User talk:Oyz

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[[User:ClockworkSoul|User:ClockworkSoul/sig]] 05:37, 1 Dec 2004 (UTC)

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---

$e^{j\pi }=1$

some notes

Complex-conjugate multiplications with complex-swap

---

Efficient Implementation for Complex-Conjugate Multiplications with Complex-Swap

Coexistence of complex and complex-conjugate multiplications

---

Householder transformation

---

• $H=I-2v{v}^{*}$
• $v=e^{j\theta }{e}_1-w$
• $Hw=e_1$
• $H{e}_1=w$
• $||v||=1$
• $||w||=1$
• $H^{*}H=I$

• $R=\mathrm{\Theta }-W$
• $W^{*}W=I,{\mathrm{\Theta }}^{*}\mathrm{\Theta }=I.$
• ${\mathrm{\Gamma }}^{*}{R}^{*}R\mathrm{\Gamma }=I$
• $R+2R\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{R}^{*}W=0$

• $R\mathrm{\Gamma }+2R\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{R}^{*}(\mathrm{\Theta }-R)\mathrm{\Gamma }=0$
• $R\mathrm{\Gamma }+2R\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{R}^{*}\mathrm{\Theta }\mathrm{\Gamma }-2R\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{R}^{*}R\mathrm{\Gamma }=0$
• $2R\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{R}^{*}\mathrm{\Theta }\mathrm{\Gamma }=R\mathrm{\Gamma }$

• $R-2R\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{R}^{*}\mathrm{\Theta }=0$

• $R\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}{R}^{*}(W+\mathrm{\Theta })=0$

• $(\mathrm{\Theta }-W)\mathrm{\Gamma }{\mathrm{\Gamma }}^{*}(\mathrm{\Theta }-W{)}^{*}(\mathrm{\Theta }+W)=0$

• ${\mathrm{\Gamma }}^{*}({\mathrm{\Theta }}^{*}W-W^{*}\mathrm{\Theta })=0$

• It implies $\mathrm{\Gamma }$ is not full-rank. It contradicts with ${\mathrm{\Gamma }}^{*}{R}^{*}R\mathrm{\Gamma }=I$.

• Therefore, ${\mathrm{\Theta }}^{*}W=W^{*}\mathrm{\Theta }$

• Since $\mathrm{\Theta }$ or $W$ can not be Hermitian matrices, the failure of the generalization is proved.

The case of one-rank modification is the only possible one for the reflection with any desired hyperplane.

• But multiple-rank reflection transform can be used for finding the basis of the null space!

Order-recursive calculation of SVD via column-wise augmentation

---

Low-latency SVD

applications to mimo detector, steering matrix gain ...

introduction

* motivation
  * real-time or massive data application: small processing resource or high data volume.
  * column- or row-wise data insertion: cache structure or memory limitation.
  * need to update inovative column information...
* enabling ideas
  * rank-one update formula: adding new column
  * solving secular equation
  * bi-digonalization for numerical stability

approach

• order-recursive formula:

${\mathbf{𝐀}}_{n+1}=\left(\begin{array}{cc}{\mathbf{𝐀}}_n& {\mathbf{𝐜}}_{n+1}\\ \end{array}\right)$

• consider the SVD of A,,n,,is available: {{{#!latex

$$
\mathbf A_n = \mathbf U_n \mathbf \Sigma_n \mathbf V_n^*
$$

}}}

* uninary matrices can be used to obtain an almost diagonalized matrix: {{{#!latex
$$
\mathbf U_n^* \ \mathbf A_{n+1} 
\begin{pmatrix}
\mathbf V_n & \mathbf 0 \\
\mathbf 0^* & 1
\end{pmatrix}
=
\begin{pmatrix}
\mathbf \Sigma_n[1:n,1:n] &  \mathbf d_{n+1}[1:n] \\
\mathbf 0_{(m-n)\times n} &  \mathbf d_{n+1}[n+1:m] \\
\end{pmatrix}
$$
where $\mathbf d_{n+1} = \mathbf U_n^* \ \mathbf  c_{n+1}$.

