Complex random variable

In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.
Some concepts of real random variables have a straightforward generalization to complex random variables—e.g., the definition of the mean of a complex random variable. Other concepts are unique to complex random variables.
Applications of complex random variables are found in digital signal processing, quadrature amplitude modulation and information theory.
Definition.
A complex random variable formula_1 on the probability space formula_2 is a function formula_3 such that both its real part formula_4 and its imaginary part formula_5 are real random variables on formula_2.
Examples.
Simple example.
Consider a random variable that may take only the three complex values formula_7 with probabilities as specified in the table. This is a simple example of a complex random variable.
The expectation of this random variable may be simply calculated:
formula_8
Uniform distribution.
Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set formula_9. This random variable is an example of a complex random variable for which the probability density function is defined. The density function is shown as the yellow disk and dark blue base in the following figure.
Complex normal distribution.
Complex Gaussian random variables are often encountered in applications. They are a straightforward generalization of real Gaussian random variables. The following plot shows an example of the distribution of such a variable.
Cumulative distribution function.
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form formula_10 make no sense. However expressions of the form formula_11 make sense. Therefore, we define the cumulative distribution formula_12 of a complex random variables via the joint distribution of their real and imaginary parts:
Probability density function.
The probability density function of a complex random variable is defined as formula_13, i.e. the value of the density function at a point formula_14 is defined to be equal to the value of the joint density of the real and imaginary parts of the random variable evaluated at the point formula_15.
An equivalent definition is given by formula_16 where formula_17 and formula_18.
As in the real case the density function may not exist.
Expectation.
Definition.
The expectation of a complex random variable is defined based on the definition of the expectation of a real random variable:
Note that the expectation of a complex random variable does not exist if formula_19 or formula_20 does not exist.
If the complex random variable formula_1 has a probability density function formula_22, then the expectation is given by formula_23.
If the complex random variable formula_1 has a probability mass function formula_25, then the expectation is given by formula_26.
Properties.
Whenever the expectation of a complex random variable exists, taking the expectation and complex conjugation commute:
The expected value operator formula_28 is linear in the sense that
for any complex coefficients formula_30 even if formula_31 and formula_32 are not independent.
Variance and pseudo-variance.
Definition of variance.
The variance is defined as:
Properties.
The variance is always a nonnegative real number. It is equal to the sum of the variances of the real and imaginary part of the complex random variable:
The variance of a linear combination of complex random variables may be calculated using the following formula:
Definition pseudo-variance.
The pseudo-variance is a special case of the pseudo-covariance and is given by
Unlike the variance of formula_31, which is always real and positive, the pseudo-variance of formula_31 is in general complex.
Covariance and pseudo-covariance.
Definition.
The covariance between two complex random variables formula_37 is defined as
Notice the complex conjugation of the second factor in the definition. In contrast to real random variables, we also define a pseudo-covariance (also called complementary variance):
The second order statistics are fully characterized by the covariance and the pseudo-covariance.
Properties.
The covariance has the following properties:
Uncorrelatedness.
Two complex random variables formula_31 and formula_32 are called uncorrelated if
Orthogonality.
Two complex random variables formula_31 and formula_32 are called orthogonal if
Circular symmetry.
Circular symmetry of complex random variables is a common assumption used in the field of wireless communication. A typical example of a circular symmetric complex random variable is the complex Gaussian random variable with zero mean and zero pseudo-covariance matrix.
Definition.
A complex random variable formula_1 is circularly symmetric if, for any deterministic formula_51, the distribution of formula_52 equals the distribution of formula_1.
Properties.
By definition, a circularly symmetric complex random variable has
formula_54 for any formula_55. 
Thus the expectation of a circularly symmetric complex random variable can only be either zero or undefined.
Additionally,
formula_56 for any formula_55. 
Thus the pseudo-variance of a circularly symmetric complex random variable can only be zero.
If formula_31 and formula_59 have the same distribution, the phase of formula_31 must be uniformly distributed over formula_61 and independent of the amplitude of formula_31.
Proper complex random variables.
The concept of proper random variables is unique to complex random variables, and has no correspondent concept with real random variables.
Definition.
A complex random variable formula_1 is called proper if the following three conditions are all satisfied:
This definition is equivalent to the following conditions. This means that a complex random variable is proper if, and only if:
Covariance matrix of the real and imaginary parts.
For a general complex random variable, the pair formula_70 has the covariance matrix
However, for a proper complex random variable, the covariance matrix of the pair formula_70 has the following simple form:
Theorem.
Every circularly symmetric complex random variable with finite variance is proper.
Cauchy-Schwarz inequality.
The Cauchy-Schwarz inequality for complex random variables, which can be derived using the Triangle inequality and Hölder's inequality, is
Characteristic function.
The characteristic function of a complex random variable is a function formula_75 defined by