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Performance Assessment of MIMO-BICM Demodulators based on System Capacity

We provide a comprehensive performance comparison of soft-output and hard-output demodulators in the context of non-iterative multiple-input multiple-output bit-interleaved coded modulation (MIMO-BICM). Coded bit error rate (BER), widely used in literature for demodulator comparison, has the drawback of depending strongly on the error correcting code being used. This motivates us to propose a code-independent performance measure in terms of system capacity, i.e., mutual information of the equivalent modulation channel that comprises modulator, wireless channel, and demodulator. We present extensive numerical results for ergodic and quasi-static fading channels under perfect and imperfect channel state information. These results reveal that the performance ranking of MIMO demodulators is rate-dependent. Furthermore, they provide new insights regarding MIMO-BICM system design, i.e., the choice of antenna configuration, symbol constellation, and demodulator for a given target rate.


A. Background
Bit-interleaved coded modulation (BICM) [2], [3] has been conceived as a pragmatic approach to coded modulation. It has received a lot of attention in wireless communications due to its bandwidth and power efficiency and its robustness against fading. For single-antenna systems, BICM with Gray labeling can approach channel capacity [2], [4]. These advantages have motivated extensions of BICM to multiple-input multiple-output (MIMO) systems [5]- [7].
In MIMO-BICM systems, the optimum demodulator is the soft-output maximum a posteriori (MAP) demodulator, which provides the channel decoder with log-likelihood ratios (LLRs) for the code bits.
Due to its high computational complexity, numerous alternative demodulators have been proposed in the literature, the most important of which are briefly describe below. Applying the max-log approximation [7] to the MAP demodulator reduces complexity without significant performance loss and leads to a search for data vectors minimizing a Euclidean norm. Efficient exact implementations of the max-log MAP detector based on sphere decoding have been presented in [8]- [10]. The implementation complexity can be further reduced by replacing the Euclidean norm with the ℓ ∞ norm [11], [12]. However, the complexity of these methods still grows exponentially with the number of transmit antennas. An alternative demodulator that approximates the max-log LLRs is based on semidefinite relaxation (SDR) and has polynomial worst-case complexity [13].
Several demodulation schemes use a list of candidate data vectors to deliver approximate LLR values.
Decreasing the size of the candidate list allows to trade off performance for complexity savings. A popular way to generate such lists are tree search techniques, with the list sphere decoder (LSD) [14] being a wellknown example. Another approach to obtain a candidate list is based on lattice reduction (LR) [15], [16].
Variants of lattice-reduction aided list demodulators were proposed in [17]- [19]. Alternatively, candidate vectors can be obtained by "flipping" some of the bits in the bit label of the data vector obtained by (hard) maximum likelihood (ML) demodulation [20] (variants are obtained by replacing the ML detector with other hard-output detectors).
MIMO demodulators with still smaller complexity consist of a linear equalizer followed by per-layer scalar soft demodulators. This approach has been studied using zero-forcing (ZF) [21], [22] and minimum mean-square error (MMSE) [23] equalization. The soft interference canceler (SoftIC) proposed in [24] iteratively performs per-layer demodulation, preceded by a subtraction of an interference estimate. This estimate is generated using soft symbols for the other layers obtained in the previous iteration. i.e., which antenna configuration, symbol constellation, and demodulator to prefer in order to achieve a certain rate with minimum SNR.

C. Paper Organization
The rest of this paper is organized as follows. Section II discusses the MIMO-BICM system model and Section III proposes system capacity as performance measure. In Sections IV and V, we assess the system capacity achievable with the MIMO-BICM demodulators referred to above for the case of fast fading. Section VI analyzes the impact of imperfect channel state information (CSI) on demodulator performance, and Section VII investigates the rate-versus-outage tradeoff of selected demodulators in quasi-static environments. In Section VIII, we summarize key observations and infer practical system design guidelines. Finally, conclusions are provided in Section IX.

A. MIMO-BICM Transmission Model
A block diagram of our MIMO-BICM model is shown in Fig. 1. A sequence of information bits b[q] is encoded using an error-correcting code and is then passed through a bitwise interleaver Π. The interleaved code bits are demultiplexed into M T antenna streams ("layers"). In each layer k = 1, . . . , M T , groups of Q code bits c  In most of what follows, we will omit the time index n for convenience. At the receiver, the optimum demodulator uses the received vector y and the channel matrix H to calculate LLRs Λ l for all code bits c l , l = 1, . . . , R 0 , carried by x. In practice, the use of suboptimal demodulators or of a channel estimateĤ will result in approximate LLRsΛ l . The LLRs are passed through the deinterleaver Π −1 and then on to the channel decoder that delivers the detected bitsb[q].

