The wideband regime of bit-interleaved coded modulation (BICM) in Gaussian channels is studied. The Taylor expansion of the coded modulation capacity for generic signal constellations at low signal-to-noise ratio (SNR) is derived and used to determine the corresponding expansion for the BICM capacity. Simple formulas for the minimum energy per bit and the wideband slope are given. BICM is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes. The minimum energy per bit using standard Gray mapping on M-PAM or M^2-QAM is given by a simple formula and shown to approach -0.34 dB as M increases. Using the low SNR expansion, a general trade-off between power and bandwidth in the wideband regime is used to show how a power loss can be traded off against a bandwidth gain.
Bit-interleaved coded modulation (BICM) was originally proposed by Zehavi [1] and further elaborated by Caire et al. [2] as a practical way of constructing efficient coded modulation schemes over non-binary signal constellations. Reference [2] defined and computed the channel capacity of BICM under a sub-optimal non-iterative decoder, and compared it to the coded modulation capacity, assuming equiprobable signalling over the constellation. When natural reflected Gray mapping was used, the BICM capacity was found to be near optimal at high signal-to-noise ratio (see Figure 1(a)). Nevertheless, plots of the BICM capacity as a function of the energy per bit for reliable communication (see Figure 1(b)) reveal the suboptimality of BICM with the non-iterative decoder of [1], [2] for low rates, that is in the power-limited or wideband regime.
Recent work by Verdú [3] presents a detailed treatment of the wideband regime. He studied the minimum bit energy-to-noise ratio E b N 0 min for reliable communication and the wideband slope, i.e., the first-order expansion of the capacity for low E b N 0 min , under a variety of channel models and channel state information (CSI) assumptions. These results are obtained by using a second-order expansion of the channel capacity at zero signal-to-noise ratio (SNR). Furthermore, using these results, he obtained a general tradeoff between data rate, power and bandwidth in the wideband regime. In particular, Verdú [3] studied the bandwidth penalty incurred by using suboptimal signal constellations in the low-power regime. An implicit assumption of this tradeoff was that the power cannot change together with the bandwidth.
Motivated by the results of Figure 1(b) and by Verdú’s analysis [3], in this paper, we give an analytical characterization of the behaviour of BICM in the low-power regime. Studying the behaviour of BICM at low rates may prove useful in the design of multi-rate communication systems where rate adaptation is carried out by modifying the binary code, while keeping the modulation unchanged. In the process, we derive a number of results of independent interest for coded modulation over the Gaussian channel. In particular, the first two coefficients of the Taylor expansion of the coded modulation capacity for arbitrary signal constellations at zero SNR are derived, and used to obtain the corresponding coefficients for BICM. We also obtain a closed form expression for the minimum E b N 0 for BICM using QAM constellations with natural reflected Gray mapping, and we show that for large constellations it approaches -0.34 dB, resulting in a 1.25 dB power loss with respect to coded modulation. Using these results, we derive the tradeoff between power and bandwidth in the wideband regime that generalizes the results of [3] to capture the effects of changing both power and bandwidth.
This paper is organized as follows. Section II introduces the system model, basic assumptions and notation. Section III defines the wideband regime, and presents the low-SNR expansion for both coded modulation and BICM. Section IV introduces the general trade-off between power and bandwidth. Concluding remarks appear in Section V. Proofs of various results are in the Appendices.
We consider a complex-valued, discrete-time additive Gaussian noise channel with fading.
The k-th channel output y k is given by
where x k is the k-th channel input, h k a fading coefficient, and z k an independent sample of circularly symmetric complex-valued Gaussian noise of unit variance; SNR denotes the average signal-to-noise ratio at the receiver. The transmitted, received, noise and fading samples, are realizations of the random variables X, Y, Z and H. The fading coefficients h k are independently drawn from a density p H (h k ) and are assumed known at the receiver. For future use we define the squared magnitudes of the fading coefficients by χ k = |h k | 2 . For a given fading realization h k , the conditional output probability density is given by
The channel inputs are modulation symbols drawn from a constellation set X with probabilities P X (x). We denote the cardinality of the constellation set by M = |X | and by m = log 2 M the number of bits required to index a modulation symbol. We define the constrained capacity C X (or coded modulation capacity) as the corresponding mutual information between channel input and output, namely
where the expectation is performed over X, Z and H. If the symbols are used with equal probabilities, i. e. P X (x) = M -1 , we refer to the constrained capacity as uniform capacity, and denote it by C u X . As we will see later, it proves convenient to consider general constellation sets with arbitrary first and second moments, respectively denoted by µ 1 (X ) and µ 2 (X ), and given by
Practical constellations have zero mean, i. e. µ 1 (X ) = 0, and unit energy, that is µ 2 (X ) = 1.
In order to transmit at rates close to the coded modulation capacity, multi-level codin
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