1.3 Special Function Matrices ●Exchange matrices: 1 J= 0 Role:Reverse all rows/columns of a matrix A Examples: 1 am aml am2 amn 2 Jm 二 .: JmA= 2 021 22 a2n Am 1 411 12 ain. (1)J=J (2)P=I 1 Special Matrices 16/60
1.3 Special Function Matrices Exchange matrices: J = 0 1 1 . . . 1 0 Role：Reverse all rows/columns of a matrix A Examples: Jm a1 a2 . . . am = am . . . a2 a1 JmA = am1 am2 · · · amn . . . . . . . . . . . . a21 a22 · · · a2n a11 a12 · · · a1n (1) J T = J (2) J 2 = I 1 Special Matrices 16 / 60
1.3 Special Function Matrices oShift matrices: 01 0 07 0 0 1 0 ... ... 0 1 L10 0 0 Role:Cyclic shift,i.e.,r←-r2,r2←s,…Tm-1← Tn,Tn←-n Examples:Apply the m x m shift matrix Pm to A: 21 022 a23 a2n a31 a32 a33 a3n PA= aml am2 am3 amn a11 a12 a13 01n 1 Special Matrices 17/60
1.3 Special Function Matrices Shift matrices: P = 0 1 0 · · · 0 0 0 1 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · 1 1 0 0 · · · 0 Role: Cyclic shift, i.e., r1 ← r2, r2 ← r3, · · · rn−1 ← rn, rn ← r1. Examples: Apply the m × m shift matrix Pm to A: PA = a21 a22 a23 · · · a2n a31 a32 a33 · · · a3n . . . . . . . . . . . . am1 am2 am3 · · · amn a11 a12 a13 · · · a1n 1 Special Matrices 17 / 60
1.3 Special Function Matrices o Generalized permutation matrices:For each row and each column of a square matrix,only one nonzero element exists. A generalized permutation matrix G can de separated into two parts:a permutation matrix P and a nonsingular di- agonal matrix D:G=PD. Example: TO 0 0 0 a TO 1 0 0 17 0 0060 0 0 0 1 0 0 Y G= 0 0 0 0 0 1 0 0 0 B 0 0 0 入 0 0 0 0 1 0 LP 000 0 11 0 00 0 BBS Application: Model x=As(=∑ aisi(t) where s(t)=[s1(t),s2(t),...,sn(t)]is the source vector, Amxn is the mixed matrix. 1 Special Matrices 18/60
1.3 Special Function Matrices Generalized permutation matrices: For each row and each column of a square matrix, only one nonzero element exists. A generalized permutation matrix G can de separated into two parts: a permutation matrix P and a nonsingular diagonal matrix D: G = PD. Example: G = 0 0 0 0 α 0 0 β 0 0 0 γ 0 0 0 0 0 0 λ 0 ρ 0 0 0 0 = 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 ρ 0 γ β λ 0 α BBS Application: Model x(t) = As(t) = Pn i=1 aisi(t) where s(t) = [s1(t),s2(t), · · · ,sn(t)]T is the source vector, Am×n is the mixed matrix. 1 Special Matrices 18 / 60
1.3 Special Function Matrices Problem:Given data(t),t=l,2,·（A unknown), recover the n-dimensional signal vector s(t). Ideal BSS:3(t)=Afx(t)(cannot be done in general) Practical BSS:Equivalent data model 0-含as0-含2a0 indicating that BBS has two uncertainties: (1)Order uncertainty of sources (2)Uncertainty of recovering sources From the viewpoint of recovering waveforms of sources,the uncertainties are permissible.So we need not the exact general inverse matrix Af,but a general mixed matrix GAT instead, 3(t)=PDAix(t)=(GAi)x(t) to propose practical BBS algorithms. 1 Special Matrices 19/60
1.3 Special Function Matrices Problem: Given data x(t), t = 1, 2, · · · (A unknown)， recover the n-dimensional signal vector s(t). Ideal BSS:ˆs(t) = A†x(t) (cannot be done in general) Practical BSS: Equivalent data model x(t) = Pn i=1 aisi(t) = Pn i=1 ai αi αisi(t) indicating that BBS has two uncertainties: (1) Order uncertainty of sources (2) Uncertainty of recovering sources From the viewpoint of recovering waveforms of sources, the uncertainties are permissible. So we need not the exact general inverse matrix A† , but a general mixed matrix GA† instead, ˆs(t) = PDA†x(t) = (GA† )x(t) to propose practical BBS algorithms. 1 Special Matrices 19 / 60
1.3 Special Function Matrices o Selective matrices:Take special elements of a vector or matrix. Example:For an m x n matrix 「1(1)(2) z1(n) X= 2(1) 2(2) z1(n) +. zm(1) 工m(2) Im(n)]mxn If J1=[Im-1,0mx1],J2= [0mx1;Im-1],then 1(1) x1(2) z1(n) JX= 2(1) 2(2) z1(n) Lm-1(1) xm-1(2) xm-1(mJ(m-1)×n 2(1) 2(2) 2(n)7 路(1) 3(2) 3(n) JX= m(1) xm(2) zm(J(m-1)×n are the fist and last m-1 rows of X,respectively. 1 Special Matrices 20/60
1.3 Special Function Matrices Selective matrices: Take special elements of a vector or matrix. Example: For an m × n matrix X = x1(1) x1(2) · · · x1(n) x2(1) x2(2) · · · x1(n) . . . . . . . . . xm(1) xm(2) · · · xm(n) m×n If J1 = [Im−1, 0m×1] , J2 = [0m×1, Im−1], then J1X = x1(1) x1(2) · · · x1(n) x2(1) x2(2) · · · x1(n) . . . . . . . . . xm−1(1) xm−1(2) · · · xm−1(n) (m−1)×n J2X = x2(1) x2(2) · · · x2(n) x3(1) x3(2) · · · x3(n) . . . . . . . . . xm(1) xm(2) · · · xm(n) (m−1)×n are the fist and last m − 1 rows of X, respectively. 1 Special Matrices 20 / 60