38th International Symposium on Multiple Valued Logic
Hybrid Reed-Muller Haar Transform and its Application in Reduction the
Spectral Representations of Logic Functions
Radomir Stankovic
Departament of Computer Science, Faculty of
Electronics,
18000 Nis, Serbia,
email: rstankovic@bankerinter.net
Susanna Minasyan, Jaakko Astola,
Karen Egiazarian
Tampere International Center of Signal
Processing, Tampere University of
Technology, P.O.Box 553, FIN-33101,
Tampere, FINLAND, email:
firstname.lastname@tut.fi
version of the Haar transform, has been constructed as
well as generalizations of this and related transforms to
functions in Galois fields of different orders.
Parametrized slant-Haar transforms [13] and Treestructured Haar transforms [14] have been introduced in
order to solve various signal processing tasks. These
transforms can be adapted to the properties of signals to
be processed by carefully adjusting certain parameters.
In this paper, we will exploit both of the above
mentioned approaches. We will define a transform as a
combination of the Reed-Muller transform [15] and the
Reed-Muller Haar transform [12], with the relative sizes
of the Reed-Muller and Haar parts as parameters. The
value of these parameters can be selected depending on
the features of the signal to be processed. This transform
will be called the hybrid Reed-Muller Reed-Muller Haar
(RM-RMH) transform. To ensure applicability of the new
transform in practice, we determine a fast calculation
algorithm based on classical theory of FFT-like
algorithms and exploiting the Kronecker product structure
of both constituent transforms.
As an illustration of possible applications of this
transform, we consider the compactness of the
corresponding spectral representations of binary valued
functions in the number of nonzero coefficients. To that
order, we also define Fixed-polarity RM-RMH
transforms.
Experimental results show that, on the average, the
proposed approach reduces the number of nonzero
coefficients in the binary spectrum compared to the RM
transform.
Abstract
In this paper we present a new hybrid transform
based on the Kronecker product of Reed-Muller and
Reed-Muller Haar transforms. The proposed transform
shares attractive properties of both Reed-Muller
transform and Reed-Muller Haar transform. An example
of application of hybrid transform for reduction of the
number of nonzero coefficients in spectra of truth vectors
of switching functions is presented. The experiments show
that the proposed approach, on average, reduces the
number of nonzero coefficients in the spectra of
benchmark functions.
1. Introduction
Different transforms have found wide applications in
many areas as signal processing, image processing,
filtering, spectral analysis, circuit synthesis, digital logic
design, communications systems, pattern recognition etc.
In particular, the Reed-Muller (RM) transform is widely
used in solving different problems in logic design, such as
AND-EXOR synthesis [1], testability [2]-[7], design of
easy testable logic networks [see in 7], etc. The same
applies to various generalizations of the Reed-Muller
transform to multiple-valued functions [8]. The Haar
transform is another classical spectral tool efficiently
applied in the same areas [9]. Combination of different
transforms is an approach to design new transforms with
properties well suited for some particular applications.
For instance, the Hadamard-Haar transform (HHT) based
on combination of Hadamard and Haar transforms has
been introduced in [10],[11]. The basis functions of HHT
are linear combinations of Haar and Walsh functions.
Such combinations have been useful in feature selection,
Wiener filtering and image processing.
Recently new classes of discrete Haar wavelets have
been derived allowing multiresolution analysis in
different algebraic structures [12], [18]. An algorithm for
derivation of Haar transform in different vector spaces has
been also provided [12]. In particular, the Reed-Muller
Haar transform (RMH), that can be viewed as a binary
0195-623X/08 $25.00 © 2008 IEEE
DOI 10.1109/ISMVL.2008.8
2. Reed-Muller Haar Transform
This section describes the Reed-Muller Haar transform
in Galois field of different orders.
We first repeat the definition of the Reed-Muller Haar
transform given in [12].
Denote by P ( G ) the space of the functions
f : G → P , where G is a finite group and P is a field that
may be the complex field C or a finite (Galois) field
232
HT ( n − 1) ⊗ [1 1]
H ( n) =
,
I ( n − 1) ⊗ [0 1]
GF(p). In this notation, the space of switching functions
of n variables will be denoted by GF2 (C2n ) , where C2 is
the cyclic group of order 2, i.e., C2 =< {0,1} , ⊕ > , and ⊕
is the addition modulo 2 or logic EXOR.
When a function f ∈ P(C pn ) is given by the truth vector
(
(1)
T
where H (1) = ( R (1) ) is the RM transform of order 2.
