Capacity Bounds for Amplitude-Constrained AWGN MIMO Channels with Fading

Abstract

We evaluate capacity bounds for multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) fading channels subject to input amplitude constraints. We focus on two practical cases, in which the transmitter: (i) employs a single antenna amplifier, which induces a constraint on the norm of the input vector, and (ii) it employs multiple amplifiers, one per antenna, which leads to independent constraints on the amplitude of each input vector entry. For both cases, we evaluate the asymptotic capacity gap between upper and lower bounds at high signal-to-noise ratio.

A Sphere Packing Bound for Vector Gaussian Fading Channels under Peak Amplitude Constraints

“…Furthermore, in [18] the capacity bounds are further generalized for n-dimensional MIMO fading channels subject to any arbitrary peak amplitude constraint. Finally, in [19] we refine the results presented in [18] for two particular constraints of practical interest.…”

“…Intuitively, we achieve better results than those in [18] by considering an upper bound that depends on a smaller constraint region S such that D ⊃ S ⊃ HX . Although the resulting asymptotic capacity gap in [19] is smaller compared to the past literature, it can be far from zero. Moreover, it widens as N grows larger and the derived upper bounds are valid just for the mentioned specific cases.…”

“…The main disadvantage of these bounds is that the more D differs from HX , the less accurate the bounds become. In [19], by extending the McKellips-Type upper bound of [15] to MIMO systems affected by fading, we improve the results of [18] for specific cases of the PA constraint and for the TA constraints. Intuitively, we achieve better results than those in [18] by considering an upper bound that depends on a smaller constraint region S such that D ⊃ S ⊃ HX .…”

“…We derive the sphere packing (SP) upper bound by extending the results in [20] to MIMO systems. We prove that the gap between the derived upper bound and the best lower bound from the literature [19], is asymptotically tight. Indeed, as the signal-to-noise ratio (SNR) goes to infinity, the bounds' gap vanishes for any channel matrix, any dimension of the MIMO system, and any convex constraint region.…”

“…In the next section, we define an upper bound that improves upon those of [19] and that is also far more general, as it is for the duality upper bounds in [18]. We prove that our upper bound provides an asymptotic gap equal to zero for any convex constraint region, any channel matrix H, and any MIMO dimension N.…”