The SIMO Block Rayleigh Fading Channel Capacity Scaling with Number of Antennas, Bandwidth and Coherence Length

TL;DR

The channel coherence block length plays a pivotal role in modulation selection and the capacity gap between coherent and non-coherent channels.

Abstract

This paper studies the capacity scaling of non-coherent Single-Input Multiple-Output (SIMO) independent and identically distributed (i.i.d.) Rayleigh block fading channels versus bandwidth ($B$), number of receive antennas ($N$) and coherence block length ($L$). In non-coherent channels (without Channel State Information --CSI) capacity scales as $Θ\left(\min(B,\sqrt{NL},N)\right)$. This is achievable using Pilot-Assisted signaling. Energy Modulation signaling rate scales as $Θ\left(\min(B,\sqrt{N})\right)$. If $L$ is fixed while $B$ and $N$ grow, the two expressions grow equally and Energy Modulation achieves the capacity scaling. However, Energy Modulation rate does not scale as the capacity with the variable $L$. The coherent channel capacity with a priori CSI, in turn, scales as $Θ\left(\min(B,N)\right)$. The coherent channel capacity scaling can be fully achieved in non-coherent channels when $L\geqΘ(N)$. In summary, the channel coherence block length plays a pivotal role in modulation selection and the capacity gap between coherent and non-coherent channels. Pilot-Assisted signaling outperforms Energy Modulation's rate scaling versus coherence block length. Only in high mobility scenarios where $L$ is much smaller than the number of antennas ($L\llΘ(\sqrt{N})$), Energy Modulation is effective in non-coherent channels.

Figures (6)

• Figure 1: Block-fading frequency-selective channel output with bandwidth $B\Delta f$, coherence length $L$ and codeword time $L/\Delta f$.
• Figure 2: Scaling of rate vs $N$, with fixed $L$ and very large $B$
• Figure 3: Achievable rate exponents with $0<\tau<1$.
• Figure 4: BER vs $N$ for different schemes.
• Figure 5: Bit rate vs $N$ for different schemes.

Theorems & Definitions (32)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Remark 1