B. Optimum Soft MAP Demodulation
Assuming i.i.d. uniform code bits (as guaranteed, e.g., by an ideal interleaver), the optimum soft MAP demodulator calculates the exact LLR for c l based on (y, H) according to [7] Λ l log p(c l = 1|y, H) p(c l = 0|y, H) = log Here, p(c l |y, H) is the probability mass function (pmf) of the code bits conditioned on y and H, X 1 l and X 0 l denote the complementary sets of transmit vectors for which c l = 1 and c l = 0, respectively (note that A MT = X 1 l ∪X 0 l ). Unfortunately, computation of (2) has complexity O(|A| MT ) = O(2 R0 ), i.e., exponential in the number of transmit antennas. For this reason, several suboptimal demodulators have been proposed which promise near-optimal performance while requiring a lower computational complexity. The aim of this work is to provide a fair performance comparison of these demodulators.

III. SYSTEM CAPACITY
In order for the information rates discussed below to have interpretations as ergodic capacities, we consider a fast fading scenario where the channel H[n] is a stationary, finite-memory process. We recall that the ergodic capacity with Gaussian inputs is given by (here, I denotes the identity matrix). The non-ergodic regime (slow fading) is discussed in Section III-D.

A. Capacity of MIMO Coded Modulation
In a coded modulation (CM) system with equally likely transmit vectors x ∈ A MT and no CSI at the transmitter, the average mutual information in bits per channel use (bpcu) is given by (cf. [5]) This expression involves the conditional probability density function (pdf) (cf. (1)) In the following, we will refer to C CM as CM capacity [2] (sometimes, C CM is alternatively termed "constellation-constrained capacity"). It is seen from (4) that C CM ≤ R 0 with the "raw" bit rate R 0 = QM T ; in fact, the last term in (4) may be interpreted as a penalty term resulting from the noise and MIMO interference.
Using the fact that the mapping between the symbol vector x and the associated bit label {c 1 , . . . , c R0 } is one-to-one and applying the chain rule for mutual information [30] to (4) leads to The single-antenna equivalent of (5) served as a motivation for multilevel coding and multistage decoding, which can indeed achieve CM capacity [4]. DRAFT

B. Capacity of MIMO-BICM
Using the assumption of i.i.d. uniform code bits, the maximum rate achievable with BICM can be shown to be given by (cf. [5]) where b ∈ {0, 1} with equal probability. Since conditioning cannot reduce mutual information, a comparison of (6) and (5) reveals that The gap C CM − C BICM increases with |A| and M T and depends strongly on the symbol labeling [7].
For single-antenna BICM systems with Gray labeling, this gap has been shown to be negligible [2], [4]; however, for MIMO it can be considerably larger (see Section IV).
It can be shown that Λ l in (2) is a sufficient statistic [30] for c l conditioned on y and H. 3 Therefore, (6) can be rewritten as Hence, C BICM can be interpreted as the capacity of an equivalent channel with input c l and output Λ l .
This channel is characterized by the conditional pdf f (Λ l |c l ), which usually is hard to obtain analytically, however.

C. System Capacity and Demodulator Performance
Motivated by the interpretation of C BICM as the system capacity of BICM using the optimum MAP demodulator, we propose to measure the performance of sub-optimal MIMO-BICM demodulators via the system capacity of the associated equivalent "modulation" channel with discrete input c l and continuous outputΛ l (cf. Fig. 1). This channel is described by the conditional pdf f (Λ l |c l ). Its system capacity is defined as the mutual information between c l andΛ l , which can be shown to equal where f (Λ l ) = 1 2 1 b=0 f (Λ l |c l = b). We emphasize that the system capacity C provides a performance measure for MIMO soft demodulators that is independent of the outer channel code. In fact, it has an intuitive operational interpretation as the highest rate achievable (in the sense of asymptotically vanishing error probability) in a BICM system (cf. [2, Section III.A]) using the specific demodulator which produces Λ l . SinceΛ l is derived from y and H, the data processing inequality [30] implies that C ≤ C BICM with equality ifΛ l is a one-to-one function of Λ l . The performance of a soft demodulator can thus be measured in terms of the gap C BICM − C. Of course, the information theoretic performance measure in (8) does not take into account complexity issues and it has to be expected that a reduction of the gap C BICM − C in general can only be achieved at the expense of increasing computational complexity.
We caution the reader that the rates in (7) and (8) are sums of mutual informations for the individual code bits c 1 , . . . , c R0 carried by one symbol vector. Indeed, the pdfs f (Λ l |c l ) and f (Λ l |c l ) in general depend on the code bit position l, even though for certain systems (e.g. 4-QAM modulation) the code bit protection and LLR statistics are independent of the bit position l for reasons of symmetry. Achieving (7) and (8) thus requires channel encoders and decoders that take the bit position into account. When the channel code fails to use this information, the rate loss is small provided that the mapping protects different code bits c l roughly equally against noise and interference.

