Bit-Interleaved Coded Modulation (BICM) Systems

• BICM is a digital communication technique that interleaves coded bits before mapping to constellations, enhancing error correction and spectral efficiency.
• Wideband analysis shows that optimal geometric shaping and NBC labeling allow BICM systems to approach the Shannon limit at low SNR.
• At medium SNR, probabilistic shaping improves throughput, offering practical performance gains despite its limited role in low-SNR regimes.

Bit-Interleaved Coded Modulation (BICM) systems are foundational in modern digital communications, providing a high level of flexibility, performance, and implementation efficiency, particularly when operating with high-order modulations and advanced coding. BICM separates channel coding and modulation by interleaving the encoded bits before mapping them onto a signal constellation, enabling robust error-correction and mitigating channel impairments such as fading. The architecture, analysis, and optimization of BICM have been the subject of extensive research, especially in the wideband regime, with major results on capacity, asymptotic behaviors, input shaping, and system design.

1. Mathematical Formulation and Capacity of BICM

BICM is characterized by transmitting sequences of coded bits that are randomly interleaved and mapped into $M=2^m$-ary constellation symbols for an additive white Gaussian noise (AWGN) or fading channel. The channel model is typically $Y=H \odot X + Z$, where $X$ is the transmitted symbol, $H$ models fading (possibly constant for AWGN), and $Z \sim \mathcal{N}(0, N_0 I)$ is Gaussian noise (Agrell et al., 2010). The general BICM capacity for input alphabet $\mathcal{X}$, labeling $L$, and symbol distribution $P$ is given by

$$I^{\textrm{BI}}_{\Omega}(\textrm{SNR}) = \sum_{k=0}^{m-1} I(C_k; Y) = \sum_{k=0}^{m-1} \mathbb{E}_{C_k, H, Y} \left[ \log_2 \frac{p_{Y|C_k, H}(Y)}{p_{Y|H}(Y)} \right],$$

where $C_k$ denotes the $k$-th bit of the labeling (Agrell et al., 2010). For uniform symbol probabilities and AWGN, the expressions can be further simplified, and the mutual information reduces to a computable sum over bit positions. The BICM capacity generally satisfies $$I^{\textrm{BI}} \leq I^{\textrm{CM}} \leq C^{\textrm{AW}}$$ (CM: coded-modulation capacity, AW: Shannon channel capacity).

At low SNR (the wideband regime), the BICM capacity admits a linear Taylor expansion (asymptotic slope) parameterized by

$I^{\textrm{BI}}(\rho) = \alpha \rho + o(\rho),$

where $\rho = E_s/N_0$ is the SNR per channel use, and $\alpha$ is the first-order coefficient governing the minimum $E_b/N_0$ for reliable communication (Agrell et al., 2012). Achieving the Shannon limit ($E_b/N_0 = \ln 2 = -1.59$ dB) requires $\alpha = 1/\ln 2$.

2. Wideband (Low-SNR) Behavior and First-Order Optimal Constellations

A core result is the explicit characterization of BICM's asymptotic behavior in the wideband regime (Agrell et al., 2010, Agrell et al., 2012). The first-order coefficient $\alpha$ for BICM with possibly nonuniform symbol probabilities $P$ is

$$\alpha = \frac{\ln 2}{E_s} \sum_{k=0}^{m-1} \left\| \mathbb{E}[X \mid C_k=0] - \mathbb{E}[X \mid C_k=1] \right\|^2,$$

where $E_s = \mathbb{E}[\|X\|^2]$. The minimum achievable $E_b/N_0$ is then $1/\alpha$, with the Shannon limit realized if and only if the input distribution and labeling satisfy stringent geometric and algebraic constraints.

First-Order Optimal (FOO) constellations for BICM—those achieving the Shannon limit—are exactly characterized as zero-mean linear projections of the $m$-dimensional hypercube, i.e., there exists an $N \times m$ real matrix $V$ such that

$$x_i = (2n(i) - \mathbf{1}_m)V, \qquad \sum_i P_i x_i = 0,$$

where $n(i)$ encodes the NBC (natural binary code) bits of symbol $i$ (Agrell et al., 2012). No other constellation and probability combination achieves the limiting performance at zero rate.

A comprehensive enumeration found, for example, that in uniform 8-PAM, only NBC labeling is FOO; none of the 26 labeling classes in 8-PSK is FOO for $M>4$ since the dimensionality is insufficient for an orthogonal hypercube embedding (Agrell et al., 2010).

