How is a complex random variable x, a complex random vector x, a complex random signal x(t ), or a complex vector-valued random signal x(t ) statistically described using probability distributions and moments? This question will occupy many of the ensuing chapters in this book. However, as a preview of our methods, we will offer a sketchy
20 The origins and uses of complex signals
account of complex second-order moments and the Gaussian probability density function for the complex scalar x= u + jv. A more general account for vector-valued x will be given in Chapter2.
1.6.1 Bivariate Gaussian distribution
The real components u andv of the complex scalar random variable x = u + jv, which may be arranged in a vector z= [u, v]T, are said to be bivariate Gaussian distributed, with mean zero and covariance matrix Rzz, if their joint probability density function (pdf) is
Here the quadratic form quv(u, v) and the covariance matrix Rzzof the composite vector z are defined as follows: In the right-most parameterization of Rzz, the terms are
Ruu= E(u2) variance of the random variable u,
√Rvv correlation coefficient of the random variables u, v.
As in (A1.38), the inverse of the covariance matrix R−1zz may be factored as R−1zz =
1.6 The bivariate Gaussian distribution 21 a zero-mean Gaussian pdf forv, with variance Rvv. The error e andv are independent.
From u = r cos θ, v = r sin θ, du dv = r dr dθ, it is possible to change variables and obtain the pdf for the polar coordinates (r, θ)
prθ(r, θ) = r · puv(u, v)|u=r cos θ,v=r sin θ. (1.49) From here it is possible to integrate over r to obtain the marginal pdf forθ and over θ to obtain the marginal pdf for r . But this sequence of steps is so clumsy that it is hard to find formulas in the literature for these marginal pdfs. There is an alternative, which demonstrates again the power of complex representations.
1.6.2 Complex representation of the bivariate Gaussian distribution Let’s code the real random variables u andv as
u
Then the quadratic form quv(u, v) in the definition of the bivariate Gaussian distribution (1.45) may be written as
where the covariance matrix Rx x and its inverse R−1x x are
Rx x = E
The new terms in this representation of the quadratic form quv(u, v) bear comment.
So let’s consider the elements of Rx x. The variance term Rx x is
Rx x = E|x|2= E[(u + jv)(u − jv)] = Ruu+ Rvv+ j0. (1.54) This variance alone is an incomplete characterization for the bivariate pair (u, v), and it carries no information at all aboutρuv, the correlation coefficient between the random variables u andv. But Rx x contains another complex second-order moment
Rx x = Ex2= E[(u + jv)(u + jv)] = Ruu− Rvv+ j2 Ruu
√Rvvρuv, (1.55)
22 The origins and uses of complex signals
which we will call the complementary variance. The complementary variance is the correlation between x and its conjugate x∗. It is zero if and only if Ruu = Rvv and ρuv= 0. This is the so-called proper case. All others are improper.
Now let’s introduce the complex correlation coefficientρ between x and x∗as ρ = Rx x
Rx x. (1.56)
Thus, we may write Rx x = Rx xρ and Rx x2 − | Rx x|2= R2x x(1− |ρ|2). The complex corre-lation coefficientρ = |ρ|ejψsatisfies|ρ| ≤ 1. If |ρ| = 1, then x = Rx xR−1x xx∗= ρx∗= ejψx∗ with probability 1. Equivalent conditions for|ρ| = 1 are Ruu = 0, or Rvv= 0, or ρuv= ±1. The first of these conditions makes the complex signal x purely imag-inary and the second makes it real. The third condition means v = tan(ψ/2)u and x = [1 + j tan(ψ/2)]u. All these cases with |ρ| = 1 are called maximally improper because the support of the pdf for the complex random variable x degenerates into a line in the complex plane.
There are three real parameters Ruu, Rvv, andρuvrequired to determine the bivariate pdf for (u, v), and these may be obtained from the three real values Rx x, Reρ, and
Now, by replacing the quadratic form quv(u, v) in (1.44) with the expression (1.51), and noting that det Rx x = 4 det Rzz, we may record the complex representation of the pdf for the bivariate Gaussian distribution or, equivalently, the pdf for complex x:
px(x) puv(u, v) = 1
This shows that the bivariate pdf for the real pair (u, v) can be written in terms of the complex variable x. Yet the formula (1.60) is not what most people expect to see when they talk about the pdf of a complex Gaussian random variable x. In fact, it is often implicitly assumed that x is proper, i.e.,ρ = 0 and x is not correlated with x∗. Then the pdf takes on the simple and much better-known form
px(x)= 1
But it is clear from our development that (1.61) models only a special case of the bivariate pdf for (u, v) where Ruu = Rvvandρuv= 0. In general, we need to incorporate both the variance Rx x and the complementary variance Rx x = Rx xρ. That is, even in this very
1.7 Analysis of the polarization ellipse 23
simple bivariate case, we need to take into account the correlation between x and its complex conjugate x∗. In Chapter2, this general line of argumentation is generalized to derive the complex representation of the multivariate pdf puv(u, v).
1.6.3 Polar coordinates and marginal pdfs
What could be the virtue of the complex representation for the real bivariate pdf? One answer is this: with the change from Cartesian to polar coordinates, the bivariate pdf takes the simple form is now a simple matter to integrate this bivariate pdf to obtain the marginal pdfs for the radius r and the angleθ. The results, to be explored more fully in Chapter2, are
pr(r )= 2r Here I0is the modified Bessel function of the first kind of order 0, defined as
I0(z)= 1 than the parameters (Ruu, Rvv,ρuv) for real (u, v), are the most natural parameterization for the joint and marginal pdfs of the polar coordinates r andθ. These marginals are illustrated in Fig.1.9forψ = π/2 and various values of |ρ|. In the proper case ρ = 0, we see that the marginal pdf for r is Rayleigh and the marginal pdf for θ is uniform.
Because of the uniform phase distribution, a proper Gaussian random variable is also called circular. The larger|ρ| the more improper (or noncircular) x becomes, and the marginal distribution forθ develops two peaks at θ = ψ/2 = π/4 and θ = ψ/2 − π. At the same time, the maximum of the pdf for r is shifted to the left. However, the change in the marginal for r is not as dramatic as the change in the marginal forθ.