D. Non-ergodic channels
In the case of quasi-static or slow fading [31], the channel H is random but constant over time, i.e., each codeword can extend over only one channel realization. In this regime, an ergodic system capacity of the equivalent modulation channel is no longer meaningful [31], [32]. Instead we consider the outage probability P out (R) P{R H < R}, where R H is a random variable defined as I H (c l ;Λ l ).
DRAFT  Here, I H (c l ;Λ l ) denotes conditional mutual information, which is evaluated with f (Λ l |c l , H) in place of f (Λ l |c l ) (cf. (8)). Note that the ergodic system capacity C in (8) equals C = E H {R H }. The outage probability P out (R) can be interpreted as the smallest probability of error achievable at rate R [32]. A closely related concept is given by the ǫ-capacity of the equivalent modulation channel, defined as The ǫ-capacity may be interpreted as the maximum rate for which a probability of error less than ǫ can be achieved. Rates smaller than C ǫ are referred to as ǫ-achievable rates [32]. If P out (R) is a continuous and increasing function of R (which is usually the case in practice), it holds that P out (C ǫ ) = ǫ.

IV. BASELINE MIMO-BICM DEMODULATORS
In this section, we first review max-log and hard ML demodulation as well as linear MIMO demodulators and then we provide results illustrating their performance in terms of system capacities. These demodulators serve as baseline systems for later demodulator performance comparisons in Section V.
We note that max-log and hard ML MIMO demodulators have the highest complexity among all soft and hard demodulation schemes, respectively, whereas linear MIMO demodulators are computationally the most efficient ones.

A. Max-Log and Hard ML Demodulator
Applying the max-log approximation to (2) simplifies the LLR computation to a minimum distance problem, resulting in the approximate LLRs [7] Eq. (11) facilitates (hardware) implementation since it avoids the logarithm and exponential functions in (2). However, computation ofΛ l in (11) still requires two searches over sets of size |A| MT /2 = 2 R0−1 .
Hard vector ML demodulation can be performed by solvinĝ The corresponding detected code bitsĉ l are then obtained via the one-to-one mapping between code bits and symbol vectors, i.e.,ĉ = (ĉ 1 . . .ĉ R0 ) T = µ −1 (x ML ). It can be shown that the code bitsĉ l obtained by the hard ML detector correspond to the sign of the corresponding max-log LLRs in (11), i.e.,ĉ l = u(Λ l ) where u(·) denotes the unit step function. When it comes to evaluating the system capacity  with hard-output demodulators, the only difference to soft-output demodulation is the discrete nature of the outputsĉ l of the equivalent "modulation" channel, which here becomes a binary symmetric channel (BSC). Consequently, the integral overΛ l in (8b) is replaced with a summation overĉ l ∈ {0, 1}.

B. Linear Demodulators
In the following, Λ k (the ith bit in the bit label of the kth symbol x k ). Soft demodulators with extremely low complexity can be obtained by using a linear (ZF or MMSE) equalizer followed by per-layer max-log LLR calculation according tõ Here, A b i ⊂ A denotes the set of (scalar) symbols whose bit label at position i equals b,x k is an estimate of the symbol in layer k provided by the equalizer, and σ 2 k is an equalizer-specific weight (see below). We emphasize that calculating LLRs separately for each layer results in a significant complexity reduction.
In fact, calculating the symbol estimatesx k using a ZF or MMSE equalizer requires O(M 3 R ) operations; furthermore, the complexity of evaluating (13) for all code bits scales as O(M T 2 Q ), i.e., linearly in the number of transmit antennas.
Equalization-based hard bit decisionsĉ (i) k can be obtained by quantization of the equalizer outputx k with respect to A (denoted by Q(·)), followed by the demapping, i.e., ĉ Again, the detected code bits correspond to the sign of the LLRs, i.e.,ĉ [21], [22]: Here, the first stage consists of ZF equalization, i.e.,

1) ZF-based Demodulator
where the post-equalization noise vectorṽ has correlation matrix Subsequently, approximate bit LLRs are obtained according to (13) with symbol estimate 4x k = (x ZF ) k and weight factor σ 2 k = (Rṽ) k,k .  [23]: Here, the first stage is an MMSE equalizer that can be written as (cf. (14) and (15)

2) MMSE-based Demodulator
Approximate LLRs are then calculated according to (13) witĥ where W k = (W) k,k . We emphasize that throughout this paper the results for soft (and hard) MMSE demodulation are based on the unbiased MMSE equalizer output (xMMSE)k Wk . In spite of similar complexity, MMSE-based demodulation outperforms ZF-based demodulation substantially [23].

