情報通信システム(特)論II Information and Communication Systems II

情報通信システム

(

)

II

Information and Communication Systems II

第２回講義資料

Lecture notes 2

複素ガウス分布

Complex Gaussian Distributions

豊橋技術科学大学

Toyohashi University of Technology

電気・電子情報工学系

Department of Electrical and Electronic Information Engineering

准教授 竹内啓悟

Associate Professor Keigo Takeuchi

実ガウス分布

(Real Gaussian distributions)

実ガウス確率変数

𝑋𝑋

Probability density function (pdf) of a real Gaussian random variable 𝑋𝑋

𝑝𝑝 _𝑋𝑋 𝑥𝑥 = 1

2𝜋𝜋𝜎𝜎 ² 𝑒𝑒 ^{− 𝑥𝑥−𝜇𝜇}

2

2𝜎𝜎 ² , 𝑥𝑥 ∈ ℝ.

𝑋𝑋 ∼ 𝒩𝒩 (𝜇𝜇, 𝜎𝜎 ² )

𝐼𝐼 = �

−∞

∞ 1

2𝜋𝜋𝜎𝜎 ² 𝑒𝑒 ^{− 𝑥𝑥−𝜇𝜇}

2

2𝜎𝜎 ² 𝑑𝑑𝑥𝑥 = 1.

規格化

(Normalization)

証明

(Proof)

𝐼𝐼 ² = 1

(Prove 𝐼𝐼 ² = 1 with a double integral.)

𝐼𝐼 ² = �

ℝ ²

1

2𝜋𝜋𝜎𝜎 ² 𝑒𝑒 ^{− 𝑥𝑥−𝜇𝜇}

2 + 𝑦𝑦−𝜇𝜇 ²

2𝜎𝜎 ² 𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑

𝑥𝑥 = 𝜇𝜇 + 𝑟𝑟 cos 𝜃𝜃

𝑑𝑑 = 𝜇𝜇 + 𝑟𝑟 sin 𝜃𝜃

To perform the change of variables, we compute the Jacobian.

𝐽𝐽 = 𝜕𝜕𝑥𝑥/𝜕𝜕𝑟𝑟 𝜕𝜕𝑥𝑥/𝜕𝜕𝜃𝜃

𝜕𝜕𝑑𝑑/𝜕𝜕𝑟𝑟 𝜕𝜕𝑑𝑑/𝜕𝜕𝜃𝜃 = cos 𝜃𝜃 −𝑟𝑟 sin 𝜃𝜃

sin 𝜃𝜃 𝑟𝑟 cos 𝜃𝜃 = 𝑟𝑟 cos ² 𝜃𝜃 + 𝑟𝑟 sin ² 𝜃𝜃 = 𝑟𝑟.

𝐼𝐼 ² = �

ℝ ²

1

2𝜋𝜋𝜎𝜎 ² 𝑒𝑒 ^{− 𝑥𝑥−𝜇𝜇}

2 + 𝑦𝑦−𝜇𝜇 ²

2𝜎𝜎 ² 𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑 = 1

2𝜋𝜋𝜎𝜎 ² �

0

∞ 𝑑𝑑𝑟𝑟 �

0

2𝜋𝜋 𝑑𝑑𝜃𝜃 𝑒𝑒 ^{− 𝑟𝑟}

2

2𝜎𝜎 ² 𝐽𝐽 = 1.

∎

ガウス分布の性質

(Properties of Gaussian distributions)

平均

(Mean)

� −∞

∞ 𝑥𝑥𝑝𝑝 _𝑋𝑋 𝑥𝑥 𝑑𝑑𝑥𝑥 = �

−∞

∞ 𝑥𝑥 1

2𝜋𝜋𝜎𝜎 ² 𝑒𝑒 ^{− 𝑥𝑥−𝜇𝜇}

2

2𝜎𝜎 ² 𝑑𝑑𝑥𝑥 = 𝜇𝜇 + �

−∞

∞ 𝜎𝜎𝜎𝜎 1

2𝜋𝜋 𝑒𝑒 ^{−𝑢𝑢 2 2} 𝑑𝑑𝜎𝜎 = 𝜇𝜇 . 𝜎𝜎 = 𝑥𝑥 − 𝜇𝜇 /𝜎𝜎

(Using the change of a variable yields)

二番目の等号で、規格化条件を使った。

In the second equality , we have used the normalization condition

最後の等号は、第二項の被積分関数の奇関数性から従う。

The last equality follows from the fact that the integrand in the second term is an even function.

