We have shown that the optimal transform is a “white” one in the sense that EHE = d20I.
When the transform order is large, however, one may not be able to find a transform that satisfies the equality exactly. Hence we seek for an “approximately white” design that gives EHE − d20I ¿ d20I.
In fact, several transformed transmission methods have been proposed in different contexts to achieve different objectives, for example, the the energy-spreading transform (EST) for MIMO transmission [47, 46] and the linear constellation precoding (LCP) for MIMO or multi-carrier CDMA transmission [102]. In our context, cyclic-prefixed block single-carrier transmission can also be treated as a transformed OFDM system. Here we present a transform design that has relatively low complexity and can help lower the complexity of the receiver.
The upper part of Fig. 8.2 illustrates the proposed transmitter structure, where the transform has a two-stage architecture. The first stage involves N orthogonal transforms and the second stage involves a space-frequency interleaving (SFI) operation. For ease of implementation, the first stage may employ a simple orthogonal transform (such as the Hadamard transform) or one for which fast computation methods exist (such as the FFT). Ideally, the orthogonal transforms should spread the coded symbols over all the subcarriers so that any differential codeword c − ˆc is not contained in a few subcarriers but is spread over the entire transmission band. Then, the SFI randomizes the distribution of the codeword energy (already spread across the frequencies by the orthogonal transform) in the spatial dimension to further exploit the spatial as well as the multipath diversity. Through this two-stage transform we attain the approximate whitening effect on the transmitted signal.
Mathematically, the transmitted signal “super-vector” x (see Section II for its definition) with the proposed SFT is given by
x =¡
WH⊗ IN¢
P (T ⊗ IN) X ,¡
WH⊗ IN¢
SX (8.21)
Figure 8.2: Proposed transform design and associated receiver structure.
where X is the modulated version of c, W and T are the DFT and the orthogonal transform matrices, respectively, and P is the SFI permutation matrix.
Fig. 8.2 also illustrates the proposed receiver structure (excluding the subsequent channel decoder and the iteration loop between the turbo DFE and the channel decoder) in its lower part. Following the OFDM receiver frontend and space-frequency deinterleaving (DeSFI), we can employ block turbo DFE for signal detection. We now turn to its discussion.
Chapter 9
Receiver Design In Transformed MIMO OFDM System
9.1 Block Turbo DFE
In this section, we first consider how block turbo DFE operates under MIMO OFDM in general.
Then in the next section, we consider how it works together with SFI. It will become clear that, unlike the conventional turbo equalizer that operates in the time domain, the proposed block turbo DFE operates in the frequency domain.
9.1.1 Block Turbo DFE as Iterative Solution to Constrained Least-Square Problem
Employing earlier notations, the signal propagation behavior of a coded MIMO OFDM system can be described as
r = Hx + n (9.1)
where (with slight abuse of the notation) x = x(c) for some codeword c and
H =
H11 H12 . . . H1N H21 H22 . . . H2N ... ... . .. ...
HM 1HM 2. . . HM N
(9.2)
with the entries Hij being circulant matrices. When n is white Gaussian and H is known, the ML detection can be formulated as a constrained least-square (CLS) problem which seeks to minimize the cost function
J (c) = kr − Hx(c)k2 (9.3)
subject to the constraint that c is a valid codeword.
A block turbo DFE estimates the transmitted signal in the following way:
ˆ
xk= Fr + B¯xk−1 (9.4)
where k is the iteration count, ¯xk−1 is the decision output of the k − 1st iteration (which could be the soft output from the channel decoder), ˆxk is the signal estimate of the kth iteration, and F and B are the feedforward filter (FFF) matrix and the feedback filter (FBF) matrix, respectively. Taking the gradient descent approach [26, 24], we obtain the DFE coefficients as
F = µHH, B = I − µHHH, (9.5)
where I is an identity matrix and µ = KN/kHk2F with kHk2F = tr{HHH}, i.e., the Frobenius norm of H. With this set of DFE coefficients, the kth iteration equalizer output can also be written as
If the decision output of the k − 1st iteration is error-free, i.e., if ¯xk−1 = x(c), then the block DFE cancels the intersymbol interference (ISI) completely.