C. Capacity Results
We next compare the performance of the above baseline demodulators (i.e., max-log, hard ML, MMSE and ZF) in terms of their associated maximum achievable rate C (ergodic system capacity) in (8). In addition, the CM capacity C CM in (4), the MIMO-BICM capacity C BICM in (6), and the Gaussian input channel capacity C G in (3) are shown as benchmarks. Throughout the paper, all capacity results have been obtained for spatially i.i.d. Rayleigh fading, with all fading coefficients normalized to unit variance.
The pdfs required for evaluating (8b) are generally hard to obtain in closed form. Thus, we measured these pdfs using Monte-Carlo simulations and then evaluated all integrals numerically. Based on the results in [33], we numerically optimized the binning (used to measure the pdfs) in order to reduce the bias and variance of the mutual information estimates. The capacity results (obtained with 10 5 fading realizations) are shown in Fig. 2 in bits per channel use versus SNR ρ E s /σ 2 v . In the following, in some of the plots we show insets that provide zooms of the capacity curves around a target rate of R 0 /2 bpcu. At a target rate of 4 bpcu, the SNR required for CM and Gaussian capacity is virtually the same, whereas that for BICM is larger by about 1.3 dB. The SNR penalty of using max-log demodulation instead of soft MAP is about 0.3 dB. Furthermore, hard ML demodulation requires 2.1 dB higher SNR to achieve this rate than max-log demodulation; for soft and hard MMSE demodulation the SNR gaps to max-log are 1.2 dB and 4.1 dB, respectively, while for soft and hard ZF demodulation they respectively equal 5.1 dB and 8.2 dB. An interesting observation in this scenario is the fact that at low rates, soft and hard MMSE demodulation essentially coincides with max-log and hard ML demodulation, respectively, whereas at high rates MMSE demodulation approaches ZF performance. Moreover, soft MMSE demodulation outperforms hard ML demodulation at low rates whereas at high rates it is the other way around. Surprisingly, at very low rates soft MMSE even performs slightly better than max-log demodulation. A similar cross-over of the capacity curves occurs with hard MMSE and soft ZF demodulation. These observations reveal the somewhat unexpected fact that the demodulator performance ranking is not universal but depends on the target rate (equivalently, the target SNR), even if the number of antennas, the symbol constellation, and the labeling are fixed. Similar observations apply to 16-QAM instead of 4-QAM and to set-partitioning labeling instead of Gray labeling (see [29]). Apart from a general shift of all curves to higher SNRs, the larger constellation and/or the different labeling strategy causes an increase of the gap between CM capacity and BICM capacity. When decreasing the antenna configuration to a 2 × 2 system, we observed that soft ZF outperforms hard ML demodulation for low-to-medium rates, e.g., by about 1.7 dB at 4 bpcu with 16-QAM [29]. In this rate regime, even hard MMSE performs slightly better than hard ML demodulation.
The situation changes for the case of a 2 × 4 MIMO system with Gray-labeled 16-QAM (again Fig. 2(b). The increased SNR gap between CM and BICM capacity implied by the larger constellation is compensated by having more receive than transmit antennas (this agrees with observations in [7]). In addition, the performance differences between the individual demodulators are significantly reduced, revealing an essential distinction being between soft and hard demodulators. Having M R > M T helps the linear demodulators approach their non-linear counterparts even at larger rates, i.e., soft ZF/MMSE perform close to max-log and hard ZF/MMSE perform close to hard ML, with an SNR gap of about 2.3 dB between hard and soft demodulators. Note that in this scenario soft MMSE and soft DRAFT March 16,2009 ZF both outperform hard ML demodulation at all rates.

D. BER Performance
Even though we advocate a demodulator comparison in terms of system capacity, the cross-over of some of the capacity curves prompts a verification in terms of the BER of soft and hard MMSE demodulation as well as max-log and hard ML demodulation. We consider a 4 × 4 MIMO-BICM system with Graylabeled 4-QAM in conjunction with regular LDPC codes 5 [34] of block length 64000. For the case of soft demodulation, the LDPC codes were designed for an additive white Gaussian noise channel whereas for the case of hard demodulators the design was for a BSC. At the receiver, message-passing LDPC decoding [34] was performed. In the case of hard demodulation, the message-passing decoder was provided with the LLRsΛ where p 0 = P{ĉ l = c l } is the cross-over probability of the equivalent BSC, which was determined via Monte-Carlo simulations. Using (17) is critical in order to provide the LDPC decoder with appropriate reliability information.
The BERs obtained are shown in Fig. 3(a) for a code rate of 1/4 (corresponding to 2 bpcu) and Fig. 3 for a code rates of 3/4 (6 bpcu). We also indicate by vertical lines the respective capacity limits, i.e., the minimum SNR required for the target rate according to Fig. 2 Fig. 3(a)); however, at rate 3/4 soft MMSE performs 2 dB poorer than hard ML and 3.6 dB poorer than max-log (see Fig. 3(b)). These results confirm the capacity-based observation that there is no universal (i.e., rate-and SNR-independent) demodulator performance ranking. We note that the block error rate simulations in [35]  in this section pertain to a 4 × 4 MIMO system with 4-QAM using Gray labeling (R 0 = 8). We note that the results for asymmetric 2 × 4 MIMO systems with 16-QAM (shown in [29] but not here) essentially confirm the general distinction between hard and soft demodulators observed in connection with Fig. 2(b).

A. List-based Demodulators
In order to save computational complexity, (11) can be approximated by decreasing the size of the search set A MT . Usually, this is achieved by generating a (non-empty) candidate list L ⊂ A MT and restricting the search in (11) to this list, i.e., The computational complexity of the metric evaluations and minima searches in (18)  For fixed list size, the performance can further strongly depend on which symbol vectors are actually included in L. In the following, we consider two types of list-based demodulation where the candidate list is obtained using sphere decoding and bit flipping, respectively.

