each subcarrier. Figure 3.3 shows the magnitude of a 32 × 32 example CFR matrix and three cross-sections on three subcarriers. It is clear that a uniform banded approximation is inappropriate, since each subcarrier faces very different ICI situations. Inspired by the observations made from (2.37), a metric |∆k/Hk| is introduced to indicate the ICI severity on the k-th subcarrier. In
0 5 10 15 20 25 30 35
Figure 3.3: The magnitude of the CFR matrix and cross-sections at selected subcarri-ers.
Fig. 3.3, the 20-th subcarrier, with a small |∆k/Hk| (|∆k/Hk| < −5 dB, SICR > 22 dB), is affected by insignificant amount of ICI. When moderate fading occurs, like on the 5-th subcarrier, |∆k/Hk| becomes higher, and moderate ICI emerges. When deep fading occurs, as on the 9-th subcarrier, |∆k/Hk| becomes significant (|∆k/Hk| > 0 dB, SICR < 12 dB), severe ICI exists and heavy ICI cancellation may be needed to maintain BER performance. The term |∆k/Hk| is able to indicate the ICI situation and the different need of ICI cancellation strategies on each subcarrier.
Our experiences show that in practical situations, the majority of subcarriers face moderate or very mild ICI situations; that is, little or no ICI cancellation needs to be done on these subcarriers and the performance will not suffer. Fig. 3.4 gives an idea of how many percentages of subcarriers face very mild (|∆k/Hk| < −5 dB) and
heavy ICI (|∆k/Hk| > 0 dB). The simulation is conducted under WiMAX standard: 10 MHz bandwidth, 2.5 GHz central frequency, and 1024 subcarriers. Two very different channel PDPs are considered: the ITU Pedestrian-B (PedB) channel and a two-path equal-gain channel. Different vehicle speeds from 60 km/h to the maximum speed of 350 km/h suggested in the LTE and WiMAX standards are simulated. The results show that as vehicle speed gets higher, more subcarriers experience severe ICI; yet even at 350 km/h, there are still 85% of subcarriers on which |∆k/Hk| < 0 dB and simple ICI cancellation strategies may be good enough for these subcarriers. An interesting observation is that the percentages of the ICI indicator remain almost the same when the channel PDP changes; therefore we expect the ICI indicator work properly for different types of channels.
50 100 150 200 250 300 350
0 10 20 30 40 50 60 70 80 90 100
Speed (km/h)
Pencentage of ICI indicator, |/H|
| /H| 0 dB, PedB
| /H| −5 dB, PedB
| /H| 0 dB, Two−path
| /H| −5 dB, Two−path
Figure 3.4: This figure shows the percentages of |∆k/Hk| that are larger than 0 dB or smaller than −5 dB at different vehicle speeds. Two channel PDPs, the ITU Pedestrian-B channel and the two-path equal-gain channel, are used.
PB ICI equalizer CE
Hk
Subtract by previous Hk
ΔK
ICI indicator
FIR Filter α
FIR Filter ΔK/Hk β
Yk
FIR Filter γ
Figure 3.5: A receiver adopting the PSA framework.
Based on the ICI indicator, a PSA receiver architecture shown in Fig. 3.5 is also proposed in [24] where the FIR filters being the realization of proposed PB ICI equaliz-ers which will be discussed in details in Section 3.3. A variety of existing ICI equalizequaliz-ers can be incorporated into this architecture and their settings can be adapted on each subcarrier according to |∆k/Hk|; in this way, a better trade-off between computational complexity and performance can be achieved. PSA ultilizes the idea of using different Q on each subcarrier to approximate the CFR matrix Hf. Figure 3.6 depicts how the CFR matrix Hf, instead of being approximated by a banded matrix with fixed bandwidth Q as shown in Fig. 3.2, is approximated by a banded matrix with variable bandwidth adaptively set at each subcarrier. The approach also works well with pop-ular wireless access technologies such as WiMAX and LTE based on OFDMA. In the following section, the PSA architecture forms the foundation for us to develop novel low-complexity ICI methods with remarkable results.
3.2.1 Incorporate the MAP ICI Equalizer
The MAP equalizer offers the optimal performance yet is very costly. In our approach, reduction in computational complexity is achieved by reducing the size of trellis at
CFR matrix H
k,m
k
m
Figure 3.6: A banded CFR matrix with variable bandwidth Q.
most subcarriers. The MAP equalizer outputs the log-likelihood ratio (LLR) of the b-th bit of S(m) via L(Sm,b) , logP (SP (Sm,bm,b=1|r)=0|r). LLRs can be efficiently calculated by the BCJR algorithm, which consists of forward recursion, backward recursion and calculating transition probabilities. The rearrangement of ICI model in (2.37) gets into calculating the branch metric of the k-th subcarrier through
P ( ˜R(k)| ˜S(k)) = − 1
2σ2k| ˜R(k) −
Q
X
l=−Q
H˜k,hk−li
N
S(hk − li˜ N)|2 (3.10) where σk2 is the variance of ˜z(k).
Consider an OFDMA system (WiMAX) with 10 MHz bandwidth, 2.5 GHz cen-tral frequency, 1024 subcarriers, QPSK modulation, and the ITU Vehicular-A channel model at 500 km/h. Fig. 3.7 shows the BER curves of MAP ICI equalizers, with fixed Q and variable Q. The result verifies that adjusting the trellis size does not hurt the performance even when the speed hits 500 km/h. The MAP ICI equalizer with a
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10−3 10−2 10−1 100
SNR (dB)
BER
Conventional one−tap equalizer MAP ICI equalizer with Q = 1
MAP ICI equalizer with variable Q 1 MAP ICI equalizer with Q = 2
MAP ICI equalizer with variable Q 2 MAP ICI equalizer with Q = 3
MAP ICI equalizer with variable Q 3 Block MMSE
Figure 3.7: BER performance comparisons for MAP ICI equalizers under 1024-point FFT, QPSK and ITU Vehicular-A channel model at 500 km/h. The BER curves of the MAP ICI equalizer (solid lines) and the variable Q MAP ICI equalizer (dash lines) essentially overlap.
variable-size trellis attains the same BER performance as that of the MAP ICI equalizer with a fixed large trellis (in fact, their BER curves overlap each other), yet the com-putational cost is significantly reduced. The complexity comparison is summarized in Table 3.3, where the number of complex floating-point operations (FLOPs) is counted.
Compared to the block MMSE and fixed Q MAP, variable Q(≤ 3) MAP significantly reduces computational complexity by two orders of magnitude without sacrificing BER performance. Moreover, the memory required to store M(2Q+1) states at each trellis stage also enjoys great reduction by adopting variable Q. In the simulation, 85% of stages use merely 4 states instead of the full 47 = 16384 states. Further reduction of
complexity is possible if techniques such as those in [18,23] can be incorporated in the future. Finally, it is interesting to note that the variable Q MAP ICI equalizer performs better than the block MMSE ICI equalizer. It is due to the extra gain provided by combining signals along different ICI paths constructively.
Table 3.1: Computational complexity comparison.
Methods Complexity Number of complex flops
Block MMSE O(N3) 4 × 108
2 × 108(Q = 3), Conventional fixed Q MAP O(M2Q+1(4Q + 2)N ) 8.6 × 106(Q = 2),
322560(Q = 1) 2.88 × 106(Q ≤ 3),
Variable Q MAP P
iO(M2Qi+1(4Qi+ 2)N ) 1.24 × 106(Q ≤ 2), 255728(Q ≤ 1)