Yilda ehtimollik nazariyasi va statistika , a murakkab tasodifiy vektor odatda a panjara ning murakkab - baholangan tasodifiy o'zgaruvchilar , va odatda a qiymatini oladigan tasodifiy o'zgaruvchidir vektor maydoni ustidan maydon kompleks sonlar. Agar Z 1 , … , Z n { displaystyle Z_ {1}, ldots, Z_ {n}} n - juftlik ( Z 1 , … , Z n ) { displaystyle chap (Z_ {1}, ldots, Z_ {n} o'ng)}
Haqiqiy tasodifiy vektorlarning ba'zi tushunchalari murakkab tasodifiy vektorlarning to'g'ridan-to'g'ri umumlashtirilishiga ega. Masalan, ning ta'rifi anglatadi murakkab tasodifiy vektorning. Boshqa tushunchalar murakkab tasodifiy vektorlarga xosdir.
Murakkab tasodifiy vektorlarning qo'llanilishi raqamli signallarni qayta ishlash .
Ta'rif
Murakkab tasodifiy vektor Z = ( Z 1 , … , Z n ) T { displaystyle mathbf {Z} = (Z_ {1}, ldots, Z_ {n}) ^ {T}} ehtimollik maydoni ( Ω , F , P ) { displaystyle ( Omega, { mathcal {F}}, P)} funktsiya Z : Ω → C n { displaystyle mathbf {Z} colon Omega rightarrow mathbb {C} ^ {n}} ( ℜ ( Z 1 ) , ℑ ( Z 1 ) , … , ℜ ( Z n ) , ℑ ( Z n ) ) T { displaystyle ( Re {(Z_ {1})}, Im {(Z_ {1})}, ldots, Re {(Z_ {n})}, Im {(Z_ {n})} ) {{T}} haqiqiy tasodifiy vektor kuni ( Ω , F , P ) { displaystyle ( Omega, { mathcal {F}}, P)} ℜ ( z ) { displaystyle Re {(z)}} z { displaystyle z} ℑ ( z ) { displaystyle Im {(z)}} z { displaystyle z} [1] :p. 292
Kümülatif taqsimlash funktsiyasi
Kümülatif taqsimlash funktsiyasini realdan murakkab tasodifiy o'zgaruvchiga umumlashtirish aniq emas, chunki shakl ifodalari P ( Z ≤ 1 + 3 men ) { displaystyle P (Z leq 1 + 3i)} P ( ℜ ( Z ) ≤ 1 , ℑ ( Z ) ≤ 3 ) { displaystyle P ( Re {(Z)} leq 1, Im {(Z)} leq 3)} F Z : C n ↦ [ 0 , 1 ] { displaystyle F _ { mathbf {Z}}: mathbb {C} ^ {n} mapsto [0,1]} Z = ( Z 1 , . . . , Z n ) T { displaystyle mathbf {Z} = (Z_ {1}, ..., Z_ {n}) ^ {T}}
F Z ( z ) = P ( ℜ ( Z 1 ) ≤ ℜ ( z 1 ) , ℑ ( Z 1 ) ≤ ℑ ( z 1 ) , … , ℜ ( Z n ) ≤ ℜ ( z n ) , ℑ ( Z n ) ≤ ℑ ( z n ) ) { displaystyle F _ { mathbf {Z}} ( mathbf {z}) = operatorname {P} ( Re {(Z_ {1})} leq Re {(z_ {1})}, Im {(Z_ {1})} leq Im {(z_ {1})}, ldots, Re {(Z_ {n})} leq Re {(z_ {n})}, Im { (Z_ {n})} leq Im {(z_ {n})})} (Tenglama 1 )
qayerda z = ( z 1 , . . . , z n ) T { displaystyle mathbf {z} = (z_ {1}, ..., z_ {n}) ^ {T}}
Kutish
Haqiqiy holatda bo'lgani kabi kutish (shuningdek, deyiladi kutilayotgan qiymat ) murakkab tasodifiy vektorning tarkibiy qismi bo'yicha olinadi.[1] :p. 293
E [ Z ] = ( E [ Z 1 ] , … , E [ Z n ] ) T { displaystyle operator nomi {E} [ mathbf {Z}] = ( operator nomi {E} [Z_ {1}], ldots, operator nomi {E} [Z_ {n}]) ^ {T}} (Ikkinchi tenglama )
Kovaryans matritsasi va psevdo-kovaryans matritsasi
Ta'riflar The kovaryans matritsasi ikkinchi markaziy moment ) K Z Z { displaystyle operatorname {K} _ { mathbf {Z} mathbf {Z}}} n × 1 { displaystyle n marta 1} n × n { displaystyle n times n} matritsa kimning ( men , j ) { displaystyle (i, j)} th element kovaryans o'rtasida men th va j th tasodifiy o'zgaruvchilar.[2] :p.372 Haqiqiy tasodifiy o'zgaruvchilardan farqli o'laroq, ikkita tasodifiy o'zgaruvchilar o'rtasidagi kovaryans quyidagilarni o'z ichiga oladi murakkab konjugat ikkinchisidan biri. Shunday qilib kovaryans matritsasi a Ermit matritsasi .[1] :p. 293
