DESIGN OF 2-DMULTIPLIERLESSFILTERS USING THE GENETIC ALGORITHM
R. Thamvichai
Tarnal Bose & Randy L. Haupt
Electrical & Computer Engineering
University of Colorado
Boulder, GO 80309
Electrical & Computer Engineering
Utah State University
Logan, UT 84322-4120
and PSV systems are given in Section 2. In Section 3, a design procedure based on the genetic algorithm is presented.
A stability test is also incorporated to ensure that the designed filter is stable. An example is presented in Section 4,
which is followed by the conclusion.
ABSTRACT
When digital filters are designed with power-of-two coefficients, the multiplications can be implemented by simple shifting operations. In this paper, the genetic algorithm
(GA) is used to design 2-D multiplierless filters. The 2D filter is designed to have periodically shift variant (PSV)
coefficients. This increases the degress of freedom for the
multiplierless coefficients so that a better approximation can
be achieved. The design involves finding the impulse response of the 2-D PSV filter in closed form and then using
the GA to find the filter coefficients. Two different types of
GA are used, namely, the Binary-GA and the Integer-GA.
Some design examples are presented to illustrate the concepts.
2. SYSTEM DESCRIPTION
Consider a causal, recursive and separable denominator 2D digital filter represented by the first model of FomasiniMarchesini [7] given by
z (h
- A l A z ~ ( hk)+Bu(h,
,
k)
y ( h ,k ) = C z ( h 1,k 1)
1. INTRODUCTION
+
During the past decade, multiplierless digital filters have
received widespread attention in the literature because of
their high computational speed and low cost implementation. In the one-dimensional (I-D) case, various techniques
have been proposed for designing multiplierless filters. Several design methods for multiplierless filters have also been
developed for two-dimensional (2-D) filters [I] -[5]. In [l]
and [2], 2-D state-space filters were designed using a genetic algorithm (GA) that optimized coefficients that were
a sum of power of two elements. These coefficients do not
need any multipliers but require shifiers and adders.
In this paper, we present a genetic algorithm for designing 2-D separable denominator filters with single powerof-two coefficients. Thus, these coefficients can be implemented using simple shifting operations and no adders. The
filters are designed directly in the state-space domain. We
design both linear shift-invariant (LSIV) and periodically
shift variant filters (PSV). T h e shift variant filters provide
more degrees of freedom in approximating a given impulse
response without increasing the filter order. The paper is organized as follows. The system descriptions for both LSIV
+
(1)
where z(h,k) E RLxl is a state vector, u(h,k), y(h, k ) E
RIX1are the input and output respectively, A1,A2 E R L x
are constant real state matrices, BE R L x l ,and C E R I x L
are constant vectors.
The impulse response h(n1,nz) of this LSIV system is
well known [SI and given by
where u(n1,n2) is the 2-D unit step function.
In [5], we proposed the design of 2-D multiplierless filters using the system of Fig. l. It consists of a Hold (which
increases the signal rate by a factor of (P,l) ), a periodicallyshift varying (PSV) system, and a decimator (which decreases the signal rate by a factor of (P,l)) connected in cascade. The use of periodic filters increases the total number
of coefficients, thereby providing the necessary degrees of
freedom for obtaining power-of-two coefficients.
The PSV system is simply the LSIV system of (I), with
periodic coefficients, given by
This work was supported in part by Raytheon and NASA Contract
#NAGS3991
0-7 803-7 147-X/01/$10.0002001 IEEE
+ 1,k + 1)= A l ~ ( hk ,+ 1 ) + A 2 ~ (+h 1, k )
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The genetic algorithm begins by defining a chromosome
or an array of parameter values to be optimized. In this
case, the parameter values are all the elements of matrices
Al(h,k) and A2, and vectors B(h,k) and C. We use two different types of GA, namely, the Binary-GA and the IntegerGA.
For the Binary-GA, each coefficient is encoded to using
B bits. The left most bit is the sign bit and the other B-1 bits
Fig. 1. PSV system.
represent the power of two. For example, consider the set
S7 =
...,*1}, where 4 bits can beemployed
to represent each element in the set. For instance, let any
coefficient be represented by 4 bits, z =[a0 a1 a2 a3]. Then
2
ai+12('-~).
x = ( ( - l ) a o ) x 2 - P where p =
i=O
For simplicity, we let Az(h,k) be a constant matrix A2, C(h,k)
be a constant vector C, Al(h,k) and B(h,k) be periodic (P,l),
i.e. Al(h+mP,k+n) = Al(h,k)andB(h+mP,k+n) =
B(h,k ) for all integers (m,n). With the system and equations above, the impulse response of the LSIV-equivalent
system can be obtained [5] as
I
For the integer GA, the coefficients are represented simply by an integer which represents the power-of-two coefficient. For example, consider the set
S7 = {f2-7, f F ,...,fl,0). This represents a set of
17 integers accounting for the sign. One way to code the
power-of-two coefficients is
(-2-7, -2-6, ..., -20,0,20, ...,2-6,2-7}
= {0,1, ...,7,8,9,
..., 15,16}.
