CN106027183A - Method capable of quickly evaluating cumulative distribution performance of composite fading channel - Google Patents

Abstract

The invention discloses a method capable of quickly evaluating a cumulative distribution performance of a composite fading channel. The method comprises the following steps: (1) employing Gamma distribution which approximates the Lognormal distribution to simulate a shadow effect, thereby constructing Gamma-Gamma distribution to be used in an approximate original Gamma-Lognormal composite fading channel model; (2) on the basis of the approximate model, deducing a closed expression of a received signal-to-noise ratio cumulative distribution function expressed by a Meijer-G function; and (3) calculating the performance of the original composite fading channel cumulative distribution from the Meijer-G function. According to the method disclosed by the invention, the proposed Gamma-Gamma distribution approximate scheme can simplify the expression of a complex infinite integral of an original accurate model; and thus, a closed form of the received signal-to-noise ratio cumulative distribution function is obtained, thereby reducing the complexity of CDF function formula computation, and being beneficial to quickly analyzing and evaluating a communication system.

Description

A kind of method that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation

Technical field

The present invention relates to a kind of method that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation, be particularly suited for Consider the complicated mobile communication system of path loss, shadow effect, multipath fading and additive white Gaussian noise.

Background technology

The communication performance of wireless communication system is limited by the radio communication channel of system work to a great extent.Wireless In communication system, decline refers to that signal amplitude that receiving terminal receives and phase place are along with different paths and the time change of transmission The random phenomenon changed occurs, and this phenomenon will greatly affect the performance of receiver.Owing to an aspect of decline reflection is Signal power problem, so according to the speed of power reduction and degree, decline is divided into large scale decline by people and little yardstick declines Fall.Multipath fading refers to the impact of the factors such as electromagnetic wave is reflected during dissemination channel, scattering, diffraction and causes connecing Receiving end received signal is the superposition of the electromagnetic wave from all directions, and this signal can cause violent fluctuation in little scope I.e. multipath fading；So-called large scale decline refer to signal in communication process by the barrier between transmitter and receiver such as Energy loss that the stop of building, high mountain and jungle etc. causes and fluctuation.In actual wireless communication system, antenna is received and dispatched It is the strongest that the multiformity of the propagation path between two ends and the landform of various complexity can cause the multipath fading of channel to have Randomness；Additionally, the antenna of sending and receiving end is spaced farther out in community in modern wireless communication systems, such as owing to accessing distance The path loss that difference makes the signal between each spaced antenna port (base station) and mobile station be experienced is different, so sky The decline that wireless signal between line cap (base station) and mobile station experiences in transmitting procedure is not only just multipath fading, Also need to consider the adverse effect of the such as large scale such as shadow fading, path loss decline factor.

Common large scale propagation model generally comprises log-distance path loss model and Lognormal shadowing model, and little chi Degree fading model then includes Nakagami decline, Rayleigh fading and Lai Si decline etc..Due in actual wireless communications environment, The signal envelope of transmission in fading channel including city can be modeled by Nakagami distribution well, and combines Pure scattering and the situations of superposition line-of-sight transmission such as conventional Rayleigh and L-S distribution, therefore this model just obtains industry once proposition It is widely recognized as and applies.

Summary of the invention

The deficiency existed for prior art, it is an object of the present invention to provide one and can quickly realize Composite Fading Channels accumulation The method of distribution performance assessment, Gamma-Gamma distribution approximate schemes proposed by the invention can simplify former accurate model The expression of complicated inifinite integral, thus draw the closed form of received signal to noise ratio cumulative distribution function, and then it is public to reduce CDF function The complexity that formula calculates, is conducive to quickly analyzing, assessing the communication system such as performance indications such as outage probability, channel capacity.

To achieve these goals, the present invention is to realize by the following technical solutions:

A kind of method that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation of the present invention, including following Step:

(1) arranging channel parameter, (approximation refers to the probability density function that Gamma is distributed to use Gamma distribution approximation (PDF) similar with the probability density function shape of Lognormal distribution, namely both probability density function curves can be preferably Overlap.For term generally in the art) Lognormal distribution is used for simulating shadow effect, and then structure Gamma-Gamma divides Cloth is used for approximating that (what this approximation referred to is exactly the PDF phase of PDF and former Gamma-Lognormal distribution of Gamma-Gamma distribution Seemingly.For term generally in the art) former Gamma-Lognormal Composite Fading Channels model；

(2) on the basis of Composite Fading Channels model (this model refers to: be used for the Gamma-Gamma distributed model approximated) On, derive the Guan Bi expression formula of the received signal to noise ratio cumulative distribution function with Meijer-G function representation；

(3) by Meijer-G function by consult formula value list or numerical computations software (such as Matlab or Mathematics) performance of former Composite Fading Channels cumulative distribution is calculated.