K Z Z = cov [ Z , Z ] = E [ ( Z − E [ Z ] ) ( Z − E [ Z ] ) H ] = E [ Z Z H ] − E [ Z ] E [ Z H ] { displaystyle { begin {aligned} & operatorname {K} _ { mathbf {Z} mathbf {Z}} = operatorname {cov} [ mathbf {Z}, mathbf {Z}] = operatorname {E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {Z} - operator nomi {E} [ mathbf {Z}])} ^ {H} ] = operator nomi {E} [ mathbf {Z} mathbf {Z} ^ {H}] - operator nomi {E} [ mathbf {Z}] operator nomi {E} [ mathbf {Z} ^ {H }] [12pt] end {hizalangan}}}
(Tenglama 3 )
K Z Z = [ E [ ( Z 1 − E [ Z 1 ] ) ( Z 1 − E [ Z 1 ] ) ¯ ] E [ ( Z 1 − E [ Z 1 ] ) ( Z 2 − E [ Z 2 ] ) ¯ ] ⋯ E [ ( Z 1 − E [ Z 1 ] ) ( Z n − E [ Z n ] ) ¯ ] E [ ( Z 2 − E [ Z 2 ] ) ( Z 1 − E [ Z 1 ] ) ¯ ] E [ ( Z 2 − E [ Z 2 ] ) ( Z 2 − E [ Z 2 ] ) ¯ ] ⋯ E [ ( Z 2 − E [ Z 2 ] ) ( Z n − E [ Z n ] ) ¯ ] ⋮ ⋮ ⋱ ⋮ E [ ( Z n − E [ Z n ] ) ( Z 1 − E [ Z 1 ] ) ¯ ] E [ ( Z n − E [ Z n ] ) ( Z 2 − E [ Z 2 ] ) ¯ ] ⋯ E [ ( Z n − E [ Z n ] ) ( Z n − E [ Z n ] ) ¯ ] ] { displaystyle operator nomi {K} _ { mathbf {Z} mathbf {Z}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1} ]) { overline {(Z_ {1} - operator nomi {E} [Z_ {1}])}}] & mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) { overline {(Z_ {2} - operator nomi {E} [Z_ {2}])}}] & cdots & mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1}]) { overline {(Z_ {n} - operator nomi {E} [Z_ {n}])}}] mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(Z_ {1} - operator nomi {E} [Z_ {1}])}}] & mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(Z_ {2} - operator nomi {E} [Z_ {2}])}}] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) { overline {(Z_ {n} - operator nomi {E} [Z_ {n}])}}] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(Z_ {1} - operator nomi {E} [Z_ {1}] )}}] & mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(Z_ {2} - operator nomi {E} [Z_ {2}] )}}] & cdots & mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(Z_ {n} - operator nomi {E} [Z_ { n}])}}] end {bmatrix}}} The psevdo-kovaryans matritsasi (munosabat matritsasi deb ham yuritiladi) quyidagicha aniqlanadi. Yuqorida belgilangan kovaryans matritsasidan farqli o'laroq Hermitian transpozitsiyasi o'rnini egallaydi transpozitsiya ta'rifda.
J Z Z = cov [ Z , Z ¯ ] = E [ ( Z − E [ Z ] ) ( Z − E [ Z ] ) T ] = E [ Z Z T ] − E [ Z ] E [ Z T ] { displaystyle operator nomi {J} _ { mathbf {Z} mathbf {Z}} = operator nomi {cov} [ mathbf {Z}, { overline { mathbf {Z}}}] = operator nomi { E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {Z} - operator nomi {E} [ mathbf {Z}])} ^ {T}] = operatorname {E} [ mathbf {Z} mathbf {Z} ^ {T}] - operatorname {E} [ mathbf {Z}] operatorname {E} [ mathbf {Z} ^ {T} ]}
(4. tenglama )
J Z Z = [ E [ ( Z 1 − E [ Z 1 ] ) ( Z 1 − E [ Z 1 ] ) ] E [ ( Z 1 − E [ Z 1 ] ) ( Z 2 − E [ Z 2 ] ) ] ⋯ E [ ( Z 1 − E [ Z 1 ] ) ( Z n − E [ Z n ] ) ] E [ ( Z 2 − E [ Z 2 ] ) ( Z 1 − E [ Z 1 ] ) ] E [ ( Z 2 − E [ Z 2 ] ) ( Z 2 − E [ Z 2 ] ) ] ⋯ E [ ( Z 2 − E [ Z 2 ] ) ( Z n − E [ Z n ] ) ] ⋮ ⋮ ⋱ ⋮ E [ ( Z n − E [ Z n ] ) ( Z 1 − E [ Z 1 ] ) ] E [ ( Z n − E [ Z n ] ) ( Z 2 − E [ Z 2 ] ) ] ⋯ E [ ( Z n − E [ Z n ] ) ( Z n − E [ Z n ] ) ] ] { displaystyle operator nomi {J} _ { mathbf {Z} mathbf {Z}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) (Z_ {1} - operator nomi {E} [Z_ {1}])] va mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1}]) (Z_ {2 } - operatorname {E} [Z_ {2}])] & cdots & mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1}]) (Z_ {n} - operator nomi {E} [Z_ {n}])] \ mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (Z_ {1} - operator nomi {E } [Z_ {1}])] & mathrm {E} [(Z_ {2} - operatorname {E} [Z_ {2}]) (Z_ {2} - operatorname {E} [Z_ {2} ])] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (Z_ {n} - operator nomi {E} [Z_ {n}])] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (Z_ {1} - operator nomi {E} [Z_ {1}])] va mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (Z_ {2} - operator nomi {E} [Z_ {2}])] & cdots & mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (Z_ {n} - operator nomi {E} [Z_ {n} ])] end {bmatrix}}} Xususiyatlari Kovaryans matritsasi a hermit matritsasi , ya'ni[1] :p. 293
K Z Z H = K Z Z { displaystyle operator nomi {K} _ { mathbf {Z} mathbf {Z}} ^ {H} = operator nomi {K} _ { mathbf {Z} mathbf {Z}}} Psevdo-kovaryans matritsasi a nosimmetrik matritsa , ya'ni
J Z Z T = J Z Z { displaystyle operator nomi {J} _ { mathbf {Z} mathbf {Z}} ^ {T} = operator nomi {J} _ { mathbf {Z} mathbf {Z}}} Kovaryans matritsasi a ijobiy yarim yarim matritsa , ya'ni
a H K Z Z a ≥ 0 Barcha uchun a ∈ C n { displaystyle mathbf {a} ^ {H} operatorname {K} _ { mathbf {Z} mathbf {Z}} mathbf {a} geq 0 quad { text {for all}} mathbf {a} in mathbb {C} ^ {n}} Haqiqiy va xayoliy qismlarning kovaryans matritsalari Tasodifiy vektorni parchalash orqali Z { displaystyle mathbf {Z}} X = ℜ ( Z ) { displaystyle mathbf {X} = Re {( mathbf {Z})}} Y = ℑ ( Z ) { displaystyle mathbf {Y} = Im {( mathbf {Z})}} Z = X + men Y { displaystyle mathbf {Z} = mathbf {X} + i mathbf {Y}} K Z Z { displaystyle operatorname {K} _ { mathbf {Z} mathbf {Z}}} J Z Z { displaystyle operatorname {J} _ { mathbf {Z} mathbf {Z}}} X { displaystyle mathbf {X}} Y { displaystyle mathbf {Y}}
K X X = E [ ( X − E [ X ] ) ( X − E [ X ] ) T ] = 1 2 Qayta ( K Z Z + J Z Z ) K X Y = E [ ( X − E [ X ] ) ( Y − E [ Y ] ) T ] = 1 2 Im ( − K Z Z + J Z Z ) K Y X = E [ ( Y − E [ Y ] ) ( X − E [ X ] ) T ] = 1 2 Im ( K Z Z + J Z Z ) K Y Y = E [ ( Y − E [ Y ] ) ( Y − E [ Y ] ) T ] = 1 2 Qayta ( K Z Z − J Z Z ) { displaystyle { begin {aligned} & operatorname {K} _ { mathbf {X} mathbf {X}} = operatorname {E} [( mathbf {X} - operatorname {E} [ mathbf {X}]) ( mathbf {X} - operatorname {E} [ mathbf {X}]) ^ { mathrm {T}}] = { tfrac {1} {2}} operatorname {Re} ( operator nomi {K} _ { mathbf {Z} mathbf {Z}} + operator nomi {J} _ { mathbf {Z} mathbf {Z}}) & operator nomi {K} _ { mathbf {X} mathbf {Y}} = operator nomi {E} [( mathbf {X} - operator nomi {E} [ mathbf {X}]) ( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ^ { mathrm {T}}] = { tfrac {1} {2}} operator nomi {Im} (- operator nomi {K} _ { mathbf {Z} mathbf {Z }} + operator nomi {J} _ { mathbf {Z} mathbf {Z}}) & operator nomi {K} _ { mathbf {Y} mathbf {X}} = operator nomi {E} [ ( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ( mathbf {X} - operator nomi {E} [ mathbf {X}]) ^ { mathrm {T}}] = { tfrac {1} {2}} operator nomi {Im} ( operator nomi {K} _ { mathbf {Z} mathbf {Z}} + operator nomi {J} _ { mathbf {Z} mathbf { Z}}) & operator nomi {K} _ { mathbf {Y} mathbf {Y}} = op eratorname {E} [( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ^ { mathrm { T}}] = { tfrac {1} {2}} operator nomi {Re} ( operator nomi {K} _ { mathbf {Z} mathbf {Z}} - operator nomi {J} _ { mathbf { Z} mathbf {Z}}) end {aligned}}} va aksincha
