We call a complex-valued random variable z=x+iy a (circular symmetric) complex Gaussian variable, or it follows complex Gaussian distribution, if its real and imaginary parts, x and y, are jointly Gaussian (i.e. (x,y) follows a two-dimensional Gaussian distribution), uncorrelated (therefore alsoindependent in this case), and they have the same variance of σ2. Denoting the mean of x and y by mx and my respectively, we call mz=mx+imy=E[z] the mean of z, and σ2=E[(x-mx)2]=E[(y-my)2]=(1/2)E[|z-mz|2] is called z's variance per real dimension.
p(x,y)=(2πσ2)-1 exp(-((x-mx)2+(y-my)2)/(2σ2)).
Noting that z=x+iy, the probability density function can also be represented in terms of z:
p(z)=(2πσ2)-1 exp(-|z-mz|2/(2σ2))
This is preferred representation of the probability density function of complex-valued random variables. Just remember to separate the real and imaginary parts, x and y, when calculating expectations involving such random variables, rather than integrating over the complex z directly. This applies to the multi-dimensional cases below as well.
The above definition can be easily extended to multi-dimensional cases, by using a 2d-dimensional real Gaussian variable to represent the real and imaginary parts of the d-dimensional complex Gaussian variable. The probability density function of a d-dimensional complex Gaussian variablez=x+iy (a column vector) is
p(z)=(2π)-d (det(Σ))-1 exp(-(z-mz)HΣ-1(z-mz)/2)
Here, mz=E[z] is the mean of the random vector z, Σ=(1/2)E[(z-mz)(z-mz)H] is its covariance matrix multipled by 1/2 (assuming it is not singular), det(Σ) is Σ's determinant, and zH means z's Hermitian transposition, or zT with its elements conjugated. Note that it looks very similar to the probability density function of a d-dimensional real Gaussian variable (see here):
p(x)=(2π)-d/2 (det(Σ))-1/2 exp(-(x-mx)TΣ-1(x-mx)/2)
Above, we have mandated that the real and imaginary parts of a complex Gaussian variable z to be uncorrelated with the same variance, in which case we call z to be circular symmetric, and the above formula for p(z) applies. If z is a d-dimensional complex Gaussian variable, circular symmetricity is more complicated, but it basically means that p(z) can be represented as above. Circular symmetricity gives complex Gaussian variables some desirable properties, as follows:
First, suppose
z is a zero-mean d-dimensional complex Gaussian variable (i.e. m
z=
0), then its distribution is
rotationally invariant, or e
iθz has the same probability density function as
z for any real θ (try proving it by yourself). This is probably why it is called "circular symmetric".
Second, let A=E[z
1z
2...z
n z
n+1*...z
n+m*], where the unique ones of z
1, ..., z
n+m (we allow for duplicates among them) are jointly Gaussian zero-mean complex random variables (i.e., they form a zero-mean multi-dimensional complex Gaussian variable
z), and * denotes
complex conjugation, then A=0 if n≠m. To prove this, let w
k=e
iθz
k, k=1,...,n+m, &theta∈
R, and define B in the same way as A only with z
k replaced by w
k. Now
w=e
iθzshould contain the unique ones of w
k, and
w and
z should have the same distribution due to circular symmetricity, therefore A=B. However, using w
k=e
iθz
k, we obtain B=e
i(n-m)θA. Since the choice of θ is arbitrary, A must be zero if n≠m. In particular, if z is a zero-mean complex Gaussian variable, then E[z
2]=0 (although E[|z|
2]=2σ
2).
By taking the modulus of a complex Gaussian variable, we get two important distributions: Rayleigh distribution and Rice distribution.
Let z=x+iy be a complex Gaussian variable, a=|E[z]| and σ2=(1/2) E[|z|2], the probability distribution function of r=|z| can be easily obtained by doing variable substitution x=r cosθ, y=r sinθ in p(x,y), then integrating over [0,2π] on θ. When a=0 (or z has zero mean), r=|z| is said to followRayleigh distribution, whose probability density function is
p(r)=(r/σ2) exp(-r2/(2σ2)), r≥0.
Here, p(r) is almost linear with r for small r, and decreases rapidly when r becomes large.
When a is not zero, r=|z| is said to follow Rice distribution, whose probability density function is
p(r)=(r/σ2) exp(-(r2+a2)/(2σ2)) I0(a r/σ2), r≥0
I0(x)=(1/2π) ∫02π exp(x cosθ) dθ.
Here, p(r) is large for r near a, and decreases rapidly for larger and smaller r.
Complex Gaussian variables are often used in
engineering. For example, in
communication theory, narrow-band Gaussian
white noise can be represented by a complex Gaussian process when using the
equivalent low-pass representation, where the modulus (absolute value) represents amplitude, and the argument (angle) represents carrier phase, therefore the integration of it over an interval, which arises in the
decision variables of many
demodulators, is a complex Gaussian variable.