分散

(Variance)

� −∞

∞ 𝑥𝑥 − 𝜇𝜇 ² 𝑝𝑝 _𝑋𝑋 𝑥𝑥 𝑑𝑑𝑥𝑥 = 𝜎𝜎 ² .

(Confirm it.)

ヒント

(Hint)

を使って積分を評価せよ。

After using the same change of a variable as in the mean case, use the integration

by parts and the normalization condition to evaluate the integral.

行列の基本演算

(Basic operations for matrices)

性質

1 (Property 1)

𝑨𝑨𝑨𝑨 ^T = 𝑨𝑨 ^T 𝑨𝑨 ^T .

性質

3 (Property 3)

𝑨𝑨𝑨𝑨 ⁻¹ = 𝑨𝑨 ⁻¹ 𝑨𝑨 ⁻¹ .

性質

4 (Property 4)

det 𝑨𝑨𝑨𝑨 = det 𝑨𝑨 det(𝑨𝑨).

性質

2 (Property 2)

𝑨𝑨𝑨𝑨 ^H = 𝑨𝑨 ^H 𝑨𝑨 ^H . 𝑨𝑨 ^H = 𝐀𝐀 ^{T ∗} (conjugate transpose)

実正定値行列

(Real positive definite matrices)

𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 > 0 for all non-zero vectors 𝒙𝒙 ∈ ℝ ^𝑛𝑛 .

𝑛𝑛

( 𝑛𝑛 -dimensional real positive definite matrix) 𝐴𝐴 ∈ ℝ ^{𝑛𝑛×𝑛𝑛}

性質

(Property)

𝑨𝑨

�𝑨𝑨 = (𝑨𝑨 + 𝑨𝑨 ^T )/2

𝐴𝐴 is positive definite if and only if the symmetric matrix �𝑨𝑨 = (𝑨𝑨 + 𝑨𝑨 ^T )/2 is positive definite.

𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 > 0

𝒙𝒙 ^T 𝑨𝑨 ^T 𝒙𝒙 > 0

Taking the transpose of both sides of 𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 > 0 yields 𝒙𝒙 ^T 𝑨𝑨 ^T 𝒙𝒙 > 0 . Thus,

必要性の証明

(Proof of the necessity)

𝒙𝒙 ^T �𝑨𝑨𝒙𝒙 = 1

2 𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 + 1

2 𝒙𝒙 ^T 𝑨𝑨 ^T 𝒙𝒙 > 0.

十分性の証明

(Proof of the sufficiency)

𝒙𝒙 ^T �𝑨𝑨𝒙𝒙 > 0

𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 + 𝒙𝒙 ^T 𝑨𝑨 ^T 𝒙𝒙 > 0

𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 ≤ 0

𝒙𝒙 ^T 𝑨𝑨 ^T 𝒙𝒙 ≤ 0

𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 > 0

The condition 𝒙𝒙 ^T �𝑨𝑨𝒙𝒙 > 0 implies 𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 + 𝒙𝒙 ^T 𝑨𝑨 ^T 𝒙𝒙 > 0 . Assume 𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 ≤ 0 . Then, we have

𝒙𝒙 ^T 𝑨𝑨 ^T 𝒙𝒙 ≤ 0 , which is a contradiction. Thus, 𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 > 0 holds.

実正定値対称行列の性質

(Properties of real positive definite symmetric matrices)

命題

2.1 (Proposition 2.1)

直交行列

(Orthogonal matrices) 𝑼𝑼

𝑼𝑼 ^T 𝑼𝑼 = 𝑰𝑰, 𝑼𝑼𝑼𝑼 ^T = 𝑰𝑰.

任意の実正定値対称行列

𝑨𝑨

𝑼𝑼

Any real positive definite symmetric matrix 𝑨𝑨 has positive eigenvalues, and can be diagonalized with an orthogonal matrix 𝑼𝑼 .

𝑼𝑼 ^T 𝑨𝑨𝑼𝑼 = 𝜦𝜦, 𝜦𝜦 = diag 𝜆𝜆 ₁ , … , 𝜆𝜆 _𝑛𝑛 𝜆𝜆 _𝑖𝑖 > 0 .

(Proof)

任意の対称行列は直交行列を使って対角化できるので、固有値が すべて正であることを確かめればよい。

Since any symmetric matrix can be diagonalized with some orthogonal matrix, it is sufficient to confirm that all eigenvalues are positive.

