1458 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 10, OCTOBER 2011
Synthesis of Integrated Passive Components for
High-Frequency RF ICs Based on Evolutionary
Computation and Machine Learning Techniques
Bo Liu, Dixian Zhao, Patrick Reynaert, and Georges G. E. Gielen, Fellow, IEEE
Abstract—State-of-the-art synthesis methods for microwave
passive components suffer from the following drawbacks. They ei-
ther have good efficiency but highly depend on the accuracy of the
equivalent circuit models, which may fail the synthesis when the
frequency is high, or they fully depend on electromagnetic (EM)
simulations, with a high solution quality but are too time consum-
ing. To address the problem of combining high solution quality
and good efficiency, a new method, called memetic machine
learning-based differential evolution (MMLDE), is presented. The
key idea of MMLDE is the proposed online surrogate model-
based memetic evolutionary optimization mechanism, whose
training data are generated adaptively in the optimization pro-
cess. In particular, by using the differential evolution algorithm as
the optimization kernel and EM simulation as the performance
evaluation method, high-quality solutions can be obtained. By
using Gaussian process and artificial neural network in the
proposed search mechanism, surrogate models are constructed
online to predict the performances, saving a lot of expensive
EM simulations. Compared with available methods with the best
solution quality, MMLDE can obtain comparable results, and has
approximately a tenfold improvement in computational efficiency,
which makes the computational time for optimized component
synthesis acceptable. Moreover, unlike many available methods,
MMLDE does not need any equivalent circuit models or any
coarse-mesh EM models. Experiments of 60 GHz syntheses and
comparisons with the state-of-art methods provide evidence of
the important advantages of MMLDE.
Index Terms—Artificial neural network, differential evolution,
gaussian process, inductor synthesis, microwave components,
surrogate model, transformer synthesis.
I. Introduction
I
N RECENT years, design methodologies for high-
frequency microwave circuits have attracted a lot of
attention. In particular, research on RF building blocks for
40 GHz to 120 GHz and beyond is increasing drastically.
On-chip passive components, e.g., inductors and transformers,
are one of the major components of the RF IC that strongly
influence the circuit performances [1]. For example, the loss
Manuscript received March 4, 2011; revised May 6, 2011; accepted June
17, 2011. Date of current version September 21, 2011. This work was
supported by a bilateral agreement scholarship of Katholieke Universiteit
Leuven, Leuven, Belgium, and Tsinghua University, Beijing, China. This
paper was recommended by Associate Editor H. E. Graeb.
The authors are with Katholieke Universiteit Leuven, Leuven 3000,
Belgium (e-mail: bo.liu@esat.kuleuven.be; dixian.zhao@esat.kuleuven.be;
patrick.reynaert@esat.kuleuven.be; georges.gielen@esat.kuleuven.be; liu
bo
765@yahoo.com.cn).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCAD.2011.2162067
of a transformer has a large impact on the power-added
efficiency and the output power of a power amplifier.
Therefore, the synthesis of passive components, including
both the sizing and the layout optimization, is a critical
problem in high-frequency RF IC design automation. High-
frequency RF component synthesis faces two challenges.
First, accurate equivalent circuit models are often not available
in literatures at these frequencies. Some designers rely on
experience and simulation verification in their design work.
Another challenge is that the performance requirements of RF
ICs keep on increasing, and therefore powerful optimization
methods are needed. Hence, the “experience and trial” method
or local optimization is often not good enough for high-
frequency RF component design. This paper focuses on these
problems.
Most RF passive components synthesis can be naturally
expressed as a constrained optimization problem [2]: the
optimization of an objective (e.g., quality factor), usually
subject to some constraints (e.g., self-resonance frequency).
The special point is that in order to obtain an accurate
result, electromagnetic (EM) simulation of the component
structure is typically necessary, especially at high frequencies.
However, EM simulations are often very CPU time expensive
[3]. This fact highly increases the need of high efficiency
of the synthesis framework. Hence, most of the state-of-the-
art methodologies [1]–[9] focus on the tradeoff between the
solution quality and the efficiency.
