2.7 The Gaussian Probability Density Function Forms of the Gaussian pdf for Real Variates

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1 .7 The Gaussian Probability Density Function Samples taken from a Gaussian process have a jointly Gaussian pdf (the definition of Gaussian process). Correlator outputs are Gaussian random variables if the input is a Gaussian process..7- A pdf of enormous importance: o Characterized completely by st and nd order moments the means and (co)variances. o Central limit theorem: sums of independent, identically distributed random variables have a pdf that looks increasingly Gaussian as their number increases [A. Papoulis, Probability, Random Variables and Stochastic Processes, Mc-Graw-Hill, 984]. o Maimum entropy. o Many, many other interesting properties..7. Forms of the Gaussian pdf for Real Variates The scalar pdf: p ( ) = ep πσ m σ

2 o Note that m ep d = πσ irrespective of m. σ.7- o Area under the tail of zero mean, unit variance Gaussian: p ( ) = e π α Q ( ) = F ( ) = e dα π o Hence the areas below are each equal to d Q σ :

3 .7-3 Bivariate Gaussian: suppose, y are real, have their own variances and are not independent. From [P..4, eq..-56], p m y m y + σ σ = ρ y σ σ y y, ( y, ) ep πσ σ y ρ ( ρ ) m y m y Here ρ is the correlation coefficient y σ ρ =. For real variates, it lies σσ y in the range ρ. o If, y are uncorrelated ( ρ = 0), then p y, y m m y ( y, ) = ep + πσ σ y σ σ y m y m y = ep ep πσ y σ πσ σy It becomes a product of individual pdfs so uncorrelated Gaussian random variables are also statistically independent. Important.

4 .7-4 The multivariate pdf in vector form is tidier. = [ ] real vectors T p( ) = ep ( ) ( ) N ( ) m R m π R where R = E ( )( ) T m m is the covariance matri.,, T N. Then for o Is this the same as bivariate Gaussian for N =? It has R σ σ, σ σ ρ = = σσ σ, σ ρ σ σ You check it. o Again, the Gaussian pdf is fully characterized by its first order (m) and second order (R) moments. o Contours of constant probability are ellipses. This sketch is typical for two correlated components. Eigenvalues of R are λ and λ.

5 Gaussian with covariance matri 0 R = 0 0.5, so ρ = side views, parallel to aes:

6 Gaussian with covariance matri 0.5 R = , so ρ = side views, parallel to aes:

7 .7. Forms of the Gaussian pdf of Comple Variates For comple random variables, we compact an, y pair into a single variable z = + jy. In general, its pdf is the joint pdf of and y. However, we don t normally bother with comple notation unless and y.7-7 y are uncorrelated ( σ = 0 ) and have the same variances (so y σ =σ ) o We define the variance of z as σ ( ) ( ) z = E z m = E m + y m y y =σ +σ = σ o The joint, y pdf is then modified to give p z ( z) = ep πσ z m z σz Pack several comple variates into = [,,, ] z z z N T z with covariance matri R= E zz, where is the Hermitian transpose. o The Gaussian pdf for comple vectors is written T { } p( ) = ep ( ) ( ) N ( π ) R m R m T { m R m } = ep ( ) ( ) N π R

8 .7-8 Summary: o For a real scalar: o For a real vector: p m ( ) = ep πσ σ T p( ) = ep ( ) ( ) N ( ) m R m. π R. o For a comple scalar z = + jy: p z ( z) = ep πσ z m z σz because of the convention that variance is σ z = E z m. T o For a comple vector: p ( ) ep { ( ) z z = z m R ( z m )} π N R..7.3 Gaussian Characteristic Function If variates u and v are independent with pdfs pu ( u ) and pv () v, then the pdf of their sum w= u+ v is the convolution of the two pdfs. The reason: o the probability that w is less than or equal to some value ω, (i.e., the cumulative distribution function F ( ω )) is the probability that u is less than or equal to ω v, averaged over v: F ( ω ) = F ( ω v) p () v dv w y v w

9 o Differentiation gives p ( ω ) = p ( ω v) p () v dv, the convolution integral. w u v.7-9 Consequently, the Fourier transform of p ( w ) equals the product of the Fourier transforms of pu ( u ) and pv () v. In statistics, it is more common to use the characteristic function Φ ( ω ), the conjugate of the Fourier transform: jνw Mw( ν ) = pw( w) = pw( w) e dw Conjugated or not, * [ ] M ( ν ) = M ( ν) M ( ν ). w u v w So: add two independent variates and the resulting characteristic function is the product of the original characteristic functions. This is the basis of the usual proof of the central limit theorem, and the property is used frequently in analysis of sophisticated modulation and coding schemes. w For a real scalar Gaussian, the characteristic function is * ( m) σ jν j M ( j [ p( ) ] e e d E ν ν )= = = e πσ jmν = e ep ( σν )

10 The characteristic function of the comple form of the multivariate Gaussian pdf is [ ] T * j M ( j p ( ) E z e z z z ν ν) = = j T = ep( jm ν) ep ν Cν.7-0 o Note that M ν k z ν= 0 = jm k and ( ν ) M j = jm ν z ν= 0 and higher derivatives give higher moments..7.4 Higher Order Mied Moments Occasionally, you need higher order moments of Gaussian random variables. If,,, N are jointly Gaussian, zero mean, then [J.M. Wozencraft and I.M Jacobs, Principles of Communication Engineering, John Wiley, 965, Problem 3.3] 0, L odd, L even i i i L = σ σ σ j j j34 j jl jl all distinct pair of subscripts o Eamples: 3 4 = σ σ σ σ σ σ 3= 0

11 .7- o If a variable is repeated, treat each occurrence as distinct. 3 = = σ σ σ σ σ σ σ σ σ σ 4 = 3 = 3 σ σ For high order moments of comple Gaussian random variables (sometimes very useful in communications), see Appendi P, which was drawn from W.F. McGee, Comple Gaussian Noise Moments, IEEE Trans. Inf. Th., vol. IT-7, no., pp , March 97. It s [Mcge7] on the Additional Readings website..7.5 Related Probability Distributions Closely related to the comple Gaussian distribution are o the Rayleigh distribution the distribution of the magnitude z of a zero mean comple Gaussian z o the eponential distribution the distribution of the squared magnitude z of a zero mean comple z o the Rice distribution the distribution of z if z has a non-zero mean o the chi-square distribution with n degrees of freedom, it is the distribution of the squared norm z of a zero mean, comple Gaussian vector z with independent components

12 .7- o the non-central chi-square distribution with n degrees of freedom, it is the distribution of the squared norm z of a non-zero mean, comple Gaussian vector z with independent components o the Nakagami-m distribution the distribution of the norm z of a zero mean, comple Gaussian vector z with independent components The distribution of Gaussian z in polar coordinates ( r, φ ) is of particular interest. Consider the non-central pdf below, with isoprobability contours shown. Standard deviation σ in real and imaginary parts. The Rice parameter K is the ratio of energy of the mean ( Kσ ) to the energy of the rest ( σ ). Pdf of radius: r σ K 0 r r K pr () r = e e I σ σ Pdf of phase (with α= 0) [Proakis95], [Stuber96]: ( ( ( ))) K Kcos ( φ) pφ( φ ) = e + 4πK cos( φ) e Q K cos φ π For more information, see [P..4] and Appendi G of the lecture notes.