Complex random variable

A complex random variable ${\displaystyle Z}$ on the probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ is a function ${\displaystyle Z\colon \Omega \rightarrow \mathbb {C} }$ such that both its real part ${\displaystyle \Re {(Z)}}$ and its imaginary part ${\displaystyle \Im {(Z)}}$ are real random variables on ${\displaystyle (\Omega ,{\mathcal {F}},P)}$.

The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form ${\displaystyle P(Z\leq 1+3i)}$ make no sense. However expressions of the form ${\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)}$ make sense. Therefore, we define the cumulative distribution ${\displaystyle F_{Z}:\mathbb {C} \to [0,1]}$ of a complex random variables via the joint distribution of their real and imaginary parts:

${\displaystyle F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})=P(\Re {(Z)}\leq \Re {(z)},\Im {(Z)}\leq \Im {(z)})}$

(Eq.1)

The probability density function of a complex random variable is defined as ${\displaystyle f_{Z}(z)=f_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})}$, i.e. the value of the density function at a point ${\displaystyle z\in \mathbb {C} }$ 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 ${\displaystyle (\Re {(z)},\Im {(z)})}$.

An equivalent definition is given by ${\displaystyle f_{Z}(z)={\frac {\partial ^{2}}{\partial x\partial y}}P(\Re {(Z)}\leq x,\Im {(Z)}\leq y)}$ where ${\displaystyle x=\Re {(z)}}$ and ${\displaystyle y=\Im {(z)}}$.

As in the real case the density function may not exist.

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

${\displaystyle \operatorname {E} [Z]=\operatorname {E} [\Re {(Z)}]+i\operatorname {E} [\Im {(Z)}]}$

(Eq.2)

Note that the expectation of a complex random variable does not exist if ${\displaystyle \operatorname {E} [\Re {(Z)}]}$ or ${\displaystyle \operatorname {E} [\Im {(Z)}]}$ does not exist.

If the complex random variable ${\displaystyle Z}$ has a probability density function ${\displaystyle f_{Z}(z)}$, then the expectation is given by ${\displaystyle \operatorname {E} [Z]=\iint _{\mathbb {C} }z\cdot f_{Z}(z)\,dx\,dy}$.

If the complex random variable ${\displaystyle Z}$ has a probability mass function ${\displaystyle p_{Z}(z)}$, then the expectation is given by ${\displaystyle \operatorname {E} [Z]=\sum _{z\in \mathbb {Z} }z\cdot p_{Z}(z)}$.

Properties

Whenever the expectation of a complex random variable exists, taking the expectation and complex conjugation commute:

${\displaystyle {\overline {\operatorname {E} [Z]}}=\operatorname {E} [{\overline {Z}}].}$

The expected value operator ${\displaystyle \operatorname {E} [\cdot ]}$ is linear in the sense that

${\displaystyle \operatorname {E} [aZ+bW]=a\operatorname {E} [Z]+b\operatorname {E} [W]}$

for any complex coefficients ${\displaystyle a,b}$ even if ${\displaystyle Z}$ and ${\displaystyle W}$ are not independent.

The variance is defined in terms of absolute squares as:[3]^: 117

${\displaystyle \operatorname {K} _{ZZ}=\operatorname {Var} [Z]=\operatorname {E} \left[\left|Z-\operatorname {E} [Z]\right|^{2}\right]=\operatorname {E} [|Z|^{2}]-\left|\operatorname {E} [Z]\right|^{2}}$

(Eq.3)

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:

${\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [\Re {(Z)}]+\operatorname {Var} [\Im {(Z)}].}$

The variance of a linear combination of complex random variables may be calculated using the following formula:

${\displaystyle \operatorname {Var} \left[\sum _{k=1}^{N}a_{k}Z_{k}\right]=\sum _{i=1}^{N}\sum _{j=1}^{N}a_{i}{\overline {a_{j}}}\operatorname {Cov} [Z_{i},Z_{j}].}$