The chromosome is an array of binary (or integer) codes of
all coefficients concatenated together. Each chromosome is
associated with a cost value found by evaluating the fitness
measure (or cost fhction) as
C A T B(O,O),
for nl= 0, n2 2 0
where Npop is the population size and m =I, ...,Npop.
3.1. Design Procedure
The impulse responses in (4) and (2) will be useful in calculating the fitness measures (or cost functions) in the next
section.
The design procedure using the GA is shown in Fig 2.
Step 0 (Initial population): Generate 2*NpOpchromosomes randomly, and rank the costs defined in (6) for all
the chromosomes from lowest cost to highest cost. The cost
evaluation is described in step 1. Only the best Npopchromosomes are kept for each iteration, while the others are
discarded.
3. GA-BASED DESIGN PROCEDURE
The design involves finding the elements of the matrices
{ Al(h,k),A2,B(h,k),C} which minimize the cost function
Step 1 (Stability test and evaluation): Test the stability
(described in the next subsection) of all chromosomes. If
the chromosome represents an unstable filter, it is given the
maximum cost (i.e. cost = 1000); otherwise its cost is evaluated using (6). Step 1 is skipped for the first iteration since
the initial population is already evaluated.
where (nl, n 2 ) E Rd, the region of support of the desired
impulse response, hd(nl,n2). The design constraint is that
Step 2 (Selection and pairing): Rank the costs from lowthe designed system must be stable. Also for a rnultiplierless realization. the elements of matrices and vectors are reest cost to highest cost and keep the best NpOp/2chromostricted to a power-of-two set S, = { 3 ~ 2 - f2-"+',
~,
..., f 1 , O ) . somes. The others are discarded. Then two-chromosomes
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33. Stability Test
II
I I
l
Stabilii test and
Testing for stability is tricky in this design process. For the
separable denominator LSIV filter, there is a simple sufficient condition for stability that can be used to guarantee
stability. For the PSV filter, several sufficient conditions
and necessary conditions are given in [SI. The sufticient
conditions given therein turn out to be too restrictive for the
multiplierless filters. Therefore we use a necessary condition (given below). This will weed out a lot of the unstable
candidate PSV filters, and will ensure that all the designed
filters satisfj. the necessary condition. However, it does not
guarantee stability. The notion we use here is that if the
given (desired) impulse response is stable and we have a
good approximation, then the designed filter is likely to be
stable. However, once we have a designed filter, we use the
sufficient conditions of [SI to assure that the filter is stable.
The following conditions for stability are now incorporated
in the GA algorithm.
LSIV system (Sufficient condition) [2]:
l
I I
evaluation
1-1
mini mu m solution
I1
I*
rninsol < = t a g e t
Selection &pairing
p(Ai)
Fig. 2. Design procedure flowchart
< 1 a n d ~ ( A 2 )< 1
where p denotes the spectral radius of a matrix.
PSV system (Necessary condition) [SI:
1
are randomly selected to produce two new offsprings using
the weighted random pairing scheme based on the cost of
each chromosome. This is called the cost weighting scheme
[6]. Pairing takes place until new offsprings are generated
to replace the discarded ones. Note that the population size,
NPOP,
is constant for all iterations.
Al(aJ9)
P(
< 1and P(Az) < 1
a= P
4. EXAMPLE
To design a 2-D multiplierless PSV filter, one idea is to
first design the LSIV system by applying an Integer-GA
or Binary-GA until we obtain a satisfactory cost value (6).
Then we use matrix A2 and C from this design to design the
PSV system using Integer-GA. This approach reduces the
length of chromosome in the PSV system design, thereby
reducing the design time.
Consider the impulse response of a "Gaussian filter"
given by
Step 3 (Crossover): Two new offspring are generated
from each pair of chromosomes in Step 2 by crossover operation with probability P,. The idea of the crossover operation is that the paired chromosomes exchange a part of them
at the crossover point, which is randomly selected between
the first and last bits of the paired chromosomes.
