B Addition and multiplication of complex numbers
B.1 Addition of two complex numbers
If two complex numbers are given in Cartesian form, then their sum can be calculated through adding the real and imaginary parts separately as follows
(m1+jn1) + (m2+jn2) = (m1+m2) +j(n1+n2). (B6)
m1
+ jn1
m2
+ jn2
+
+
m1+m2
j(n1+n2)
Figure B1: Addition of two complex numbers in Cartesian form
B.2.3 Solution in exponential form
To calculate the product of two complex numbers in exponential form, it is enough to use an adder and a multiplier as follows
m1ejφ1·m2ejφ2 =m1m2ej(φ1+φ2). (B12)
m1
+ jn1
m2
+ jn2
×
×
×
×
+
+
m1m2−n1n2
j(m1n2+m2n1)
Figure B2: Multiplication of two complex numbers in Cartesian form
m1
n1
m2
n2
+
+
×
×
×
+
+
m1m2−n1n2
m1n2+m2n1
−
−
−
Figure B3: Multiplication of two complex numbers by Karatsuba-algorithm
m1
φ1
m2
φ2
×
+
m1m2
φ1+φ2
Figure B4: Multiplication of two complex numbers in exponential form
C Inverse of a hypermatrix with 4 submatrices
In this section the inverse of a matrixMwith four submatrices is derived. The following linear equation system is considered for formulating the problem
Mq=r, (C13)
which can be further rewritten with the splitting ofM in four submatrices as M11 M12
M21 M22
! q1
q2
!
= r1
r2
!
. (C14)
Assuming that the matrix M22 is regular, and the diagonal elements are quadratic matrices, the second row of the equation can multiplied by−M12M−122, and than it is added to the rst row. These steps can be formulated using a transformation matrixTthat is dened as
T= I11 −M12M−122 0 I22
!
. (C15)
Both sides of the equation system presented in (C13) are multiplied byT; thus, the resulting equation is formulated as
TMq=Tr. (C16)
The modied equation system using the transformed and reduced matrixMr=TM which is a lower triangular Toeplitz matrix can be given as
Mrq= M11,r 0 M21,r M22,r
! q1
q2
!
= r1,r
r2
!
(C17)
where
M11,r =M11−M12M−122M21, (C18)
M21,r =M21, (C19)
M22,r =M22, (C20)
r1,r =r1−M12M22r2. (C21)
The submartixM11,r is also known as Schur complement of the blockM22. As a further step, the inverse of the reduced matrixMr denoted byKr can be expressed as
M−1r =Kr= K11,r 0 K21,r K22,r
!
(C22)
where
K11,r=M−111,r, (C23)
K21,r=−M−122M21M−111, (C24)
K22,r=M−122. (C25)
Finally, the inverse of the matrixM denoted byK can be expressed with the matricesTandMras
K=M−1= (T−1Mr)−1=M−1r T= M−111,r 0
−M−122M21M−111,r M−122
! I11 −M12M−122 0 I22
!
= K11 K12
K12 K22
!
(C26) where the elements of the matrixK can be expressed with the aid of (C15) and (C22) as [92]
K11=M−111,r, (C27)
K12=−M−111,rM12M−122, (C28)
K21=−M−122M21M−111,r, (C29)
K22=M−122 +M−122M21M−111,rM12M−122. (C30)
D The DurbinLevinson-algorithm
Iterative algorithm
Prediction χ[n]
ξ[n] + + ε[n]
+ x[n]
h[k] −
y[n]
Figure D5: Block diagram of lter design by DurbinLevinson-algorithm
The DurbinLevinson-algorithm was introduced in 1946 to determine iterative the weighting function of a linear lter. Later, this procedure was used by Durbin in 1960 to solve the one-dimensional Yule-Walker equations [93, 94, 95].
Suppose that the observed signalx[n]is a noisy measurement of χ[n]
x[n] =χ[n] +ξ[n] (D31)
while the original signalχ[n]is predicted from the observation as follows
y[n] =
K−1
X
k=0
x[n−k]·h[k]. (D32)
The error of the predictionε[n]can be written as
ε[n] =χ[n]−y[n] (D33)
that has to be minimised in mean squared error sense; thus, using (D32) and (D33), the mean square errorσM2 [K]has this form
σM2 [K] = E{ε[n]}= E (
χ[n]−
K−1
X
k=0
x[n−k]·h[k]
)
. (D34)
Let evaluate the derivative ofσM2 with respect to h[k]
∂σ2M[K]
∂h[k] =−2Rx[n] + 2
K−1
X
k=0
h[k]Rx[k−n] (D35)
whereRxdenotes the auto-correlation function ofx[n]. To nd the optimal settings, the following equa-tion has to be solved
K−1
X
k=0
h[k]Rx[k−n] =Rx[n]. (D36)
Let denote one set of solution forh[k]as(αK[0], αK[1], . . . , αK[k−1])whereKdenotes in subscript that the size of set depends on the current value ofK. Using this, (D36) can be substituted as
K−1
X
k=0
αK[k]Rx[k−n] =Rx[n]. (D37)
Levinson's iterative procedure start with an auxiliary sequenceβK[k][95]
β0[0] = Rx[1]
Rx[0], (D38)
βK[0] =Rx[0]−
K−1
X
k=0
βK−1[k]Rx[K−k], (D39)
βK[k] =βK−1[k−1] =βK[0]βK−1[K−k]. (D40) From (D38)-(D40), the solutions forαK[k]can be obtained
α0[0] =Rx[0], (D41)
αK+1[K+ 1] =Rx[0]−
K−1
X
k=0
βK[k]Rx[K+ 1−k], (D42) αK+1[k] =αK[k]−βK[k]αK+1[K+ 0]. (D43) After having an optimal settingsαK[k]for coecients, the inverse auto-correlation matrixR[n]can be constructed through the Gohberg-Semencul formula for lter sizeK [96]
R−1[n] = 1 σ2M[K]
1 0 . . . 0
αK[0] 1 . . . 0 ... ... ... ...
