CN106027183B - A kind of method fast implementing Composite Fading Channels cumulative distribution Performance Evaluation - Google Patents
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Abstract
The invention discloses a kind of methods fast implementing Composite Fading Channels cumulative distribution Performance Evaluation, including following steps：(1) it uses Gamma distribution approximation Lognormal distributions for simulating shadow effect, and then builds Gamma Gamma distributions for approximate original Gamma Lognormal Composite Fading Channels models；(2) on the basis of this approximate model, the closure expression formula of the received signal to noise ratio cumulative distribution function indicated with Meijer G-functions is derived；(3) performance of former Composite Fading Channels cumulative distribution is calculated by Meijer G-functions.Gamma Gamma distributions approximate schemes proposed by the invention can simplify the expression of the complicated inifinite integral of former accurate model, to obtain the closed form of received signal to noise ratio cumulative distribution function, and then the complexity that CDF function formulas calculate is reduced, be conducive to quickly analysis, assessment communication system.
Description
Technical Field
The invention relates to a method for rapidly realizing the performance evaluation of cumulative distribution of a composite fading channel, which is particularly suitable for a complex mobile communication system comprehensively considering path loss, shadow effect, small-scale fading and additive white Gaussian noise.
Background
The communication performance of a wireless communication system is largely limited by the wireless communication channel in which the system operates. In a wireless communication system, fading refers to a phenomenon that the amplitude and phase of a signal received by a receiving end are randomly changed along with different paths and time changes of transmission, and the phenomenon greatly affects the performance of a receiver. Since one aspect of fading reflection is signal power issues, one divides fading into large-scale fading and small-scale fading, depending on how fast and how much power is reduced. The small-scale fading refers to the superposition of electromagnetic waves from all directions on signals received by a receiving end due to the influence of factors such as reflection, scattering, diffraction and the like on the electromagnetic waves in the channel propagation process, and the signals can cause severe fluctuation, namely multipath fading, in a small range; large scale fading refers to the loss and fluctuation of energy in a signal during propagation caused by obstructions between a transmitter and a receiver, such as buildings, mountains, jungles, and the like. In an actual wireless communication system, the diversity of propagation paths between the transmitting end and the receiving end of an antenna and various complex terrains can cause the small-scale fading of a channel to have very strong randomness; in addition, antennas at the transmitting and receiving ends in modern wireless communication systems are spaced far apart in a cell, for example, due to different access distances, the path loss experienced by signals between each distributed antenna port (base station) and a mobile station is different, so that the fading experienced by wireless signals between the antenna ports (base stations) and the mobile stations during transmission is not only small-scale fading, but also needs to consider the adverse effects of large-scale fading factors such as shadow fading, path loss, and the like.
Common large-scale propagation models generally include logarithmic distance path loss and lognormal shadowing models, while small-scale fading models include Nakagami fading, rayleigh fading, leis fading, and the like. Since the Nakagami distribution can well model the envelope of signals transmitted in fading channels including cities in an actual wireless communication environment, and combines the common situations of pure scattering and superposition line-of-sight transmission such as rayleigh and leis distribution, the model is widely accepted and applied in the industry once being proposed.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide a method for rapidly realizing the performance evaluation of the cumulative distribution of the composite fading channel, and the Gamma-Gamma distribution approximation scheme provided by the invention can simplify the expression of the complex infinite integral of the original accurate model, thereby obtaining the closed form of the cumulative distribution function of the received signal-to-noise ratio, further reducing the complexity of the calculation of the CDF function formula, and being beneficial to rapidly analyzing and evaluating the performance indexes of the communication system, such as the interruption probability, the channel capacity and the like.
