Tight Upper and Lower Bounds on the Symmetric Capacity and Outage Probability for Bandwidth-efficient Qam Transmissions over Rayleigh-faded Channels

In this contribution novel, quickly computable analytical upper and lower bounds are presented on the symmetric capacity and the outage probability of flat-faded Rayleigh-channels with 2D QAM constellations. The proposed bounds are tight both for high and low SNRs so that in this respect they outperform the usual ones based on the simplifying assumption of continuous Gaussian-distributed channel input-alphabet. I. SUMMARY OF THE WORK Performance of data-systems over radio and satellite links can be impaired by fading phenomena which induce severe losses on the ultimate available channel throughputs. In this contribution we focus on Shannon’s capacity and outage probability of these links. Since the actual computation of the capacity formulas requires multiple cumbersome numerical integrations [1,Sect.4.6], we present analytical lower and upper bounds for a more simple evaluation of the capacity and outage probability of the considered systems. In particular, moving from known results about the capacity of unfaded links [1,Sect.4.5], we prove that for the symmetric capacity C of a Rayleigh-faded channel the following chain of upper and lower bounds holds: LB2 ≤ LB1 ≤ C ≤ UB1 ≤ UB2. (1) The indicated bounds are obtained through a reiterated application of Jensen's inequality and refer to systems with ML soft-decoding and perfect CSI. The illustrative plots of Fig.1 show that the proposed bounds are tight and asymptotically exact both for high and low SNR values γ c ; furthermore, they outperform (for medium and high SNRs) the conventional Gaussian upper-bound UBG [2,Sect.2] derived under the assumption of Gaussian-distributed inputsymbols for the transmission channel. An examination of Fig.1 also shows that the bounds LB1 and UB1 differ from C * within 2-2.6 dB and approach C for γ c below 5 dB and over 15 dB; moreover, the simpler bounds LB2 and UB2 differ from the corresponding LB1 and UB1 ones within 1-1.5 dB and typically approach C* for γ c below 2-3 dB and over 16-17 dB. Moreover, by indicating as g the Rayleigh distributed random channel-gain and as C(g) the corresponding conditioned channel capacity, we can bound the resulting outage probability P P C g o ( ) ( ( ) ) * δ δ ≡ ≤ of the underlying transmission system as below reported:LB3(δ) ≤Po(δ) ≤UB3(δ).(2)The above indicated limits exhibit simple exponential-likedependences on the SNR γ c and on the key-parameterscharacterizing the employed QAM constellation. Theillustrative plots of Fig.2 show that the bounds in (2) areasymptotically exact for δ→0, δ→logq and for γ c → 0 ,γ c → ∞ , thus outperforming the correspondingconventional limits [2,Sect.2.C] derived under the abovementioned assumption of Gaussian-distributed channel inputalphabet. REFERENCES[1] S.G. Wilson, Digital Modulation and Coding, PrenticeHall 1996.[2] L.H. Ozarow, S. Shamay, A.D. Wyner, “InformationTheoretic Considerations for Cellular Mobile Radio”,IEEE Trans. on Vehic. Techn., vol.43, no.2, pp.359-378,May 1994. 0,00,51,01,52,02,53,0 -5 5 15 25 35 45γc (dB)bits/symbol LB2LB1C*UB1UB2UBG4PSK