2. HB Formulation Based on the Cubic Spline Wavelets and Daubechies Wavelets
2.1. Generalized HB Formulation
The harmonic balance (HB) method is a powerful technique for the analysis of high-frequency nonlinear circuits such as mixers, power amplifiers, and oscillators. The basic idea of HB is to expand the unknown state variable
x
(
t
)
in electrical circuit equations by some series
x
(
t
)
=
∑
X
k
v
k
(
t
)
. Then the problem is transformed into the frequency domain focusing on the coefficients
X
k
.
Let us consider the general approach of HB which assumes obtaining the solution
x
(
t
)
of the nonlinear modified nodal analysis (MNA) equation in [8]
(1)
C
x
˙
+
G
x
+
f
(
x
)
+
u
=
0
,
which satisfies the following periodical boundary condition:
(2)
x
(
t
+
L
)
=
x
(
t
)
,
where
C
and
G
are
N
x
×
N
x
matrices,
x
is a
N
x
dimensional column vector of unknown circuit variables, and
u
is a
N
x
dimensional column vector of independent sources. Let
{
v
k
}
be the basis; then the unknown function
x
(
t
)
can be expanded
x
(
t
)
=
∑
X
k
v
k
(
t
)
. To solve (1) with periodic boundary condition (2), assume that the expansion basis is periodic with period
τ
and
[
x
l
]
is a discrete vector containing values of
x
(
t
)
sampled in the time domain at time points
[
t
l
]
,
l
=
1
,
…
,
N
t
. Then (1) can be written in the transform domain as a nonlinear algebraic equation system:
(3)
Φ
(
X
)
=
(
C
^
D
+
G
^
)
X
+
F
(
X
)
+
U
=
0
,
where
(4)
X
=
T
x
,
x
=
T
-
1
X
,
U
=
T
u
,
C
^
,
D
, and
G
^
are
N
t
N
x
×
N
t
N
x
matrices, especially, the matrix
D
is a representation matrix of the derivative operator
d
/
d
t
in expansion basis
{
v
i
}
(5)
[
D
i
,
j
]
=
〈
d
d
t
v
i
,
v
j
〉
,
and, finally,
T
and
T
-
1
are the matrices associated with the forward and inverse transform arising from the chosen expansion basis. The nonlinear matrix system (3) can be solved by Newton iterative method
(6)
J
(
X
(
i
)
)
(
X
(
i
+
1
)
-
X
(
i
)
)
=
-
Φ
(
X
(
i
)
)
,
where
X
(
i
)
is the solution of the
i
th iteration and
J
(
X
)
is the Jacobian matrix of
Φ
(
X
)
(7)
J
(
X
)
=
[
J
k
l
(
X
)
]
=
[
∂
Φ
k
∂
X
l
]
=
∂
Φ
∂
X
=
C
^
D
+
G
^
+
T
[
∂
f
k
∂
x
l
]
T
-
1
,
l
l
l
l
l
l
l
m
k
,
l
=
1
,
…
,
(
N
t
N
x
)
.
Hence, the sparsity of this Jacobian matrix
J
(
X
)
affects the computational cost of iterative method. Because these matrices
C
^
and
G
^
have a rather sparse structure due to the MNA formulation and
[
(
∂
f
k
)
/
(
∂
x
l
)
]
for time-invariant systems is just a block matrix consisting of diagonal blocks, the sparsity of the Jacobian matrix
J
(
X
)
is determined by three matrices
T
,
T
-
1
, and the representation matrix
D
of the differential operator
d
/
d
t
.
Given the base
{
v
k
}
k
=
1
N
, the matrices
D
,
T
, and
T
-
1
are constructed before those iterative methods are used. So the sparsity of the Jacobian matrix based on these different basis functions indicates how to solve the nonlinear algebraic system. Next, we give the formulation for two kinds of wavelet bases.
2.2. Description of the Periodic Daubechies Wavelets
Two functions
ψ
and
ϕ
are the wavelet function and its corresponding scaling function described by Daubechies [9]. They are defined in the frame of the wavelet theory and can be constructed with finite spatial support under the following conditions:
(8)
ψ
(
t
)
=
2
∑
k
=
0
M
-
1
g
k
+
1
ϕ
(
2
t
-
k
)
,
ϕ
(
t
)
=
2
∑
k
=
0
M
-
1
h
k
+
1
ϕ
(
2
t
-
k
)
,
∫
-
∞
+
∞
ϕ
(
t
)
d
t
=
1
,
where the coefficients
{
h
k
}
k
=
0
M
-
1
and
{
g
k
}
k
=
0
M
-
1
are the quadrature mirror filters (QMFs) of length
L
M
. The quadrature mirror filters
{
h
k
}
and
{
g
k
}
are defined
(9)
g
k
=
(
-
1
)
k
h
M
-
k
-
1
,
k
=
0,1
,
…
,
M
-
1
.
The function
ψ
has
p
vanishing moments; that is,
(10)
∫
-
∞
∞
ψ
(
t
)
t
m
d
t
=
0
,
0
≤
m
≤
p
-
1
.