In order to achieve the purpose, the invention is realized by the following technical scheme:
the invention relates to a method for rapidly realizing the performance evaluation of cumulative distribution of a composite fading channel, which comprises the following steps:
(1) setting channel parameters, adopting Gamma distribution approximation (approximation means that the shapes of Probability Density Functions (PDFs) of the Gamma distribution and the Lognormal distribution are similar, namely that the probability density function curves of the Gamma distribution and the Lognormal distribution can be well overlapped with each other, which is a general term in the field) Lognormal distribution for simulating a shadow effect, and further constructing a Gamma-Gamma distribution for approximating (approximation means that PDFs of the Gamma-Gamma distribution and PDFs of the original Gamma-Lognormal distribution are similar, which is a general term in the field) an original Gamma-Lognormal composite fading channel model;
(2) deriving a closed expression of a received signal-to-noise ratio cumulative distribution function expressed by a Meijer-G function on the basis of a composite fading channel model (the model refers to a Gamma-Gamma distribution model for approximation);
(3) the Meijer-G function calculates the performance of the cumulative distribution of the original composite fading channel by referring to a formula value list or numerical calculation software (such as Matlab or Mathematics).
In step (1), the method for constructing the composite fading channel model comprises the following steps:
under a small-scale Nakagami fading channel, the envelope α of a wireless communication system transmission signal follows Nakagami distribution, and the PDF is:
in the above formula, m and ω are two important parameters of the Nakagami distribution, and the expressions are respectively:
wherein, E [. cndot. ] represents the mean value, Var [. cndot. ] represents the variance, Γ (. cndot.) represents the gamma function, ω is the mean square value of the fading amplitude α, m is called the form factor or fading index, which represents the severity of the small-scale fading at the moment, and the value of m is more than or equal to 1/2;
there are several special cases of different values of the fading index m: when m is 1/2, the degradation is unilateral Gaussian distribution; when m is 1, it is exactly the rayleigh distribution; when m is>At 1, the Nakagami distribution may be equivalent to a Rice factor ofThe rice distribution of (a);
under the condition that additive white Gaussian noise exists in the Nakagami fading channel, the average receiving signal-to-noise ratio corresponding to each symbol of the receiving end is consideredAnd the instantaneous received signal-to-noise ratio γ have the following relationship:
wherein N is0And EsPower spectral density and signal transmission power of Gaussian white noise respectively;
according to the above formula, the following relationship exists between the probability density functions of the instantaneous received signal-to-noise ratio γ and the received signal envelope α of a single symbol:
according to the Jacobian determinant transformation rule between the random variable PDF and the new random variable PDF obtained by the function of the random variable PDF, the PDF for receiving the single symbol instantaneous signal-to-noise ratio gamma can be obtained as
This expression clearly illustrates that the random variable γ follows a Gamma distribution;
if large scale path loss and shadow fading exist in the channel at the same time, the average receiving signal-to-noise ratioObey a lognormal distribution having a PDF of
in the above formula, ξ is 10/ln10 as a fixed constant, mu and sigma are the average path loss and the random fluctuation standard deviation generated by the obstacle to the envelope power of the received signal respectively;
under the condition of comprehensively considering Nakagami fading, path loss and shadow fading, the PDF of the received signal-to-noise ratio gamma of the composite fading channel model at the moment can be obtained as follows:
wherein,the average received signal-to-noise ratio is referred to, and is also an independent variable of the infinite integral, and as can be seen from the expression form of the above equation (7), the received signal-to-noise ratio of the composite fading channel model in the simulated actual complex communication environment follows the Gamma-Lognnorm distribution.
In the step (2), the closed expression of the cumulative distribution function of the received signal-to-noise ratio is as follows:
the CDF of the received signal-to-noise ratio γ obtained by integrating the PDF obtained by equation (7) is:
the log-normal distribution in expression (7) is replaced by a Gamma distribution to model logarithmic shadow fading, i.e. the PDF of the average received signal-to-noise ratio expressed by the Gamma distribution is:
in the above formula, n is the order of Gamma distribution; χ represents the average power; between the approximate formula (9) obtained from the Gamma distribution and the formula (6) obtained from the original exact Lognormal distribution, the transformation relationship between the core parameters is:
in the above formula, ψ (-) and ψ' () are digamma and trigamma functions, respectively; therefore, the PDF of the received signal-to-noise ratio γ in the approximated composite fading channel can be obtained as:
as can be seen from the expression form of the above formula (11), the constructed approximate composite fading channel model has the received signal-to-noise ratio obeying the Gamma-Gamma distribution, and t ═ s/χ and η ═ m/χ are derived to obtain the final product
Wherein, K(m-n)(. h) is a modified Bessel function of the second class of order (m-n); the CDF of the received signal-to-noise ratio gamma is obtained by integrating the above formula
Wherein,is a Meijer-G function, k is more than or equal to 0 and less than or equal to q, l is more than or equal to 0 and less than or equal to p and less than or equal to q; k, l, p and q are integers;
equation (13) shows a table function (which is a term commonly used in engineering mathematics and is called "Tabulated function" in english, and is translated into the table function "), and a closed expression of the composite fading channel received signal-to-noise ratio CDF is calculated by numerical simulation software.
