Analytical Characterization of the Operational Diversity Order in Fading Channels

Abstract

We introduce and characterize the operational diversity order (ODO) in fading channels, as a proxy to the classical notion of diversity order at any arbitrary operational signal-to-noise ratio (SNR). Thanks to this definition, relevant insights are brought up in a number of cases: (i) We quantify that in dominant line-of-sight scenarios an increased diversity order is attainable compared to that achieved asymptotically, even in the single-antenna case; (ii) this effect is attenuated, but still visible, in the presence of an additional dominant specular component; (iii) the decay slope in Rayleigh product channels increases very slowly, never fully achieving unitary slope for a finite SNR.

Figures (6)

• Figure 1: ODO for the Rician fading channel, as a function of $\Omega_{\rm 0}$, for different values of $K$. Solid lines are obtained from \ref{['ODO_Rice']}, markers correspond to Monte Carlo (MC) simulations using diff over the empirical log-CDF. Conventional DO is included as an asymptotic reference (black dashed line) to highlight that the ODO converges to it as $\Omega_{\rm 0}\rightarrow\infty$.
• Figure 2: ODO-based linear approximations to the OP for the Rician fading channel at different operational points (each indicated by a colored + mark). Parameter values are $R=1.7$ bps/Hz and $K=15$. Solid black line corresponds to the theoretical expression for the OP. .
• Figure 3: ODO under Rician fading with SC and MRC diversity schemes, for different values of $K$. Solid lines obtained from \ref{['ODO_RiceMRC']}, markers indicate MC simulations using diff over the empirical log-CDF. Conventional DO is included as asymptotic reference (black dashed line) to highlight that the ODO converges to it as $\Omega_{\rm 0}\rightarrow\infty$.
• Figure 4: Power increase (in dB) required to achieve a 1 order of magnitude improvement in OP for the TWDP fading channel, as a function $\Omega_{\rm 0}$, for different values of $\Delta$. Parameter values are $K=12$ and $R=1.7$ bps/Hz. Solid lines are obtained from \ref{['ODO_TWDP']}and \ref{['power']}, markers correspond to MC simulations using diff over the empirical log-CDF. The value of $c$ predicted by the conventional DO is included as an asymptotic reference (black dashed line) as $\Omega_{\rm 0}\rightarrow\infty$.
• Figure 5: ODO for the cascaded and single Rayleigh fading channels, as a function of $\Omega_{\rm 0}$. Solid lines are obtained from \ref{['ODO_cas']}, markers correspond to MC simulations using diff over the empirical log-CDF.

Theorems & Definitions (3)

• Lemma 1: CDF
• proof
• Remark 1: How to use the ODO for system design?