K Z Z = K X X + K Y Y + men ( K Y X − K X Y ) J Z Z = K X X − K Y Y + men ( K Y X + K X Y ) { displaystyle { begin {aligned} & operatorname {K} _ { mathbf {Z} mathbf {Z}} = operatorname {K} _ { mathbf {X} mathbf {X}} + operatorname {K} _ { mathbf {Y} mathbf {Y}} + i ( operatorname {K} _ { mathbf {Y} mathbf {X}} - operatorname {K} _ { mathbf {X} mathbf {Y}}) & operatorname {J} _ { mathbf {Z} mathbf {Z}} = operatorname {K} _ { mathbf {X} mathbf {X}} - operatorname {K} _ { mathbf {Y} mathbf {Y}} + i ( operator nomi {K} _ { mathbf {Y} mathbf {X}} + operator nomi {K} _ { mathbf {X} mathbf {Y}}) end {hizalangan}}} Kross-kovaryans matritsasi va psevdo-kross-kovaryans matritsasi
Ta'riflar The kovaryans matritsasi ikkita murakkab tasodifiy vektor o'rtasida Z , V { displaystyle mathbf {Z}, mathbf {W}}
K Z V = cov [ Z , V ] = E [ ( Z − E [ Z ] ) ( V − E [ V ] ) H ] = E [ Z V H ] − E [ Z ] E [ V H ] { displaystyle operator nomi {K} _ { mathbf {Z} mathbf {W}} = operator nomi {cov} [ mathbf {Z}, mathbf {W}] = operator nomi {E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {W} - operator nomi {E} [ mathbf {W}])} ^ {H}] = operator nomi {E} [ mathbf {Z} mathbf {W} ^ {H}] - operator nomi {E} [ mathbf {Z}] operator nomi {E} [ mathbf {W} ^ {H}]} (5-tenglik )
K Z V = [ E [ ( Z 1 − E [ Z 1 ] ) ( V 1 − E [ V 1 ] ) ¯ ] E [ ( Z 1 − E [ Z 1 ] ) ( V 2 − E [ V 2 ] ) ¯ ] ⋯ E [ ( Z 1 − E [ Z 1 ] ) ( V n − E [ V n ] ) ¯ ] E [ ( Z 2 − E [ Z 2 ] ) ( V 1 − E [ V 1 ] ) ¯ ] E [ ( Z 2 − E [ Z 2 ] ) ( V 2 − E [ V 2 ] ) ¯ ] ⋯ E [ ( Z 2 − E [ Z 2 ] ) ( V n − E [ V n ] ) ¯ ] ⋮ ⋮ ⋱ ⋮ E [ ( Z n − E [ Z n ] ) ( V 1 − E [ V 1 ] ) ¯ ] E [ ( Z n − E [ Z n ] ) ( V 2 − E [ V 2 ] ) ¯ ] ⋯ E [ ( Z n − E [ Z n ] ) ( V n − E [ V n ] ) ¯ ] ] { displaystyle operator nomi {K} _ { mathbf {Z} mathbf {W}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1} ]) { overline {(W_ {1} - operator nomi {E} [W_ {1}])}}] & mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) { overline {(W_ {2} - operatorname {E} [W_ {2}])}}] & cdots & mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1}]) { overline {(W_ {n} - operator nomi {E} [W_ {n}])}}] mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(W_ {1} - operator nomi {E} [W_ {1}])}}] & mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(W_ {2} - operatorname {E} [W_ {2}])}}]] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) { overline {(W_ {n} - operatorname {E} [W_ {n}])}}] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(W_ {1} - operator nomi {E} [W_ {1}] )}}] & mathrm {E} [(Z_ {n} - operatorname {E} [Z_ {n}]) { overline {(W_ {2} - operatorname {E} [W_ {2}] )}}] & cdots & mathrm {E} [(Z_ {n} - operatorname {E} [Z_ {n}]) { overline {(W_ {n} - operatorname {E} [W_ {) n}])}}] end {bmatrix}}} Va pseudo-cross-kovaryans matritsasi quyidagicha aniqlanadi:
J Z V = cov [ Z , V ¯ ] = E [ ( Z − E [ Z ] ) ( V − E [ V ] ) T ] = E [ Z V T ] − E [ Z ] E [ V T ] { displaystyle operatorname {J} _ { mathbf {Z} mathbf {W}} = operator nomi {cov} [ mathbf {Z}, { overline { mathbf {W}}}] = operator nomi { E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {W} - operator nomi {E} [ mathbf {W}])} ^ {T}] = operatorname {E} [ mathbf {Z} mathbf {W} ^ {T}] - operatorname {E} [ mathbf {Z}] operatorname {E} [ mathbf {W} ^ {T} ]} (6-tenglik )