3. Impact of Probabilistic and Geometric Shaping

A principal insight is that, at asymptotically low SNR, probabilistic shaping does not introduce additional degrees of freedom for optimizing the wideband mutual information of BICM systems beyond what is possible via geometric (constellation) shaping. This result is formalized via a new linear transform (dependent on the bit probabilities) that maps any shaped constellation $[\mathcal{X}, P]$ to a geometrically shaped uniform-probability constellation $[\mathcal{S}, 1/M]$ with identical low-SNR parameters $(\mu, E_s, \alpha)$ (Agrell et al., 2012):

$S_i = \sum_{j=0}^{M-1} g_{i,j} \sqrt{P_j} x_j,$

with $g_{i,j}$ determined only by the bit probabilities and labeling.

Thus, in the wideband regime, all first-order optimality is determined by the projected geometry; probabilistic shaping can only influence the higher-order, medium-SNR regime. At moderate SNR, shaped constellations with suitably chosen $P \neq 1/M$ can yield strict generalized mutual information (GMI) improvements over pure uniform-QAM or QPSK (Agrell et al., 2012).

4. Binary Labeling, Hadamard Transform, and Classification

The choice of binary labeling—mapping bits to constellation symbols—is critical. For $M$-PAM, the NBC is unique in achieving FOO status (maximal $\alpha$); Gray or folded binary labeling incurs a fixed penalty, with $\alpha = (3M^2)/(4(M^2-1)) \cdot \ln 2$ (Agrell et al., 2010). Via the Hadamard transform, a geometric criterion emerges: a BICM constellation is FOO if and only if all Hadamard coefficients vanish except at the "single-bit" positions $j\in\{1,2,4,\dots,2^{m-1}\}$.

The complete classification for 8-PAM reveals 72 classes of binary labelings (only one of which is FOO), and 26 for 8-PSK (none FOO for $M>4$). This high granularity is crucial for analyzing sensitivity to labeling at low rates.

5. Shaping and Mutual Information at Medium SNR

Although probabilistic shaping confers no first-order gain at zero SNR, at operational SNRs, numerical studies demonstrate that shaping the bit or symbol probabilities can recover nearly the full 1 dB canonical gap between BICM with uniform bits and full AWGN capacity at modest rates (up to about 2 bits/symbol), e.g., in 8-PAM with Gray or NBC labeling (Agrell et al., 2010, Agrell et al., 2012).

The optimal constellation, labeling, and bit or symbol probabilities at given SNR can be found via information-maximizing procedures or low-complexity optimization techniques such as BACM (bit-alternating convex-concave method) for higher-order constellations. This is particularly pronounced for higher-order QAM or PSK, where designed probabilities yield significant practical throughput gains over uniform input (Böcherer et al., 2012).

6. Design Guidelines and System Implications

Key design implications stemming from the above theoretical results include:

• Labeling: For PAM/QAM in BICM under AWGN or mild fading, use NBC labeling at low rates to achieve the Shannon limit. For QAM, this extends via the Kronecker product structure of PAM axes (Agrell et al., 2010).
• Probabilistic shaping: Employ probabilistic shaping only to improve medium-SNR performance; recognize that no shaping can enable BICM to reach the Shannon limit at zero rate unless the constellation is a zero-mean projection of the binary hypercube (Agrell et al., 2012).
• PSK: For $M$-PSK, it is impossible to construct a FOO BICM constellation for $M>4$. This reflects geometric limitations of constant-energy, low-dimensional constellations (Agrell et al., 2010).
• Low-SNR evaluation: The first-order asymptotic coefficient $\alpha$ (as derived) should be explicitly computed for candidate BICM constellations to benchmark their distance to the Shannon limit.

7. Broader Context and Related Results

The asymptotic equivalence between geometric and probabilistic shaping for low-SNR BICM is consistent with foundational results on mutual information and coded modulation in the wideband regime (0710.4046). Key approaches such as the derivative-of-BICM mutual information formalize the sensitivity of mutual information to SNR and further tie low-SNR performance directly to the statistical moments of subconstellations for each binary labeling (0708.2026).

For practical system design, optimization of labeling and input distribution often combines exhaustive search (for moderate $M$) with algorithmic methods for larger alphabets, acknowledging the exponential growth in labeling classes. Theoretical classification guides constellation choice and bit mapping for capacity-approaching designs, especially in highly spectral- or power-constrained scenarios.

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References:

• "Signal Shaping for BICM at Low SNR" (Agrell et al., 2012)
• "On the BICM Capacity" (Agrell et al., 2010)
• "Derivative of BICM Mutual Information" (0708.2026)
• "Bit-interleaved coded modulation in the wideband regime" (0710.4046)
• "An Efficient Algorithm to Calculate BICM Capacity" (Böcherer et al., 2012)