Step 4 (Mutation): Keep the best (lowest cost) chromosome and apply mutation to the rest of the population with
probability P,. The idea of mutation is to change the value
of the bit (or integer) of the chromosome at the mutation
point. The mutation point is randomly selected from the
(Npop- 1) x L, total number of points where L, is the
length of chromosome. For the Binary-GA, changing the
value of the bit is achieved by simply complementing the
bit. For the Integer-GA, changing the value of the integer is
done by replacing it with a new integer in the same set.
h d ( i , j ) = 0.256322ezp(-O.l03203{(Z
- 4)2 + ( j - 4)2})
for(0,O) <(ij)<(lO,lO). Thegoal istodesigna(4,4)thorder
filter in the Fornasini-Marchesinistate-space form. The GA
parameters are selected as follows: population size NpOp=
128, crossover probability P,=O. 6 , and mutation probability P, =0.15. Four binary bits are used to encode each coefficient for Binary-GA and integers 0 to 16 are used for the
Integer-GA. We run the following design cases: (a) LSIV
design using Binary-GA; (b) LSIV design using IntegerGA, (c) Using A2 and C from the LSIV filter and then designing the PSV filter using Integer-GA. Table l shows the
Step 5: Repeat Steps 1-4. The procedure stops if the
minimum solution (minsol) meets the target value.
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error values. These results are obtained by running the GA
for a total of about 400 times.
Table 1. Design results
The best result is obtained by using the Integer-GA for
the LSIV filter and then using that design to optimize the
PSV filter. The frequency responses of the ideal and designed filters are given in Fig. 3(a) and (b), for the best case
example. It is conjectured that the GA parameters can be
further adjusted to further improve the results. We are currently working on developing techniques to automatically
tune the GA parameters for this problem.
In our experiments with the proposed design method,
the error values range from 0.1 to 0.2, which is very comparable to those obtained in [I] and [2]. It should be noted that
in these references, the filter coefficients were not strictly
power-of-two but sum of powers-of-two numbers. However, in this paper we have strictly power-of-two coefficients
and yet we can achieve comparable error values. Also, we
see from Fig. 3(a) and (b) that there is a clearly a good
match between the two magnitude responses.
hpurr
-F
Fig.3(b) Magnitude response of designed filter.
6. REFERENCES
[l] Y.-H. Lee, M. Kawamata and T. Higuchi, ”GA-based design
ofmultiplierless 2-D state-space digital filters with low roundoff noise,” IEEE Proc.-Circuits Devices Syst., vol. 145, no. 2,
pp. 118-124, Apr. 1998.
[2]
-
” Design of Multiplierless 2-D State-space Digital filters over a Powers-of-Two Coefficient Space,” IEICE Trans.
Fundamentals, vol. E79-A, 110.3, pp.374-377, Mar.1996.
[3] S. Pei and S. Jaw, ” Efficient design of 2-D multiplierless FIR
5. CONCLUSION
filters by transformation,” Proceedings ICASSP 87, vol. 3, pp.
1669-72, 1987.
In this paper, we developed a method of designing 2-D multiplierless filters. T h e method is based on the use of periodic
[4] H.K. Kwan and C.L. Chan, ”Circularly symmetric two-
filters and the genetic algorithm for optimization. Two different types of genetic algorithms have been used, namely,
the Binary-GA and the Integer-GA. The use of the latter decreases the chromosome size and makes the algorithm much
faster. However, the Binary-GA is slower but tends to get
closer to the solution, thereby yielding a lower cost function. This paper provides preliminary work on CA based
2-D filter design. Further work needs to be done in reducing the search time of the GA and also reducing the final
design error.
dimensional multiplierless FIR digital filter design using an
enhanced McClellan transformation,” IEE Proceedings, vol.
136, no.3, pp. 129-134, June 1989.
[5] R. Thamvichai, T. Bose, and D.M. Etter, “Design of multiplierless 2-D filters,” Proc. IEEE Intl. Symposium on Intelligent
Signal Processing and Communication Systems, Nov. 2000.
[6] R.L. Haupt and S.E. Haupt, ”Practical genetic algorithms,”
John Wiley & Sons, Inc., 1998.
[7] E. Fornasini and G . Marchesini, ”State-space realization theory of two-dimensional filters,” IEEE Trans.
Automat. Contr., vol. AC-2 I, pp.484-492, Aug. 1976.
[SI R. Thamvichai and T. Bose, ”Stability of 2-D periodically
shift variant filters,” IEEE Truns. Circuit Syst.-II (submitted).
6-
.,
_I
FC.Q-(*
Fig.3(a) Magnitude response of Gaussian filter.
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