J Z V = [ E [ ( Z 1 − E [ Z 1 ] ) ( V 1 − E [ V 1 ] ) ] E [ ( Z 1 − E [ Z 1 ] ) ( V 2 − E [ V 2 ] ) ] ⋯ E [ ( Z 1 − E [ Z 1 ] ) ( V n − E [ V n ] ) ] E [ ( Z 2 − E [ Z 2 ] ) ( V 1 − E [ V 1 ] ) ] E [ ( Z 2 − E [ Z 2 ] ) ( V 2 − E [ V 2 ] ) ] ⋯ E [ ( Z 2 − E [ Z 2 ] ) ( V n − E [ V n ] ) ] ⋮ ⋮ ⋱ ⋮ E [ ( Z n − E [ Z n ] ) ( V 1 − E [ V 1 ] ) ] E [ ( Z n − E [ Z n ] ) ( V 2 − E [ V 2 ] ) ] ⋯ E [ ( Z n − E [ Z n ] ) ( V n − E [ V n ] ) ] ] { displaystyle operator nomi {J} _ { mathbf {Z} mathbf {W}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) (W_ {1} - operator nomi {E} [W_ {1}])] va mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1}]) (W_ {2 } - operatorname {E} [W_ {2}])] & cdots & mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1}]) (W_ {n} - operator nomi {E} [W_ {n}])] \ mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (W_ {1} - operator nomi {E } [W_ {1}])] & mathrm {E} [(Z_ {2} - operatorname {E} [Z_ {2}]) (W_ {2} - operatorname {E} [W_ {2}) ])] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (W_ {n} - operator nomi {E} [W_ {n}])] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (W_ {1} - operator nomi {E} [W_ {1}])] va mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (W_ {2} - operator nomi {E} [W_ {2}])] & cdots & mathrm {E} [(Z_ {n} - operatorname {E} [Z_ {n}]) (W_ {n} - operatorname {E} [W_ {n} ])] end {bmatrix}}} Aloqasizlik Ikki murakkab tasodifiy vektor Z { displaystyle mathbf {Z}} V { displaystyle mathbf {W}} aloqasiz agar
K Z V = J Z V = 0 { displaystyle operator nomi {K} _ { mathbf {Z} mathbf {W}} = operator nomi {J} _ { mathbf {Z} mathbf {W}} = 0} Mustaqillik
Ikki murakkab tasodifiy vektor Z = ( Z 1 , . . . , Z m ) T { displaystyle mathbf {Z} = (Z_ {1}, ..., Z_ {m}) ^ {T}} V = ( V 1 , . . . , V n ) T { displaystyle mathbf {W} = (W_ {1}, ..., W_ {n}) ^ {T}} mustaqil agar
F Z , V ( z , w ) = F Z ( z ) ⋅ F V ( w ) Barcha uchun z , w { displaystyle F _ { mathbf {Z, W}} ( mathbf {z, w}) = F _ { mathbf {Z}} ( mathbf {z}) cdot F _ { mathbf {W}} ( mathbf {w}) quad { text {for all}} mathbf {z}, mathbf {w}} (Tenglama 7 )
qayerda F Z ( z ) { displaystyle F _ { mathbf {Z}} ( mathbf {z})} F V ( w ) { displaystyle F _ { mathbf {W}} ( mathbf {w})} Z { displaystyle mathbf {Z}} V { displaystyle mathbf {W}} Tenglama 1 F Z , V ( z , w ) { displaystyle F _ { mathbf {Z, W}} ( mathbf {z, w})} Z { displaystyle mathbf {Z}} V { displaystyle mathbf {W}} Z ⊥ ⊥ V { displaystyle mathbf {Z} perp ! ! ! perp mathbf {W}} Z { displaystyle mathbf {Z}} V { displaystyle mathbf {W}}
F Z 1 , … , Z m , V 1 , … , V n ( z 1 , … , z m , w 1 , … , w n ) = F Z 1 , … , Z m ( z 1 , … , z m ) ⋅ F V 1 , … , V n ( w 1 , … , w n ) Barcha uchun z 1 , … , z m , w 1 , … , w n { displaystyle F_ {Z_ {1}, ldots, Z_ {m}, W_ {1}, ldots, W_ {n}} (z_ {1}, ldots, z_ {m}, w_ {1}, ldots, w_ {n}) = F_ {Z_ {1}, ldots, Z_ {m}} (z_ {1}, ldots, z_ {m}) cdot F_ {W_ {1}, ldots, W_ {n}} (w_ {1}, ldots, w_ {n}) quad { text {for all}} z_ {1}, ldots, z_ {m}, w_ {1}, ldots, w_ {n}} Dumaloq simmetriya
Ta'rif Murakkab tasodifiy vektor Z { displaystyle mathbf {Z}} φ ∈ [ − π , π ) { displaystyle varphi in [- pi, pi)} e men φ Z { displaystyle e ^ { mathrm {i} varphi} mathbf {Z}} Z { displaystyle mathbf {Z}} [3] :500-501 betlar