In step (1), the construction method of described Composite Fading Channels model is as follows:

Under little yardstick Nakagami fading channel, the envelope α of wireless communication system transmission signal obeys Nakagami and divides Cloth, its PDF is:

$f_{\mathrm{\α}}(\mathrm{\α};m,\mathrm{\ω})=\frac{2m^m}{\mathrm{\Γ}\left(m\right){\mathrm{\ω}}^m}{\mathrm{\α}}^{2m-1}\exp (-\frac{m}{\mathrm{\ω}}{\mathrm{\α}}^2),---\left(1\right)$

In above formula, m and ω is two important parameters of Nakagami distribution, and expression formula is respectively as follows:

$\{\begin{array}{c}m=E^2\[{\mathrm{\α}}^2\]/Var\[{\mathrm{\α}}^2\]\\ \mathrm{\ω}=E\[{\mathrm{\α}}^2\]\end{array},---\left(2\right)$

Wherein, E [] expression is averaged, and variance is sought in Var [] expression, and Γ () represents gamma function, and ω is decline width The mean-square value of degree α, m is referred to as form factor or decline index, represents the order of severity of now multipath fading, and its value meets m≥1/2；

There are several special circumstances in the different values of decline exponent m: as m=1/2, it deteriorates to monolateral Gauss distribution；Work as m=1 Time, exactly rayleigh distributed；As m > 1 time, Nakagami distribution can be equivalent to Rice factor and is L-S distribution；

In the case of there is additive white Gaussian noise in considering this Nakagami fading channel, each symbol pair of receiving terminal The average received signal to noise ratio answeredFollowing relation is there is with instantaneous received signal to noise ratio γ:

$\left\{\begin{array}{c}\mathrm{\γ}={\mathrm{\α}}^2{E}_s/N_0\\ \stackrel{\‾}{\mathrm{\γ}}={\mathrm{\ωE}}_s/N_0\end{array}\right.,---\left(3\right)$

Wherein, N₀And E_sIt is respectively power spectral density and the signal transmitting power of white Gaussian noise；

Understand according to above formula, the instantaneous received signal to noise ratio γ of single symbol and reception signal envelope α probability density letter Following relation is there is between number:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\frac{f_{\mathrm{\α}}\left(\sqrt{\mathrm{\ω}\mathrm{\γ}/\stackrel{\‾}{\mathrm{\γ}}}\right)}{2\sqrt{\mathrm{\γ}\stackrel{\‾}{\mathrm{\γ}}/\mathrm{\ω}}},---\left(4\right)$

According to the Jacobian transformation rule between stochastic variable PDF and its function gained new stochastic variable PDF, can The PDF that must receive single symbol instantaneous signal-to-noise ratio γ is

$f_{\mathrm{\γ}}(\mathrm{\γ};m,\stackrel{\‾}{\mathrm{\γ}})=\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right){\stackrel{\‾}{\mathrm{\γ}}}^m}\exp (-\frac{m\mathrm{\γ}}{\stackrel{\‾}{\mathrm{\γ}}}),---\left(5\right)$

This expression formula clearly demonstrates stochastic variable γ and obeys Gamma distribution；

If channel exists large scale path loss and shadow fading, then its average received signal to noise ratio simultaneouslyIt is right to obey Number normal distribution, its PDF is

$f_{\stackrel{\‾}{\mathrm{\γ}}}\left(\stackrel{\‾}{\mathrm{\γ}}\right)=\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}\stackrel{\‾}{\mathrm{\γ}}}\exp \[-\frac{{(10lg\stackrel{\‾}{\mathrm{\γ}}-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\],---\left(6\right)$

In above formula, ξ=10/ln10 is a fixed constant；μ and σ is respectively barrier docking and receives signal envelope power generation Average path loss and random fluctuation standard deviation；

In the case of considering Nakagami decline, path loss and shadow fading, now composite fading letter can be obtained The PDF of the received signal to noise ratio γ of road model is:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\underset{0}{\overset{\mathrm{\∞}}{\∫}}\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right){\stackrel{\‾}{\mathrm{\γ}}}^m}\exp (-\frac{m\mathrm{\γ}}{\stackrel{\‾}{\mathrm{\γ}}})\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}\stackrel{\‾}{\mathrm{\γ}}}\exp \[-\frac{{(10lg\stackrel{\‾}{\mathrm{\γ}}-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\]d\stackrel{\‾}{\mathrm{\γ}},\mathrm{\γ}>0,---\left(7\right)$

Wherein,Refer to average received signal to noise ratio, be also the independent variable of this inifinite integral, from the expression of above formula (7) simultaneously In form it can be seen that its received signal to noise ratio of Composite Fading Channels model in simulation actual complex communication environment obeys Gamma- Lognormal is distributed.