In beginning iterations, however, ¯xk−1 may contain many or large decision errors. Such errors would affect adversely the signal estimate ˆxk in the kth iteration, especially if kBk2F is large. To alleviate such effects, therefore, we consider the generalized gradient descent approach which improves the convergence property by “conditioning” the iterative updates with a shaping filter or precondition filter as [82]
ˆ
In other words, we modify the DFE filters to
F = CHH, B = I − CHHH, (9.8)
According to [82], C should be Hermitian symmetric, positive semi-definite, and commute with HHH. An example is given by the quasi-Newton method [50] where
C = µ¡
HHH + αI¢−1
, (9.9)
with α being some constant.
9.1.2 MMSE Shaping Filtering
We consider using a shaping filter that minimizes the mean-square error. Let ek−1= ¯xk−1−x(c).
Then the error in equalizer output is given by
wk , ˆxk− x(c) = CHHn +¡
I − CHHH¢
ek−1. (9.10)
In minimum mean-square error (MMSE) shaping filtering, we seek to minimize E{kwkk2}. In addition, in order to avoid direct error feedback, we should constrain the average of the diagonal elements of B to zero.
To proceed, let α = σ2n/σ2e, i.e., the ratio of noise variance to the variance of the decision error ek−1. Note that both σ2e and α are functions of k. But for notational convenience we have omitted explicit indication of this dependence. In the Appendix, we show that the MMSE shaping filter is given by
C = µ¡
We see that the MMSE shaping filter has the same form as the quasi-Newton shaping filter.
In the case of single-input single-output systems, the above result is the same as that in [13].
But our result is more general in that it applies to MIMO systems. Note, in addition, that when the various quantities converge with iterations, σe2 reduces to zero and α approaches infinity.
Then the shaped DFE coefficients also approach that without shaping.
A major drawback of MMSE shaped turbo DFE is, of course, the heavy complexity burden associated with changing the filter coefficients with each iteration. We consider reducing the computational complexity in the next subsection.
9.1.3 Employing Fixed Shaping Filter for Reduced Complexity
To reduce the computational complexity, we consider fixing the shaping filter during the iter-ations. This results in a three-stage algorithm: 1) Initialization: Perform MMSE block linear equalization (i.e., let B = 0), because no decision output is available at this time. 2) Middle stage: Use a fixed shaping filter that can tolerate a large range of decision error power. 3) Final stage: Use the unshaped DFE filters.
Based on the foregoing results, the determination of the shaping filter for the second stage reduces to the choice of a suitable operating value for α. For this, we do not have a theoretically optimal formula, but only some rules of thumb. Experience shows that underestimation of α would not cause significant enhancement of total MSE when the true α is large enough. In contrast, if the true α is small, then overestimation of it would cause great increase of the MSE. This phenomenon is heuristically reasonable, because (true) α is defined to be equal to σ2n/σ2e. Assuming a smaller value for α than its true value is tantamount to assuming a less converged state, which may result in some slowdown in the convergence speed but would not likely cause stability problems. On the other hand, assuming a larger value for α than its true value means being over-optimistic on the convergence status, which would more likely cause performance degradation. Therefore, we choose to use a reasonably small value in the place of α. By experiment, we find that a suitable range of its values is 0.5–2, with the unity value being a good choice.
To decide whether to switch from the second stage to the final stage, we observe the variance of the likelihood ratio (LLR) of the decoded codeword. When the variation in the variance over two successive iterations is small, we assume that the equalizer has converged sufficiently and make the switch. Note that the LLR variance indicates the reliability of the decoded bits [59]. When its value over a number of iterations in stage 2 (shaped block turbo DFE) is even smaller than that in stage 1 (block linear equalization), we may safely conclude that the channel condition is very bad and the turbo DFE may not provide any advantage. In this case, we may simply use the linear equalizer output for decoding.
9.1.4 Benefit of Whitening Transform to Turbo DFE Performance
Similar to the EST (energy-spreading transform) [47], the proposed MIMO OFDM with trans-form can benefit the noise pertrans-formance of block turbo DFE by reducing error propagation. This is due to its ability to spread the symbol energy over the whole block. As a result, any sym-bol decision error is also so spread. This reduces the interference contribution of each symsym-bol decision error to all other symbols, thereby lowering the probability of error propagation and benefiting the convergence of turbo DFE. Note that the benefit applies not only to uncoded systems, but also to coded systems, because in typical channel codes the difference between two nearby codewords (in Hamming distance or Euclidean distance) is concentrated in only a few locations rather than having its energy spread over a large signal block. The mechanism can be compared to how coded MIMO OFDM with transform excels over coded MIMO OFDM without transform as discussed in an earlier section.