Xususiyatlari Dumaloq simmetrik kompleks tasodifiy vektorlarning kutilishi nolga teng yoki u aniqlanmagan.[3] :p. 500 Dumaloq simmetrik kompleks tasodifiy vektorlarning psevdo-kovaryans matritsasi nolga teng.[3] :p. 584 To'g'ri murakkab tasodifiy vektorlar
Ta'rif Murakkab tasodifiy vektor Z { displaystyle mathbf {Z}} to'g'ri agar quyidagi uchta shart bajarilsa:[1] :p. 293
E [ Z ] = 0 { displaystyle operatorname {E} [ mathbf {Z}] = 0} var [ Z 1 ] < ∞ , … , var [ Z n ] < ∞ { displaystyle operatorname {var} [Z_ {1}] < infty, ldots, operatorname {var} [Z_ {n}] < infty} E [ Z Z T ] = 0 { displaystyle operator nomi {E} [ mathbf {Z} mathbf {Z} ^ {T}] = 0} Ikki murakkab tasodifiy vektor Z , V { displaystyle mathbf {Z}, mathbf {W}} birgalikda to'g'ri kompozit tasodifiy vektor ( Z 1 , Z 2 , … , Z m , V 1 , V 2 , … , V n ) T { displaystyle (Z_ {1}, Z_ {2}, ldots, Z_ {m}, W_ {1}, W_ {2}, ldots, W_ {n}) ^ {T}}
Xususiyatlari Murakkab tasodifiy vektor Z { displaystyle mathbf {Z}} v ∈ C n { displaystyle mathbf {c} in mathbb {C} ^ {n}} v T Z { displaystyle mathbf {c} ^ {T} mathbf {Z}} [1] :p. 293 Tegishli murakkab tasodifiy vektorlarning chiziqli o'zgarishlari to'g'ri keladi, ya'ni Z { displaystyle mathbf {Z}} n { displaystyle n} A { displaystyle A} m × n { displaystyle m marta n} A Z { displaystyle A mathbf {Z}} [1] :p. 295 Barcha doiraviy nosimmetrik kompleks tasodifiy vektor, uning barcha tarkibiy qismlari cheklangan dispersiyasiga ega.[1] :p. 295 Dumaloq nosimmetrik bo'lmagan tegishli murakkab tasodifiy vektorlar mavjud.[1] :p. 504 Haqiqiy tasodifiy vektor, agar u doimiy bo'lsa, mos keladi. Ikkala birgalikda murakkab kompleks tasodifiy vektorlar o'zaro bog'liq emas, agar ularning kovariace matritsasi nolga teng bo'lsa, ya'ni K Z V = 0 { displaystyle operator nomi {K} _ { mathbf {Z} mathbf {W}} = 0} Koshi-Shvarts tengsizligi
The Koshi-Shvarts tengsizligi murakkab tasodifiy vektorlar uchun
| E [ Z H V ] | 2 ≤ E [ Z H Z ] E [ | V H V | ] { displaystyle left | operator nomi {E} [ mathbf {Z} ^ {H} mathbf {W}] o'ng | ^ {2} leq operator nomi {E} [ mathbf {Z} ^ {H } mathbf {Z}] operatorname {E} [| mathbf {W} ^ {H} mathbf {W} |]} Xarakterli funktsiya
The xarakterli funktsiya murakkab tasodifiy vektorning Z { displaystyle mathbf {Z}} n { displaystyle n} C n → C { displaystyle mathbb {C} ^ {n} to mathbb {C}} [1] :p. 295
φ Z ( ω ) = E [ e men ℜ ( ω H Z ) ] = E [ e men ( ℜ ( ω 1 ) ℜ ( Z 1 ) + ℑ ( ω 1 ) ℑ ( Z 1 ) + ⋯ + ℜ ( ω n ) ℜ ( Z n ) + ℑ ( ω n ) ℑ ( Z n ) ) ] { displaystyle varphi _ { mathbf {Z}} ( mathbf { omega}) = operatorname {E} left [e ^ {i Re {( mathbf { omega} ^ {H} mathbf {Z})}} o'ng] = operator nomi {E} chap [e ^ {i ( Re {( omega _ {1})} Re {(Z_ {1})} + Im {( omega _ {1})} Im {(Z_ {1})} + cdots + Re {( omega _ {n})} Re {(Z_ {n})} + Im {( omega _ {n})} Im {(Z_ {n})})} o'ng]} Shuningdek qarang
Adabiyotlar
^ a b v d e f g h men j Lapidot, Amos (2009). Raqamli aloqa asoslari . Kembrij universiteti matbuoti. ISBN 978-0-521-19395-5 ^ Gubner, Jon A. (2006). Elektr va kompyuter muhandislari uchun ehtimollik va tasodifiy jarayonlar . Kembrij universiteti matbuoti. ISBN 978-0-521-86470-1 ^ a b v Tse, Devid (2005). Simsiz aloqa asoslari . Kembrij universiteti matbuoti. 
