Complex Gaussian Ratio Distribution with Applications for Error Rate Calculation in Fading Channels with Imperfect CSI

Abstract

Communications systems rarely have perfect channel state information (PCSI) when demodulating received symbols. This paper shows that the symbol error rate (SER) of a flat fading communications system can be expressed in closed form by expressing the demodulator outputs as random variable (RVs) that have a complex ratio distribution, which is the ratio of two correlated complex Gaussian RVs. To complete the analysis, the complex ratio probability density function (PDF) and cumulative distribution function (CDF) are both derived. Finally, using several scenarios based on M-QAM signaling, the SER performance of imperfect channel state information (ICSI) systems is analyzed.

Closed-form capacity expression for low complexity BICM with uniform inputs

“…The proof of Lemma 1 can straightforwardly be obtained by inserting our system model in the general expression of the complex Gaussian ratio distribution [3].…”

“…Note that the pmf in (21) can be expressed by Q-functions, the pdf in (22) can be found in [3] and the expectations E{·} are evaluated by Monte Carlo simulation. Figure 2 compares our closed-form capacity expression given by (19) with the usually considered BICM capacity for the case of 4-QAM.…”

“…Closedform expressions are important because they provide analytical insights and allow efficient performance optimization with respect to specific parameters such as the pilot symbol design or the optimal Orthogonal Frequency Division Multiplexing (OFDM) subcarrier spacing in a doubly-selective channel. Our derivations are based on recent results of the complex Gaussian ratio distribution [3]. However, the cumulative distribution function (cdf) in [3] requires multiple functions, making it rather hard to use.…”

“…Our derivations are based on recent results of the complex Gaussian ratio distribution [3]. However, the cumulative distribution function (cdf) in [3] requires multiple functions, making it rather hard to use. By assuming a specific system model, we are able to provide a compact expression for the cdf, see Lemma 1.…”

An Explicit Distribution to Model the Proportion of Heating Degree Day and Cooling Degree Day

“…[Nadarajah and Kibria (2006)] apply the distribution of the ratio, when X and Y are independent generalized Pareto random variables, in environmental sciences to assess the relative extremity of rainfall for different locations in Florida whereas [Manas (2011)] uses the Paretian ratio distribution to model the granular volatility of firms in GDP. [Baxley et al (2010)] calculate the symbol error rate of a flat communication system with imperfect channel state information by using the ratio of complex Gaussian distributions. ] derive the distribution of the ratio when X and Y are Fréchet independent random variables.…”