For p=3 the RMH of order 3n is the following:
THT ( n − 1) ⊗3 [1 1 1]
TH ( n ) = I ( n − 1) ⊗3 [0 1 2] ,
I ( n − 1) ⊗3 [0 1 1]
)
F = [ f ( 0 ) ,..., f p n − 1 ]T then the spectrum and the
series expansion for f are defined as
S f = Q −1 ( n ) F ,
(2)
T
where TH (1) = ( GF3 (1) ) is the RMH transform of order
F = Q (n) S f ,
3, and ⊗3 is the Kronecker product modulo 3.
where S f = [S(0),..., S( p n − 1)]T is the vector of spectral
coefficients in the expansion with respect to the basis Q
defined by columns of the matrix
3. Proposed Hybrid Transforms
n
Q ( n ) = ⊗ Q (1) ,
In this section we define a new hybrid transform by
combining the RM and RMH transforms. We give also a
fast calculation algorithm for the hybrid transform.
Here due to the performance efficiency in chosen
application discussed in Section 5, we use the following
version of RMH transform of order 2n:
H ( n − 1) ⊗ [1 0]
(3)
H ( n) =
.
I ( n − 1) ⊗ [1 1]
i =1
where Q (1) is the Fourier transform matrix in P (C p ) or
the matrix defining the product terms in polynomial
expressions in P (C p ) .
Definition 1 (Haar functions): Haar functions in P (C np )
are defined by the columns of the following matrix:
H ( n − 1) ⊗ q 0
I ( n − 1) ⊗ q1
,
H ( n) =
#
I ( n − 1) ⊗ q p −1
where q i , i = 0,..., p − 1 are rows of QT (1) for P=GF(p),
and of Q −1 (1) for P = C , and I ( r ) is the identity matrix
of order r .
For a function f ∈ P(C pn ) given by its truth vector F the
Haar spectrum and the series expansion are defined as:
S f = H −1 ( n ) F,
F = H (n)S f ,
,
where S f is the vector of Haar spectral coefficients with
respect to basis Q (1) .
Example 1: For p=2, 3, 4 the basic Q (1) matrices used to
define the Haar functions are the following:
1 0 0 0
1 0 0
1 0
1 1 1 , GF (1) = 1 1 1 1 .
=
R (1) =
GF
,
(1)
3
4
1 2 3 1
1 1
1 2 1
1 3 2 1
For p=2 the Reed-Muller Haar (RMH) transform of order
2n has the following form:
233
The difference between transforms (1) and (3) is that
we use in (3) the non-transposed RM transform.
Example 2: For n=3 the modified RMH transform H(3) is
1 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0
1 0 1 0 0 0 0 0
0 0 0 0 1 0 1 0
.
H (3) =
1 1 0 0 0 0 0 0
0 0 1 1 0 0 0 0
0 0 0 0 1 1 0 0
0 0 0 0 0 0 1 1
Definition 2 (0-polarity hybrid transform): The forward
RMH transform of order N = 2n with an integer parameter
r is defined as
T ( n, r ) = R ( r ) ⊗ H ( n − r ) , 0 ≤ r ≤ n ,
(4)
r
where R ( r ) is the Reed-Muller transform of order 2
and H ( n − r ) is the Reed-Muller Haar transform of order
n-r
2 .
The parameter r can be selected appropriately to best
exploit the features of the function to be represented in
order to get the smallest number of nonzero coefficients.
In particular, we will consider the following cases of
this hybrid transform:
1. If r = 0, then T(n, 0) = H(n).
2. If r = 1 and r = 2, then
T ( n,1) = R (1) ⊗ H ( n − 1) , T ( n, 2 ) = R ( 2 ) ⊗ H ( n − 2 ) .
3.1 Ternary Hybrid Transform
3. If r = n, then T(n, n) = R (n) .
In the case 1 the hybrid transform coincides with the
RMH transform, while in the case 3 the hybrid transform
coincides with the RM transform of order 2n .
Example 3: For n = 3 and r = 1, the hybrid transform
matrix T ( 3,1) and its inverse T−1 (3,1) are
The hybrid transform may be extended also from binary
to higher order fields. Here we give briefly the definition
of the hybrid RM-RMH transform over GF(3), thus, the
addition and multiplication operations are done modulo 3.
Definition 3 The ternary hybrid RM-RMH transform of
order N=3n with an integer parameter r is defined as
M ( n, r ) = TR ( r ) ⊗3 TH ( n − r ) ,
where TR(r) is the ternary RM transform of order 3r,
TH(n-r) is the ternary RMH transform of order 3n-r, and
⊗3 is the Kronecker product modulo 3.
Example 4: For n=2, r=1 the ternary hybrid transform is
M ( 2,1) = TR (1) ⊗3 TH (1) =
T ( 3,1) = R (1) ⊗ H ( 2 ) =
1
1
1
0
=
1
1
1
0
0 0 0 0 0 0 0
0 1 0 0 0 0 0
1 0 0 0 0 0 0
0 1 1 0 0 0 0
0 0 0 1 0 0 0
0 1 0 1 0 1 0
1 0 0 1 1 0 0
0 1 1 0 0 1 1
1
1
1
1
= 1
1
1
1
1
and
T−1 (3,1) = R (1) ⊗ H −1 ( 2 ) =
1 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0
1 1 0 0 0 0 0 0
0 0 1 1 0 0 0 0
=
.
