https://youtu.be/H3kmlsXJUfA

Covariance matrix

• 어떤 랜덤 vector의 covariance matrix $K_{z}$는 다음과 같다.
  • $K_{z} = E\{[z(u) - m_{z}] [z(u) - m_{z}]^{\dagger}\} \\ = E\{z(u)z^{\dagger}(u) - m_{z}z^{\dagger}(u) - z(u)m_{z}^{\dagger} + m_{z}m_{z}^{\dagger} \} \\ = E\{z(u)z^{\dagger}(u)\} - m_{z}E\{z^{\dagger}(u)\} - E\{z(u)\}m_{z}^{\dagger} + m_{z}m_{z}^{\dagger} \\ = R_{z} - m_{z} m_{z}^{\dagger}$
    • $m_{z}$는 평균벡터
    • $\dagger$는 conjugate transpose로 켤레전치행렬이 된다. (복소수의 부호를 바꾸고 전치시킨 행렬)

Pseudo-correlation, Pseudo-covariance matrix

• Pseudo-correlation
  • $\tilde{T}_{z} = E \{z(u)z^{t}(u)\}$
• Pseudo-covariance
  • $\tilde{K}{z} = E \{ [z(u) - m{z}][z(u) - m_{z}]^{t} \}$
• 임의의 벡터 $a \in C^{n}$ 에 대하여
  • $a^{\dagger}K_{z}a \geq 0$
  • 이러한 성질에 때문에 Covariance matrix $K_{z}$는 non-negative definite라고 한다. 이는 correlation matrix $R_{z}$도 마찬가지다.
    • (non-negative definite은 positive semi definite(양의 준정부호)이라고도 한다)

Theorem

• Correlation function은 non-negative definite이다.
• 증명)
  • $w(u) \triangleq \sum_{j=1}^{n} a_{j} z(u, t_{j})$
  • $\mathbb{E}\{|w(u)|^{2}\} = \mathbb{E} \{ | \sum_{j=1}^{n} a_{j} z(u, t_{j}) |^{2} \} \\ = \sum_{j=1}^{n} \sum_{k=1}^{n} a_{j} \mathbb{E}\{ z(u, t_{j}) z^{} (u, t_{k})\} a_{k}^{} \\ = \sum_{j=1}^{n} \sum_{k=1}^{n} a_{j} R_{z}(t_{j}, t_{k}) a_{k}^{*} \geq 0$

Linear transformation or random vectors

• $y(u) = \left[ \begin{array}{rrrr} y(u, 1) \\ y(u, 2) \\ ... \\ y(u, n) \end{array} \right]$
  • $y(u)$는 랜덤 벡터
• $= \left[ \begin{array}{rrrr} h_{11} & h_{12} & ... & h_{1n} \\ h_{21} & h_{22} & ... & h_{2n} \\ ... \\ h_{m1} & h_{m2} & ... & h_{mn} \end{array} \right] \left[ \begin{array}{rrrr} z(u, 1) \\ z(u, 2) \\ ... \\ z(u, n) \end{array} \right] =Hz(u)$
  • $z(u)$는 랜덤 벡터
  • $H$ 는 선형변환 (행렬)
• $E \{ y(u, t) \} = E \{ \sum_{\tau = 1}^{n} h_{t \tau} z(u, \tau) \} = \sum_{\tau =1}^{n} h_{t \tau} E\{z(u, \tau) \} = \sum_{\tau =1}^{n} h_{t \tau} m_{z}(\tau)$
• $E\{y(u, t_{1}) y^{*}(u, t_{2})\}$
  • $= E \{ [ \sum_{\tau_{1} = 1}^{n} h_{t_{1} \tau_{1}} z(u, \tau_{1}) ] [ \sum_{\tau_{2} = 1}^{n} h_{t_{2} \tau_{2}}^{} z^{}(u, \tau_{2}) ] \} \\ = \sum_{\tau_{1} = 1}^{n} \sum_{\tau_{2} = 1}^{n} h_{t_{1} \tau_{1}} h_{t_{2} \tau_{2}}^{} E[ z(u, \tau_{1}) z^{}(u, \tau_{2}) ]$
• $E\{ y(u, t_{1}) y^{*}(u, t_{2}) \}$
  • $= \sum_{\tau_{1} = 1}^{n} \sum_{\tau_{2} = 1}^{n} h_{t_{1} \tau_{1}} h_{t_{2} \tau_{2}}^{*} R_{z} (\tau_{1}, \tau_{2})$
• Thus,
  • $m_{y} = E\{y(u)\} = E \{Hz(u)\} = HE\{z(u)\} = Hm_{z}$
  • $R_{y} = E\{y(u) y^{\dagger}(u) \}$
    • $= E\{Hz(u)(Hz(u))^{\dagger}\} \\ = E\{Hz(u) z^{\dagger}(u) H^{\dagger}\} = HE\{z(u) z^{\dagger}(u)\}H^{\dagger} \\ = HR_{z} H^{\dagger}$
  • $\tilde{R}_{y} = E \{ Hz(u) z^{t}(u) H^{t} \}$
    • $HE\{z(u) z^{t}(u)\}H^{t} = H \tilde{R}_{z} H^{t}$
• Centered output vector
  • $y_{0}(u) = y(u) - m_{y} = Hz(u) - Hm_{z} = H[z(u) - m_{z}] = Hz_{0}(u)$
• Covariance matrix
  • $K_{y} = HK_{z}H^{\dagger}$
• Pseudo-covariance matrix
  • $\tilde{K}{y} = H \tilde{K}{z} H^{t}$