1) List Sphere Decoder (LSD):
The LSD proposed in [14] uses a simple modification of the harddecision sphere decoder [36] (12)). If the lth bit in the labels of all x ∈ L equals 1, the set L ∩ X 0 l is empty and (18) cannot be evaluated. Since in this case there is strong evidence for c l = 1 (at least if |L| is not too small), the corresponding LLRΛ l is set to a prescribed positive valueΛ ≫ 0. Analogously,Λ l = −Λ in case L ∩ X 1 l equals the empty set.
While the LSD may offer significant complexity savings compared to max-log demodulation, statements about its computational complexity are difficult and depend strongly on the actual implementation of the sphere decoder as well as the choice of the list size. We note that the case |L| = 2 R0 = |A MT | implies L = A MT ; thus, L ∩ X b l = X b l such that (18) equals the max-log demodulator in (11). The other extreme is a list size of one, i.e., L = {x ML } (cf. (12)), in which case either L∩X 0 l or L∩X 1 l is empty (depending on the bit label ofx ML ); here,Λ l = (2ĉ l − 1)Λ whereĉ = (ĉ 1 . . .ĉ R0 ) T = µ −1 (x ML ) and thus the LSD output is equivalent to hard ML demodulation (except for the choice ofΛ, which is irrelevant, however, for capacity).
Capacity Results. Fig. 4 shows the maximum rates achievable with an LSD for various list sizes. BICM and soft MMSE capacity are shown for comparison. Note that with 4-QAM and M T = 4, |L| = 256 and |L| = 1 correspond to max-log and hard ML demodulation, respectively. It is seen that with increasing list size the gap between LSD and max-log decreases rapidly, specifically at high rates. In particular, the LSD with list sizes of |L| ≥ 8 is already quite close to max-log performance. However, even with large list sizes LSD is outperformed by soft MMSE demodulation at sufficiently low rates. Specifically, below 3.6 dB, 1.5 dB, and −0.2 dB the system capacity of soft MMSE demodulation is higher than that of LSD with list size 2, 4, and 8, respectively. Similar observations apply to other antenna configurations and symbol constellations (see [29]).

2) Bit Flipping Demodulators:
Another way of generating the candidate list L, proposed in [20], is to flip some of the bits in the label of the hard ML symbol vector estimatex ML in (12). More generally, the ML solutionx ML can be replaced by a symbol vectorx ∈ A MT obtained with an arbitrary hard-output demodulator (e.g., hard ZF and MMSE demodulation). Letĉ = µ −1 (x) denote the bit label ofx. The that bit flipping demodulation performs close to max-log at low rates and that hard ML and hard MMSE initialization are very close to each other for rates up to 6 bpcu (SNR up to 7 dB); in fact, below 5 bpcu hard MMSE initialization performs slightly better than hard ML initialization while at higher rates ML initialization gives significant gains. To maintain this behavior for larger constellations and more antennas, the maximum Hamming distance D has to increase with increasing R 0 (see [29]).

B. Lattice-Reduction-Aided Demodulation
Lattice reduction (LR) is an important technique for improving the performance or complexity of MIMO demodulators [15], [16]    an alternative basis which amounts to a transformation of the system model (1) prior to demodulation; the advantage of such an approach is that the transformed channel matrix (i.e., the reduced basis) has improved properties (e.g., smaller condition number). An efficient algorithm to obtain a reduced basis was proposed by Lenstra, Lenstra, and Lovász (LLL) [37]. The overall computational complexity of LR-aided demodulation depends strongly on the complexity of the LLL algorithm which has been assessed in [38].
A comparison of different LR methods in the context of MIMO hard demodulation was provided in [39].
LR is usually formulated for the equivalent real-valued model; hence, for now we assume all quantities to be real-valued. Any lattice basis transformation can be formulated in terms of a unimodular transformation matrix T, i.e., a matrix with integer entries and det(T) = ±1. Denoting the "reduced channel" byH = HT and defining z = T −1 x, the system model (1) can be rewritten as Under the assumption x ∈ Z MT (which for QAM can be ensured by an appropriate offset and scaling), the unimodularity of T guarantees z ∈ Z MT and hence any demodulator can be applied to the better-behaved transformed system model on the right-hand side of (19). Here, we adopt the LR-aided hard-output MMSE demodulator developed in [16]. LR-aided soft demodulators (cf. [17]) are essentially list-based [18], [19]. In fact, many of these methods apply bit flipping (cf. Section V-A2) to an LR-aided hard demodulator output.
Capacity Results. Fig. 6 shows the capacity results for hard and soft LR-aided MMSE demodulation. demodulator output (cf. Section V-A2). It is seen that with hard MMSE demodulation, LR is beneficial only at SNRs above 8 dB (rates higher than 4.4 bpcu). At rates higher than about 6.8 bpcu, LR-aided hard demodulation even outperforms soft MMSE demodulation. Bit flipping is helpful particularly at low rates. Thus, at low rates LR-aided soft MMSE demodulation approaches max-log performance and outperforms hard ML. When flipping up to D = 2 bits, LR-aided soft demodulation reveals a significant performance advantage over soft MMSE demodulation without LR which is especially pronounced in the high-rate regime.