In step (2), the Guan Bi expression formula of described received signal to noise ratio cumulative distribution function is as follows:

The PDF that (7) formula is obtained be integrated the CDF of received signal to noise ratio γ is:

$\begin{array}{c}{F}_{\mathrm{\γ}}\left(X\right)={\∫}_0^X{f}_{\mathrm{\γ}}\left(\mathrm{\γ}\right)d\mathrm{\γ}\\ =1-\frac{1}{\mathrm{\Γ}\left(m\right)}{\∫}_0^{\mathrm{\∞}}\mathrm{\Γ}(m,mX/s)\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}s}\exp \[-\frac{{(10\lg s-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\]ds\end{array},---\left(8\right)$

By the logarithm normal distribution in Gamma distribution replacement expression formula (7) so that logarithm shadow fading is modeled, namely The PDF being distributed the average received signal to noise ratio represented with Gamma is:

$f_{\stackrel{\‾}{\mathrm{\γ}}}\left(\stackrel{\‾}{\mathrm{\γ}}\right)=\frac{{\stackrel{\‾}{\mathrm{\γ}}}^{n-1}}{\mathrm{\Γ}\left(n\right){\mathrm{\χ}}^n}\exp (-\frac{\stackrel{\‾}{\mathrm{\γ}}}{\mathrm{\χ}}),\stackrel{\‾}{\mathrm{\γ}}>0,---\left(9\right)$

In above formula, n is the exponent number of Gamma distribution；χ represents mean power；The approximate formula (9) obtained it is distributed by Gamma And between the formula (6) that former accurate Lognormal distribution obtains, the transformation relation between core parameter is:

$\left\{\begin{array}{c}\mathrm{\μ}=\mathrm{\ξ}\[ln\mathrm{\χ}+\mathrm{\ψ}\left(n\right)\]\\ {\mathrm{\σ}}^2={\mathrm{\ξ}}^2{\mathrm{\ψ}}^{\′}\left(n\right)\end{array}\right.,---\left(10\right)$

In above formula, ψ () and ψ ' () is digamma and trigamma function respectively；So answering after can being approximated Closing the PDF of received signal to noise ratio γ in fading channel is:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)={\∫}_0^{\mathrm{\∞}}\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right)s^m}\exp (-\frac{m\mathrm{\γ}}{s})\frac{s^{n-1}}{\mathrm{\Γ}\left(n\right){\mathrm{\χ}}^n}\exp (-\frac{s}{\mathrm{\χ}})ds,\mathrm{\γ}>0,---\left(11\right)$

From the expression-form of above formula (11) it can be seen that approximation its received signal to noise ratio of Composite Fading Channels model built takes It is distributed from Gamma-Gamma；Make t=s/ χ and η=m/ χ, can obtain through deriving

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\frac{2{\mathrm{\η}}^{\frac{m+n}{2}}}{\mathrm{\Γ}\left(m\right)\mathrm{\Γ}\left(n\right)}{\mathrm{\γ}}^{\frac{m+n}{2}-1}{K}_{(m-n)}\left(2\sqrt{\mathrm{\η}\mathrm{\γ}}\right),---\left(12\right)$

Wherein, K_(m-_n)() is (m-n) rank Equations of The Second Kind modified Bessel functions；It is integrated above formula again receiving The CDF of signal to noise ratio γ is

$F_{\mathrm{\γ}}\left(X\right)={\∫}_0^X{f}_{\mathrm{\γ}}\left(\mathrm{\γ}\right)d\mathrm{\γ}=\frac{1}{\mathrm{\Γ}\left(m\right)\mathrm{\Γ}\left(n\right)}{G}_{13}^{21}\left(\mathrm{\η}X{|}_{m,n,0}^1\right)---\left(13\right)$

Wherein,It is Meijer-G function and 0≤k≤q, 0≤l≤p≤q；K, l, p, q are integer；

Formula (13) give a kind of can with tabular function conventional in engineering mathematics (be technical term, English entitled " Tabulated function ", translation comes " tabular function ") represent, Composite Fading Channels received signal to noise ratio The Guan Bi expression formula of CDF passes through numerical simulation computed in software.

Above-mentioned numerical simulation software specifically uses Matlab or Mathematic.

A kind of method quickly realizing Composite Fading Channels cumulative distribution function Performance Evaluation that the present invention proposes, is allowed to just In analyzing the system such as performance indications such as outage probability, channel capacity.The program be build composite fading model time, still with The multipath fading of Nakagami distribution reflection transmission signal, and use Gamma distribution approximate log normal state (Lognormal) point Cloth describes the large scale fading characteristic of path loss and shadow effect, draws the receiving terminal each symbol noise after approximation The probability density function of ratio obeys Gamma-Gamma distribution.Based on this, the cumulative distribution function of this Composite Fading Channels can be changed Letter be one with the Guan Bi expression formula of so-called " tabular function " Meijer-G function representation, it is quick that this is beneficial to carry out computer Numerical simulation is to analyze the characteristic of Composite Fading Channels.

Accompanying drawing explanation

Fig. 1 is wireless communication system universal model figure based on beehive area structure；

Fig. 2 is a kind of method workflow that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation of the present invention Figure；

Fig. 3, for making variable with decline exponent m, uses Gamma-Gamma approximation Composite Fading Channels received signal to noise ratio accumulation Distribution character and accurate Gamma-lognormal cumulative distribution function curve comparison figure；

Fig. 4, for making variable with shadow fading degree σ, uses Gamma-Gamma to approximate Composite Fading Channels received signal to noise ratio Cumulative distribution characteristic and accurate cumulative distribution function curve comparison figure；

Fig. 5, for making variable with average path loss μ, uses Gamma-Gamma to approximate Composite Fading Channels received signal to noise ratio Cumulative distribution characteristic and accurate cumulative distribution function curve comparison figure.

