As indicated, in this chapter we consider LMMSE and iterative LMMSE signal detection, with partial whitening of additive disturbance (i.e., the residual I+N) to lower the error floor and with soft decision feedback.
Consider a vector of 2q + 1 frequency-domain signal samples centered at sample m, where q need not be equal to K:
ym = [Ym−q · · · Ym · · · Ym+q]′ = Hmxm+ zm (5.1) where xm = [Xm−p · · · Xm · · · Xm+p]′ for some integer p, Hm is a (2q + 1) ×(2p + 1) submatrix of channel matrix H of bandwidth K, and zm collects all the right-hand-side (RHS) terms in (1.2) (or (1.4)) associated with Yk, m − q ≤ k ≤ m + q, that do not appear in Hmxm. The elements of zm include both residual ICI and channel noise. To avoid clogging the mathematical expressions, we have omitted explicit indexing of various quantities in (5.1) with the parameters K, p, and q, understanding that their dimensions and contents depend on these parameters. As
examples, with {K = 1, q = 1, p = 1} we have
A tradeoff between complexity and performance can be achieved by judicious choice of {K, q, p}.
Let Kuv denote the covariance matrix of random vectors u and v, i.e., Kuv = E[(u−E{u})(v−E{v})H] where superscript H denotes Hermitian transpose. When u = v, we simply write Ku. For simplicity, rather than performing whitening over a complete sequence, we do blockwise whitening of the residual I+N over windows of size 2q + 1 as
where we have omitted the subscript m in Kz due to its invariance over m. The quantity K−
1
z2 may be defined in more than one way; for example, we may let K−z12 = UΛ−12UH where U is the matrix of orthonormal eigenvectors of Kz and Λ is the diagonal matrix of corresponding eigenvalues of Kz.
If we treat each signal block (window) separately without regard to their par-tially overlapping relationship, then the LMMSE estimate of some Xd in xm (where m − p ≤ d ≤ m + p), conditioned on prior estimates of all other elements of xm, is
estimates) and the notation “|¯x(d)m ” indicates conditioning on ¯x(d)m . The term ˜Hmx¯(d)m
gives the contribution of the priors (except that at subcarrier d) in the received signal
˜ contexts than the present work, that ignoring the nondiagonal terms of Kx
m|¯x(d)m only results in minor performance loss [27,28]. Previous works on iterative LMMSE ICI equalization have also adopted a diagonal approximation to the conditional signal covariance matrix [22,23]. Therefore, we also employ such a diagonal approximation for simplicity: Kx
m|¯x(d)m ≈ V(d)m ≈ diag(vm−p, . . . , vd−1, Es, vd+1, . . . , vm+p) where vk = E[|Xk|2| ¯Xk] − | ¯Xk|2, k = m − p, . . . , d − 1, d + 1, . . . , m + p. Carrying out the above estimation for each Xd in each xm would yield 2p + 1 estimates for each signal sample.
Based on the above, our LMMSE detector considers each Xd in each xm in se-quence and conducts conditional LMMSE estimation as described, with the needed priors formed by soft-combining the most recent 2p + 1 estimates of Xk, k 6= d. Af-ter completing the estimation of all Xd in all xm, the process may be repeated over the same signal samples, resulting in iterative LMMSE detection. The above proce-dure resembles the “sequential iterative estimation (SIE)” method of [22] except for multiple (i.e., 2p + 1) estimations of each signal sample and their soft combination.
Simulation results show that these modifications can yield significant performance gain. We now explain the method of soft combination.
First, we set up a buffer of (2p + 1)N entries to hold ˆXd(m) ∀d ∀m. The buffer entries are initialized to zero. A new estimate ˆXd(m) immediately overwrites the previous value recorded in the corresponding entry and is used in subsequent soft combination. In soft-combining the multiple estimates, we take the average of signal values over the posterior probability distribution as
X¯d=X
function of ˆxd for Xd = ξ. For simplicity, assume that the likelihood function observes a jointly circularly Gaussian distribution as
f (ˆxd|Xd = ξ) = 1
ce−12(ˆxd−uξ)HK−1ξ (ˆxd−uξ) (5.7) where c is an inconsequential constant, uξ = E[ˆxd|Xd = ξ], and Kξ denotes the covariance matrix of ˆxd conditioned on Xd= ξ.
To avoid the complexity of working with a full Kξmatrix, we approximate it by
1
sKˆξ where ˆKξ is a diagonal matrix that has the same diagonal elements as Kξ and s is a subunity factor to compensate for the (statistically) over-optimistic likelihood characterization arising from omission of the nondiagonal terms in Kξ. This is similar to what has been considered in turbo decoding [29,30], and a factor s = 0.7 is suggested in [30] based on simulation. We also let s = 0.7 in our simulation.
Concerning the elements of uξ and ˆKξ, we have
E[ ˆXd(m)|Xd = ξ] = gm(d)Hh˜(d)m ξ (5.8) and
Kˆξ = 1
2diag([σd(d−p)(ξ)]2, . . . , [σd(d+p)(ξ)]2) (5.9) with
[σd(m)(ξ)]2 = g(d)m Hh˜(d)m (1 − ˜h(d)m Hgm(d)), (5.10) where d − p ≤ m ≤ d + p and g(d)m , ( ˜HmV(d)m H˜Hm+ I)−1h˜(d)m . Also for simplicity, for QAM the update (5.6) is carried out in the I and Q directions separately, which is particularly appropriate under Gray coding.