𝒆𝒆 _𝑖𝑖

ℝ ^𝑛𝑛

𝑖𝑖

( 𝑰𝑰 _𝑛𝑛

𝑖𝑖

)

𝒙𝒙 = 𝑼𝑼𝒆𝒆 _𝑖𝑖

Let 𝒆𝒆 _𝑖𝑖 denote the 𝑖𝑖 th standard basis of ℝ ^𝑛𝑛 ( 𝑖𝑖 th column of 𝑰𝑰 _𝑛𝑛 ). For 𝒙𝒙 = 𝑼𝑼𝒆𝒆 _𝑖𝑖 ,

0 < 𝒙𝒙 ^T 𝑨𝑨𝒙𝒙 = 𝒆𝒆 _𝑖𝑖 ^T 𝑼𝑼 ^T 𝑨𝑨𝑼𝑼𝒆𝒆 _𝑖𝑖 = 𝒆𝒆 _𝑖𝑖 ^T 𝜦𝜦𝒆𝒆 _𝑖𝑖 = 𝜆𝜆 _𝑖𝑖 . ∎

𝑛𝑛

( 𝑛𝑛 -dimensional real Gaussian distributions)

𝑝𝑝 _𝑿𝑿 𝒙𝒙 = 1

2𝜋𝜋 ^𝑛𝑛/2 det 𝚺𝚺 ^1/2 𝑒𝑒 ^{−12 𝒙𝒙−𝝁𝝁 T 𝚺𝚺 −1 (𝒙𝒙−𝝁𝝁)} , 𝒙𝒙 ∈ ℝ ^𝑛𝑛 .

(Joint pdf)

任意の

𝝁𝝁 ∈ ℝ ^𝑛𝑛

𝚺𝚺

For any 𝝁𝝁 ∈ ℝ ^𝑛𝑛 and positive definite symmetric matrix 𝚺𝚺 ,

規格化

(Normalization)

� ℝ ^𝑛𝑛 𝑝𝑝 _𝑿𝑿 𝒙𝒙 𝑑𝑑𝑥𝑥 = 1.

∵

𝑼𝑼 ^T 𝚺𝚺𝑼𝑼 = 𝜦𝜦

det 𝑼𝑼 = 1

𝒚𝒚 = 𝑼𝑼 ^T 𝒙𝒙

�𝝁𝝁 = 𝑼𝑼 ^T 𝝁𝝁

Since | det 𝑼𝑼| = 1 holds in the diagonalization 𝑼𝑼 ^T 𝚺𝚺𝑼𝑼 = 𝜦𝜦 , with �𝝁𝝁 = 𝑼𝑼 ^T 𝝁𝝁 , the change of variables 𝒚𝒚 = 𝑼𝑼 ^T 𝒙𝒙 implies

二番目の等号を確かめよ。

(Confirm the second equality .)

� ℝ ^𝑛𝑛 𝑝𝑝 _𝑿𝑿 𝒙𝒙 𝑑𝑑𝒙𝒙 = �

ℝ ^𝑛𝑛 𝑝𝑝 _𝑿𝑿 𝑼𝑼𝒚𝒚 𝑑𝑑𝒚𝒚 = �

𝑖𝑖=1

𝑛𝑛 �

ℝ

1

2𝜋𝜋𝜆𝜆 _𝑖𝑖 𝑒𝑒 ^{− 𝑦𝑦 𝑖𝑖 −�𝜇𝜇 𝑖𝑖}

2

2𝜆𝜆 _𝑖𝑖 𝑑𝑑𝑑𝑑 _𝑖𝑖 = 1.

∎

𝑛𝑛

( 𝑛𝑛 -dimensional real Gaussian distributions)

平均

(Mean)

𝔼𝔼 𝒙𝒙 = �

ℝ ^𝑛𝑛 𝒙𝒙𝑝𝑝 _𝑿𝑿 𝒙𝒙 𝑑𝑑𝒙𝒙 = �

ℝ ^𝑛𝑛 (𝝁𝝁 + 𝒚𝒚) 𝑒𝑒 ^{−12𝒚𝒚 T 𝚺𝚺 −1 𝒚𝒚}

2𝜋𝜋 ^𝑛𝑛/2 det 𝚺𝚺 ^1/2 𝑑𝑑𝒚𝒚 = 𝝁𝝁.

𝔼𝔼 𝒙𝒙 = 𝝁𝝁.

∵ 𝒚𝒚 = 𝒙𝒙 − 𝝁𝝁

(The change of variables 𝒚𝒚 = 𝒙𝒙 − 𝝁𝝁 implies)

共分散

(Covariance) 𝔼𝔼 𝒙𝒙 − 𝝁𝝁 𝒙𝒙 − 𝝁𝝁 ^T = 𝚺𝚺.