Detailed description of the invention

For the technological means making the present invention realize, creation characteristic, reach purpose and be easy to understand with effect, below in conjunction with Detailed description of the invention, is expanded on further the present invention.

Embodiments of the invention can be widely applied to exist the modern wireless communication systems of complex communication environment, i.e. passes through A kind of method proposing rapid evaluation channel cumulative distribution performance is such as wireless communication system outage probability and channel capacity Analyze and provide strong foundation with assessment.Comprise the universal model schematic diagram such as figure of the modern wireless communication systems of Composite Fading Channels Shown in 1: in the wireless communications environment of a similar honeycomb texture, the wireless signal of Base Transmitter at the regional level in carry out remote During Distance Transmission, occur big owing to signal can run into the stop of the such as barrier such as pile, jungle in transmitting procedure Yardstick declines；And signal is when arriving near receiving terminal, owing to wireless signal is by the shadow of the factors such as surrounding scattering, reflection environment Ring and produce multipath multipath fading.Send so the signal that the receiving terminal of the mobile devices such as similar mobile phone receives is exactly transmitting terminal Primary signal by obtaining after there is the channel of the strongest randomness, the most therefore, the Modeling Research one to radio communication channel Straight is all the emphasis in mobile communication research process.

The present invention consider the large scale declines such as path loss and shadow effect, Nakagami multipath fading and Construct a Composite Fading Channels model meeting practical communication transmission conditions in the case of white noise, obtain based on this model The probability density function (Probability Density Function, PDF) of snr of received signal be a kind of accurate Signal power statistics characteristic expression formula, the cumulative distribution function (Cumulative obtained according to this PDF expression formula on this basis Distribution Function, CDF) then can assess such as outage probability, channel capacity in Modern Communication System further Etc. important system performance indications, therefore the calculating to this CDF function is just particularly important with assessment.Owing to above-mentioned composite fading is believed The PDF expression formula of channel receiving signal signal to noise ratio is typically a complicated inifinite integral form, therefore cannot be carried out abbreviation and closed The cumulative distribution function expression formula of type, so being unfavorable for carrying out further correlational study, it has become Modern Communication System performance The difficult point analyzed.The present invention i.e. provides a kind of simplification approximate model for above-mentioned Composite Fading Channels such that it is able to fast Speed realizes the performance evaluation to Composite Fading Channels cumulative distribution function characteristic and research.

Seeing Fig. 2, under little yardstick Nakagami fading channel, the envelope α of wireless communication system transmission signal obeys Nakagami is distributed, and its PDF is:

$f_{\mathrm{\α}}(\mathrm{\α};m,\mathrm{\ω})=\frac{2m^m}{\mathrm{\Γ}\left(m\right){\mathrm{\ω}}^m}{\mathrm{\α}}^{2m-1}\exp (-\frac{m}{\mathrm{\ω}}{\mathrm{\α}}^2),---\left(1\right)$

In above formula, m and ω is two important parameters of Nakagami distribution, and expression formula is respectively as follows:

$\{\begin{array}{c}m=E^2\[{\mathrm{\α}}^2\]/Var\[{\mathrm{\α}}^2\]\\ \mathrm{\ω}=E\[{\mathrm{\α}}^2\]\end{array},---\left(2\right)$

Wherein, E [] expression is averaged, and variance is sought in Var [] expression.Γ () represents gamma function, and ω is amplitude of fading α Mean-square value, m be referred to as form factor or decline index, represent the order of severity of now multipath fading, its value meets m >=1/2. There are several special circumstances in the different values of decline exponent m: as m=1/2, it deteriorates to monolateral Gauss distribution；As m=1, just It it is rayleigh distributed；As m > 1 time, Nakagami distribution can be equivalent to Rice factor and is's L-S distribution.In the case of there is additive white Gaussian noise in considering this Nakagami fading channel, each symbol of receiving terminal Corresponding average received signal to noise ratioFollowing relation is there is with instantaneous received signal to noise ratio γ:

$\left\{\begin{array}{c}\mathrm{\γ}={\mathrm{\α}}^2{E}_s/N_0\\ \stackrel{\‾}{\mathrm{\γ}}={\mathrm{\ωE}}_s/N_0\end{array}\right.,---\left(3\right)$

Wherein, N₀And E_sIt is respectively power spectral density and the signal transmitting power of white Gaussian noise.Understand according to above formula, single The instantaneous received signal to noise ratio γ of individual symbol and receive the following relation that exists between signal envelope α probability density function:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\frac{f_{\mathrm{\α}}\left(\sqrt{\mathrm{\ω}\mathrm{\γ}/\stackrel{\‾}{\mathrm{\γ}}}\right)}{2\sqrt{\mathrm{\γ}\stackrel{\‾}{\mathrm{\γ}}/\mathrm{\ω}}},---\left(4\right)$

According to the Jacobian transformation rule between stochastic variable PDF and its function gained new stochastic variable PDF, can The PDF that must receive single symbol instantaneous signal-to-noise ratio γ is

$f_{\mathrm{\γ}}(\mathrm{\γ};m,\stackrel{\‾}{\mathrm{\γ}})=\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right){\stackrel{\‾}{\mathrm{\γ}}}^m}\exp (-\frac{m\mathrm{\γ}}{\stackrel{\‾}{\mathrm{\γ}}}),---\left(5\right)$

This expression formula clearly demonstrates stochastic variable γ and obeys Gamma distribution.

