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.[1] 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,[2] quadrature amplitude modulation and information theory.

DefinitionEdit

A complex random variable   on the probability space   is a function   such that both its real part   and its imaginary part   are real random variables on  .

ExamplesEdit

Simple exampleEdit

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.

Probability   Value  
   
   
   

The expectation of this random variable may be simply calculated:  

Uniform distributionEdit

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 normal distributionEdit

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 functionEdit

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:

 

 

 

 

 

(Eq.1)

Probability density functionEdit

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.

ExpectationEdit

DefinitionEdit

The expectation of a complex random variable is defined based on the definition of the expectation of a real random variable:[3]:p. 112

 

 

 

 

 

(Eq.2)

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  .

PropertiesEdit

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-varianceEdit

Definition varianceEdit

The variance is defined as:[3]:p. 117

 

 

 

 

 

(Eq.3)

PropertiesEdit

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-varianceEdit

The pseudo-variance is a special case of the pseudo-covariance and is given by

 

 

 

 

 

(Eq.4)

Unlike the variance of  , which is always real and positive, the pseudo-variance of   is in general complex.

Covariance and pseudo-covarianceEdit

DefinitionEdit

The covariance between two complex random variables   is defined as[3]:p. 119

 

 

 

 

 

(Eq.5)

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):

 

 

 

 

 

(Eq.6)

The second order statistics are fully characterized by the covariance and the pseudo-covariance.

PropertiesEdit

The covariance has the following properties:

  •   (Conjugate symmetry)
  •   (Sesquilinearity)
  •  
  •  
  •  
  •  

UncorrelatednessEdit

Two complex random variables   and   are called uncorrelated if

 

OrthogonalityEdit

Two complex random variables   and   are called orthogonal if

 .

Circular symmetryEdit

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.

DefinitionEdit

A complex random variable   is circularly symmetric if, for any deterministic  , the distribution of   equals the distribution of  .

PropertiesEdit

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  .[4]

Proper complex random variablesEdit

The concept of proper random variables is unique to complex random variables, and has no correspondent concept with real random variables.

DefinitionEdit

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:

  •  
  •  
  •  

Covariance matrix of the real and imaginary partsEdit

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:

 .

TheoremEdit

Every circularly symmetric complex random variable with finite variance is proper.

Cauchy-Schwarz inequalityEdit

The Cauchy-Schwarz inequality for complex random variables, which can be derived using the Triangle inequality and Hölder's inequality, is

 .

Characteristic functionEdit

The characteristic function of a complex random variable is a function   defined by

 

See alsoEdit

ReferencesEdit

  1. ^ Eriksson, Jan; Ollila, Esa; Koivunen, Visa (2009). "Statistics for complex random variables revisited". Cite journal requires |journal= (help)
  2. ^ Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955.
  3. ^ a b c Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3.
  4. ^ Peter J. Schreier, Louis L. Scharf (2011). Statistical Signal Processing of Complex-Valued Data. Cambridge University Press. ISBN 9780511815911.