5.2 Simulation Results
Consider an OFDM system with DFT size N = 128, subcarrier spacing fs = 10.94 kHz, and sample period Tsa = 1/(Nfs) = 714 ns, which are some of the Mobile WiMAX parameters. Let there be no channel coding. The modulations employ Gray-coded bit-to-symbol mapping. The channel is WSSUS with PDP as shown in Table 4.1 and with each path subject to Rayleigh fading. Assume that the
10 15 20 25 30
Proposed MMSE, K=2, p=q=1, iteration #=0 Proposed MMSE, K=2, p=q=1, iteration #=1 Whitening MLSE, K=p=q=1
Whitening MLSE, K=2, p=q=1 Residual ICI−free bound, K=1
Figure 5.1: Bit error rate of different detection methods in the TU6 channel, with N = 128, Tsa = 714 ns, fd= 1500 Hz (normalized peak Doppler frequency fdTsaN = 0.1371) and QPSK subcarrier modulation.
receiver has perfect channel state information (CSI), which includes the channel matrix within band K and the covariance matrix Kz of the residual I+N.
First, consider QPSK subcarrier modulation in a relatively high fd = 1500 Hz (normalized peak Doppler frequency ≈ 0.14). Let {K = 2, p = q = 1}. From the theory outlined in Sec. III, it can be derived that the normalized autocorrelation matrix of the out-of-band (i.e., residual) ICI is given by
The first four curves in Fig.5.1compare the performance of the proposed technique with that of MLSE which treats the residual ICI as white [4]. They show that the proposed method can yield a substantial gain compared to nonwhitening MLSE at
500 1000 1500 2000 2500 3000 3500
Peak Doppler Frequency f
d(Hz)
B it E rr o r R a te
0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32 Normalized Peak Doppler Frequency (f
dT
saN)
Proposed method,
K=2,q=1,p=1, QSPK iterate 3 times iterate 1 time QPSK MLSE, residual ICI treated as white, K=1, K=2
Figure 5.2: Bit error rate floor versus Doppler spread of different detection methods in the TU6 channel with N = 128, Tsa = 714 ns, and QPSK subcarrier modulation.
both K = 1 and K = 2. For additional comparison, we also show the performance of the whitened MLSE of [25] and a benchmark, namely, the matched-filter bound (MFB). The MFB does MLSE with perfect knowledge of the interfering symbols and with the residual ICI outside band K fully cancelled. Not surprisingly, the whitened MLSE has a better performance, but the proposed LMMSE technique has a much lower complexity and thus provides a good complexity-performance tradeoff.
We now examine the ICI-induced error floors of different techniques. Fig. 5.2 shows the results of the proposed technique (with soft-combined feedback) and that of MLSE which treats the residual ICI as white [4], over a large range of peak Doppler frequencies under QPSK subcarrier modulation. For the proposed tech-nique, we again let {K = 2, p = q = 1}. The proposed method (three bottom curves) shows a remarkable gain of roughly two to three orders of magnitude com-pared to treating the residual ICI as white (two top curves). The error floor can be
500 1000 1500 2000 2500 3000 3500
Peak Doppler Frequency f
d(Hz)
B it E rr o r R a te
0.0457 0.0914 0.1371 0.1829 0.2286 0.2743 0.32 Normalized Peak Doppler Frequency (f
dT
saN)
iterate 1 time
Figure 5.3: Bit error rate floor versus Doppler spread of different detection methods in the TU6 channel with N = 128, Tsa= 714 ns, and 16QAM subcarrier modulation.
driven to well below 10−4 even at the very high normalized peak Doppler frequency of 0.32. The results also shows that, under the simulated conditions, one iteration of the proposed method may already provide close to what more iterations can provide in performance. For comparison, we also show the performance of LMMSE without soft combination in feedback, i.e., with ¯Xk= ˆXk(k) ∀k (two middle curves). There is obvious gain from performing soft combination.
In Fig. 5.3, we look at the ICI-induced error floors of different methods under 16QAM subcarrier modulation. As MLSE-based techniques appear too complicated with high-order modulations, we only consider LMMSE methods. We compare the performance of the proposed method with the sequential iterative LMMSE (without whitening of residual I+N) of [22]. In the proposed method, we again let {K = 2, p = q = 1}. We see that there is still an order-of-magnitude performance gain with the proposed technique under 16QAM, with soft-combined feedback. This can be highly
beneficial when coupled with channel coding.
5.3 Summary of Results
In this chapter, we considered LMMSE signal detection with blockwise whiten-ing of residual ICI plus noise. After whitenwhiten-ing, the method performed conditional LMMSE equalization of each signal sample in a sequential manner, enlisting previ-ously equalized samples at nearby subcarriers in soft-combined feedback to enhance detection performance. We presented some simulation results based on 3 × 3 block whitening and three-sample equalization. The results showed that a good tradeoff between complexity and performance could be achieved. They also showed that the proposed technique could attain a substantially lower ICI-induced error floor than conventional MLSE and iterative LMMSE, not only under QPSK but also under a higher-order modulation such as 16QAM.
Chapter 6
Thesis Conclusions and Potential Future Topics
In time-varying channels, OFDM transmission suffers from ICI. In a system with-out ICI, the channel frequency response matrix that relates the inputs of the inverse discrete Fourier transform (IDFT) and the outputs of the DFT is diagonal. Fast channel variation introduces sizable off-diagonal elements in the matrix, thus re-sulting in ICI. As stated previously, a band approximation to channel matrix that retains only the dominant terms about the diagonal may ease receiver design, but also results in an irreducible noise floor at receiver.
In this thesis, we exploit the correlation of the residual ICI outside the band of channel matrix to attain a significantly enhanced signal detection performance.