The numerical simulation software specifically adopts Matlab or Mathematic.
The method for rapidly realizing the performance evaluation of the cumulative distribution function of the composite fading channel provided by the invention is convenient for analyzing the performance indexes of the system, such as the outage probability, the channel capacity and the like. According to the scheme, when a composite fading model is constructed, the small-scale fading of a transmission signal is reflected by Nakagami distribution, the large-scale fading characteristics of path loss and shadow effect are described by adopting Gamma distribution approximate Lognormal (Lognnormal) distribution, and the probability density function of each symbol signal-to-noise ratio of a receiving end subjected to approximation obeys the Gamma-Gamma distribution. Based on this, the cumulative distribution function of the composite fading channel can be reduced to a closed expression expressed by a so-called "tabulated function" Meijer-G function, which is beneficial to performing computer fast numerical simulation to analyze the characteristics of the composite fading channel.
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Fig. 1 is a general model diagram of a wireless communication system based on a cell structure;
FIG. 2 is a flowchart illustrating the operation of a method for fast performance evaluation of cumulative distribution of a composite fading channel according to the present invention;
FIG. 3 is a graph comparing the received signal-to-noise ratio cumulative distribution characteristic of the composite fading channel approximated by Gamma-Gamma with the accurate Gamma-lognormal cumulative distribution function curve, using the fading index m as the variable;
FIG. 4 is a graph comparing the received signal-to-noise ratio cumulative distribution characteristic and the accurate cumulative distribution function curve of a composite fading channel approximated by Gamma-Gamma with the shadow fading degree sigma as a variable;
fig. 5 is a graph comparing the received signal-to-noise ratio cumulative distribution characteristic and the accurate cumulative distribution function curve of the composite fading channel approximated by Gamma-Gamma with the average path loss mu as a variable.
Detailed Description
In order to make the technical means, the creation characteristics, the achievement purposes and the effects of the invention easy to understand, the invention is further described with the specific embodiments.
The embodiment of the invention can be widely applied to modern wireless communication systems with complex communication environments, namely, a method for rapidly evaluating the cumulative distribution performance of the channel is provided to provide powerful basis for analyzing and evaluating the outage probability and the channel capacity of the wireless communication system. A general model diagram of a modern wireless communication system including a composite fading channel is shown in fig. 1: in a wireless communication environment similar to a cellular structure, when a wireless signal transmitted by a base station is transmitted in a long distance in a regional range, the signal can encounter the blockage of obstacles such as tall buildings, jungles and the like in the transmission process to generate large-scale fading; when the signal reaches the vicinity of the receiving end, multipath small-scale fading is generated because the wireless signal is influenced by factors such as surrounding scattering, reflection environment and the like. Therefore, signals received by a receiving end of mobile equipment such as a mobile phone are obtained after original signals sent by a sending end pass through a channel with strong randomness, and therefore, modeling research on wireless communication channels is always the key point in the research process of mobile communication.
The invention constructs a composite fading channel model which accords with the actual communication transmission condition under the condition of comprehensively considering large-scale fading such as path loss, shadow effect and the like, Nakagami small-scale fading and white noise, a Probability Density Function (PDF) of a received signal-to-noise ratio obtained based on the model is an accurate signal power statistical characteristic expression, and important system performance indexes such as interruption Probability, channel capacity and the like in a modern communication system can be further evaluated according to a Cumulative Distribution Function (CDF) obtained by the PDF expression on the basis, so the calculation and evaluation of the CDF Function are particularly important. Because the PDF expression of the signal-to-noise ratio of the received signal of the composite fading channel is usually a complex infinite integral form, it cannot be simplified to obtain a closed cumulative distribution function expression, so it is not favorable for further carrying out related research, and it has become a difficulty in performance analysis of modern communication systems. The invention provides a simplified approximate model aiming at the composite fading channel, thereby being capable of rapidly realizing the performance analysis and research of the cumulative distribution function characteristic of the composite fading channel.