}}}

* Using Householder transformation, the upper-triangular form can be obtained (tall matrix assumed.): {{{#!latex
$$
\begin{pmatrix}
\mathbf I_n & \mathbf O \\
\mathbf O   & \mathbf H_{m-n}
\end{pmatrix}
\begin{pmatrix}
\mathbf \Sigma_n[1:n,1:n] &  \mathbf d_{n+1}[1:n] \\
\mathbf 0_{(m-n)\times n} &  \mathbf d_{n+1}[n+1:m] \\
\end{pmatrix}
=
\begin{pmatrix}
\mathbf \Sigma_n[1:n,1:n] &  \mathbf d_{n+1}[1:n] \\
\mathbf 0_{n}^* &  f_{n+1} \\
\mathbf 0_{(m-n-1)\times n} &  \mathbf 0_{m-n-1} \\
\end{pmatrix}
$$
where $\mathbf d_{n+1} = \mathbf U_n^* \ \mathbf  c_{n+1}$.

}}}

* The almost diagonal matrix can be diagonalized by means of the previous approaches.
* Among them, the secular equation solving is the best for rank-one update: 
  * it leads to finding simple zeros of polynomials. 
  * linear interplation/iterations are enough.

solving secular equation

• Summary:

1. move zero sigmas right-most: column-swap
2. move up zero d's: column-and-row swap
3. make a square part by householder transforming residual d.
4. apply secular equation for the square part of dimension r-q+1.
5. merge diagonal parts and unitary matrices: singular values are not ordered for calculation speed.
6. sort the diagonal

• re-visit formula:

${\mathbf{𝐔}}_n^{*}{\mathbf{𝐀}}_{n+1}\left(\begin{array}{cc}{\mathbf{𝐕}}_n& \mathbf{𝟎}\\ {\mathbf{𝟎}}^{*}& 1\end{array}\right)=\left(\begin{array}{ccc}{\mathrm{\Sigma }}_n[1:r,1:r]& {\mathbf{𝐎}}_{r\times (n-r)}& {\mathbf{𝐝}}_{n+1}[1:r]\\ {\mathbf{𝐎}}_{(n-r)\times r}& {\mathbf{𝐎}}_{n-r}& {\mathbf{𝐝}}_{n+1}[r+1:n]\\ {\mathbf{𝐎}}_{(m-n)\times r}& {\mathbf{𝐎}}_{(m-n)\times (n-r)}& {\mathbf{𝐝}}_{n+1}[n+1:m]\\ \end{array}\right)$

where $r$ is rank of ${\mathrm{\Sigma }}_n$. Note that ${\mathbf{𝐝}}_{n+1}[1:r]$ may include zeros.

• more swapping rows and columns for zero singular values and diagonal parts.

1. move zero sigmas right-most: column-swap {{{#!latex

$$
\mathbf U_n^* \ \mathbf A_{n+1}
\begin{pmatrix}
\mathbf V_n & \mathbf 0 \\
\mathbf 0^* & 1
\end{pmatrix}
\mathbf P_{\mathbf\Sigma_n}
=
\begin{pmatrix}
\mathbf \Sigma_n[1:r,1:r]    &  \mathbf d_{n+1}[1:r]   & \mathbf O_{r\times(n-r)}\\
\mathbf O_{(n-r)\times r}   &  \mathbf d_{n+1}[r+1:n] & \mathbf O_{n-r}        \\
\mathbf O_{(m-n)\times r}    & \mathbf d_{n+1}[n+1:m]  & \mathbf O_{(m-n)\times(n-r)} \\
\end{pmatrix}
$$
where $\mathbf P_{\mathbf \Sigma_n}$ is a proper permutation matrix.

