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

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 (<inline-formula> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula>), number of receive antennas (<inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>) and coherence block length (<inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>). In non-coherent channels (without Channel State Information&#x2013;CSI) capacity scales as <inline-formula> <tex-math notation="LaTeX">$\Theta (\min (B,\sqrt {NL},N))$ </tex-math></inline-formula>. This is achievable using Pilot-Assisted signaling. Energy Modulation signaling rate scales as <inline-formula> <tex-math notation="LaTeX">$\Theta (\min (B,\sqrt {N}))$ </tex-math></inline-formula>. If <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> is fixed while <inline-formula> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. The coherent channel capacity with a priori CSI, in turn, scales as <inline-formula> <tex-math notation="LaTeX">$\Theta (\min (B,N))$ </tex-math></inline-formula>. The coherent channel capacity scaling can be fully achieved in non-coherent channels when <inline-formula> <tex-math notation="LaTeX">$L\geq \Theta (N)$ </tex-math></inline-formula>. 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&#x2019;s rate scaling versus coherence block length. Only in high mobility scenarios where <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> is much smaller than the number of antennas (<inline-formula> <tex-math notation="LaTeX">$L\ll \Theta (\sqrt {N})$ </tex-math></inline-formula>), Energy Modulation is effective in non-coherent channels.