C. Semidefinite Relaxation (SDR) Demodulation
Semidefinite relaxation allows to approximately solve the hard ML demodulation problem in (12) [40], [41] with polynomial worst-case complexity based on convex optimization techniques. In this paper, we focus on the quasi-ML hard-output demodulator and its soft extension proposed in [13] that approximates max-log demodulation (with an overall worst-case complexity of O(R 4.5 0 )). We note that this approach applies only to BPSK or 4-QAM alphabets and employs a randomization procedure described in detail in [41].
Capacity Results. In Fig. 7 we show the system capacity for hard and soft SDR demodulation (as described in [13]) using randomization with 25 trials. It can be seen that hard SDR demodulation coincides with hard ML at low rates; at high rates it performs worse than hard ML but still better than soft MMSE.
Soft SDR performs consistently better than hard SDR; while at low rates it coincides with soft MMSE demodulation, at rates higher than 6.2 bpcu its capacity lies between hard ML and soft MMSE.
Specifically, the use of the ℓ ∞ norm avoids expensive squaring operations. The hard ℓ ∞ -norm demodulator delivers the symbol vector estimatex Here, Q and R are the M R ×M T unitary and M T ×M T upper triangular factors in the QR decomposition H = QR of the channel matrix. Note that the solution to (20) may not be unique, in which case one solution is selected at random.
We propose to generate soft outputs by using the ℓ ∞ -norm sphere decoder to determinẽ 1} and then evaluating the approximate LLRs using the ℓ 2 norm: Capacity Results. Fig. 8 shows the system capacity for hard and soft ℓ ∞ -norm demodulation. For the 4 × 4 case with 4-QAM in Fig. 8(a), hard and soft ℓ ∞ -norm demodulation perform within 1 dB of hard ML and max-log, respectively, particularly for high rates. However, at low rates (below 4 bpcu) ℓ ∞ -norm demodulation is even outperformed by its corresponding MMSE counterpart. An interesting observation  applies to the 2 × 4 case with 16-QAM depicted in Fig. 8(b). All soft-output baseline demodulators perform almost identical and the same is true for all hard-output baseline demodulators, i.e., there is only a distinction between soft and hard demodulation (cf. Fig. 2(b)). However, soft and hard ℓ ∞ -norm demodulation perform significantly worse in this asymmetric setup, specifically at low to medium rates.
At 2 bpcu, soft ℓ ∞ -norm demodulation requires 1.75 dB higher SNR than max-log and soft MMSE and hard ℓ ∞ -norm demodulation requires 2.6 dB higher SNR than hard ML/MMSE.

E. Successive and Soft Interference Cancelation
Successive interference cancelation (SIC) is a hard-output demodulation approach that became popular with the V-BLAST (Vertical Bell Labs Layered Space-Time) system [26]. Within one SIC iteration, only that layer with the largest post-equalization SNR is detected and its contribution to the receive signal is subtracted (canceled). A SIC implementation that replaces the ZF-based algorithm from [26] with an MMSE-based scheme and performs efficient layer ordering according to signal-to-interference-plusnoise-ratio (SINR) was presented in [27]. Suboptimal but more efficient SIC schemes are discussed in [28].
In order to mitigate the error propagation inherent to SIC, [24] proposed a parallel soft interference cancelation (SoftIC) scheme. SoftIC is an iterative method that alternatingly performs (i) parallel MIMO interference cancelation based on soft symbols and (ii) computation of improved soft symbols using the output of the interference cancelation stage. Each iteration of this scheme has a complexity that scales linearly with the number of antennas. Here, we use a modification that builds upon bit-LLRs. LetΛ March 16,2009 denote the LLR for the ith bit in layer k obtained in the jth iteration. Symbol probabilities can then be obtained as , with b i (x) denoting the ith bit in the label of x ∈ A, leading to the soft symbol estimatê Soft interference cancelation for each layer then yields where h k denotes the kth column of H. Finally, updated LLRsΛ are calculated from (21) based on a Gaussian assumption for the residual interference plus noise (for implementation details we refer to [24]). In contrast to [24], we suggest to initialize the scheme with the LLRs obtained by a low-complexity soft demodulator, e.g., the soft ZF detector described in Section IV-B. The complexity per iteration of the above SoftIC algorithm is given by O(2 Q M T (Q + M R )) (plus the operations required for the initial LLRs).
Capacity Results. In Fig. 9, we display capacity results for (hard) MMSE-SIC with detection ordering as in [28] (there referred to as 'MMSE-BLAST') and for SoftIC demodulation (initialized using soft ZF demodulation, also shown for reference). Hard MMSE-SIC demodulation is seen to perform similarly to hard ML demodulation at low rates. While at high rates MMSE-SIC shows a noticeable gap to hard ML, it can outperform both soft MMSE and SoftIC in this regime. When using 16-QAM instead of 4-QAM, at very low rates MMSE-SIC tends to even slightly outperform hard ML demodulation (see [29]). SoftIC (with 8 iterations) is superior to MMSE-SIC up to rates of 6.7 bpcu. At low rates, SoftIC even performs slightly better than max-log demodulation and essentially coincides with BICM capacity. For the chosen system parameters, SoftIC furthermore beats soft MMSE over the whole SNR range shown.
This statement does not hold in general, however. For example, with 16-QAM the SoftIC performance drops below that of soft MMSE demodulation at high rates (see [29]).