In this paper, we propose a new framework, the memetic
machine learning-based differential evolution (MMLDE)
method, focusing on optimized RF passive component syn-
thesis at high frequencies. Compared to available methods
with the best solution quality, MMLDE can obtain comparable
results, but has approximately a tenfold improvement in com-
putational efficiency. A high-performance passive component
for RF ICs can be synthesized in a very reasonable time, which
is in the order of a few hours clock time on a single CPU
node.
The remainder of this paper is organized as follows.
Section II reviews the related works and motivates the strategy
of MMLDE. Section III introduces the components and the
general framework of MMLDE. Section IV tests MMLDE on
practical examples at 60 GHz. Comparisons with the state-of-
the-art methods are also performed. Concluding remarks are
presented in Section V.
0278-0070/$26.00
c
2011 IEEE
LIU et al.: SYNTHESIS OF INTEGRATED PASSIVE COMPONENTS FOR HIGH-FREQUENCY RF ICS 1459
II. Related Works and Motivations
The available computer-aided design optimization method-
ologies for microwave components can be classified into four
categories: 1) equivalent circuit model and global optimization
algorithm based (ECGO) methods [4], [5]; 2) EM-simulation
and global optimization algorithm based (EMGO) methods
[1]; 3) off-line surrogate model, EM-simulation and global
optimization algorithm based (SEMGO) methods [2]; and
4) surrogate model and local optimization algorithm based
(SMLO) methods [3], [6]–[9]. These will now be described
in more detail.
1) The ECGO methods [4], [5] depend on the equivalent
circuit model to obtain the performances of the
microwave structure. Their advantage is high efficiency.
The synthesis of a 5 GHz inductor considering
process variations, which requires many performance
evaluations, has been achieved successfully and
efficiently by ECGO [4]. On the other hand, when the
frequency is high, equivalent circuit models available in
the microwave area are typically not accurate enough or
difficult to find. Hence, even with global optimization
algorithms, the synthesis of high-frequency components
may also fail as the used equivalent circuit models may
not reflect well the performances of the microwave
structures.
2) The EMGO methods [1] can provide an accurate
performance analysis of the microwave structure
because they use EM simulations. Combined with global
optimization algorithms, the quality of the solution is
the best among all the available methods, especially in
high-frequency RF component synthesis. However, its
major bottleneck is the high computational cost of the
EM simulations limiting their use in practice [3].
3) Reference [2] represented a surrogate-model EMGO
(SEMGO), which is an important progress of EMGO.
SEMGO uses an off-line artificial neural network (ANN)
model to enhance the speed of the standard EMGO. In
[2], the surrogate model is first trained to approximate
the performance of the microwave structure before opti-
mization. Then, the optimization algorithm uses this sur-
rogate model as the performance evaluator to find the op-
timal design. The training data are generated uniformly
in the design space and the corresponding performances
are obtained by EM simulations. When combined with
global optimization algorithms, this method has the
ability of global search. However, the training data gen-
eration process in this method is expensive and we found
that the constructed ANN model is not always reliable in
our 60 GHz inductor synthesis example (see Section IV).
4) The SMLO methods [3], [6]–[9] combine the efficiency
of ECGO with the accuracy of the EM simulations from
EMGO. Fig. 1 shows the general flow. First, a coarse
model, either an equivalent circuit model or a model
evaluated by EM simulation but with coarse meshes, is
constructed and optimized. Then, some base vectors in
the vicinity of the optimal point of the coarse model are
selected as the base points to train a surrogate model,
Fig. 1. Flow of the SMLO methods.
whose purpose is to predict the performances of the
microwave structure. At last, the surrogate model is used
to optimize the microwave component, whose result is
verified by the fine model using expensive high-fidelity
EM simulations. The data received by the fine simula-
tions will update the surrogate model to make it more ac-
curate. In the development of the SMLO methods, some
works have been presented focusing on selecting the
coarse model [3], [7] and the surrogate model [3], [8].
SMLO, however, highly depends on the accuracy of the
coarse model, which leads to two significant challenges
for high-frequency RF passive component synthesis.
