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Complex random variable
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<h2 id="definition">Definition</h2> <p>A complex random variable Z {\displaystyle Z} on the <a href="/facts/Probability_space/dEcv2VrU">probability space</a> ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> Z : Ω → C {\displaystyle Z\colon \Omega \rightarrow \mathbb {C} } such that both its real part ℜ ( Z ) {\displaystyle \Re {(Z)}} and its imaginary part ℑ ( Z ) {\displaystyle \Im {(Z)}} are real <a href="/facts/Random_variables/TwTBXnLT">random variables</a> on ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} . </p> <h2 id="examples">Examples</h2> <h3>Simple example</h3> <p>Consider a random variable that may take only the three complex values 1 + i , 1 − i , 2 {\displaystyle 1+i,1-i,2} with probabilities as specified in the table. This is a simple example of a complex random variable. </p> <table><tbody><tr><th>Probability P ( z ) {\displaystyle P(z)} </th><th>Value z {\displaystyle z} </th></tr><tr><td> 1 4 {\displaystyle {\frac {1}{4}}} </td><td> 1 + i {\displaystyle 1+i} </td></tr><tr><td> 1 4 {\displaystyle {\frac {1}{4}}} </td><td> 1 − i {\displaystyle 1-i} </td></tr><tr><td> 1 2 {\displaystyle {\frac {1}{2}}} </td><td> 2 {\displaystyle 2} </td></tr></tbody></table> <p>The expectation of this random variable may be simply calculated: E [ Z ] = 1 4 ( 1 + i ) + 1 4 ( 1 − i ) + 1 2 2 = 3 2 . {\displaystyle \operatorname {E} [Z]={\frac {1}{4}}(1+i)+{\frac {1}{4}}(1-i)+{\frac {1}{2}}2={\frac {3}{2}}.} </p> <h3>Uniform distribution</h3> <p>Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set { z ∈ C ∣ | z | ≤ 1 } {\displaystyle \{z\in \mathbb {C} \mid |z|\leq 1\}} . 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. </p> <h3>Complex normal distribution</h3> <p class="note">Main article: <a href="/facts/Complex_normal_distribution/KVUmoU1L">Complex normal distribution</a></p> <p>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. </p> <h2 id="cumulative-distribution-function">Cumulative distribution function</h2> <p>The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form P ( Z ≤ 1 + 3 i ) {\displaystyle P(Z\leq 1+3i)} make no sense. However expressions of the form P ( ℜ ( Z ) ≤ 1 , ℑ ( Z ) ≤ 3 ) {\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)} make sense. Therefore, we define the cumulative distribution F Z : C → [ 0 , 1 ] {\displaystyle F_{Z}:\mathbb {C} \to [0,1]} of a complex random variables via the <a href="/facts/Joint_probability_distribution/klX2ksGY">joint distribution</a> of their real and imaginary parts: </p> <table><tbody><tr><td> F Z ( z ) = F ℜ ( Z ) , ℑ ( Z ) ( ℜ ( z ) , ℑ ( z ) ) = P ( ℜ ( Z ) ≤ ℜ ( z ) , ℑ ( Z ) ≤ ℑ ( z ) ) {\displaystyle F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})=P(\Re {(Z)}\leq \Re {(z)},\Im {(Z)}\leq \Im {(z)})} </td> <td></td> <td>Eq.1</td></tr></tbody></table> <h2 id="probability--density-function">Probability density function</h2> <p>The probability density function of a complex random variable is defined as f Z ( z ) = f ℜ ( Z ) , ℑ ( Z ) ( ℜ ( z ) , ℑ ( z ) ) {\displaystyle f_{Z}(z)=f_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})} , i.e. the value of the density function at a point z ∈ C {\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 ( ℜ ( z ) , ℑ ( z ) ) {\displaystyle (\Re {(z)},\Im {(z)})} . </p><p>An equivalent definition is given by f Z ( z ) = ∂ 2 ∂ x ∂ y P ( ℜ ( Z ) ≤ x , ℑ ( Z ) ≤ y ) {\displaystyle f_{Z}(z)={\frac {\partial ^{2}}{\partial x\partial y}}P(\Re {(Z)}\leq x,\Im {(Z)}\leq y)} where x = ℜ ( z ) {\displaystyle x=\Re {(z)}} and y = ℑ ( z ) {\displaystyle y=\Im {(z)}} . </p><p>As in the real case the density function may not exist. </p> <h2 id="expectation">Expectation</h2> <p>The expectation of a complex random variable is defined based on the definition of the expectation of a real random variable:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: p. 112 </p> <table><tbody><tr><td> E [ Z ] = E [ ℜ ( Z ) ] + i E [ ℑ ( Z ) ] {\displaystyle \operatorname {E} [Z]=\operatorname {E} [\Re {(Z)}]+i\operatorname {E} [\Im {(Z)}]} </td> <td></td> <td>Eq.2</td></tr></tbody></table> <p>Note that the expectation of a complex random variable does not exist if E [ ℜ ( Z ) ] {\displaystyle \operatorname {E} [\Re {(Z)}]} or E [ ℑ ( Z ) ] {\displaystyle \operatorname {E} [\Im {(Z)}]} does not exist. </p><p>If the complex random variable Z {\displaystyle Z} has a probability density function f Z ( z ) {\displaystyle f_{Z}(z)} , then the expectation is given by E [ Z ] = ∬ C z ⋅ f Z ( z ) d x d y {\displaystyle \operatorname {E} [Z]=\iint _{\mathbb {C} }z\cdot f_{Z}(z)\,dx\,dy} . </p><p>If the complex random variable Z {\displaystyle Z} has a <a href="/facts/Probability_mass_function/LhurokRt">probability mass function</a> p Z ( z ) {\displaystyle p_{Z}(z)} , then the expectation is given by E [ Z ] = ∑ z ∈ Z z ⋅ p Z ( z ) {\displaystyle \operatorname {E} [Z]=\sum _{z\in \mathbb {Z} }z\cdot p_{Z}(z)} . </p> Properties <p>Whenever the expectation of a complex random variable exists, taking the expectation and <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugation</a> commute: </p> E [ Z ] ¯ = E [ Z ¯ ] . {\displaystyle {\overline {\operatorname {E} [Z]}}=\operatorname {E} [{\overline {Z}}].} <p>The expected value operator E [ ⋅ ] {\displaystyle \operatorname {E} [\cdot ]} is <a href="/facts/Linear_operator/Q1PBKNVV">linear</a> in the sense that </p> E [ a Z + b W ] = a E [ Z ] + b E [ W ] {\displaystyle \operatorname {E} [aZ+bW]=a\operatorname {E} [Z]+b\operatorname {E} [W]} <p>for any complex coefficients a , b {\displaystyle a,b} even if Z {\displaystyle Z} and W {\displaystyle W} are not <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a>. </p> <h2 id="variance-and-pseudo-variance">Variance and pseudo-variance</h2> <p>The variance is defined in terms of <a href="/facts/Absolute_square/Jzpq0R0z">absolute squares</a> as:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>: 117 </p> <table><tbody><tr><td> K Z Z = Var [ Z ] = E [ | Z − E [ Z ] | 2 ] = E [ | Z | 2 ] − | E [ Z ] | 2 {\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}} </td> <td></td> <td>Eq.3</td></tr></tbody></table> Properties <p>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: </p> Var [ Z ] = Var [ ℜ ( Z ) ] + Var [ ℑ ( Z ) ] . {\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [\Re {(Z)}]+\operatorname {Var} [\Im {(Z)}].