VI. IMPERFECT CHANNEL STATE INFORMATION
We next investigate the ergodic system capacity (8) for the case of imperfect channel state information (CSI). In particular, we consider training-based estimation of the channel matrix H and the noise variance σ 2 v and assess how the amount of training influences the performance of the various demodulators. Training-based Channel Estimation. In order to estimate the channel, the transmitter sends N p > M T training vectors 6 which are arranged into a full rank M T × N p training matrix X p . We assume that the transmit power per channel use for training and actual data is the same such that the Frobenius norm [42] of X p equals X p 2 F = N p E s . Assuming that the channel stays constant for the duration of one block (which contains training and actual data), the M R × N p receive matrix Y p induced by the training is given by Here, the M R × N p matrix V contains the noise received during the training period.
Using (22), the least-squares channel estimate (identical to the ML estimate under a Gaussian i.i.d. assumption for the noise) is computed as [43] This estimate is unbiased and its mean square error equals where the lower bound is attained with orthogonal training sequences, i.e., X p X H p = NpEs MT I (we recall that here ρ = E s /σ 2 v denotes the SNR). The estimated channel matrixĤ is then used to obtain the noise variance estimatê which essentially amounts to measuring the mean power of Y p in the (N p −M T )-dimensional orthogonal complement of the range space of X H p . The noise variance estimate is also unbiased and has MSE which should be noted to be independent of the transmit power.
The estimatesĤ in (23) andσ 2 v in (24) then replace the true channel matrix and noise variance in the computation of the demodulator outputs, i.e., we consider mismatched demodulation. One could also change the metric in order to take into account the channel imperfections as in [44]; however, this is beyond the scope of this paper.
Capacity Results. The section provides numerical results for the ergodic system capacity of maxlog, hard ML, and soft MMSE demodulation with perfect and estimated CSI (using orthogonal training sequences). Throughout, a 4 × 4 MIMO system with 4-QAM and Gray labeling is considered (R 0 = 8).
Results for other demodulators with imperfect CSI are provided in [29]. Fig. 10 shows the maximum achievable rates versus SNR for a fixed training sequence length of N p = 5 (this corresponds to the worst case with minimum amount of training). It is seen that for all three detectors imperfect CSI results in a significant performance loss, e.g., at 4 bpcu the SNR loss for max-log, hard ML, and soft MMSE is 3.9 dB, 3.1 dB, and 4.95 dB, respectively. In fact, in the considered worst case imperfect CSI setup the performance advantage of soft MMSE demodulation over hard ML demodulation at low rates is much less pronounced; note that the cross-over between hard ML and soft MMSE performance at an SNR of about 6.3 dB shifts from 4.9 bpcu for perfect CSI to 3 bpcu for the case of imperfect CSI. The performance losses for all demodulators tend to be smaller at high rates, which may be partly attributed to the fact that CSI becomes more accurate with increasing SNR whereas this is not the case for the noise variance estimate. In general it can be observed that the performance loss of hard ML is the smallest while soft MMSE and max-log performance deteriorates much stronger; note that hard ML does not use the noise variance and hence is more robust to estimation errors in σ 2 v . Here, hard ML comes within less than 1.3 dB of max-log performance. In contrast, soft MMSE uses the imperfect channel and the noise variance estimate in the MMSE equalization stage (cf. (16)) and in the LLR calculation and is thus most strongly affected.
To investigate the impact of the amount of training, Fig. 11(a) and (b) depict the minimum SNRs required by the individual demodulators to achieve target rates of 2 bpcu and 6 bpcu, respectively, versus the number of training vectors N p . It is seen that for all demodulators, the required SNR decreases rapidly with increasing amount of training. Yet, even for N p = 20 there is a significant gap of 1 to 2 dB to perfect  CSI performance (indicated by corresponding horizontal, gray lines). Here, soft MMSE outperforms hard ML at 2 bpcu and gets closer to max-log performance with a larger amount of training. In contrast, at 6 bpcu hard ML performs consistently better than soft MMSE.
We note that further capacity results for the demodulators discussed in Section V are provided in [29].
One of the more interesting observations obtained from those results is that for training duration N p = 5, the LSD with list size |L| ≥ 8 consistently outperforms max-log (at least in the MIMO setup considered).
Although this is not the case for larger training duration, LSD with |L| = 8 performs mostly very close to max-log. DRAFT March 16,2009 VII. QUASI-STATIC FADING In this section we provide a demodulator performance comparison for quasi-static fading MIMO channels based on the outage probability P out (R) in (9) and the ǫ-capacity C ǫ in (10). The setup considered (4 × 4 MIMO with Gray-labeled 4-QAM) is the same as before apart from the spatially i.i.d. Rayleigh fading channel which now is assumed to be quasi-static. The outage probability P out (R) was measured using 10 5 blocks (affected by independent fading realizations), each consisting of 10 5 symbol vectors.
To keep the presentation concise, we restrict to the baseline demodulators from Section IV. Fig. 12(a) shows the outage probability versus SNR ρ for target rates of R = 2 bpcu and R = 6 bpcu. The degradation of soft MMSE with increasing rate is clearly visible in Fig. 12(b), which shows ǫ-capacity versus SNR for a maximum outage probability of ǫ = 10 −2 . Some of the behavior of the ǫcapacity curves is qualitatively very similar to the results obtained for the ergodic capacity (cf. Fig. 2(a)): soft MMSE demodulation outperforms hard ML demodulation (by up to 2.2 dB for rates less than 3.8 bpcu) while at high rates it is the other way round. Furthermore, for low rates soft MMSE demodulation essentially coincides with max-log whereas at high rates it approaches soft ZF performance.