First, the optimal solution of the coarse model defines
the search space and the constructed surrogate model is
only accurate in that space, because the base points are
selected around it [3]. The success of SMLO comes from
the basic assumption that the optimal point of the coarse
and fine models are not far away in the design space, as
shown in [3] and [6]–[9]. However, this assumption only
holds when the coarse model is accurate enough. Al-
though it has been shown that SMLO can solve RF com-
ponent synthesis well at comparatively low frequencies
[3], [6]–[9] (e.g., 10 GHz), for passive components in
high-frequency RF ICs (e.g., 60 GHz), this assumption is
often not true. In many cases, a workable equivalent cir-
cuit model is even difficult to find, and the mesh density
of the coarse-mesh EM model is difficult to decide. The
second challenge is that SMLO can only do local search,
which is not suited for synthesis with strong require-
ments. This is not only because of the fact that the cur-
rent SMLO methods use local optimization algorithms,
but also because of the fact that the search space is
decided first by the coarse model [3], [6]–[9]. Therefore,
using global optimization algorithms makes little sense.
In summary, ECGO and SMLO work well in comparatively
low-frequency RF component synthesis, but their high depen-
dence on the accuracy of the equivalent circuit or coarse model
limits their use for the synthesis of high-frequency microwave
structures. EMGO can provide high-quality results even when
the frequency is high, but is too CPU time intensive. Although
SEMGO [2] makes a great progress on EMGO, to the best
of our knowledge, the development of sufficiently effective
and efficient synthesis methods for high-frequency microwave
components is still in great need.
To address these problems, we propose a new framework,
the MMLDE method. The key idea of MMLDE is the
proposed online surrogate-model-based memetic evolutionary
optimization mechanism, whose training data are generated
adaptively in the optimization process. The efficiency versus
quality targets aimed at with MMLDE are shown in Fig. 2.
1460 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 10, OCTOBER 2011
Fig. 2. Review of the available methods in HIGH-FREQUENCY component
synthesis and the targets of MMLDE.
In addition, MMLDE does not need any coarse model nor the
complex tuning of the parameters.
III. The MMLDE Algorithm
A. Key Ideas of MMLDE
Two conclusions can be drawn from the available methods:
1) global optimization and EM simulations are the keys to
obtain high-quality solutions, and 2) machine learning tech-
niques, or the surrogate model in this application, are the keys
to enhance the efficiency.
Hence, the question becomes: how to integrate the machine
learning techniques with the global optimization and the
EM simulation-based algorithm? The answer, however, is not
trivial. In the literature, there are mainly two kinds of methods
that use surrogate models in the synthesis of RF components.
The first one is the method used in SMLO. The second one is
the off-line surrogate model in SEMGO [2]. More information
of these two methods has been presented in Section II.
In MMLDE we propose a new framework. After a small
Latin-hypercube sampling (LHS) of the design space and
using EM simulations to evaluate these samples, we train
the initial surrogate model as a rough estimation of the
performances of the microwave structure. Then, we use our
online surrogate-model-based evolutionary algorithm. In each
iteration, the candidate solutions are generated by the memetic
evolutionary computation algorithm, whose performances are
evaluated by the surrogate model. We perform EM simulation
to the candidate solution with the possible best potential
to improve the objective function. Note that the candidate
solution with the best potential is not simply the one with the
best predicted value, like SMLO. In MMLDE, the potential
is calculated based on the used machine learning technique
and the corresponding potential measurement method. We then
update the surrogate model by including the new candidate
with EM simulation result. There is only one EM simulation
in each iteration.
The MMLDE mechanism is different from the mechanism
of SEMGO [2]. SEMGO first constructs a good surrogate
model which covers the whole design space and then uses
it. In the optimization process, there are no EM simulations
and updating. To obtain a reliable surrogate model, the training
data need to cover the whole design space with a reasonably
high density. Hence, a lot of EM simulations are necessary.
On the other hand, only a small part of the design space is
useful in the optimization. The reason is that the optimization
algorithm in [2] is not based on enumeration, but based on
iteration, so many of these expensive EM simulations are
wasted. In contrast, MMLDE holds to the idea of “in the
deep darkness of the design space, there is no need to lighten
the whole world but rather the close vicinity of the path to
the destination.” MMLDE, therefore, first constructs a very
rough surrogate model, and then improves it online but only
in the necessary area of the design space, which is determined
by the optimization algorithm and the updating technique.