If channel exists large scale path loss and shadow fading, then its average received signal to noise ratio simultaneouslyIt is right to obey Number normal distribution, its PDF is

$f_{\stackrel{\‾}{\mathrm{\γ}}}\left(\stackrel{\‾}{\mathrm{\γ}}\right)=\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}\stackrel{\‾}{\mathrm{\γ}}}\exp \[-\frac{{(10\lg \stackrel{\‾}{\mathrm{\γ}}-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\].---\left(6\right)$

In above formula, ξ=10/ln10, μ and σ (all in units of dB), respectively barrier docking receive signal envelope power and produce Raw average path loss and random fluctuation standard deviation.Considering Nakagami decline, path loss and shadow fading In the case of, the PDF of the received signal to noise ratio γ that can obtain now Composite Fading Channels model is:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\underset{0}{\overset{\mathrm{\∞}}{\∫}}\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right){\stackrel{\‾}{\mathrm{\γ}}}^m}\exp (-\frac{m\mathrm{\γ}}{\stackrel{\‾}{\mathrm{\γ}}})\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}\stackrel{\‾}{\mathrm{\γ}}}\exp \[-\frac{{(10\lg \stackrel{\‾}{\mathrm{\γ}}-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\]d\stackrel{\‾}{\mathrm{\γ}},\mathrm{\γ}>0,---\left(7\right)$

Composite Fading Channels mould from the expression-form of above formula (7) it can be seen that in simulation actual complex communication environment Its received signal to noise ratio of type obeys Gamma-Lognormal distribution.The PDF obtained above formula again is integrated receiving noise Than the CDF of γ it is:

$\begin{array}{c}{F}_{\mathrm{\γ}}\left(X\right)={\∫}_0^X{f}_{\mathrm{\γ}}\left(\mathrm{\γ}\right)d\mathrm{\γ}\\ =1-\frac{1}{\mathrm{\Γ}\left(m\right)}{\∫}_0^{\mathrm{\∞}}\mathrm{\Γ}(m,mX/s)\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}s}\exp \[-\frac{{(10\lg s-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\]ds\end{array}.---\left(8\right)$

From formula (8), the infinite product fraction that CDF is a complexity of the received signal to noise ratio γ of Composite Fading Channels, nothing Method is write as the form of closed-form expression general in engineering mathematics, and this will be highly detrimental to carry out based on this cumulative distribution function table In the modern wireless communication systems reaching formula and set up the such as performance indications such as outage probability, channel capacity assessment and research work Make.So the present invention intends providing a kind of approximation method that simplifies to solve this problem, i.e. replace in expression formula (8) with Gamma distribution Logarithm normal distribution so that logarithm shadow fading is modeled, namely the average received signal to noise ratio represented with Gamma distribution PDF is:

$f_{\stackrel{\‾}{\mathrm{\γ}}}\left(\stackrel{\‾}{\mathrm{\γ}}\right)=\frac{{\stackrel{\‾}{\mathrm{\γ}}}^{n-1}}{\mathrm{\Γ}\left(n\right){\mathrm{\χ}}^n}\exp (-\frac{\stackrel{\‾}{\mathrm{\γ}}}{\mathrm{\χ}}),\stackrel{\‾}{\mathrm{\γ}}>0.---\left(9\right)$

In above formula, n is the exponent number of Gamma distribution；χ represents mean power.The approximate formula (9) obtained it is distributed by Gamma And between the formula (6) that former accurate Lognormal distribution obtains, the transformation relation between core parameter is:

$\left\{\begin{array}{c}\mathrm{\μ}=\mathrm{\ξ}\[ln\mathrm{\χ}+\mathrm{\ψ}\left(n\right)\]\\ {\mathrm{\σ}}^2={\mathrm{\ξ}}^2{\mathrm{\ψ}}^{\′}\left(n\right)\end{array}\right.,---\left(10\right)$

In above formula, ψ () and ψ ' () is digamma and trigamma function respectively.So answering after can being approximated Closing the PDF of received signal to noise ratio γ in fading channel is:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)={\∫}_0^{\mathrm{\∞}}\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right)s^m}\exp (-\frac{m\mathrm{\γ}}{s})\frac{s^{n-1}}{\mathrm{\Γ}\left(n\right){\mathrm{\χ}}^n}\exp (-\frac{s}{\mathrm{\χ}})ds,\mathrm{\γ}>0.---\left(11\right)$