1 0 0 0 1 0 0 0
1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0
0 0 1 1 0 0 1 1
Both the RM and RMH transforms have fast
calculation algorithms. The fast algorithm for the RM
transform has the structure similar to that of the
Hamadard transform, while the flowgraph of the RMH
transform is similar to that of the fast Haar transform.
Figure 1 illustrates the butterfly diagram of the fast RMH
transform. As for many fast transforms, the fast hybrid
RM-RMH transform may be calculated by a product of
sparse factor matrices. In Figure 2, the fast hybrid RMRMH transform flowgraph is shown for the case r=1,
N=8. The butterfly diagram of hybrid transform has a
structure of fast Hadamard-Haar transform [3],[4]. The
number of additions required to implement the fast RM is
( N 2 ) ⋅ log 2 N . The computational cost of the fast RMH
transform is N – 1. For the hybrid RM-RMH transform of
order N and the parameter r , the number of additions in
0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0
2 1 0 0 0 0 0 0
0 0 1 0 0 1 0 0
1 1 1 1 1 1 1 1 ,
2 1 1 2 1 1 2 1
0 0 2 2 2 1 0 0
1 1 2 2 2 1 1 1
2 1 2 1 2 1 2 1
1 0 0
where TR (1) = TH (1) = 1 1 1 is the ternary RM
1 2 1
transform of order 3. For that case, the ternary hybrid
transform coincides with the ternary RM transform of the
same order.
The ternary hybrid transform based on ternary RMH
transform (2) can be also considered and may be useful in
many applications where similar transforms over Galois
field have already found interesting applications [2],[9],
[16], [18].
4. Fixed-polarity Hybrid Transform
In this section the fixed polarity hybrid RM-RMH
transforms based on both fixed-polarity RM and fixed
polarity RMH are proposed. First, we define the fixedpolarity RMH transform.
Definition 4 The fixed polarity RMH transform
fast implementation is 2r ⋅ ( N / 2r − 1) + r ⋅ ( N / 2) .
It should be noticed that the computational costs are
different for different values of r. In other words, for
r=1,2,...,n-1 the number of additions in the fast algorithm
is between the number of additions corresponding to the
RMH and RM transforms.
(FRMH)
H
hn
( n)
with
the
polarity
h n = ( h1 , h2 ,..., hn ) , where hi ∈ {0,1} , is defined as
234
vector
0 0 0
0 1 0
0 1 0
0 0 0
=
1 1 0
0 0 1
0 0 0
0 0 0
We now define the
transform.
Definition 5 For an
Figure 1. Flowgraph of fast RMH transform
of order N=8.
0 0 1 0 0
0 0 1 0 0
1 0 0 0 0
0 0 1 0 1
.
0 0 0 0 0
1 0 0 0 0
0 1 1 0 0
0 0 0 1 1
fixed-polarity hybrid RM-RMH
n-variable switching function, the
fixed-polarity hybrid RM-RMH transform Thn ( n, r )
with the parameter r and the polarity vector
h n = ( h1 , h2 ,..., hn ) , hi ∈ {0,1} , is defined as:
Th n ( n, r ) = R h r ( r ) ⊗ H h n − r ( n − r ) =
h
r
H n − r −1 ( n − r − 1) ⊗ C w
(6)
= ⊗ R hr (1) ⊗
,
I n − r −1 ⊗ [1 1]
i =1
0≤r≤n,
where R hr ( r ) is the RM transform of order 2r for the
polarity vector h r = ( h1 , h2 ,..., hr ) , and the H hn − r ( n − r )
Figure 2. Flowgraph for fast 0-polarity hybrid
RM-RMH transform with parameter r=1 and
order N=8.
H
hn
where H
w
hn −1
( n ) = H ( n − 1) ⊗ C ,
I ( n − 1) ⊗ [1 1]
h n −1
is the fixed-polarity RMH of order 2n − r for the polarity
vector h n − r = ( hr +1 , hr + 2 ,..., hn ) and parameter r. The
polarities for the RM and RMH transforms of order 2 are
1 0
, hi = 0
1 1
hi
hi
.
R (1) = H (1) =
0 1 , h = 1
1 1 i
(5)
( n − 1) is FRMH for the polarity vector
h n −1 = ( h1 , h2 ,..., hn −1 ) , and C
polarity number w, that is,
C
w
=
{
w
is a vector with the
Notice that this definition of the hybrid fixed-polarity
RM-RMH transform is similar to the corresponding
definitions of other fixed-polarity transforms, as for
instance, the recently defined multi-polarity generalized
Haar and arithmetic-Walsh transforms [16],[17].