Consequently, MMLDE is more efficient in terms of the
number of EM simulations than SEMGO. Moreover, because
all the performances that have potential to be used as the final
result are evaluated by EM simulations, rather than by the
surrogate model, MMLDE is also more accurate.
Although there also exists an updating process in SMLO,
this updating in MMLDE is largely different from the updating
in SMLO. The main purpose of the updating in SMLO is
to improve the local accuracy and to help local search. One
reason is that the search space is defined by the optimal
point of the coarse model and the surrogate model is only
accurate in the vicinity of that point. Moreover, even when
the coarse model is accurate and the global optimal point
is included in the newly defined search space, the updating
which only considering the predicted value also causes a
low probability to achieve global optimization [10], [11]. The
reason is that the updating mechanism only using the predicted
value puts too much emphasis on exploiting the predictor
and no emphasis on exploring points where we are uncertain.
In contrast, the updating in MMLDE can both guide the
global and local search, which is achieved by the memetic
evolutionary algorithm and the method to decide the candidate
with the possible best potential. For the Gaussian process-
based surrogate model, we use the expected improvement (EI)
[10] to measure the potential of the candidate. For the ANN-
based surrogate model, we directly use the predicted value to
measure the potential. The EI measurement has the ability to
judge the potential for global search for a candidate because
the uncertainty of the Gaussian process prediction is consid-
ered. Hence, the quality of a candidate point is considered in a
global picture. When combined with evolutionary algorithms,
global optimization can therefore be achieved. On the other
hand, the potential measurement used for the ANN surrogate
model is more powerful in local refinement compared with
the EI measurement. Hence, we combine the two machine
learning and potential measurement techniques to construct a
memetic evolutionary algorithm with enhanced search ability
and efficiency.
In the following, the basic components of MMLDE will be
introduced first. The key techniques and the general framework
will be presented afterward.
B. Using Gaussian Process in MMLDE
Gaussian process (GP) machine learning [12]–[14] is one
of the chief techniques to construct the surrogate model
in MMLDE. GP machine learning not only has very good
LIU et al.: SYNTHESIS OF INTEGRATED PASSIVE COMPONENTS FOR HIGH-FREQUENCY RF ICS 1461
prediction ability, but also can provide a meaningful un-
certainty measurement for a prediction. This is very impor-
tant when combined with optimization. For online surrogate-
model-based optimization, the accuracy and reliability of the
GP model is improved gradually in the process of optimiza-
tion, as more additional training data are provided throughout
the optimization process. This leads to a problem that some
data predicted by the GP model may have large differences
compared with the real EM simulation results, especially when
the training data are not sufficient. Hence, if we only use
the predicted values, it is very easy to be trapped in a local
optimum. To prevent this, we use the EI measurement [10] to
call for a balance between exploration and exploitation, which
are computed by the predicted value and the standard error
(uncertainty measurement).
Here, we provide an intuitive introduction and the main
formulas for the technique of GP machine learning [15].
GP predicts a function value y(x) at some design point x
by modeling y(x) as a stochastic variable with mean μ and
variance σ. If the function is continuous, the function values
of two points x
i
and x
j
should be close if they are highly
correlated. In this paper, we use the Gaussian correlation
function to describe the correlation between two variables
Corr(x
i
,x
j
)=exp(−
d
l=1
θ
l
|x
il
− x
jl
|
2
) (1)
where d is the dimension of x and θ
l
is the correlation
parameter which determines how fast the correlation decreases
when x
il
moves in the l direction. The formulas to decide θ
l
can be found in [16]. The values of μ, σ and θ are determined
by maximizing the likelihood function of the observed data.