∵ 𝒚𝒚 = 𝑼𝑼 ^T (𝒙𝒙 − 𝝁𝝁)

(The change of variables 𝒚𝒚 = 𝑼𝑼 ^T (𝒙𝒙 − 𝝁𝝁) implies)

𝔼𝔼 𝒙𝒙 − 𝝁𝝁 𝒙𝒙 − 𝝁𝝁 ^T = �

ℝ ^𝑛𝑛 𝑼𝑼𝒚𝒚𝒚𝒚 ^T 𝑼𝑼 ^T 𝑒𝑒 ^{−12𝒚𝒚 T 𝚲𝚲 −1 𝒚𝒚}

2𝜋𝜋 ^𝑛𝑛/2 det 𝚲𝚲 ^1/2 𝑑𝑑𝒚𝒚

= 𝑼𝑼 �

ℝ ^𝑛𝑛 𝒚𝒚𝒚𝒚 ^T �

𝑖𝑖=1

𝑛𝑛 𝑒𝑒 ^{−𝑦𝑦 𝑖𝑖 2 /(2𝜆𝜆 𝑖𝑖 )}

2𝜋𝜋𝜆𝜆 _𝑖𝑖 𝑑𝑑𝑑𝑑 _𝑖𝑖 𝑼𝑼 ^T

= 𝑼𝑼𝚲𝚲𝑼𝑼 ^T = 𝚺𝚺.

円対称複素分布

(Circularly symmetric complex distributions)

複素確率変数

𝑍𝑍

𝜃𝜃 ∈ [0, 2𝜋𝜋)

𝑒𝑒 ^j𝜃𝜃 𝑍𝑍 ∼ 𝑍𝑍

円対称と呼ばれる。

A complex random variable 𝑍𝑍 is called circularly symmetric if 𝑒𝑒 ^j𝜃𝜃 𝑍𝑍 ∼ 𝑍𝑍 holds for any 𝜃𝜃 ∈ [0, 2𝜋𝜋) .

ℜ 𝑍𝑍 = (𝑍𝑍 + 𝑍𝑍 ^∗ )/2

ℑ 𝑍𝑍 = (𝑍𝑍 − 𝑍𝑍 ^∗ )/(2j)

𝑍𝑍

𝑍𝑍

𝑍𝑍 ^∗ (

)

From ℜ 𝑍𝑍 = (𝑍𝑍 + 𝑍𝑍 ^∗ )/2 and ℑ 𝑍𝑍 = (𝑍𝑍 − 𝑍𝑍 ^∗ )/(2j) , the distribution of a complex random variable 𝑍𝑍 is determined by the joint distribution of 𝑍𝑍 and 𝑍𝑍 ^∗ (complex conjugate).

補題

2.1 (Lemma 2.1)

任意の円対称確率変数

𝑍𝑍

(For any circularly symmetric random variable 𝑍𝑍 ,)

𝔼𝔼 𝑍𝑍 ^𝑚𝑚 𝑍𝑍 ^{∗ 𝑛𝑛} = 0, for 𝑚𝑚 ≠ 𝑛𝑛.

𝔼𝔼 𝑍𝑍 ^𝑚𝑚 𝑍𝑍 ^{∗ 𝑛𝑛} = 𝔼𝔼 𝑒𝑒 ^j𝜃𝜃 𝑍𝑍 ^𝑚𝑚 𝑒𝑒 ^−j𝜃𝜃 𝑍𝑍 ^{∗ 𝑛𝑛} = 𝑒𝑒 ^{j(𝑚𝑚−𝑛𝑛)𝜃𝜃} 𝔼𝔼 𝑍𝑍 ^𝑚𝑚 𝑍𝑍 ^{∗ 𝑛𝑛} .

𝑚𝑚 ≠ 𝑛𝑛

𝑒𝑒 ^{j 𝑚𝑚−𝑛𝑛 𝜃𝜃} ≠ 1

𝜃𝜃

𝔼𝔼 𝑍𝑍 ^𝑚𝑚 𝑍𝑍 ^{∗ 𝑛𝑛} = 0

For 𝑚𝑚 ≠ 𝑛𝑛 , there is some 𝜃𝜃 such that 𝑒𝑒 ^{j 𝑚𝑚−𝑛𝑛 𝜃𝜃} ≠ 1. Thus, 𝔼𝔼 𝑍𝑍 ^𝑚𝑚 𝑍𝑍 ^{∗ 𝑛𝑛} = 0 holds.