αK[k−1] . . . αK[0] 1
1 αK[0] . . . αK[k−1]
0 1 . . . αK[k−2]
... ... ... ...
0 0 . . . 1
− (D44)
− 1 σM2 [K]
0 0 . . . 0
αK[k−1] 0 . . . 0
... ... ... ...
αK[0] . . . αK[k−1] 0
0 αK[k−1] . . . αK[0]
0 0 . . . αK[1]
... ... ... ...
0 0 . . . 0
. (D45)
E Hermitian transpose of the auto-correlation matrix
In this section, the inverse of a complex-valued auto-correlation matrix is investigated to derivate that it is Hermitian; therefore, its elements of the main diagonal are real-valued. Firstly, it is shown that the auto-correlation matrix is Hermitian, then it is demonstrated that its inverse is also Hermitian.
E.1 Hermitian auto-correlation matrix
As is it given in (43), the auto-correlation matrix of a complex-valued matrix M can be calculated in the following form
R=MHM. (E46)
E.1.1 Using matrix multiplication
The matrix Mcontainsprows,qcolumns, and the following elements
M=
m11+jn11 m12+jn12 . . . m1q+jn1q
m21+jn21 m22+jn22 . . . m2q+jn2q
... ... ... ...
mp1+jnp1 mp2+jnp2 . . . mpq+jnpq
. (E47)
After Hermitian transposing ofM, the auto-correlation matrix can be expressed from (E46) as
R=MHM= (E48)
=
m11−jn11 m21−jn21 . . . mp1−jnp1
m12−jn12 m22−jn22 . . . mp2−jnp2
... ... ... ...
m1q−jn1q m2q−jn2q . . . mpq−jnpq
m11+jn11 m12+jn12 . . . m1q+jn1q
m21+jn21 m22+jn22 . . . m2q+jn2q
... ... ... ...
mp1+jnp1 mp2+jnp2 . . . mpq+jnpq
=
=
p
P
i=0
(mi1−jni1) (mi1+jni1)
p
P
i=0
(mi1−jni1) (mi2+jni2) . . .
p
P
i=0
(mi1−jni1) (miq+jniq)
p
P
i=0
(mi2−jni2) (mi1+jni1)
p
P
i=0
(mi2−jni2) (mi2+jni2) . . .
p
P
i=0
(mi2−jni2) (miq+jniq)
... ... ... ...
p
P
i=0
(miq−jniq) (mi1+jni1)
p
P
i=0
(miq−jniq) (mi2+jni2) . . .
p
P
i=0
(miq−jniq) (miq+jniq)
=
=
p
P
i=0
m2i1+n2i1
Pp
i=0
(mi1mi2+ni1ni2+j(mi1ni2−mi2ni1)) . . .
p
P
i=0
(mi1mi2+ni1ni2−j(mi1ni2−mi2ni1))
p
P
i=0
m2i2+n2i2
. . .
... ... ...
.
Investigating the result of (E48),
• there are the sums of squares of the real and imaginary components in the main diagonal without imaginary part,
• while the other elements are complex conjugate to each with respect to the main diagonal;
therefore,Rij=Rji=⇒R=RH; thus, the auto-correlation matrix is Hermitian.
E.1.2 Using the denition of the auto-correlation matrix
The auto-correlation matrix of a complex-valued vector m = (m1+jn1, m2+jn2, . . . , mp+jnp)is the following
R= E{mmH}= (E49)
=
E{(m1+jn1) (m1+jn1)} E{(m1+jn1) (m2+jn2)} . . . E{(m1+jn1) (mp+jnp)}
E{(m2+jn2) (m1+jn1)} E{(m2+jn2) (m2+jn2)} . . . E{(m2+jn2) (mp+jnp)}
... ... ... ...
E{(mp+jnp) (m1+jn1)} E{(mp+jnp) (m2+jn2)} . . . E{(mp+jnp) (mp+jnp)}
=
=
E{m21+n21} E{m1m2+n1n2−j(m1n2−m2n1)} . . . E{m1m2+n1n2+j(m1n2−m2n1)} E{m22+n22} . . .
... ... ...
.
As in Sec. E.1.1, the expected values of the main diagonal are purely real, while the other elements are complex conjugate to each with respect to the main diagonal; therefore, the auto-correlation matrix is Hermitian.
E.2 Inverse of a Hermitian matrix
As it is shown in Appendix E.1, the auto-correlation matrix R is Hermitian. If R−1R = I, then IH = R−1RH
= RH R−1H
. Considering the fact that the identity matrix is Hermitian as well, R−1= R−1H satises.
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