From the expression-form of above formula (11) it can be seen that approximation its received signal to noise ratio of Composite Fading Channels model built takes It is distributed from Gamma-Gamma.Make t=s/ χ and η=m/ χ, can obtain through deriving

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\frac{2{\mathrm{\η}}^{\frac{m+n}{2}}}{\mathrm{\Γ}\left(m\right)\mathrm{\Γ}\left(n\right)}{\mathrm{\γ}}^{\frac{m+n}{2}-1}{K}_{(m-n)}\left(2\sqrt{\mathrm{\η}\mathrm{\γ}}\right),---\left(12\right)$

Wherein, K_(m-_n)() is (m-n) rank Equations of The Second Kind modified Bessel functions.It is integrated above formula again receiving The CDF of signal to noise ratio γ is

$F_{\mathrm{\γ}}\left(X\right)={\∫}_0^X{f}_{\mathrm{\γ}}\left(\mathrm{\γ}\right)d\mathrm{\γ}=\frac{1}{\mathrm{\Γ}\left(m\right)\mathrm{\Γ}\left(n\right)}{G}_{13}^{21}\left(\mathrm{\η}X{|}_{m,n,0}^1\right)---\left(13\right)$

Wherein,It is Meijer-G function and 0≤k≤q, 0≤l≤p≤q；K, l, p, q are integer.Formula (13) Give and a kind of can represent with " tabular function " conventional in engineering mathematics, relevant Composite Fading Channels received signal to noise ratio The Guan Bi expression formula of CDF, this expression formula can by such as numerical simulation software Matlab, Mathematic etc. quickly, accurately Calculate.

Fig. 3 is that the exponent m that declines makees variable, and the Composite Fading Channels using Gamma-Gamma approximation to obtain receives noise Than cumulative distribution function and former accurate cumulative distribution function curve comparison figure.From this figure, it can be seen that in given typical case's large scale On the premise of the influence of fading factor such as path loss μ=20dB and shadow fading degree σ=8dB, change multipath fading shape Curve and accurate cumulative distribution function performance that approximation cumulative distribution function CDF of factor m gained changes with received signal to noise ratio X are bent Reasonable propinquity effect, i.e. Gamma-Gamma distribution is had can preferably to reflect the accumulation of true Composite Fading Channels between line Distribution character.

Fig. 4 is to make variable with shadow fading degree σ, uses Gamma-Gamma to approximate Composite Fading Channels received signal to noise ratio Cumulative distribution function and accurate cumulative distribution function curve comparison figure.Refer in given typical case's multipath fading factor of influence such as decline In the case of the factor of influence path loss μ=20dB of number m=1 and large scale decline, change shadow fading random fluctuation standard Propinquity effect figure between approximation and the accurate cumulative distribution function performance curve of difference σ gained.No matter from this figure, it can be seen that with Machine fluctuation standard deviation sigma how value, between cumulative distribution function curve chart and the former accurate model curve chart after approximate processing Error is less, and two class curve evolution trend are basically identical, i.e. this approximate processing has the highest accuracy.

Fig. 5 is to make variable with average path loss μ, uses Gamma-Gamma to approximate Composite Fading Channels received signal to noise ratio Cumulative distribution function and accurate cumulative distribution function curve comparison figure.Given typical case the large scale influence of fading factor i.e. decline with Machine fluctuation standard deviation sigma=8dB and multipath fading factor of influence i.e. decline exponent m=1 when, change average path loss The approximation cumulative distribution function performance curve of μ gained also can good approximate exact cumulative distribution function performance curve.

The specific implementation process be given by the above present invention is it can be seen that Composite Fading Channels cumulative distribution performance is entered by this The approximate evaluation method of row rapid evaluation all has the strongest suitability with accurate under the conditions of the different influence of fading factors Property: it is for determining the decline exponent m of the multipath fading order of severity and determining the random wave of the large scale decline order of severity Dynamic standard deviation sigma, the typical value of average path loss μ, the method all can relatively accurately approximate exact received signal to noise ratio accumulation Distribution function curve, and then the cumulative distribution characteristic of reflection compound channel.

In a word, Gamma-Gamma distribution approximate schemes proposed by the invention can simplify the complexity nothing of former accurate model The expressing thus draw the closed form of received signal to noise ratio cumulative distribution function of poor integration, and then reduce CDF function formula and calculate Complexity, be conducive to quickly analyzing, the assessment communication system such as performance indications such as outage probability, channel capacity.

The ultimate principle of the present invention and principal character and advantages of the present invention have more than been shown and described.The technology of the industry Personnel, it should be appreciated that the present invention is not restricted to the described embodiments, simply illustrating this described in above-described embodiment and description The principle of invention, without departing from the spirit and scope of the present invention, the present invention also has various changes and modifications, and these become Change and improvement both falls within scope of the claimed invention.Claimed scope by appending claims and Equivalent defines.

