# Entropy and Channel Capacity under Optimum Power and Rate Adaptation over Generalized Fading Conditions

###### Abstract

Accurate fading characterization and channel capacity determination are of paramount importance in both conventional and emerging communication systems. The present work addresses the nonlinearity of the propagation medium and its effects on the channel capacity. Such fading conditions are first characterized using information theoretic measures, namely, Shannon entropy, cross entropy and relative entropy. The corresponding effects on the channel capacity with and without power adaptation are then analyzed. Closed-form expressions are derived and validated through computer simulations. It is shown that the effects of nonlinearities are significantly larger than those of fading parameters such as the scattered-wave power ratio, and the correlation coefficient between the in-phase and quadrature components in each cluster of multipath components.

Adaptation policies, channel capacity, entropy.

## I Introduction

Multipath fading characterization is considered a critical task in the effective analysis of the performance of wireless communication systems. To this end, the , , and distributions have been considered relatively suitable models as they also include as special cases the widely known Rayleigh, Nakagami, Hoyt, Weibull and Rice distributions, see [1, 2, 4, 5, 12, 20, 3, 16, 9, 11, 21, 18, 6, 7, 10, 14, 13, 19, 8, 15, 17] and the references therein.

It is also known that accurate determination of the channel capacity for different communication scenarios has been of core importance in wireless communications, including requirements on optimum rate and transmit power constraints. To this end, the average channel capacity over generalized fading channels was thoroughly analyzed in [4], whereas the channel capacity under different adaptation policies was thoroughly investigated in [22, 23, 24].

Nevertheless, in spite of the usefulness of the aforementioned distributions, their accuracy and generality are limited, since the non-linearity parameter in these models is not related to the fading parameters , and [25].
As a consequence, the detrimental effects of non-linear propagation medium are not considered in the majority of investigations. Motivated by this, the authors in [25] proposed the and distributions which constitute a generalization of the , and models and were shown to provide remarkably accurate fitting to measurement results in various communication scenarios.
In this context, useful statistical results and properties for the and models were reported in [26, 27, 28] whereas the corresponding outage probability was analyzed in [29]. A simple and remarkably accurate random sequence generator for and variates was proposed in [30] while the corresponding symbol error rate and channel capacity was addressed in [31, 32].

However, none of the reported analyses address the corresponding channel capacity with optimum power and rate adaptation. Motivated by this, the aim of the present work is twofold: we firstly derive novel analytic expressions for the Shannon entropy of and distributions. Capitalizing on this, we derive analytic expressions for the cross-entropy and relative entropy for these distributions with respect to the and distributions, respectively.
Secondly, we derive novel closed-form expressions for the corresponding channel capacity under optimum rate adaptation as well as optimum power and rate adaptation.
The derived expressions are subsequently used in evaluating the corresponding performance, which indicates that the effects of fading non-linearities are larger than those of popular and widely considered fading parameters.

## Ii Non-Linear Fading Channels and Entropies

### Ii-a The and Models

The SNR PDF of the distribution is given by [25]

(1) |

where denotes the non-linearity of the propagation medium, is the scattered wave power ratio between the in-phase and quadrature components and is related to the number of multipath clusters. Likewise, the SNR PDF of the fading model is expressed as follows:

(2) |

where denotes the correlation coefficient between the in-phase and quadrature components of the fading signal [26]. It is recalled that the and distributions include as special cases the , the and the fading models for , and , respectively [25].

### Ii-B Shannon Entropy

The Shannon entropy is a fundamental metric which denotes the amount of information contained in a signal and indicates the number of bits required for encoding this signal.

###### Lemma 1.

For and , the Shannon entropy of the fading distribution is expressed as

(3) |

in bits/message where

(4) |

and with denoting the Euler-Mascheroni constant.

###### Proof.

The Shannon entropy for continuous random variables with PDF is given by . Thus, for the case of fading in (1) and with the aid of standard logarithmic identities it follows that

(5) |

By recalling that , one obtains

(6) |

The first two integrals in (6) can be expressed in closed-form with the aid of [33, eq. (4.352.1)] and [33, eq. (8.310.1)] and involve the digamma function, . By also noticing that in the logarithm of (6) becomes practically negligible as , recalling that and after some algebraic manipulations, one obtains (3), which completes the proof. ∎

Importantly, the Shannon entropy can quantify the detrimental effects of fading non-linearity by determining the variation of required bits per message for each value of . This demonstrates the difference between the and distributions with the popular and distributions, respectively, which is also reflected by the considered channel capacity measures. This is subsequently analyzed along with the derivation of useful analytic expressions for the corresponding cross entropy and relative entropy metrics.

### Ii-C Cross Entropy

The cross entropy measures the average number of bits required to encode a message when a distribution is replaced by a distribution . In the present analysis this can exhibit the number of bits required to encode and when they are replaced by and distributions, respectively, which corresponds to the case where the non-linearity parameter is not taken into account.

###### Lemma 2.

For and , the cross entropy between and distributions is given by

(7) |

###### Proof.

The cross entropy between two continuous distributions with PDFs and is given by . Therefore, for the case of and distributions, respectively, it follows that

(8) |

where , and correspond to the parameters of distribution. It is evident that the algebraic representation of (8) is similar to (3). Based on this, the proof follows immediately using Lemma 1. ∎

In the next Section, it is shown that the value of the considered varies substantially at even slight variations of . It is also shown that this variation is larger than that from widely used fading parameters such as and . This justifies the usefulness of the and distributions.