注意

(Remark) 2

( 𝑚𝑚 + 𝑛𝑛 ≤ 2 )

𝔼𝔼[ 𝑍𝑍 ² ]

𝔼𝔼[ 𝑍𝑍 ² ] is the only non-zero moment up to 2nd order.

∵

円対称複素ガウス分布

(Circularly symmetric complex Gaussian (CSCG) distributions)

円対称確率変数

𝑍𝑍

Z

𝜎𝜎 ² = 𝔼𝔼 𝑍𝑍 ²

A circularly symmetric random variable 𝑍𝑍 is called CSCG if the real and imaginary parts of 𝑍𝑍 are jointly Gaussian-distributed. In particular, 𝜎𝜎 ² = 𝔼𝔼 𝑍𝑍 ² is called variance.

定理

2.1 (Theorem 2.1)

分散

𝜎𝜎 ²

𝑍𝑍

0

𝜎𝜎 ² /2

証明

(Proof)

0

𝔼𝔼 𝑍𝑍 = 0

(The zero mean follows from 𝔼𝔼 𝑍𝑍 = 0 .)

さらに、

𝑍𝑍 ² = ℜ 𝑍𝑍 ² − ℑ 𝑍𝑍 ² + 2jℜ 𝑍𝑍 ℑ[𝑍𝑍]

𝔼𝔼 𝑍𝑍 ² = 0

Furthermore, 𝑍𝑍 ² = ℜ 𝑍𝑍 ² − ℑ 𝑍𝑍 ² + 2jℜ 𝑍𝑍 ℑ[𝑍𝑍] and 𝔼𝔼 𝑍𝑍 ² = 0 imply that the real and imaginary parts are identically distributed and uncorrelated (i.e. independent).

分散

𝜎𝜎 ² /2

𝑍𝑍 ² = ℜ 𝑍𝑍 ² + ℑ 𝑍𝑍 ²

𝔼𝔼 𝑍𝑍 ² = 𝜎𝜎 ²

The variance 𝜎𝜎 ² /2 follows from 𝑍𝑍 ² = ℜ 𝑍𝑍 ² + ℑ 𝑍𝑍 ² and 𝔼𝔼 𝑍𝑍 ² = 𝜎𝜎 ² . ∎

複素ガウス分布

(Complex Gaussian distributions)

複素確率変数

𝑍𝑍

𝑍𝑍 − 𝜇𝜇

𝜎𝜎 ²

𝜇𝜇

𝜎𝜎 ²

𝒞𝒞𝒩𝒩 (𝜇𝜇, 𝜎𝜎 ² )

We say in communications that a complex random variable 𝑍𝑍 follows the complex Gaussian distribution 𝒞𝒞𝒩𝒩(𝜇𝜇, 𝜎𝜎 ² ) with mean 𝜇𝜇 and variance 𝜎𝜎 ² , if 𝑍𝑍 − 𝜇𝜇 is CSCG with variance 𝜎𝜎 ² .

確率密度関数

(Pdf)

𝑝𝑝 _𝑍𝑍 𝑧𝑧 = 1

𝜋𝜋𝜎𝜎 ² 𝑒𝑒 ^{− 𝑧𝑧−𝜇𝜇}

2

𝜎𝜎 ² , 𝑧𝑧 ∈ ℂ.

𝑍𝑍 ∼ 𝒞𝒞𝒩𝒩(𝜇𝜇, 𝜎𝜎 ² )

∵

2.1

(From Theorem 2.1, we have)

𝑝𝑝 _𝑍𝑍 𝑧𝑧 = 1

2𝜋𝜋𝜎𝜎 ² /2 𝑒𝑒 − ℜ 𝑧𝑧 −ℜ 𝜇𝜇 2𝜎𝜎 ² /2 ² 1

2𝜋𝜋𝜎𝜎 ² /2 𝑒𝑒 − ℑ 𝑧𝑧 −ℑ 𝜇𝜇 2𝜎𝜎 ² /2 ² = 1

𝜋𝜋𝜎𝜎 ² 𝑒𝑒 ^{− 𝑧𝑧−𝜇𝜇}

2

𝜎𝜎 ² .