![{ displaystyle F _ { mathbf {Z}}: mathbb {C} ^ {n} mapsto [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8661e6d29a4d779324e5c3978506838f8c244b65)
![{ displaystyle operator nomi {E} [ mathbf {Z}] = ( operator nomi {E} [Z_ {1}], ldots, operator nomi {E} [Z_ {n}]) ^ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bad8687a134b34189596765316018f79fdca495)
![{ displaystyle { begin {aligned} & operatorname {K} _ { mathbf {Z} mathbf {Z}} = operatorname {cov} [ mathbf {Z}, mathbf {Z}] = operatorname {E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {Z} - operator nomi {E} [ mathbf {Z}])} ^ {H} ] = operator nomi {E} [ mathbf {Z} mathbf {Z} ^ {H}] - operator nomi {E} [ mathbf {Z}] operator nomi {E} [ mathbf {Z} ^ {H }] [12pt] end {hizalangan}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd5472da7a2129480754c26bf1527c36e0e1aef)
![{ displaystyle operator nomi {K} _ { mathbf {Z} mathbf {Z}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1} ]) { overline {(Z_ {1} - operator nomi {E} [Z_ {1}])}}] & mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) { overline {(Z_ {2} - operator nomi {E} [Z_ {2}])}}] & cdots & mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1}]) { overline {(Z_ {n} - operator nomi {E} [Z_ {n}])}}] mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(Z_ {1} - operator nomi {E} [Z_ {1}])}}] & mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(Z_ {2} - operator nomi {E} [Z_ {2}])}}] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) { overline {(Z_ {n} - operator nomi {E} [Z_ {n}])}}] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(Z_ {1} - operator nomi {E} [Z_ {1}] )}}] & mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(Z_ {2} - operator nomi {E} [Z_ {2}] )}}] & cdots & mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(Z_ {n} - operator nomi {E} [Z_ { n}])}}] end {bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/445eafa0ab8d1d761c1c34d3bb993b4598ec4b15)
![{ displaystyle operator nomi {J} _ { mathbf {Z} mathbf {Z}} = operator nomi {cov} [ mathbf {Z}, { overline { mathbf {Z}}}] = operator nomi { E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {Z} - operator nomi {E} [ mathbf {Z}])} ^ {T}] = operatorname {E} [ mathbf {Z} mathbf {Z} ^ {T}] - operatorname {E} [ mathbf {Z}] operatorname {E} [ mathbf {Z} ^ {T} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/073e290548d019170600567c2a29b7feaaf98a51)
![{ displaystyle operator nomi {J} _ { mathbf {Z} mathbf {Z}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) (Z_ {1} - operator nomi {E} [Z_ {1}])] va mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1}]) (Z_ {2 } - operatorname {E} [Z_ {2}])] & cdots & mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1}]) (Z_ {n} - operator nomi {E} [Z_ {n}])] \ mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (Z_ {1} - operator nomi {E } [Z_ {1}])] & mathrm {E} [(Z_ {2} - operatorname {E} [Z_ {2}]) (Z_ {2} - operatorname {E} [Z_ {2} ])] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (Z_ {n} - operator nomi {E} [Z_ {n}])] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (Z_ {1} - operator nomi {E} [Z_ {1}]) & mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (Z_ {2} - operator nomi {E} [Z_ {2}])] & cdots & mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (Z_ {n} - operator nomi {E} [Z_ {n} ])] end {bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50248ec68758163e693b5fa474537f0d953110ee)
![{ displaystyle { begin {aligned} & operatorname {K} _ { mathbf {X} mathbf {X}} = operatorname {E} [( mathbf {X} - operatorname {E} [ mathbf {X}]) ( mathbf {X} - operatorname {E} [ mathbf {X}]) ^ { mathrm {T}}] = { tfrac {1} {2}} operatorname {Re} ( operatorname {K} _ { mathbf {Z} mathbf {Z}} + operatorname {J} _ { mathbf {Z} mathbf {Z}}) & operatorname {K} _ { mathbf {X} mathbf {Y}} = operator nomi {E} [( mathbf {X} - operator nomi {E} [ mathbf {X}]) ( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ^ { mathrm {T}}] = { tfrac {1} {2}} operator nomi {Im} (- operator nomi {K} _ { mathbf {Z} mathbf {Z }} + operator nomi {J} _ { mathbf {Z} mathbf {Z}}) & operator nomi {K} _ { mathbf {Y} mathbf {X}} = operator nomi {E} [ ( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ( mathbf {X} - operator nomi {E} [ mathbf {X}]) ^ { mathrm {T}}] = { tfrac {1} {2}} operator nomi {Im} ( operator nomi {K} _ { mathbf {Z} mathbf {Z}} + operator nomi {J} _ { mathbf {Z} mathbf { Z}}) & operator nomi {K} _ { mathbf {Y} mathbf {Y}} = op eratorname {E} [( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ( mathbf {Y} - operator nomi {E} [ mathbf {Y}]) ^ { mathrm { T}}] = { tfrac {1} {2}} operator nomi {Re} ( operator nomi {K} _ { mathbf {Z} mathbf {Z}} - operator nomi {J} _ { mathbf { Z} mathbf {Z}}) end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3067c56dfb03faec313d1cd2dbf2258779bedc)