Claims (4)

1. the method that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation, it is characterised in that include following several Individual step:

(1) channel parameter is set, uses Gamma distribution approximation Lognormal distribution to be used for simulating shadow effect, and then build Gamma-Gamma distribution is used for approximating former Gamma-Lognormal Composite Fading Channels model；

(2) on the basis of Composite Fading Channels model, the received signal to noise ratio iterated integral with Meijer-G function representation is derived The Guan Bi expression formula of cloth function；

(3) gone out former Composite Fading Channels by Meijer-G function tire out by consulting formula value list or numerical computations computed in software The performance of integration cloth.

The method that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation the most according to claim 1, its feature Being, in step (1), the construction method of described Composite Fading Channels model is as follows:

Under little yardstick Nakagami fading channel, the envelope α of wireless communication system transmission signal obeys Nakagami distribution, its PDF is:

$f_{\mathrm{\α}}(\mathrm{\α};m,\mathrm{\ω})=\frac{2m^m}{\mathrm{\Γ}\left(m\right){\mathrm{\ω}}^m}{\mathrm{\α}}^{2m-1}\exp (-\frac{m}{\mathrm{\ω}}{\mathrm{\α}}^2),---\left(1\right)$

In above formula, m and ω is two important parameters of Nakagami distribution, and expression formula is respectively as follows:

$\{\begin{array}{c}m=E^2\[{\mathrm{\α}}^2\]/Var\[{\mathrm{\α}}^2\]\\ \mathrm{\ω}=E\[{\mathrm{\α}}^2\]\end{array},---\left(2\right)$

Wherein, E [] expression is averaged, and variance is sought in Var [] expression, and Γ () represents gamma function, and ω is amplitude of fading α Mean-square value, m is referred to as form factor or decline index, represents the order of severity of now multipath fading, and its value meets m >=1/ 2；

There are several special circumstances in the different values of decline exponent m: as m=1/2, it deteriorates to monolateral Gauss distribution；As m=1, Exactly rayleigh distributed；As m > 1 time, Nakagami distribution can be equivalent to Rice factor and is L-S distribution；

In the case of there is additive white Gaussian noise in considering this Nakagami fading channel, each symbol of receiving terminal is corresponding Average received signal to noise ratioFollowing relation is there is with instantaneous received signal to noise ratio γ:

$\left\{\begin{array}{c}\mathrm{\γ}={\mathrm{\α}}^2{E}_s/N_0\\ \stackrel{\‾}{\mathrm{\γ}}={\mathrm{\ωE}}_s/N_0\end{array}\right.,---\left(3\right)$

Wherein, N₀And E_sIt is respectively power spectral density and the signal transmitting power of white Gaussian noise；

According to above formula understand, the instantaneous received signal to noise ratio γ of single symbol and receive signal envelope α probability density function it Between there is following relation:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\frac{f_{\mathrm{\α}}\left(\sqrt{\mathrm{\ω}\mathrm{\γ}/\stackrel{\‾}{\mathrm{\γ}}}\right)}{2\sqrt{\mathrm{\γ}\stackrel{\‾}{\mathrm{\γ}}/\mathrm{\ω}}},---\left(4\right)$

According to the Jacobian transformation rule between stochastic variable PDF and its function gained new stochastic variable PDF, can connect The PDF receiving single symbol instantaneous signal-to-noise ratio γ is

$f_{\mathrm{\γ}}(\mathrm{\γ};m,\stackrel{\‾}{\mathrm{\γ}})=\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right){\stackrel{\‾}{\mathrm{\γ}}}^m}\exp (-\frac{m\mathrm{\γ}}{\stackrel{\‾}{\mathrm{\γ}}}),---\left(5\right)$

This expression formula clearly demonstrates stochastic variable γ and obeys Gamma distribution；

If channel exists large scale path loss and shadow fading, then its average received signal to noise ratio simultaneouslyJust obey logarithm State is distributed, and its PDF is

$f_{\stackrel{\‾}{\mathrm{\γ}}}\left(\stackrel{\‾}{\mathrm{\γ}}\right)=\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}\stackrel{\‾}{\mathrm{\γ}}}\exp \[-\frac{{(10lg\stackrel{\‾}{\mathrm{\γ}}-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\],---\left(6\right)$

In above formula, ξ=10/ln10 is a fixed constant；μ and σ is respectively barrier docking and receives the flat of signal envelope power generation All path loss and random fluctuation standard deviations；

In the case of considering Nakagami decline, path loss and shadow fading, now Composite Fading Channels mould can be obtained The PDF of the received signal to noise ratio γ of type is:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\underset{0}{\overset{\mathrm{\∞}}{\∫}}\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right){\stackrel{\‾}{\mathrm{\γ}}}^m}\exp (-\frac{m\mathrm{\γ}}{\stackrel{\‾}{\mathrm{\γ}}})\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}\stackrel{\‾}{\mathrm{\γ}}}\exp \[-\frac{{(10lg\stackrel{\‾}{\mathrm{\γ}}-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\]d\stackrel{\‾}{\mathrm{\γ}},\mathrm{\γ}>0,---\left(7\right)$

Wherein,Refer to average received signal to noise ratio, be also the independent variable of this inifinite integral, from the expression-form of above formula (7) simultaneously It is upper it can be seen that its received signal to noise ratio of Composite Fading Channels model in simulation actual complex communication environment obeys Gamma- Lognormal is distributed.