referring to fig. 2, under a small-scale Nakagami fading channel, an envelope α of a transmission signal of the wireless communication system follows a Nakagami distribution, whose PDF is:
in the above formula, m and ω are two important parameters of the Nakagami distribution, and the expressions are respectively:
wherein, E [. C]Meaning the mean value, Var [. cndot]the method is characterized in that the variance is solved, gamma (·) is shown, omega is the mean square value of the fading amplitude α, m is called a shape factor or a fading index and shows the severity of the small-scale fading at the moment, the value of the gamma is more than or equal to 1/2, and different values of the fading index m have several special conditions, namely when m is 1/2, the gamma is degraded into unilateral Gaussian distribution, when m is 1, the gamma is exactly Rayleigh distribution, and when m is 1, the gamma is not changed into the unilateral Gaussian distribution>At 1, the Nakagami distribution may be equivalent to a Rice factor ofThe rice distribution of (a). Under the condition that additive white Gaussian noise exists in the Nakagami fading channel, the average receiving signal-to-noise ratio corresponding to each symbol of the receiving end is consideredAnd the instantaneous received signal-to-noise ratio γ have the following relationship:
wherein N is0And Esthe following relationship exists between the instantaneous received signal-to-noise ratio γ and the probability density function of the received signal envelope α for a single symbol, as can be seen from the above equation:
according to the Jacobian determinant transformation rule between the random variable PDF and the new random variable PDF obtained by the function of the random variable PDF, the PDF for receiving the single symbol instantaneous signal-to-noise ratio gamma can be obtained as
This expression clearly illustrates that the random variable γ follows a Gamma distribution.
If large scale path loss and shadow fading exist in the channel at the same time, the average receiving signal-to-noise ratioObey a lognormal distribution having a PDF of
in the above equation, ξ ═ 10/ln10, μ and σ (both in dB), are the average path loss and the standard deviation of random fluctuation, respectively, due to the envelope power of the received signal caused by the obstacle, and when the Nakagami fading, the path loss and the shadow fading are considered together, the PDF of the received signal-to-noise ratio γ of the composite fading channel model at this time can be found as:
from the expression of the above equation (7), it can be seen that the received signal-to-noise ratio of the complex fading channel model in the actual complex communication environment is simulated to follow the Gamma-Lognormal distribution. And integrating the PDF obtained by the above formula to obtain the CDF of the received signal-to-noise ratio gamma as follows:
as can be seen from equation (8), the CDF of the received snr γ of the composite fading channel is a complex infinite integral equation, and cannot be written into a form of a general closed expression in engineering mathematics, which is very unfavorable for performing evaluation and research work of performance indexes such as outage probability, channel capacity and the like in a modern wireless communication system established based on the cumulative distribution function expression. Therefore, the present invention intends to provide a simplified approximation method to solve the problem, i.e. using Gamma distribution to replace the lognormal distribution in expression (8) to model logarithmic shadow fading, i.e. the PDF of the average received signal-to-noise ratio expressed by Gamma distribution is:
in the above formula, n is the order of Gamma distribution; χ represents the average power. Between the approximate formula (9) obtained from the Gamma distribution and the formula (6) obtained from the original exact Lognormal distribution, the transformation relationship between the core parameters is:
in the above equation, ψ (-) and ψ' () are digamma and trigamma functions, respectively. Therefore, the PDF of the received signal-to-noise ratio γ in the approximated composite fading channel can be obtained as:
as can be seen from the expression form of the above equation (11), the constructed approximate complex fading channel model has the received snr obeying the Gamma-Gamma distribution, let t ═ s/χ and η ═ m/χ, and can be derived to obtain the
Wherein, K(m-n)(. cndot.) is a modified Bessel function of the second class (m-n). The CDF of the received signal-to-noise ratio gamma is obtained by integrating the above formula
Wherein,is a Meijer-G function, k is more than or equal to 0 and less than or equal to q, l is more than or equal to 0 and less than or equal to p and less than or equal to q; k, l, p, q are integers. Equation (13) gives a closed expression for the CDF, which can be expressed by a "tabulation function" commonly used in engineering mathematics, and can be quickly and accurately calculated by, for example, numerical simulation software Matlab, Mathematic, etc.
