REAL VERSUS COMPLEX GAUSSIAN DISTRIBUTIONS FOR DIGITAL COMMUNICATION.
ABSTRACT: Concept of both real and complex Gaussian Random Variables (GRVs) and their corresponding Gaussian Random processes (GRPs) is very important and critical for understanding and designing of a real Communication System. In this paper, we thus discuss and compute the variousparameters involved in characterizing the real and complex Gaussian distributions completely for one andmultiple Random Variables (RVs). We first carry out our analysis for one real GRV and then extend our work to multiple real GRVs. Furthermore, we apply our technique for studying one complex GRV and subsequently extend it to the analysis of multiple complex GRVs also known as multivariate complex Gaussian random vector. Finally, we make the necessary comparison between real and complex Gaussian Distributions which are the key components of Additive White Gaussian Noise (AWGN) for baseband and bandpass transmission of the digital data through non ideal channels of digital Communication Systems.
Key Words: Complex Gaussian, Cumulative Density Function, MGF and CF, Probability Density Function, RealGaussian 1 INTRODUCTIONThe study of both real and complex Gaussian distributions isvery essential and critical in further understanding of Gaussian Random and Stochastic Processes. The transmission of information through non-ideal channel involves addition of additive white Gaussian Noise (AWGN) which plays significant role in distorting the information. This noise carries the properties and characteristics either of a real or a complex Gaussian Random Process depending upon the type of transmission, i.e., baseband or bandpass. The power spectral density (PSD) of both white and thermal noises is practically flat over a very large band (up to 1000GHz at room temperature), .Bandpass transmission contains the PSD of a random process to be confined to a certain passband while basebandtransmission allows the PSD of the process to be defined around the low frequency spectrum region. Bandpass random processes can be used effectively to modelmodulated communication signals and bandpass noises. Likebandpass signals, we can also model the bandpass noise in terms of its in phase and quadrature components using the two famous carriers. The PSD of this process can easily be expressed in terms of the PSD of the baseband random noise process .The Gaussian random process is perhaps the single most important random process in the area of communication. This process has a uniform PSD over large range offrequency spectrum and is thus known as a white Gaussian random process. The envelope of Gaussian noise processbehaves like Rayleigh density distribution. This distribution can easily be fabricated from two independent real GRVs orone scalar complex GRV having zero mean and the same variance. The magnitude of scalar complex GRV thus follows Rayleigh density function and phase of scalar complex GRV follows the uniform distribution. On the otherhand, when a sinusoidal signal is buried in this narrowbandGaussian noise, then the envelope of this transmission over the communication channel follows Ricean density instead frequency spectrum and is thus known as a white Gaussian random process. The envelope of Gaussian noise processbehaves like Rayleigh density distribution. This distribution can easily be fabricated from two independent real GRVs orone scalar complex GRV having zero mean and the same variance. The magnitude of scalar complex GRV thus follows Rayleigh density function and phase of scalar complex GRV follows the uniform distribution. On the otherhand, when a sinusoidal signal is buried in this narrowbandGaussian noise, then the envelope of this transmission over the communication channel follows Ricean density instead
approaches to Gaussian density with certain a mean and variance for the case when amplitude of the signal is much larger than variance of the noise. The phase of the noise in this case does not follow the uniform distribution due to non-linear terms involved into its density expression [4 7]. This paper is organized as follows: Section 2 computes the parameters required for complete description of Gaussian distribution for one real random variable. Extension to more than one real Gaussian random variable is carried out in section 3. Section 4 and 5 describe the detailed analysis for one and more than one complex Gaussian Random variables respectively. We make the important comparison between real and complex Gaussian Distributions in section 6. Finally, we present our conclusions in section 7.
2 COMPUTATION OF THE PARAMETERS NEEDED FOR REAL GAUSSIAN DISTRIBUTION IN CASE OF ONE RANDOM VARIABLE
This is a very important type of continuous distribution which is extensively used in the performance analysis of many communication systems because of the addition of white Gaussian Noise (AWGN) in the transmitted signal when it is received by the receiver. A real Gaussian Random Variable (RV), X is said to be normally distributed with mean, (mx) and variance, (sx2), if its probability density function, (PDF) can be expressed in the following form
We'll soon show that the area bounded by fX(x) over the entire x-axis is unity. Since linear transformation applied to Gaussian Random variables does not change its property , so we introduce such type of transformation as, Y = (X mx)/sx in order to convert this PDF into standard Gaussian PDF , whose mean is zero and variance equal to unity. The PDF of the RV, Y using the results of linear transformation thus becomes1
7 COMPARISONOFREALVSCOMPLEX GAUSSIAN DISTRIBUTIONSReal Gaussian distribution, X is normally used for basebandtransmission while complex Gaussian distribution, Z is used for bandpass transmission in digital communication. Mean of X is a real quantity while it is complex for the case of Z. Likewise, variance of Z is computed as E(|Z mZ|)2 in order to make it real in comparison to the variance of X which is simply equal to E(X mX)2 and is always real. For a scalar complex Distribution, its covariance matrix, CZ = Var(Z) =2Var(X) = 2Var(Y) and its PDF, fZ(z) is obtained bysubstituting n = 2 in eq. (11) along with using the relationsalready established in section 4. For multivariate complex random vector Z, we already mentioned that CZ = 2CX = 2CY and CXY = -CXY for proper and circular Gaussian RVs , . This relation leads to det2(CZ) = 22ndet(CX) and (X mX)t(CX)-1 (X mX) = 2(Z mZ)H(CZ)-1(Z mZ) in this case. Linear transformation applied to multivariate complex Gaussian vector, Z modifies the covariance matrix from ACXAt in case of multivariate real Gaussian vector, X to ACZAH. Mean vector also becomes complex due to complex nature of the random vector, Z. While computing MGF and CF in case of multivariate complex Gaussian distribution, the inner product term (i.e., first) defined in eq. (20), and the2nd term (i.e., quadratic form) of the same equation have to be modified accordingly (i.e., as per eq. (22)). Multivariatecomplex Gaussian distribution is quite general in nature since multivariate real Gaussian distribution expressed in
We are extremely grateful to the Department of Electrical Engineering of COMSATS Institute of Information Technology (CIIT), Lahore for carrying out this work. Moreover, we are also thankful to the anonymous reviewers for their valuable suggestions towards the improvement with respect to the quality of the paper.
1. John G. Proakis and Masoud Salehi, Digital3.LeonW.Couch,II,DigitalandCommunication Systems, Pearson Edition, 7th2009. Analogedition,
4.B. P. Lathi and Zhi Ding, Modern Digital and Analog Communication Systems, 4th international edition, Oxford University Press, 2010.5.A. B. Carlson, P. B. Crilly and J. C. Rutledge,Communication Systems. 42002, New York, NY 10020. Edition, McGraw Hill
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|Author:||Ansari, Ejaz A.; Akhtar, Saleem|
|Date:||Mar 31, 2014|
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