### Ii-D Relative Entropy

The relative entropy, which is also known as the Kullback-Leibler divergence, is a measure of the distance between two probability distributions. It accounts for the inefficiency of assuming that a true distribution with PDF is replaced by a distribution with PDF . In the present analysis, this measure quantifies the information loss encountered when the non-linearity parameter is not taken into account, which occurs when and distributions are replaced by the simpler and distributions, respectively.

###### Lemma 3.

###### Proof.

By recalling that the relative entropy is defined as

(10) |

the proof follows with the aid of Lemma 1 and Lemma 2. ∎

Notably, similar expressions for the Shannon entropy of distribution as well as the cross entropy and relative entropy between and distributions are obtained by setting in (3), (7) and (9), respectively. Extensive results for all considered entropies for both and fading conditions are provided in Section IV, indicating the non-negligible effects of fading non-linearity.

## Iii Capacity Under Different Adaptation Policies

### Iii-a Capacity with Optimum Rate Adaptation

The Shannon capacity of a channel constitutes a theoretical upper bound for the maximum rate of data transmission and is also used as a benchmark for comparisons of the channel capacity under different adaptation schemes [1]. In what follows, we derive a closed-form expression for the average Shannon capacity over fading channels.

###### Theorem 1.

For and , the average channel capacity over fading channels with optimum rate adaptation is expressed as follows:

(11) |

where is the corresponding bandwidth, is the Meijer Gfunction and

###### Proof.

Recalling the definition of the average ergodic capacity

(12) |

it immediately follows that

(13) |

With the aid of [33, eq. (8.467)], it follows that

(14) |

Importantly, the exponential and logarithmic terms in (14) can be expressed in terms of the Meijer G-function in [34] yielding

(15) |

The above integral can be expressed in closed-form in terms of [34, eq. (21)]. To this effect and after long but basic algebraic manipulations, one obtains (11). This concludes the proof. ∎

Equation (11) is expressed in closed-form and can be computed in software packages such as Maple and Mathematica.

### Iii-B Capacity with Optimum Power and Rate Adaptation

Wireless systems are often subject to average transmit power constraints. Based on this, the maximum capacity for optimum power and rate adaptation, , can be determined [1].

###### Theorem 2.

For and , the channel capacity over fading channels with optimum power and rate adaptation is expressed as follows:

(16) |

where is the SNR threshold that determines transmission.

###### Proof.

By firstly recalling the following generic expression

(17) |

as well as substituting (1) and expanding the logarithm using [33, eq. 8.467], one obtains (19) (top of the next page). Notably, the two integrals that emerge from the second term of the numerator of (19) can be expressed in terms of [33, eq. (8.310.1)] which yields

(18) |

## Iv Numerical Results

The offered results are employed in quantifying the effects of the non-linearity of the propagation medium. To this end, Table I depicts the , and measures for different values of the fading parameters at dB. It is shown that the Shannon entropy is inversely proportional to the parameter which indicates that the more severe the incurred non-linearity, the more bits are required to encode the corresponding message. To this effect, it is shown that an extra bit is required for encoding the message correctly when , with , and , compared to the similar case with . On the contrary, it is observed that the lowest value of cross entropy in both cases is for , i.e., when and reduce to and , respectively. From this point, the value of increases as the value of both increases or decreases. For example, while 7 bits/message are practically required when , this value increases to 10 bits/message and 14 bits/message for and , respectively. Finally, the relative entropy measure demonstrates the information lost when is not considered. It is evident that when and increases as decreases and particularly when it increases.

The effect of fading non-linearity on and is demonstrated in Table II and Fig. 1, respectively. It is shown that the corresponding spectral efficiency is, as expected, proportional to the value of in both and fading conditions. Interestingly, it is shown that the effect of is more significant than the overall effect of and parameters together with . This illustrates the importance of considering the effects of fading non-linearity in the design and performance analysis of wireless communication systems.

-- | H(p) | H(p,q) | D(pq) | -- | H(p) | H(p,q) | D(pq) |
---|---|---|---|---|---|---|---|

-- | -- | ||||||

-- | -- | ||||||

-- | -- | ||||||

-- | -- | ||||||

-- | -- | ||||||

-- | -- |

-- / (dB) | -- / (dB) | ||
---|---|---|---|

-- / -5dB | -- / -5dB | ||

-- / 15dB | -- / 15dB | ||

-- / 35dB | -- / 35dB | ||

-- / -5dB | -- / -5dB | ||

-- / 15dB | -- / 15dB | ||

-- / 35dB | -- / 35dB | ||

-- / -5dB | -- / -5dB | ||

-- / 15dB | -- / 15dB | ||

-- / 35dB | -- / 35dB | ||

-- / -5dB | -- / -5dB | ||

-- / 15dB | -- / 15dB | ||

-- / 35dB | -- / 35dB | ||

-- / -5dB | -- / -5dB | ||

-- / 15dB | -- / 15dB | ||

-- / 35dB | -- / 35dB |

## V Conclusion

This work quantified the effects of non-linearity, , of the propagation medium in wireless transmission by means of the Shannon entropy, cross entropy and relative entropy measures. The average channel capacity over generalized non-linear fading conditions was also evaluated for the cases of optimum rate and optimum power and rate adaptations. It was shown that the effects of are more pronounced than those of the commonly used fading parameters , and combined.

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