規格化

(Normalization)

� ℂ 𝑝𝑝 _𝑍𝑍 𝑧𝑧 𝑑𝑑𝑧𝑧 = 1,

ここで、

𝑑𝑑𝑧𝑧 = 𝑑𝑑ℜ 𝑧𝑧 𝑑𝑑ℑ[𝑧𝑧]

ℝ ²

where the integration is defined as the double integral over ℝ ² for 𝑑𝑑𝑧𝑧 = 𝑑𝑑ℜ 𝑧𝑧 𝑑𝑑ℑ[𝑧𝑧] .

正定値行列

(Positive definite matrices)

𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 > 0 for all non-zero vectors 𝒛𝒛 ∈ ℂ ^𝑛𝑛 .

𝑛𝑛

( 𝑛𝑛 -dimensional complex positive definite matrix) 𝐴𝐴 ∈ ℂ ^{𝑛𝑛×𝑛𝑛}

性質

(Property)

𝑨𝑨

ℑ 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 = 0

�𝑨𝑨 = (𝑨𝑨 + 𝑨𝑨 ^H )/2

𝐴𝐴 is positive definite if and only if ℑ 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 = 0 holds and the Hermitian matrix �𝑨𝑨 = (𝑨𝑨 + 𝑨𝑨 ^H )/2 is positive definite.

ℑ 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 = 0

𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 > 0

𝒛𝒛 ^H 𝑨𝑨 ^H 𝒛𝒛 > 0

ℑ 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 = 0 is trivial. Taking the conjugate transpose of both sides of 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 > 0 yields 𝒛𝒛 ^H 𝑨𝑨 ^H 𝒛𝒛 > 0 . Thus,

必要性の証明

(Proof of the necessity)

𝒛𝒛 ^H �𝑨𝑨𝒛𝒛 = (𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 + 𝒛𝒛 ^H 𝑨𝑨 ^H 𝒛𝒛)/2 > 0.

十分性の証明

(Proof of the sufficiency)

𝒛𝒛 ^H �𝑨𝑨𝒛𝒛 > 0

𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 + 𝒛𝒛 ^H 𝑨𝑨 ^H 𝒛𝒛 > 0

ℑ 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 = 0

𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 ≤ 0

𝒛𝒛 ^H 𝑨𝑨 ^H 𝒛𝒛 ≤ 0

𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 > 0

The condition 𝒛𝒛 ^H �𝑨𝑨𝒛𝒛 > 0 implies 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 + 𝒛𝒛 ^H 𝑨𝑨 ^H 𝒛𝒛 > 0 . Assume 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 ≤ 0 , since ℑ 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 = 0 holds.

Then, we have 𝒛𝒛 ^H 𝑨𝑨 ^H 𝒛𝒛 ≤ 0 , which is a contradiction. Thus, 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 > 0 holds.

正定値エルミート行列の性質

(Properties of positive definite Hermitian matrices)

命題

2.1’ (Proposition 2.1’)

ユニタリ行列

(Unitary matrices) 𝑼𝑼

𝑼𝑼 ^H 𝑼𝑼 = 𝑰𝑰, 𝑼𝑼𝑼𝑼 ^H = 𝑰𝑰.

任意の正定値エルミート行列

𝑨𝑨

𝑼𝑼

Any positive definite Hermitian matrix 𝑨𝑨 has positive eigenvalues, and can be diagonalized with a unitary matrix 𝑼𝑼 .

𝑼𝑼 ^H 𝑨𝑨𝑼𝑼 = 𝜦𝜦, 𝜦𝜦 = diag 𝜆𝜆 ₁ , … , 𝜆𝜆 _𝑛𝑛 𝜆𝜆 _𝑖𝑖 > 0 .

(Proof)

任意のエルミート行列はユニタリ行列を使って対角化できるので、固 有値がすべて正であることを確かめればよい。

Since any Hermitian matrix can be diagonalized with some unitary matrix, it is sufficient to confirm that all eigenvalues are positive.

𝒆𝒆 _𝑖𝑖

ℂ ^𝑛𝑛

𝑖𝑖

( 𝑰𝑰 _𝑛𝑛

𝑖𝑖

)

𝒛𝒛 = 𝑼𝑼𝒆𝒆 _𝑖𝑖

Let 𝒆𝒆 _𝑖𝑖 denote the 𝑖𝑖 th standard basis of ℂ ^𝑛𝑛 ( 𝑖𝑖 th column of 𝑰𝑰 _𝑛𝑛 ). For 𝒛𝒛 = 𝑼𝑼𝒆𝒆 _𝑖𝑖 ,