Pseudo-variance

The pseudo-variance is a special case of the pseudo-covariance and is defined in terms of ordinary complex squares, given by:

${\displaystyle \operatorname {J} _{ZZ}=\operatorname {E} [(Z-\operatorname {E} [Z])^{2}]=\operatorname {E} [Z^{2}]-(\operatorname {E} [Z])^{2}}$

(Eq.4)

Unlike the variance of ${\displaystyle Z}$, which is always real and positive, the pseudo-variance of ${\displaystyle Z}$ is in general complex.

For a general complex random variable, the pair ${\displaystyle (\Re {(Z)},\Im {(Z)})}$ has a covariance matrix of the form:

${\displaystyle {\begin{bmatrix}\operatorname {Var} [\Re {(Z)}]&\operatorname {Cov} [\Im {(Z)},\Re {(Z)}]\\\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]&\operatorname {Var} [\Im {(Z)}]\end{bmatrix}}}$

The matrix is symmetric, so ${\displaystyle \operatorname {Cov} [\Re {(Z)},\Im {(Z)}]=\operatorname {Cov} [\Im {(Z)},\Re {(Z)}]}$

Its elements equal:

${\displaystyle {\begin{aligned}&\operatorname {Var} [\Re {(Z)}]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{ZZ}+\operatorname {J} _{ZZ})\\&\operatorname {Var} [\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{ZZ}-\operatorname {J} _{ZZ})\\&\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{ZZ})\\\end{aligned}}}$

Conversely:

${\displaystyle {\begin{aligned}&\operatorname {K} _{ZZ}=\operatorname {Var} [\Re {(Z)}]+\operatorname {Var} [\Im {(Z)}]\\&\operatorname {J} _{ZZ}=\operatorname {Var} [\Re {(Z)}]-\operatorname {Var} [\Im {(Z)}]+i2\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]\end{aligned}}}$

The covariance between two complex random variables ${\displaystyle Z,W}$ is defined as[3]^: 119

${\displaystyle \operatorname {K} _{ZW}=\operatorname {Cov} [Z,W]=\operatorname {E} [(Z-\operatorname {E} [Z]){\overline {(W-\operatorname {E} [W])}}]=\operatorname {E} [Z{\overline {W}}]-\operatorname {E} [Z]\operatorname {E} [{\overline {W}}]}$

(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):

${\displaystyle \operatorname {J} _{ZW}=\operatorname {Cov} [Z,{\overline {W}}]=\operatorname {E} [(Z-\operatorname {E} [Z])(W-\operatorname {E} [W])]=\operatorname {E} [ZW]-\operatorname {E} [Z]\operatorname {E} [W]}$

(Eq.6)

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

Properties

The covariance has the following properties:

• ${\displaystyle \operatorname {Cov} [Z,W]={\overline {\operatorname {Cov} [W,Z]}}}$ (Conjugate symmetry)
• ${\displaystyle \operatorname {Cov} [\alpha Z,W]=\alpha \operatorname {Cov} [Z,W]}$ (Sesquilinearity)
• ${\displaystyle \operatorname {Cov} [Z,\alpha W]={\overline {\alpha }}\operatorname {Cov} [Z,W]}$
• ${\displaystyle \operatorname {Cov} [Z_{1}+Z_{2},W]=\operatorname {Cov} [Z_{1},W]+\operatorname {Cov} [Z_{2},W]}$
• ${\displaystyle \operatorname {Cov} [Z,W_{1}+W_{2}]=\operatorname {Cov} [Z,W_{1}]+\operatorname {Cov} [Z,W_{2}]}$
• ${\displaystyle \operatorname {Cov} [Z,Z]={\operatorname {Var} [Z]}}$
• Uncorrelatedness: two complex random variables ${\displaystyle Z}$ and ${\displaystyle W}$ are called uncorrelated if ${\displaystyle \operatorname {K} _{ZW}=\operatorname {J} _{ZW}=0}$ (see also: uncorrelatedness (probability theory)).
• Orthogonality: two complex random variables ${\displaystyle Z}$ and ${\displaystyle W}$ are called orthogonal if ${\displaystyle \operatorname {E} [Z{\overline {W}}]=0}$.