The number
M
of the filter coefficients is related to the number of vanishing moments
p
, and
M
=
2
p
for the wavelets constructed in [9].
We observe that once the filter
{
h
k
}
has been chosen, the functions
ϕ
and
ψ
can be confirmed. Moreover, due to the recursive definition of the wavelet bases, via the two-scale equation, all of the manipulations are performed with the quadrature mirror filters
{
h
k
}
and
{
g
k
}
. Especially, the wavelet transform matrix
T
and the derivative matrix
D
for the differential operator
d
/
d
t
can be obtained by the filters.
In HB method the wavelets on the interval
[
0
,
L
]
are required. Hence, periodic Daubechies wavelets on the interval
[
0
,
L
]
are constructed by periodization. Here, we describe the discrete wavelet transform matrix by the periodic Daubechies wavelets. The discrete wavelet transform with the period
L
=
2
n
can be considered as a linear transformation taking the vector
f
J
∈
V
J
determined by its sampling data into the vector
(11)
d
=
(
c
0
,
d
0
,
d
1
,
d
2
,
d
3
,
…
,
d
J
-
1
)
T
,
where
c
j
stands for the scaling coefficients of the function
x
(
t
)
and
d
j
for the wavelet coefficients.
This linear transform can be represented by the
N
=
2
n
dimensional matrix
T
Daube
such that
(12)
T
Daube
f
J
=
d
.
If the level of the DWT is
J
≤
n
, then the DWT of the sequence has exactly
2
n
coefficients. The transform matrix
T
Daube
is composed of QMFs coefficients
{
h
k
}
and
{
g
k
}
as follows:
(13)
T
Daube
=
(
h
0
h
1
⋯
h
M
-
1
0
⋯
0
⋯
0
g
0
g
1
⋯
g
M
-
1
0
⋯
0
⋯
0
0
0
h
0
h
1
⋯
h
M
-
1
0
⋯
0
0
0
g
0
g
1
⋯
g
M
-
1
0
⋯
0
⋯
h
2
h
3
⋯
h
M
-
1
⋯
⋯
0
h
1
h
2
g
2
g
3
⋯
g
M
-
1
⋯
⋯
0
g
1
g
2
)
,
where
M
is the length of the filters.
The periodized Daubechies wavelet HB formulation has been formulated in [4], so we have
D
Daube
=
T
Daube
R
T
Daube
-
1
, where
R
is a band limited circulant matrix with its diagonals filled by
r
m
in [10], where with the following properties:
(14)
r
m
≠
0
,
for
-
M
+
2
≤
m
≤
M
-
2
,
r
0
=
0
,
r
-
m
=
-
r
m
,
∑
m
m
r
m
=
-
1
,
r
m
=
2
[
r
2
m
+
1
2
∑
k
=
1
M
/
2
a
2
k
-
1
(
r
2
m
-
2
k
+
1
+
r
2
m
+
2
k
-
1
)
]
,
in which
a
i
are autocorrelation coefficients of the QMFs
(15)
a
i
=
2
∑
m
=
0
M
-
i
-
1
h
~
m
h
m
+
1
,
i
=
1
,
…
,
M
-
1
.
And the matrix
T
Daube
-
1
is the inverse matrix of the forward transform matrix
T
Daube
which satisfies
T
Daube
-
1
=
T
Daube
T
due to the orthogonality of the matrix
T
Daube
.
2.3. The Cubic Spline Wavelet Basis
Consider the cubic spline wavelets as the expansion base in HB technique. The cubic spline wavelets are constructed in [7], which are semiorthogonal wavelets. The high approximation rate and the interpolation property can be inherited from spline functions. Therefore, the cubic spline wavelet transform matrix
T
cubic
and the differential operator representation matrix
D
cubic
have the following properties which are suitable for HB method.
Due to the periodic condition
x
(
t
+
L
)
=
x
(
t
)
, we must use the periodization functions of the cubic spline wavelets on the interval
[
0
,
L
]
,
L
>
4
. For convenience, we still denote by
v
i
(
t
)
the periodic function. Let us assume that expansion bases are
(16)
{
v
i
}
i
=
1
N
s
=
{
i
2
ϕ
0
,
-
1
,
ϕ
0
,
k
(
0
≤
k
≤
L
-
4
)
,
ϕ
0
,
L
-
3
,
m
ψ
j
,
k
(
0
≤
j
≤
J
-
1
,
-
1
≤
k
≤
n
j
-
2
)
}
,
where
n
j
=
2
j
L
,
N
s
=
2
J
L
-
1
. Correspondingly, the unknown state variable
x
(
t
)
is approximated by the bases of these spaces
(17)
V
J
=
V
J
-
1
⊕
W
J
-
1
⋮
=
V
0
⊕
W
0
⊕
⋯
⊕
W
J
-
1
,
where
(18)
V
0
=
span
{
ϕ
-
1
,
-
1
(
t
)
,
…
,
ϕ
-
1
,
L
-
4
(
t
)
,
ϕ
-
1
,
L
-
3
(
L
-
t
)
}
,
W
i
=
span
{
ψ
i
,
-
1
(
t
)
,
ψ
i
,
0
(
t
)
,
…
,
ψ
i
,
n
i
-
2
(
t
)
}
,
ψ
i
,
-
1
(
t
)
,
ψ
i
,
0
(
t
)
,
…
iiiiiiiiiiii
,
0
≤
i
≤
J
-
1
.