0 < 𝒛𝒛 ^H 𝑨𝑨𝒛𝒛 = 𝒆𝒆 _𝑖𝑖 ^H 𝑼𝑼 ^H 𝑨𝑨𝑼𝑼𝒆𝒆 _𝑖𝑖 = 𝒆𝒆 _𝑖𝑖 ^H 𝜦𝜦𝒆𝒆 _𝑖𝑖 = 𝜆𝜆 _𝑖𝑖 . ∎

円対称複素確率ベクトル

(Circularly symmetric complex random vectors)

複素確率ベクトル

𝒁𝒁

𝜃𝜃 ∈ [0, 2𝜋𝜋)

𝑒𝑒 ^j𝜃𝜃 𝒁𝒁 ∼ 𝒁𝒁

A complex random vector 𝒁𝒁 is called circularly symmetric if 𝑒𝑒 ^j𝜃𝜃 𝒁𝒁 ∼ 𝒁𝒁 holds for any 𝜃𝜃 ∈ [0, 2𝜋𝜋) .

補題

2.1’ (Lemma 2.1’)

任意の円対称確率ベクトル

𝒁𝒁 = 𝑍𝑍 ₁ , … , 𝑍𝑍 _𝑛𝑛 ^T

For any circularly symmetric random vector 𝒁𝒁 ,

𝔼𝔼 � 𝑖𝑖=1

𝑛𝑛 𝑍𝑍 _𝑖𝑖 ^{𝑚𝑚 𝑖𝑖} 𝑍𝑍 _𝑖𝑖 ^{∗ 𝑛𝑛 𝑖𝑖} = 𝑒𝑒 ^{j ∑ 𝑖𝑖 𝑚𝑚 𝑖𝑖 −𝑛𝑛 𝑖𝑖 𝜃𝜃} 𝔼𝔼 �

𝑖𝑖=1

𝑛𝑛 𝑍𝑍 _𝑖𝑖 ^{𝑚𝑚 𝑖𝑖} 𝑍𝑍 _𝑖𝑖 ^{∗ 𝑛𝑛 𝑖𝑖} .

注意

(Remark) 2

𝔼𝔼[𝑍𝑍 _𝑖𝑖 𝑍𝑍 _𝑖𝑖 ^{∗ ′} ]

𝔼𝔼[𝑍𝑍 _𝑖𝑖 𝑍𝑍 _𝑖𝑖 ∗

] is the only non-zero moment up to 2nd order.

∵

𝔼𝔼 � 𝑖𝑖=1

𝑛𝑛 𝑍𝑍 _𝑖𝑖 ^{𝑚𝑚 𝑖𝑖} 𝑍𝑍 _𝑖𝑖 ^{∗ 𝑛𝑛 𝑖𝑖} = 0 for �

𝑖𝑖=1 𝑛𝑛

𝑚𝑚 _𝑖𝑖 ≠ �

𝑖𝑖=1 𝑛𝑛

𝑛𝑛 _𝑖𝑖 .

複素ガウス確率ベクトル

(Complex Gaussian random vectors)

円対称確率ベクトル

𝒁𝒁

𝒁𝒁

𝚺𝚺 = 𝔼𝔼 𝒁𝒁𝒁𝒁 ^H

A circularly symmetric random vector 𝒁𝒁 is called CSCG if the real and imaginary parts of 𝒁𝒁 are jointly Gaussian-distributed. 𝚺𝚺 = 𝔼𝔼 𝒁𝒁𝒁𝒁 ^H is called covariance matrix.

定理

2.2 (Theorem 2.2)

𝑝𝑝 _𝒁𝒁 𝒛𝒛 = 1

𝜋𝜋 ^𝑛𝑛 det 𝚺𝚺 𝑒𝑒 ^{− 𝒛𝒛−𝝁𝝁}

H 𝚺𝚺 ⁻¹ (𝒛𝒛−𝝁𝝁) , 𝒛𝒛 ∈ ℂ ^𝑛𝑛 . 𝒁𝒁 ∼ 𝒞𝒞𝒩𝒩(𝝁𝝁, 𝚺𝚺 )

複素確率ベクトル

𝒁𝒁

𝒁𝒁 − 𝝁𝝁

𝚺𝚺

𝝁𝝁

𝚺𝚺

𝒞𝒞𝒩𝒩(𝝁𝝁, 𝚺𝚺)

We say in communications that a complex random vector 𝒁𝒁 follows the complex Gaussian distribution 𝒞𝒞𝒩𝒩(𝝁𝝁, 𝚺𝚺) with mean 𝝁𝝁 and covariance matrix 𝚺𝚺 , if 𝒁𝒁 − 𝝁𝝁 is CSCG.