}}}

1. move up zero d's: column-and-row swap

${\mathbf{𝐏}}_{{\mathbf{𝐝}}_{n+1}}^{*}{\mathbf{𝐔}}_n^{*}{\mathbf{𝐀}}_{n+1}\left(\begin{array}{cc}{\mathbf{𝐕}}_n& \mathbf{𝟎}\\ {\mathbf{𝟎}}^{*}& 1\end{array}\right){\mathbf{𝐏}}_{{\mathrm{\Sigma }}_n}{\mathbf{𝐏}}_{{\mathbf{𝐝}}_{n+1}}=\left(\begin{array}{cccc}{\mathrm{\Sigma }}_{n,0}& {\mathbf{𝐎}}_{q\times (r-q)}& {\mathbf{𝟎}}_q& {\mathbf{𝐎}}_{q\times (n-r)}\\ {\mathbf{𝐎}}_{(r-q)\times q}& {\mathrm{\Sigma }}_{n,1}& {\mathbf{𝐟}}_{n+1}[q+1:r]& {\mathbf{𝐎}}_{(r-q)\times (n-r)}\\ {\mathbf{𝐎}}_{(m-r)\times q}& {\mathbf{𝐎}}_{(m-r)\times (r-q)}& {\mathbf{𝐝}}_{n+1}[r+1:m]& {\mathbf{𝐎}}_{(m-r)\times (n-r)}\\ \end{array}\right)$

where ${\mathbf{𝐏}}_{{\mathbf{𝐝}}_{n+1}}$ is a proper permutation matrix s.t. the non-zero elements of ${\mathbf{𝐝}}_{n+1}[1:r]$ form a new vector ${\mathbf{𝐟}}_{n+1}[q+1:r]$.

1. make a square part by householder transforming residual d. {{{#!latex

\begin{*align}
\ &
\begin{pmatrix}
\mathbf I_r & \mathbf O \\
\mathbf O   & \mathbf H_{m-r}
\end{pmatrix}
\mathbf P_{\mathbf d_{n+1}}^*
\mathbf U_n^* \ \mathbf A_{n+1}
\begin{pmatrix}
\mathbf V_n & \mathbf 0 \\
\mathbf 0^* & 1
\end{pmatrix}
\mathbf P_{\mathbf \Sigma_n}
\mathbf P_{\mathbf d_{n+1}}
\\
&=
\begin{pmatrix}
\mathbf \Sigma_{n,0}      & \mathbf O_{q\times(r-q)}   &  \mathbf 0_q                & \mathbf O_{q\times(n-r)}\\
\mathbf O_{(r-q)\times q}  & \mathbf \Sigma_{n,1}    &  \mathbf f_{n+1}[q+1:r]     & \mathbf O_{(r-q)\times(n-r)}\\
\mathbf 0_{q}^*           &  \mathbf 0_{r-q}^*}   &  -\mathbf f_{n+1}[r+1] e^{j\angle \mathbf d_{n+1}[r+1]} & \mathbf 0_{n-r}^*        \\
\mathbf O_{(m-r-1)\times q} &  \mathbf O_{(m-r-1)\times(r-q)}    & \mathbf 0_{m-r-1}  & \mathbf O_{(m-r-1)\times(n-r)} \\
\end{pmatrix}
\end{*align}
\\
where $\mathbf f_{n+1}[r+1]=||\mathbf d_{n+1}[r+1:m]||$.

}}}

  1. apply secular equation for the square part of dimension r-q+1. {{{#!latex
$$
\begin{pmatrix}
\mathbf \Sigma_{n,1}    &  \mathbf f_{n+1}[q+1:r] \\
\mathbf 0_{r-q}^*}      &  \mathbf f_{n+1}[r+1]   \\
\end{pmatrix}
=
\mathbf U_{n+\tfrac{1}{2}} \mathbf \Sigma_{n+\tfrac{1}{2}} \mathbf V_{n+\tfrac{1}{2}}^*
$$

}}}

     Note that the coefficients of the secular equation will be non-zero. It leads to easy non-generic soluation.

1. merge diagonal parts and unitary matrices: singular values are not ordered for calculation speed.

