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.
Consider a random variable that may take only the three complex values 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:
Uniform distribution
Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set. 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 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.
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form make no sense. However expressions of the form make sense. Therefore, we define the cumulative distribution 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, i.e. the value of the density function at a point 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. An equivalent definition is given by where and. 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 or does not exist. If the complex random variable has a probability density function, then the expectation is given by. If the complex random variable has a probability mass function, then the expectation is given by.
Properties
Whenever the expectation of a complex random variable exists, taking the expectation and complex conjugation commute: The expected value operator is linear in the sense that for any complex coefficients even if and are not independent.
Variance and pseudo-variance
Definition 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, which is always real and positive, the pseudo-variance of is in general complex.
Covariance and pseudo-covariance
Definition
The covariance between two complex random variables 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 : 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 and are called uncorrelated if
Orthogonality
Two complex random variables and are called orthogonal if
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 is circularly symmetric if, for any deterministic, the distribution of equals the distribution of.
Properties
By definition, a circularly symmetric complex random variable has for any. Thus the expectation of a circularly symmetric complex random variable can only be either zero or undefined. Additionally, for any. Thus the pseudo-variance of a circularly symmetric complex random variable can only be zero. If and have the same distribution, the phase of must be uniformly distributed over and independent of the amplitude of.
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 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:
For a general complex random variable, the pair has the covariance matrix However, for a proper complex random variable, the covariance matrix of the pair has the following simple form:
Theorem
Every circularly symmetric complex random variable with finite variance is proper.