Suppose that there are n observed data x =(x
1
,x
2
···,x
n
), and
their corresponding function values are y =(y
1
,y
2
···,y
n
),
then the optimal values of μ and σ can be found by setting
the derivatives of the likelihood function (2) to 0
h =
1
(2π)
n/2
(σ
2
)
n/2
|R|
1/2
exp(−
1
2σ
2
(y − Iμ)
T
R
−1
(y − Iμ))
(2)
where I is a n ×1 vector of ones, R is the correlation matrix
and
R
i,j
= Corr(x
i
,x
j
),i,j=1, 2, ···n. (3)
By solving the equations, the ˆμ and
ˆ
σ
2
are as follows:
ˆμ =(I
T
R
−1
I)
−1
I
T
R
−1
y (4)
ˆ
σ
2
=(y − I ˆμ)
T
R
−1
(y − I ˆμ)n
−1
. (5)
Using the GP model, the function value y(x
∗
) at a new point
x
∗
can be predicted as (x
∗
should be added in R, r)
ˆy(x
∗
)= ˆμ + r
T
R
−1
(y − I ˆμ) (6)
where
r =[Corr(x
∗
,x
1
), Corr(x
∗
,x
2
), ···, Corr(x
∗
,x
n
)]
T
. (7)
Fig. 3. Solid line represents an objective function that has been sampled at
the five points shown as dots. The dotted line is a DACE predictor fit to these
points (from [10]).
The measurement of the uncertainty of the prediction, i.e.,
the mean square error (MSE), which is used to assess the
model accuracy, can be described as
MSE(x
∗
)=
ˆ
σ
2
[I − r
T
R
−1
r +(I − r
T
R
−1
r)
2
(I
T
R
−1
I)
−1
]. (8)
In this paper, we use the DACE toolbox [16] to implement
the Gaussian process machine learning.
Besides the above basic principles from GP machine learn-
ing, we introduce another important concept, the expected
improvement EI [10], which is calculated as
E[I(x)]=(f
min
− y(x))(
f
min
− y(x)
√
MSE(x)
)
+
√
MSE(x)φ(
f
min
− y(x)
√
MSE(x)
)
(9)
where f
min
is the current best function value in the population
(the population with EM simulation results, not the generated
population after evolutionary operators). φ(·) is the standard
normal density function, and (·) is the standard normal
distribution function. I(x) is the improvement of f .
EI measures the potential of a candidate solution in
MMLDE, which considers both global search and local search.
EI is the part of the curve of the standard error in the model
that lies below the best function value sampled so far. Figs. 3
and 4 provide an example. As shown in Fig. 3, the function
value of x = 8 is better than that of x =3,butx = 8 cannot
be selected when directly using the GP prediction values.
However, the point x = 8 is possible to be selected when
using the EI measurement. In Fig. 4, the probability density
of the prediction uncertainty at the point x = 8 in the curve of
the DACE predictor is shown by curve B. We can find that at
the tail of the density function (area A), the EI value of x =8
is better than the EI of the current f
min
(near x = 3), so it is
possible that the true value at x = 8 is better than the current
f
min
. Mathematically, the potential is calculated by (9). More
details are in [10].
C. Using Artificial Neural Network in MMLDE
An ANN is a computational mechanism, the structure
of which essentially mimics the process of knowledge
acquisition, information processing and organizational skills of
a human brain. An ANN has the capability of learning complex
nonlinear relationships and associations from a large volume
of data, and enables the analysis of a wide range of pattern
recognition [17]. An ANN is composed of a number of highly
interconnected neurons, usually arranged in several layers.
These layers generally include an input layer, a number of
1462 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 10, OCTOBER 2011
Fig. 4. Uncertainty about the function’s value at a point (such as x =8
above) can be treated as if there were a realization of a normal random
variable with mean and standard deviation given by the DACE predictor and
its standard error (from [10]).
hidden layers, and an output layer. Signals generated from the
input layer propagate through the network on a layer-by-layer
basis in the forward direction. Neurons in the hidden layers are
used to find associations between the input data and to extract
patterns that can provide meaningful outputs. The output of
each neuron that responds to a particular combination of inputs
has an impact on the overall output. The weight is controlled
by the level of the activation of each neuron, and the strength
of the connections between the individual neurons. Patterns of
activation and interconnections are adjusted through a training
process to achieve the desired output for the training data. If
the averaged error is within a predefined tolerance, the training
is stopped and the weights are locked in; the network is then
ready to be used [18]. In MMLDE we use a feed-forward
ANN with one hidden layer and the predicted value of the
ANN model is used to measure the potential of the microwave
structure.