証明

(Proof)

�𝒁𝒁 = 𝒁𝒁 − 𝝁𝝁

𝝁𝝁 = 𝟎𝟎

Without loss of generality, we assume 𝝁𝝁 = 𝟎𝟎 by letting �𝒁𝒁 = 𝒁𝒁 − 𝝁𝝁 .

円対称複素ガウス分布が満たすべき性質

𝔼𝔼 𝒁𝒁 = 𝟎𝟎

𝔼𝔼 𝒁𝒁𝒁𝒁 ^T = 𝑶𝑶

𝔼𝔼 𝒁𝒁𝒁𝒁 ^H = 𝚺𝚺

We prove the properties 𝔼𝔼 𝒁𝒁 = 0, 𝔼𝔼 𝒁𝒁𝒁𝒁 ^T = 𝑶𝑶 , and 𝔼𝔼 𝒁𝒁𝒁𝒁 ^H = 𝚺𝚺 which the CSCG distribution

should satisfy.

定理

2.2

(Proof of Theorem 2.2)

平均

(Mean)

𝚺𝚺 = 𝑼𝑼𝜦𝜦𝑼𝑼 ^H

𝒗𝒗 = 𝑼𝑼 ^H 𝒛𝒛

Using the change of variables 𝒗𝒗 = 𝑼𝑼 ^H 𝒛𝒛 for the diagonalization 𝚺𝚺 = 𝑼𝑼𝜦𝜦𝑼𝑼 ^H yields

𝔼𝔼 𝒁𝒁 = �

ℂ ^𝑛𝑛 𝒛𝒛𝑝𝑝 _𝒁𝒁 𝒛𝒛 𝑑𝑑𝒛𝒛 = 𝑼𝑼 �

ℂ ^𝑛𝑛 𝒗𝒗 �

𝑖𝑖=1

𝑛𝑛 1

𝜋𝜋𝜆𝜆 _𝑖𝑖 𝑒𝑒 ^{− 𝑣𝑣 𝑖𝑖}

2

𝜆𝜆 _𝑖𝑖 𝑑𝑑𝑣𝑣 _𝑖𝑖 = 𝟎𝟎.

共分散

(Covariance)

𝔼𝔼 𝒁𝒁𝒁𝒁 ^H = �

ℂ ^𝑛𝑛 𝒛𝒛𝒛𝒛 ^H 𝑝𝑝 _𝒁𝒁 𝒛𝒛 𝑑𝑑𝒛𝒛 = 𝑼𝑼 �

ℂ ^𝑛𝑛 𝒗𝒗𝒗𝒗 ^H �

𝑖𝑖=1

𝑛𝑛 1

𝜋𝜋𝜆𝜆 _𝑖𝑖 𝑒𝑒 ^{− 𝑣𝑣 𝑖𝑖}

2

𝜆𝜆 _𝑖𝑖 𝑑𝑑𝑣𝑣 _𝑖𝑖 𝑼𝑼 ^H

= 𝑼𝑼𝜦𝜦𝑼𝑼 ^H = 𝚺𝚺.

𝔼𝔼 𝒁𝒁𝒁𝒁 ^T = �

ℂ ^𝑛𝑛 𝒛𝒛𝒛𝒛 ^T 𝑝𝑝 _𝒁𝒁 𝒛𝒛 𝑑𝑑𝒛𝒛 = 𝑼𝑼 �

ℂ ^𝑛𝑛 𝒗𝒗𝒗𝒗 ^T �

𝑖𝑖=1

𝑛𝑛 1

𝜋𝜋𝜆𝜆 _𝑖𝑖 𝑒𝑒 ^{− 𝑣𝑣 𝑖𝑖}

2

𝜆𝜆 _𝑖𝑖 𝑑𝑑𝑣𝑣 _𝑖𝑖 𝑼𝑼 ^T = 𝑶𝑶.

最後の等号の導出で、

𝑉𝑉 _𝑖𝑖 ∼ 𝒞𝒞𝒩𝒩(0, 𝜆𝜆 _𝑖𝑖 )

𝔼𝔼 𝑉𝑉 _𝑖𝑖 ² = 0

In the derivation of the last equality , we have used 𝔼𝔼 𝑉𝑉 _𝑖𝑖 ² = 0 for 𝑉𝑉 _𝑖𝑖 ∼ 𝒞𝒞𝒩𝒩(0, 𝜆𝜆 _𝑖𝑖 ) .

∎