} <p>The variance of a linear combination of complex random variables may be calculated using the following formula: </p> Var [ ∑ k = 1 N a k Z k ] = ∑ i = 1 N ∑ j = 1 N a i a j ¯ Cov [ Z i , Z j ] . {\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}].} <h3>Pseudo-variance</h3> <p>The pseudo-variance is a special case of the pseudo-covariance and is defined in terms of ordinary <a href="/facts/Complex_square/2w7mqI7e">complex squares</a>, given by: </p> <table><tbody><tr><td> J Z Z = E [ ( Z − E [ Z ] ) 2 ] = E [ Z 2 ] − ( E [ Z ] ) 2 {\displaystyle \operatorname {J} _{ZZ}=\operatorname {E} [(Z-\operatorname {E} [Z])^{2}]=\operatorname {E} [Z^{2}]-(\operatorname {E} [Z])^{2}} </td> <td></td> <td>Eq.4</td></tr></tbody></table> <p>Unlike the variance of Z {\displaystyle Z} , which is always real and positive, the pseudo-variance of Z {\displaystyle Z} is in general complex. </p> <h2 id="covariance-matrix-of-real-and-imaginary-parts">Covariance matrix of real and imaginary parts</h2> <p class="note">Further information: <a href="/facts/Complex_random_vector/orfynJII">Complex random vector § Covariance matrices of real and imaginary parts</a></p> <p>For a general complex random variable, the pair ( ℜ ( Z ) , ℑ ( Z ) ) {\displaystyle (\Re {(Z)},\Im {(Z)})} has a <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a> of the form: </p> [ Var [ ℜ ( Z ) ] Cov [ ℑ ( Z ) , ℜ ( Z ) ] Cov [ ℜ ( Z ) , ℑ ( Z ) ] Var [ ℑ ( Z ) ] ] {\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}}} <p>The matrix is symmetric, so Cov [ ℜ ( Z ) , ℑ ( Z ) ] = Cov [ ℑ ( Z ) , ℜ ( Z ) ] {\displaystyle \operatorname {Cov} [\Re {(Z)},\Im {(Z)}]=\operatorname {Cov} [\Im {(Z)},\Re {(Z)}]} </p><p>Its elements equal: </p> Var [ ℜ ( Z ) ] = 1 2 Re ( K Z Z + J Z Z ) Var [ ℑ ( Z ) ] = 1 2 Re ( K Z Z − J Z Z ) Cov [ ℜ ( Z ) , ℑ ( Z ) ] = 1 2 Im ( J Z Z ) {\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}}} <p>Conversely: </p> K Z Z = Var [ ℜ ( Z ) ] + Var [ ℑ ( Z ) ] J Z Z = Var [ ℜ ( Z ) ] − Var [ ℑ ( Z ) ] + i 2 Cov [ ℜ ( Z ) , ℑ ( Z ) ] {\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}}} <h2 id="covariance-and-pseudo-covariance">Covariance and pseudo-covariance</h2> <p>The covariance between two complex random variables Z , W {\displaystyle Z,W} is defined as<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>: 119 </p> <table><tbody><tr><td> K Z W = Cov [ Z , W ] = E [ ( Z − E [ Z ] ) ( W − E [ W ] ) ¯ ] = E [ Z W ¯ ] − E [ Z ] E [ W ¯ ] {\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}}]} </td> <td></td> <td>Eq.5</td></tr></tbody></table> <p>Notice the complex conjugation of the second factor in the definition. </p><p>In contrast to real random variables, we also define a pseudo-covariance (also called complementary variance): </p> <table><tbody><tr><td> J Z W = Cov [ Z , W ¯ ] = E [ ( Z − E [ Z ] ) ( W − E [ W ] ) ] = E [ Z W ] − E [ Z ] E [ W ] {\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]} </td> <td></td> <td>Eq.6</td></tr></tbody></table> <p>The second order statistics are fully characterized by the covariance and the pseudo-covariance. </p> Properties <p>The covariance has the following properties: </p> <ul><li> Cov [ Z , W ] = Cov [ W , Z ] ¯ {\displaystyle \operatorname {Cov} [Z,W]={\overline {\operatorname {Cov} [W,Z]}}} (Conjugate symmetry)</li> <li> Cov [ α Z , W ] = α Cov [ Z , W ] {\displaystyle \operatorname {Cov} [\alpha Z,W]=\alpha \operatorname {Cov} [Z,W]} (Sesquilinearity)</li> <li> Cov [ Z , α W ] = α ¯ Cov [ Z , W ] {\displaystyle \operatorname {Cov} [Z,\alpha W]={\overline {\alpha }}\operatorname {Cov} [Z,W]} </li> <li> Cov [ Z 1 + Z 2 , W ] = Cov [ Z 1 , W ] + Cov [ Z 2 , W ] {\displaystyle \operatorname {Cov} [Z_{1}+Z_{2},W]=\operatorname {Cov} [Z_{1},W]+\operatorname {Cov} [Z_{2},W]} </li> <li> Cov [ Z , W 1 + W 2 ] = Cov [ Z , W 1 ] + Cov [ Z , W 2 ] {\displaystyle \operatorname {Cov} [Z,W_{1}+W_{2}]=\operatorname {Cov} [Z,W_{1}]+\operatorname {Cov} [Z,W_{2}]} </li> <li> Cov [ Z , Z ] = Var [ Z ] {\displaystyle \operatorname {Cov} [Z,Z]={\operatorname {Var} [Z]}} </li> <li>Uncorrelatedness: two complex random variables Z {\displaystyle Z} and W {\displaystyle W} are called uncorrelated if K Z W = J Z W = 0 {\displaystyle \operatorname {K} _{ZW}=\operatorname {J} _{ZW}=0} (see also: <a href="/facts/Uncorrelatedness_(probability_theory)/Ub2XK1WJ">uncorrelatedness (probability theory)</a>).</li> <li>Orthogonality: two complex random variables Z {\displaystyle Z} and W {\displaystyle W} are called orthogonal if E [ Z W ¯ ] = 0 {\displaystyle \operatorname {E} [Z{\overline {W}}]=0} .</li></ul> <h2 id="circular-symmetry">Circular symmetry</h2> <p>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 <a href="/facts/Complex_normal_distribution/KVUmoU1L">complex Gaussian random variable</a> with zero mean and zero pseudo-covariance matrix. </p><p>A complex random variable Z {\displaystyle Z} is circularly symmetric if, for any deterministic ϕ ∈ [ − π , π ] {\displaystyle \phi \in [-\pi ,\pi ]} , the distribution of e i ϕ Z {\displaystyle e^{\mathrm {i} \phi }Z} equals the distribution of Z {\displaystyle Z} . </p> Properties <p>By definition, a circularly symmetric complex random variable has E [ Z ] = E [ e i ϕ Z ] = e i ϕ E [ Z ] {\displaystyle \operatorname {E} [Z]=\operatorname {E} [e^{\mathrm {i} \phi }Z]=e^{\mathrm {i} \phi }\operatorname {E} [Z]} for any ϕ {\displaystyle \phi } . </p><p>Thus the expectation of a circularly symmetric complex random variable can only be either zero or undefined. </p><p>Additionally, E [ Z Z ] = E [ e i ϕ Z e i ϕ Z ] = e 2 i ϕ E [ Z Z ] {\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 ϕ {\displaystyle \phi } . </p><p>Thus the pseudo-variance of a circularly symmetric complex random variable can only be zero. </p><p>If Z {\displaystyle Z} and e i ϕ Z {\displaystyle e^{\mathrm {i} \phi }Z} have the same distribution, the phase of Z {\displaystyle Z} must be uniformly distributed over [ − π , π ] {\displaystyle [-\pi ,\pi ]} and independent of the amplitude of Z {\displaystyle Z} .<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h2 id="proper-complex-random-variables">Proper complex random variables</h2> <p>The concept of proper random variables is unique to complex random variables, and has no correspondent concept with real random variables. </p><p>A complex random variable Z {\displaystyle Z} is called proper if the following three conditions are all satisfied: </p> <ul><li> E [ Z ] = 0 {\displaystyle \operatorname {E} [Z]=0} </li> <li> Var [ Z ] < ∞ {\displaystyle \operatorname {Var} [Z]<\infty } </li> <li> E [ Z 2 ] = 0 {\displaystyle \operatorname {E} [Z^{2}]=0} </li></ul> <p>This definition is equivalent to the following