Fig. 3 is a graph comparing the cumulative distribution function of the received signal-to-noise ratio of the composite fading channel obtained by using the fading index m as a variable and using Gamma-Gamma approximation with the original precise cumulative distribution function curve. It can be seen from the figure that, on the premise of giving typical large-scale fading influence factors such as 20dB of path loss μ and 8dB of shadow fading degree σ, there is a relatively good approximation effect between the curve of the approximate cumulative distribution function CDF varying with the received signal-to-noise ratio X obtained by changing the small-scale fading shape factor m and the accurate cumulative distribution function performance curve, that is, the Gamma-Gamma distribution can better reflect the cumulative distribution characteristic of the real composite fading channel.
Fig. 4 is a graph comparing the received signal-to-noise ratio cumulative distribution function and the accurate cumulative distribution function curve of the shadow fading degree sigma by adopting Gamma-Gamma approximate composite fading channel. And changing an approximate effect graph between an approximation obtained by the random fluctuation standard deviation sigma of the shadow fading and an accurate cumulative distribution function performance curve under the condition that the typical small-scale fading influence factors such as the fading index m is 1 and the influence factor path loss mu of the large-scale fading is 20 dB. It can be seen from the figure that no matter how the random fluctuation standard deviation sigma takes a value, the error between the cumulative distribution function curve graph after the approximation processing and the original accurate model curve graph is small, and the development trends of the two types of curves are basically consistent, namely the approximation processing has high accuracy.
Fig. 5 is a graph comparing the average path loss mu as a variable and using Gamma-Gamma approximate composite fading channel received signal-to-noise ratio cumulative distribution function with the accurate cumulative distribution function curve. Under the conditions that the standard deviation sigma of the fading random fluctuation which is a typical large-scale fading influence factor is given to be 8dB and the fading index m which is a small-scale fading influence factor is given to be 1, the approximate cumulative distribution function performance curve obtained by changing the average path loss mu can also well approximate the precise cumulative distribution function performance curve.
It can be seen from the above specific implementation process that the approximate processing method for rapidly evaluating the cumulative distribution performance of the composite fading channel has very strong applicability and accuracy under different fading influence factors: namely, for the typical values of the fading index m determining the severity of the small-scale fading, the random fluctuation standard deviation sigma determining the severity of the large-scale fading and the average path loss mu, the method can accurately approximate the cumulative distribution function curve of the accurate receiving signal-to-noise ratio, and further reflect the cumulative distribution characteristic of the composite channel.
In conclusion, the Gamma-Gamma distribution approximation scheme provided by the invention can simplify the expression of the complex infinite integral of the original precise model so as to obtain the closed form of the cumulative distribution function of the received signal-to-noise ratio, further reduce the complexity of the calculation of the CDF function formula, and is beneficial to rapidly analyzing and evaluating the performance indexes of the communication system, such as the interruption probability, the channel capacity and the like.