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.

A complex random variable ${\displaystyle Z}$ is circularly symmetric if, for any deterministic ${\displaystyle \phi \in [-\pi ,\pi ]}$, the distribution of ${\displaystyle e^{\mathrm {i} \phi }Z}$ equals the distribution of ${\displaystyle Z}$.

Properties

By definition, a circularly symmetric complex random variable has

$${\displaystyle \operatorname {E} [Z]=\operatorname {E} [e^{\mathrm {i} \phi }Z]=e^{\mathrm {i} \phi }\operatorname {E} [Z]}$$

for any ${\displaystyle \phi }$.

Thus the expectation of a circularly symmetric complex random variable can only be either zero or undefined.

Additionally,

$${\displaystyle \operatorname {E} [ZZ]=\operatorname {E} [e^{\mathrm {i} \phi }Ze^{\mathrm {i} \phi }Z]=e^{\mathrm {2} i\phi }\operatorname {E} [ZZ]}$$

for any ${\displaystyle \phi }$.

Thus the pseudo-variance of a circularly symmetric complex random variable can only be zero.

If ${\displaystyle Z}$ and ${\displaystyle e^{\mathrm {i} \phi }Z}$ have the same distribution, the phase of ${\displaystyle Z}$ must be uniformly distributed over ${\displaystyle [-\pi ,\pi ]}$ and independent of the amplitude of ${\displaystyle Z}$.[4]

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

A complex random variable ${\displaystyle Z}$ is called proper if the following three conditions are all satisfied:

• ${\displaystyle \operatorname {E} [Z]=0}$
• ${\displaystyle \operatorname {Var} [Z]<\infty }$
• ${\displaystyle \operatorname {E} [Z^{2}]=0}$

This definition is equivalent to the following conditions. This means that a complex random variable is proper if, and only if:

• ${\displaystyle \operatorname {E} [Z]=0}$
• ${\displaystyle \operatorname {E} [\Re {(Z)}^{2}]=\operatorname {E} [\Im {(Z)}^{2}]\neq \infty }$
• ${\displaystyle \operatorname {E} [\Re {(Z)}\Im {(Z)}]=0}$

For a proper complex random variable, the covariance matrix of the pair ${\displaystyle (\Re {(Z)},\Im {(Z)})}$ has the following simple form:

${\displaystyle {\begin{bmatrix}{\frac {1}{2}}\operatorname {Var} [Z]&0\\0&{\frac {1}{2}}\operatorname {Var} [Z]\end{bmatrix}}}$.

I.e.:

${\displaystyle {\begin{aligned}&\operatorname {Var} [\Re {(Z)}]=\operatorname {Var} [\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Var} [Z]\\&\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]=0\\\end{aligned}}}$

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

${\displaystyle \left|\operatorname {E} \left[Z{\overline {W}}\right]\right|^{2}\leq \left|\operatorname {E} \left[\left|Z{\overline {W}}\right|\right]\right|^{2}\leq \operatorname {E} \left[|Z|^{2}\right]\operatorname {E} \left[|W|^{2}\right]}$.

The characteristic function of a complex random variable is a function ${\displaystyle \mathbb {C} \to \mathbb {C} }$ defined by

${\displaystyle \varphi _{Z}(\omega )=\operatorname {E} \left[e^{i\Re {({\overline {\omega }}Z)}}\right]=\operatorname {E} \left[e^{i(\Re {(\omega )}\Re {(Z)}+\Im {(\omega )}\Im {(Z)})}\right].}$

• Central moment
• Complex random vector