conditions. This means that a complex random variable is proper if, and only if: </p> <ul><li> E [ Z ] = 0 {\displaystyle \operatorname {E} [Z]=0} </li> <li> E [ ℜ ( Z ) 2 ] = E [ ℑ ( Z ) 2 ] ≠ ∞ {\displaystyle \operatorname {E} [\Re {(Z)}^{2}]=\operatorname {E} [\Im {(Z)}^{2}]\neq \infty } </li> <li> E [ ℜ ( Z ) ℑ ( Z ) ] = 0 {\displaystyle \operatorname {E} [\Re {(Z)}\Im {(Z)}]=0} </li></ul> <p>Theorem—Every circularly symmetric complex random variable with finite variance is proper. </p> <p>For a proper complex random variable, the covariance matrix of the pair ( ℜ ( Z ) , ℑ ( Z ) ) {\displaystyle (\Re {(Z)},\Im {(Z)})} has the following simple form: </p> [ 1 2 Var [ Z ] 0 0 1 2 Var [ Z ] ] {\displaystyle {\begin{bmatrix}{\frac {1}{2}}\operatorname {Var} [Z]&0\\0&{\frac {1}{2}}\operatorname {Var} [Z]\end{bmatrix}}} . <p>I.e.: </p> Var [ ℜ ( Z ) ] = Var [ ℑ ( Z ) ] = 1 2 Var [ Z ] Cov [ ℜ ( Z ) , ℑ ( Z ) ] = 0 {\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}}} <h2 id="cauchy-schwarz-inequality">Cauchy-Schwarz inequality</h2> <p>The <a href="/facts/Cauchy-Schwarz_inequality/poMzQUIY">Cauchy-Schwarz inequality</a> for complex random variables, which can be derived using the <a href="/facts/Triangle_inequality/KubLjkwr">Triangle inequality</a> and <a href="/facts/H%C3%B6lder%27s_inequality/7I1ubTWw">Hölder's inequality</a>, is </p> | E [ Z W ¯ ] | 2 ≤ | E [ | Z W ¯ | ] | 2 ≤ E [ | Z | 2 ] E [ | W | 2 ] {\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]} . <h2 id="characteristic-function">Characteristic function</h2> <p>The <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> of a complex random variable is a function C → C {\displaystyle \mathbb {C} \to \mathbb {C} } defined by </p> φ Z ( ω ) = E [ e i ℜ ( ω ¯ Z ) ] = E [ e i ( ℜ ( ω ) ℜ ( Z ) + ℑ ( ω ) ℑ ( Z ) ) ] . {\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].} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Central_moment/geXcMwyv">Central moment</a></li> <li><a href="/facts/Complex_random_vector/orfynJII">Complex random vector</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Eriksson, Jan; Ollila, Esa; Koivunen, Visa (2009). Statistics for complex random variables revisited. 2009 IEEE International Conference on Acoustics, Speech and Signal Processing. Taipei, Taiwan: Institute of Electrical and Electronics Engineers. pp. 3565–3568. doi:10.1109/ICASSP.2009.4960396. <a href="/wiki/Institute_of_Electrical_and_Electronics_Engineers" target="_blank">/wiki/Institute_of_Electrical_and_Electronics_Engineers</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955. <a href="9780521193955" target="_blank">9780521193955</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3. <a href="978-3-319-68074-3" target="_blank">978-3-319-68074-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3. <a href="978-3-319-68074-3" target="_blank">978-3-319-68074-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3. <a href="978-3-319-68074-3" target="_blank">978-3-319-68074-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Peter J. Schreier, Louis L. Scharf (2011). Statistical Signal Processing of Complex-Valued Data. Cambridge University Press. ISBN 9780511815911. <a href="9780511815911" target="_blank">9780511815911</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>