The foregoing shows and describes the general principles and broad features of the present invention and advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (3)
1. A method for rapidly realizing performance evaluation of cumulative distribution of a composite fading channel is characterized by comprising the following steps:
(1) setting channel parameters, adopting Gamma distribution to approximate Lognormal distribution for simulating shadow effect, and further constructing Gamma-Gamma distribution for approximating the original Gamma-Lognormal composite fading channel model;
(2) deriving a closed expression of a receiving signal-to-noise ratio cumulative distribution function expressed by a Meijer-G function on the basis of a composite fading channel model;
(3) calculating the performance of the cumulative distribution of the original composite fading channel by the Meijer-G function by consulting a formula value list or numerical calculation software; in step (1), the method for constructing the composite fading channel model comprises the following steps:
under a small-scale Nakagami fading channel, the envelope α of a transmission signal of a wireless communication system follows Nakagami distribution, and the probability density function PDF of the envelope α is as follows:
in the above formula, m and ω are two important parameters of the Nakagami distribution, and the expressions are respectively:
wherein, E [. cndot. ] represents the mean value, Var [. cndot. ] represents the variance, Γ (. cndot.) represents the gamma function, ω is the mean square value of the fading amplitude α, m is called the form factor or fading index, which represents the severity of the small-scale fading at the moment, and the value of m is more than or equal to 1/2;
there are several special cases of different values of the fading index m: when m is 1/2, the degradation is unilateral Gaussian distribution; when m is 1, it is exactly the rayleigh distribution; when m is>At 1, the Nakagami distribution may be equivalent to a Rice factor ofThe rice distribution of (a);
under the condition that additive white Gaussian noise exists in the Nakagami fading channel, the average receiving signal-to-noise ratio corresponding to each symbol of the receiving end is consideredAnd the instantaneous received signal-to-noise ratio γ have the following relationship:
wherein N is0And EsPower spectral density and signal transmission power of Gaussian white noise respectively;
according to the above formula, the following relationship exists between the probability density functions of the instantaneous received signal-to-noise ratio γ and the received signal envelope α of a single symbol:
according to the Jacobian determinant transformation rule between the random variable PDF and the new random variable PDF obtained by the function of the random variable PDF, the PDF for receiving the single symbol instantaneous signal-to-noise ratio gamma can be obtained as
This expression clearly illustrates that the random variable γ follows a Gamma distribution;
if large scale path loss and shadow fading exist in the channel at the same time, the average receiving signal-to-noise ratioObey a lognormal distribution having a PDF of
in the above formula, ξ is 10/ln10 as a fixed constant, mu and sigma are the average path loss and the random fluctuation standard deviation generated by the obstacle to the envelope power of the received signal respectively;
under the condition of comprehensively considering Nakagami fading, path loss and shadow fading, the PDF of the received signal-to-noise ratio gamma of the composite fading channel model at the moment can be obtained as follows:
wherein,the average received signal-to-noise ratio is referred to, and is also an independent variable of the infinite integral, and as can be seen from the expression form of the above equation (7), the received signal-to-noise ratio of the composite fading channel model in the simulated actual complex communication environment follows the Gamma-Lognnorm distribution.
2. The method according to claim 1, wherein in step (2), the closed expression of the cumulative distribution function of received signal-to-noise ratio is as follows:
the CDF of the received signal-to-noise ratio γ obtained by integrating the PDF obtained by equation (7) is:
in the formula, symbol X represents an integral value upper limit of the received signal-to-noise ratio γ; symbol s, the average received signal-to-noise ratio in equation (7)For the sake of simple symbol description and distinction from gamma, it is denoted by s;
the log-normal distribution in expression (7) is replaced by a Gamma distribution to model logarithmic shadow fading, i.e. the PDF of the average received signal-to-noise ratio expressed by the Gamma distribution is:
in the above formula, n is the order of Gamma distribution; χ represents the average power; between the approximate formula (9) obtained from the Gamma distribution and the formula (6) obtained from the original exact Lognormal distribution, the transformation relationship between the core parameters is:
in the above formula, ψ (-) and ψ' () are digamma and trigamma functions, respectively; therefore, the PDF of the received signal-to-noise ratio γ in the approximated composite fading channel can be obtained as:
as can be seen from the expression form of the above formula (11), the constructed approximate composite fading channel model has the received signal-to-noise ratio obeying the Gamma-Gamma distribution, and t ═ s/χ and η ═ m/χ are derived to obtain the final product
Wherein, K(m-n) (· is a modified Bessel function of the second class of order (m-n); the cumulative distribution function CDF of the received signal-to-noise ratio gamma is obtained by integrating the above formula
Wherein,is a Meijer-G function, k is more than or equal to 0 and less than or equal to q, l is more than or equal to 0 and less than or equal to p and less than or equal to q; k, l, p and q are integers;
equation (13) gives a closed expression of the received signal-to-noise ratio CDF of the composite fading channel, which can be represented by a tabulated function commonly used in engineering mathematics, and can be calculated by numerical simulation software.
3. The method of claim 2, wherein the numerical simulation software specifically employs Matlab or Mathematic.
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| CN107994965B (en) * | 2018-01-30 | 2020-08-28 | 合肥工业大学 | Lognormal channel model verification system |
| CN109309536B (en) * | 2018-10-23 | 2020-07-17 | 河海大学 | Method for reducing approximation complexity of Nakagami inverse CDF function |
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