${\mathbf{𝐀}}_{n+1}={\mathbf{𝐔}}_{n+1}\left(\begin{array}{c}{\mathrm{\Sigma }}_{n+1}\\ {\mathbf{𝐎}}_{(m-n-1)\times (n+1)}\end{array}\right){\mathbf{𝐕}}_{n+1}^{*}$

where the unordered diagonal matrix is

${\mathrm{\Sigma }}_{n+1}=\left(\begin{array}{cc}\begin{array}{cc}{\mathrm{\Sigma }}_{n,0}& {\mathbf{𝐎}}_{q\times (r-q+1)}\\ {\mathbf{𝐎}}_{(r-q+1)\times q}& {\mathrm{\Sigma }}_{n+\frac{1}{2}}\\ \end{array}|& {\mathbf{𝐎}}_{(n+1)\times (n-r)}\end{array}\right)$

and the unitary matrices are calculated by multiplying the intermediate unitary matrices:

${\mathbf{𝐔}}_{n+1}={\mathbf{𝐔}}_n\left(\begin{array}{cc}{\mathbf{𝐈}}_r& \mathbf{𝐎}\\ \mathbf{𝐎}& {\mathbf{𝐇}}_{m-r}\end{array}\right){\mathbf{𝐏}}_{{\mathbf{𝐝}}_{n+1}}\left(\begin{array}{ccc}{\mathbf{𝐈}}_r& \mathbf{𝟎}& \mathbf{𝐎}\\ {\mathbf{𝟎}}^{*}& -e^{j\mathrm{\angle }{\mathbf{𝐝}}_{n+1}[n+1]}& {\mathbf{𝟎}}^{*}\\ \mathbf{𝐎}& \mathbf{𝟎}& {\mathbf{𝐈}}_{m-r-1}\\ \end{array}\right)\left(\begin{array}{ccc}{\mathbf{𝐈}}_q& \mathbf{𝐎}& \mathbf{𝐎}\\ \mathbf{𝐎}& {\mathbf{𝐔}}_{n+\frac{1}{2}}& {\mathbf{𝐎}}_{(r-q+1)\times (m-r-1)}\\ \mathbf{𝐎}& {\mathbf{𝐎}}_{(m-r-1)\times (r-q+1)}& {\mathbf{𝐈}}_{m-r-1}\end{array}\right)$

and

${\mathbf{𝐕}}_{n+1}=\left(\begin{array}{cc}{\mathbf{𝐕}}_n& \mathbf{𝟎}\\ {\mathbf{𝟎}}^{*}& 1\end{array}\right){\mathbf{𝐏}}_{{\mathrm{\Sigma }}_n}{\mathbf{𝐏}}_{{\mathbf{𝐝}}_{n+1}}\left(\begin{array}{ccc}{\mathbf{𝐈}}_q& \mathbf{𝐎}& \mathbf{𝐎}\\ \mathbf{𝐎}& {\mathbf{𝐕}}_{n+\frac{1}{2}}& \mathbf{𝐎}\\ \mathbf{𝐎}& \mathbf{𝐎}& {\mathbf{𝐈}}_{n-r-1}\\ \end{array}\right).$

1. sort the diagonal

example

* mx2 case
  * formula: {{{#!latex
$$
\mathbf A_{2} =
\begin{pmatrix}
\mathbf a_1 & \mathbf  c_{2} \\
\end{pmatrix}
$$

}}}

  * trivial SVD of a,,1,,: {{{#!latex
$$
\mathbf a_1 = \mathbf u_1 \cdot \sigma_1 \cdot 1
$$

}}}

  * almost digonalization: {{{#!latex
$$
\mathbf U_n^* \ \mathbf A_{2} 
\begin{pmatrix}
          1 & 0 \\
          0 & 1
\end{pmatrix}
=
\begin{pmatrix}
\mathbf \sigma_1 &  d_{1} \\
\mathbf 0_{m-1}        &  \mathbf d[2:m] 
\end{pmatrix}
$$
where $d_{1} = \mathbf u_1^* \ \mathbf  c_{2}$ and 

$\mathbf d[2:m] = \mathbf U_1[2:m]^* \ \mathbf c_{2}$. }}}

  * upper-triangular form is good for numerical stability and compact calculation as well:
* mx3 case
* mx4 case

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