The method that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation the most according to claim 2, its feature Being, in step (2), the Guan Bi expression formula of described received signal to noise ratio cumulative distribution function is as follows:

The PDF that (7) formula is obtained be integrated the CDF of received signal to noise ratio γ is:

$\begin{array}{c}{F}_{\mathrm{\γ}}\left(X\right)={\∫}_0^X{f}_{\mathrm{\γ}}\left(\mathrm{\γ}\right)d\mathrm{\γ}\\ =1-\frac{1}{\mathrm{\Γ}\left(m\right)}{\∫}_0^{\mathrm{\∞}}\mathrm{\Γ}(m,mX/s)\frac{\mathrm{\ξ}}{\sqrt{2\mathrm{\π}}\mathrm{\σ}s}\exp \[-\frac{{(10\lg s-\mathrm{\μ})}^2}{2{\mathrm{\σ}}^2}\]ds\end{array},---\left(8\right)$

By the logarithm normal distribution in Gamma distribution replacement expression formula (7) so that logarithm shadow fading is modeled, namely with The PDF of the average received signal to noise ratio that Gamma distribution represents is:

$f_{\stackrel{\‾}{\mathrm{\γ}}}\left(\stackrel{\‾}{\mathrm{\γ}}\right)=\frac{{\stackrel{\‾}{\mathrm{\γ}}}^{n-1}}{\mathrm{\Γ}\left(n\right){\mathrm{\χ}}^n}\exp (-\frac{\stackrel{\‾}{\mathrm{\γ}}}{\mathrm{\χ}}),\stackrel{\‾}{\mathrm{\γ}}>0,---\left(9\right)$

In above formula, n is the exponent number of Gamma distribution；χ represents mean power；The approximate formula (9) obtained by Gamma distribution is with former Between the formula (6) that accurately Lognormal distribution obtains, the transformation relation between core parameter is:

$\left\{\begin{array}{c}\mathrm{\μ}=\mathrm{\ξ}\[ln\mathrm{\χ}+\mathrm{\ψ}\left(n\right)\]\\ {\mathrm{\σ}}^2={\mathrm{\ξ}}^2{\mathrm{\ψ}}^{\′}\left(n\right)\end{array}\right.,---\left(10\right)$

In above formula, ψ () and ψ ' () is digamma and trigamma function respectively；So compound after can being approximated declines The PDF of received signal to noise ratio γ in channel that falls is:

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)={\∫}_0^{\mathrm{\∞}}\frac{m^m{\mathrm{\γ}}^{m-1}}{\mathrm{\Γ}\left(m\right)s^m}\exp (-\frac{m\mathrm{\γ}}{s})\frac{s^{n-1}}{\mathrm{\Γ}\left(n\right){\mathrm{\χ}}^n}\exp (-\frac{s}{\mathrm{\χ}})ds,\mathrm{\γ}>0,---\left(11\right)$

From the expression-form of above formula (11) it can be seen that approximation its received signal to noise ratio of Composite Fading Channels model built is obeyed Gamma-Gamma is distributed；Make t=s/ χ and η=m/ χ, can obtain through deriving

$f_{\mathrm{\γ}}\left(\mathrm{\γ}\right)=\frac{2{\mathrm{\η}}^{\frac{m+n}{2}}}{\mathrm{\Γ}\left(m\right)\mathrm{\Γ}\left(n\right)}{\mathrm{\γ}}^{\frac{m+n}{2}-1}{K}_{(m-n)}\left(2\sqrt{\mathrm{\η}\mathrm{\γ}}\right),---\left(12\right)$

Wherein, K_(m-n)() is (m-n) rank Equations of The Second Kind modified Bessel functions；It is integrated above formula again obtaining received signal to noise ratio The CDF of γ is

$F_{\mathrm{\γ}}\left(X\right)={\∫}_0^X{f}_{\mathrm{\γ}}\left(\mathrm{\γ}\right)d\mathrm{\γ}=\frac{1}{\mathrm{\Γ}\left(m\right)\mathrm{\Γ}\left(n\right)}{G}_{13}^{21}\left(\mathrm{\η}X{|}_{m,n,0}^1\right)---\left(13\right)$

Wherein,It is Meijer-G function and 0≤k≤q, 0≤l≤p≤q；K, l, p, q are integer；

Formula (13) gives and a kind of can represent with tabular function conventional in engineering mathematics, and Composite Fading Channels receives The Guan Bi expression formula of signal to noise ratio CDF passes through numerical simulation computed in software.

The method that can quickly realize Composite Fading Channels cumulative distribution Performance Evaluation the most according to claim 3, its feature Being, described numerical simulation software specifically uses Matlab or Mathematic.