D. Optimization Kernel: The DE Algorithm
For the optimization core we choose an evolutionary com-
putation (EC) algorithm. It may seem that they may cost
more function evaluations compared with non-population-
based algorithms. However, choosing EC is motivated by the
following three considerations: 1) EC algorithms can achieve
global optimization, which is the aim of this paper; 2) although
a group of candidates is generated in each iteration, we
only perform one EM simulation for the candidate with the
possible best potential; and 3) the evaluations of individuals
in the EC algorithms are independent of each other in a
population, so it is very suited for parallel computation to
enhance the efficiency. On the other hand, non-population-
based optimization algorithms can do this not so easily, hence
many of them cannot be combined with parallel computation.
Although our current implementation in this paper does not
yet use parallel computation techniques, powerful parallel
computation techniques are available.
The DE algorithm [19] is selected as the search engine
in MMLDE. The DE algorithm outperforms many other EC
algorithms in terms of solution quality and convergence speed
[19]. DE uses a simple differential operator to create new
candidate solutions and a one-to-one competition scheme to
greedily select new candidates.
The ith candidate solution in the d-dimensional search space
at generation t can be represented as
x
i
(t)=[x
i,1
,x
i,2
, ···,x
i,d
]. (10)
At each generation t, the mutation and crossover operators
are applied to the candidate solutions, and a new population
arises. Then, selection takes place, and the corresponding
candidate solutions from both populations compete to com-
prise the next generation. The operators are now explained in
detail.
For each target candidate solution, according to the mutation
operator, a mutant vector is built
V
i
(t +1)=[v
i,1
(t +1),...,v
i,d
(t + 1)]. (11)
It is generated by adding the weighted difference between a
given number of candidate solutions randomly selected from
the previous population to another candidate solution. The
mutation operation is therefore described by the following
equation (DE/best/1/bin [19]):
V
i
(t +1)=x
best
(t)+F (x
r1
(t) − x
r2
(t)) (12)
where indices r
1
and r
2
(r
1
,r
2
∈{1, 2,...,NP}) are randomly
chosen and mutually different, and also different from the
current index i. Parameter F ∈ (0, 2] is a constant called
the scaling factor, which controls the amplification of the
differential variation x
r1
(t) − x
r2
(t). The base vector to be
perturbed x
best
(t) is the best member of the current popula-
tion, so that the best information can be shared among the
population. To avoid stagnation and to improve the balance
between exploration and exploitation, we use the random-
scale search DE mutation operator. In this mutation, for the
scaling factor we use a vector
ˆ
F composed of Gaussian-
distributed random variables with mean value μ and variance
σ:
ˆ
F
i,j
= norm(μ, σ),i=1, 2,...NP,j =1, 2,...d. Equation
(12) is therefore changed to (13). For more details please refer
to [19]
V
i
(t +1)=x
best
(t)+
ˆ
F
i
(x
r1
(t) − x
r2
(t)). (13)
After the mutation phase, the crossover operator is applied to
increase the diversity of the population. Thus, for each target
candidate solution, a trial vector is generated as follows:
U
i
(t +1)=[u
i,1
(t +1),...,u
i,d
(t + 1)] (14)
u
i,j
(t +1)=
v
i,j
(t +1), if (rand(i, j ) ≤ CR)orj = randn(i)
x
i,j
(t), otherwise
(15)
where rand(i, j ) is an independent random number uniformly
distributed in the range [0, 1]. Parameter randn(i) is a ran-
domly chosen index from the set {1, 2,...,d}. Parameter
CR ∈ [0, 1] is a constant called the crossover parameter, which
controls the diversity of the population.
Following the crossover operation, the selection operation
decides on the population of the next generation (t + 1).
In standard DE, U
i
(t + 1) is compared to the initial target
candidate solution x
i
(t) by a one-to-one-based greedy selection
criterion. However, in MMLDE, we do not use this selection
operator, because we need to minimize the number of EM
simulations. Instead, we select the best solution (or solution
with the possible best potential) among all the trial solutions
U(t + 1) and then perform EM simulation to it. Then, we add
![Fig. 3. Solid line represents an objective function that has been sampled at the five points shown as dots. The dotted line is a DACE predictor fit to these points (from [10]).](/figures/fig-3-solid-line-represents-an-objective-function-that-has-2w7nxuye.webp)