![{ displaystyle operator nomi {K} _ { mathbf {Z} mathbf {W}} = operator nomi {cov} [ mathbf {Z}, mathbf {W}] = operator nomi {E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {W} - operator nomi {E} [ mathbf {W}])} ^ {H}] = operator nomi {E} [ mathbf {Z} mathbf {W} ^ {H}] - operator nomi {E} [ mathbf {Z}] operator nomi {E} [ mathbf {W} ^ {H}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9844313bcca0b6e7e67e2c61bae3559a8714b4e4)
![{ displaystyle operator nomi {K} _ { mathbf {Z} mathbf {W}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) { overline {(W_ {1} - operator nomi {E} [W_ {1}])}}] & mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) { overline {(W_ {2} - operator nomi {E} [W_ {2}])}}] & cdots & mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1}]) { overline {(W_ {n} - operator nomi {E} [W_ {n}])}}] mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(W_ {1} - operator nomi {E} [W_ {1}])}}] & mathrm {E} [(Z_ {2} - operator nomi {E } [Z_ {2}]) { overline {(W_ {2} - operatorname {E} [W_ {2}])}}]] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) { overline {(W_ {n} - operator nomi {E} [W_ {n}])}}] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) { overline {(W_ {1} - operator nomi {E} [W_ {1}] )}}] & mathrm {E} [(Z_ {n} - operatorname {E} [Z_ {n}]) { overline {(W_ {2} - operatorname {E} [W_ {2}] )}}] & cdots & mathrm {E} [(Z_ {n} - operatorname {E} [Z_ {n}]) { overline {(W_ {n} - operatorname {E} [W_ {) n}])}}] end {bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9941690a38b08ff9f3b9be4bd1de0bb2fb9a67b5)
![{ displaystyle operator nomi {J} _ { mathbf {Z} mathbf {W}} = operator nomi {cov} [ mathbf {Z}, { overline { mathbf {W}}}] = operator nomi { E} [( mathbf {Z} - operator nomi {E} [ mathbf {Z}]) {( mathbf {W} - operator nomi {E} [ mathbf {W}])} ^ {T}] = operatorname {E} [ mathbf {Z} mathbf {W} ^ {T}] - operatorname {E} [ mathbf {Z}] operatorname {E} [ mathbf {W} ^ {T} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6a151a8db17da379cb32babce0e737bca9a8b9)
![{ displaystyle operator nomi {J} _ { mathbf {Z} mathbf {W}} = { begin {bmatrix} mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1} ]) (W_ {1} - operator nomi {E} [W_ {1}])] va mathrm {E} [(Z_ {1} - operator nomi {E} [Z_ {1}]) (W_ {2 } - operatorname {E} [W_ {2}])] & cdots & mathrm {E} [(Z_ {1} - operatorname {E} [Z_ {1}]) (W_ {n} - operator nomi {E} [W_ {n}])] \ mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (W_ {1} - operator nomi {E } [W_ {1}])] & mathrm {E} [(Z_ {2} - operatorname {E} [Z_ {2}]) (W_ {2} - operatorname {E} [W_ {2}) ])] & cdots & mathrm {E} [(Z_ {2} - operator nomi {E} [Z_ {2}]) (W_ {n} - operator nomi {E} [W_ {n}])] \ vdots & vdots & ddots & vdots \ mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (W_ {1} - operator nomi {E} [W_ {1}]]] va mathrm {E} [(Z_ {n} - operator nomi {E} [Z_ {n}]) (W_ {2} - operator nomi {E} [W_ {2}])] & cdots & mathrm {E} [(Z_ {n} - operatorname {E} [Z_ {n}]) (W_ {n} - operatorname {E} [W_ {n} ])] end {bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/240fc2c9717f84911d95a271d7ca87595b185cc5)
![{ displaystyle operatorname {E} [ mathbf {Z}] = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d136e6dc4d72ba213940846421b1f8c00160bc8d)
![{ displaystyle operatorname {var} [Z_ {1}] < infty, ldots, operatorname {var} [Z_ {n}] < infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca262a1fa11c0e897e89da432119076df0575dc4)
![{ displaystyle operator nomi {E} [ mathbf {Z} mathbf {Z} ^ {T}] = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e567ff4fa3785973cc8d53106ec6a0e075d969e9)
![{ displaystyle left | operator nomi {E} [ mathbf {Z} ^ {H} mathbf {W}] o'ng | ^ {2} leq operator nomi {E} [ mathbf {Z} ^ {H } mathbf {Z}] operatorname {E} [| mathbf {W} ^ {H} mathbf {W} |]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6675b4efe08db65ef119042110564db71520472)
![{ displaystyle varphi _ { mathbf {Z}} ( mathbf { omega}) = operatorname {E} left [e ^ {i Re {( mathbf { omega} ^ {H} mathbf {Z})}} o'ng] = operator nomi {E} chap [e ^ {i ( Re {( omega _ {1})} Re {(Z_ {1})} + Im {( omega _ {1})} Im {(Z_ {1})} + cdots + Re {( omega _ {n})} Re {(Z_ {n})} + Im {( omega _ {n})} Im {(Z_ {n})})} o'ng]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ef4b68890cb6b51eea0081d9a757a4c9f88849)