VIII. KEY OBSERVATIONS AND DESIGN GUIDELINES
Based on the results of the previous sections, we summarize some key observations and provide practical guidelines for system design.
Soft MMSE demodulation was seen to approach max-log performance for low rates, both in the ergodic and the quasi-static regime and for various system configurations (see also [29]). Moreover, soft MMSE demodulation is very attractive since it has the lowest complexity among all the suboptimal demodulators discussed in this work. Therefore, soft MMSE demodulation is arguably the demodulator of choice when designing MIMO-BICM systems with outer codes of rate less than 1/2. For the case of imperfect CSI, however, care should be taken to ensure a sufficient amount of training since otherwise soft MMSE may suffer severely from inaccurate estimates of the channel matrix and the noise variance. We note that soft ZF demodulation performed consistently poorer than soft MMSE at the same computational cost; thus, there appear to be no reasons to prefer soft ZF in practical implementations.
Soft MMSE has an even stronger case against its competitors for asymmetric MIMO systems (i.e., M T < M R ), where it performs close to BICM capacity for all rates. Fig. 13 compares a 4 × 4 MIMO system using 4-QAM (system I) and a 2 × 4 system using 16-QAM (system II), both using Gray labeling and achieving R 0 = 8. Whereas at low rates soft MMSE demodulation performs better with system I than with system II, it is the other way around for high rates. For example, at 6 bpcu system II requires soft MMSE demodulation below 4 bpcu system I is preferable, whereas above 4 bpcu it is advantageous to deactivate two transmit antennas and switch to the 16-QAM constellation (system II). We note that with max-log and hard ML demodulation, system II performs consistently worse than system I.
The only regime where soft MMSE suffers from a noticeable performance loss is symmetric systems at high rates, where no low-complexity demodulation scheme is able to achieve max-log performance.
These observations apply to the ergodic and quasi-static scenario. However, with perfect CSI the LSD and soft ℓ ∞ -norm demodulation come reasonably close to max-log. The LSD has the additional advantages of being able to trade off performance for complexity reduction. Furthermore, note that for system I hard ML (i.e., the LSD with list size 1) outperforms most suboptimal soft demodulators for rates larger than 6 bpcu.
As a general rule of thumb, we conclude that at low rates linear soft demodulation is sufficient and generally preferable due to its low computational complexity. At high rates non-linear demodulator structures provide better performance, even when they deliver hard rather than soft outputs. If complexity is no issue, max-log demodulation is mostly superior to all other demodulators since it yields the highest rates over a wide range of system parameters and SNRs. Notable exceptions are SoftIC (which is a low complexity demodulator) and LSD which have the potential to outperform max-log at low rates and for imperfect CSI, respectively.  DRAFT

IX. CONCLUSION
We provided a comprehensive performance assessment and comparison of soft and hard demodulators for non-iterative MIMO-BICM systems. Our comparison is based on the information-theoretic notion of system capacity, which can be interpreted as the maximum achievable rate of the equivalent "modulation" channel that comprises modulator, physical channel, and demodulator. As a performance measure, system capacity has the main advantage of being independent of any outer code. Extensive simulation results show that a universal demodulator performance ranking is not possible and that the demodulator performance can depend strongly on the rate (or equivalently the SNR) at which a system operates. In addition to ergodic capacity results, we investigated the non-ergodic fading scenario in terms of outage probability and ǫ-capacity and analyzed the robustness of certain demodulators under imperfect channel state information. Our observations provide new insights into the design of MIMO-BICM systems (i.e., choice of demodulator, number of antennas, and symbol constellation). Moreover, our approach sheds light on issues that have not been apparent in the previously prevailing BER performance comparisons for specific outer codes. For example, an important key observation is that with low-rate outer codes soft MMSE demodulation is preferable over other demodulators since it has low complexity but nonetheless comes very close to max-log performance.

ACKNOWLEDGMENT
The authors thank Gottfried Lechner for kindly providing his LDPC decoder implementation.