Based on the interpolation property of the cubic spline wavelets, we have
(19)
P
V
J
x
(
t
)
=
I
V
b
x
(
t
)
+
∑
j
=
0
J
-
1
I
W
j
x
(
t
)
=
x
^
-
1
,
-
3
η
1
(
t
)
+
x
^
-
1
,
-
2
η
2
(
t
)
+
x
^
-
1
,
-
1
ϕ
b
(
t
)
+
∑
k
=
0
L
-
4
x
^
-
1
,
k
ϕ
k
(
t
)
+
x
^
-
1
,
L
-
3
ϕ
b
(
L
-
t
)
+
x
^
-
1
,
L
-
2
η
2
(
L
-
t
)
+
x
^
-
1
,
L
-
1
η
1
(
L
-
t
)
+
∑
j
=
0
J
-
1
[
∑
k
=
-
1
n
j
-
2
x
^
j
,
k
ψ
j
,
k
(
t
)
]
.
Denote the expansion coefficients by a
N
s
×
1
dimensional vector
x
^
J
,
(20)
x
^
J
=
(
x
^
-
1
,
-
3
,
…
,
x
^
-
1
,
L
-
1
,
x
^
0
,
-
1
,
…
,
x
^
0
,
n
0
-
2
,
…
,
x
^
J
-
1
,
-
1
,
m
…
,
x
^
J
-
1
,
k
,
…
,
x
^
J
-
1
,
n
J
-
2
)
T
,
that will be determined by satisfying the collocation conditions,
N
s
=
2
J
L
+
3
. Interpolate
P
V
J
at the collocation points
(21)
{
t
1
(
-
1
)
=
0
,
t
2
(
-
1
)
=
1
2
,
m
t
k
(
-
1
)
=
k
-
2
,
k
=
3
,
…
,
L
+
1
;
m
t
L
+
2
(
-
1
)
=
L
-
1
2
,
t
L
+
3
(
-
1
)
=
L
}
,
{
t
-
1
(
j
)
=
1
2
j
+
2
,
t
k
(
j
)
=
k
+
1.5
2
j
,
0
≤
k
≤
n
j
-
3
,
m
t
n
j
-
2
(
j
)
=
L
-
1
2
j
+
2
}
,
as follows:
(22)
P
V
J
x
(
t
k
-
1
)
=
x
(
t
k
-
1
)
,
1
≤
k
≤
L
+
3
,
P
V
J
x
(
t
k
j
)
=
x
(
t
k
j
)
,
j
≥
0
,
-
1
≤
k
≤
n
j
-
2
,
0
≤
j
≤
J
-
1
.
Substituting the expressions into (1), we obtain nonlinear discrete algebraic systems.
Denote by
T
cubic
the cubic spline wavelet transform matrix. We introduce an inverse wavelet transform (IWT)
T
cubic
-
1
which maps its wavelet coefficients
x
^
J
to discrete sample values
f
J
with length
N
s
; that is
T
cubic
-
1
x
^
J
=
f
J
. The inverse transform matrix
T
cubic
-
1
is
(23)
T
cubic
-
1
=
(
B
M
0
M
1
⋮
M
J
-
1
)
,
where
B
denotes a tridiagonal matrix with dimension
L
+
2
and
M
j
is a tridiagonal matrix with dimension
2
j
L
.
We obtain the derivative matrix
D
cubic
in [11] as follows:
(24)
D
cubic
=
H
1
-
1
H
2
,
where
(25)
H
1
=
[
λ
1
1
λ
1
2
μ
1
λ
2
2
μ
2
·
·
·
λ
i
2
μ
i
·
·
·
λ
N
s
-
1
2
μ
N
s
-
1
1
μ
N
s
-
1
]
(
N
s
+
1
)
×
(
N
s
+
1
)
,
H
2
=
[
a
1
a
2
a
3
c
1
d
1
e
1
·
·
·
·
·
·
c
i
d
i
e
i
·
·
·
c
N
s
-
1
d
N
s
-
1
e
N
s
-
1
b
3
b
2
b
1
]
(
N
s
+
1
)
×
(
N
s
+
1
)
,
and these constants in these matrices
H
1
and
H
2
can be referenced from the formulae (2.20a)–(2.20d) in [11].
For the whole nonlinear equation system
(26)
(
C
^
H
1
-
1
H
2
+
G
^
+
T
cubic
[
∂
f
k
∂
x
l
]
T
cubic
-
1
)
(
X
(
i
+
1
)
-
X
(
i
)
)
=
-
Φ
(
X
(
i
)
)
,
where
H
1
,
H
2
, and
T
cubic
-
1
are tridiagonal matrices, the triangular decomposition of the tridiagonal matrix can be used to decompose the Jacobian iterative matrix.