[1 0], w = 0
[0 1], w = 1
where the polarity w = hn is the last component of the
polarity vector hn .
Example 5: For n=3, and the polarity vector h3={1,0,1}
the FRMH transform of order N=2n is
H (1,0) ( 2 ) ⊗ C (1)
h
H 3 ( 3) =
=
I ( 2 ) ⊗ [1 1]
0 1
⊗ [1 0]
1 1
⊗
= I 1 ⊗ [1 1]
( )
I
2
⊗
( )
5. Application of the Hybrid RM-RMH
Transform to the Reduction of Spectral
Representations
In this section, we consider the application of the
hybrid RM-RMH transform for the 0-polarity and the
fixed-polarity to the reduction of the number of nonzero
coefficients in spectral representations of n-variable
switching functions.
A switching function f defined by the truth-vector F is
transferred into the spectral domain by T(n, r ) transforms
[0 1]
=
1
1
[ ]
235
spectrum. In fifth column the number of nonzero elements
in the spectrum of fixed-polarity RMH transform
(FRMH) is given. As we see, the FRMH transform
increases the number of nonzero spectral coefficients,
which is a justification to consider the hybrid transforms.
In the last column denoted by FHybr we give the
results of applying the method based on fixed-polarity
hybrid RM-RMH transform. This method is described by
the Algorithm 2 in the previous section. The method
outperforms the FPRM for at about 32%.
Table 3 compares the number of 4-variable functions
requiring a given number of products t, where t = 0,…,16,
in the minimum expressions.
The average number of products per each function for
different expressions is given in the last row of the table.
In the columns 2 and 4, the results for the positive
polarity RM (PRM) and fixed polarity RM expressions
are given, respectively. The number of functions requiring
t products for RMH transform is not given in this table
since it coincides with the case of PRM. The fifth
column, FRMH, shows the results for fixed polarity
RMH expressions. In columns 3 and 6, the number of
functions requiring t products for 0-polarity hybrid
transform and fixed-polarity hybrid transform expressions
are given, respectively.
As one can see from Table 3, the PRM requires, on
average, 8.00 products to realize an arbitrary function,
while the Hybr require only 6.71 products. At the same
time, the fixed-polarity hybrid transform (FHybr), on
average, requires fewer number of products than the
FRMH and FRM expressions.
of order 2n , r=0,1…,n, defined according to (4), (6)
resulting in vector of spectral coefficients:
S = T(n, r ) ⋅ F ,
(7)
The approaches utilizing the proposed spectral
transforms may be described by two algorithms.
Algorithm 1:
1) Given an n-variable switching function by its truth
vector F.
For r = 0,1,...,n, calculate the spectra
Sr = T(n, r ) ⋅ F, where, according to (4), the T(n, r ) is
the 0-polarity hybrid RM-RMH transform. Then,
calculate the number of nonzero binary spectral
coefficients.
2) Find an optimal transform T∗ (n, r ) that gives the
minimal number of nonzero spectral coefficients. In this
approach n+1 transforms are considered.
Algorithm 2:
1) Given an n-variable switching function by the truthvector F. Then, for every r =0,1,...,n, calculate the fixedpolarity hybrid RM-RMH transforms (6), combining the
fixed-polarity RM and fixed-polarity RMH transforms. In
other words, for every polarity vector hr all
combinations of polarity vectors h n − r of
FRMH
transform are considered. In that case, for each r, we
consider 2n transforms. Then, calculate the number of
nonzero coefficients Numr in spectra of thus obtained
transforms.
2) Find an optimal transform having the corresponding
optimal polarity vector h*n ={ hr , hn - r } that gives the
Table 1. The number of nonzero coefficients in
the spectra of RM and hybrid transforms.
minimal number of nonzero spectral coefficients.
6. Experiments
The proposed approaches were tested via experiments
on a set of different benchmark functions and results were
averaged. Tables 1, 2 and 3 illustrate the results for a
sample of 10 benchmark functions randomly selected. In
Tables 1 and 2 the second column denoted by in shows
the number of input variables of an n-variable binary
function. The third column shows the number of 1’s in the
original domain.
In Table 1 the columns RM and RMH, give the
number of nonzero elements in spectrum of the RM and
RMH transforms. The last column denoted by Hybr
shows the results of applying the Algorithm 1, that is, the
number of nonzero elements in the spectra of hybrid RMRMH transforms for the 0-polarity. The results show that
on average the proposed approach reduces the number of
nonzero coefficients to 76.1 which is about 49.02% better
than in the case of the RM transform.
In Table 2 the fourth column shows the number of
nonzero coefficients in the fixed-polarity RM (FRM)
Bench.
in
#1
RM
RMH
Hybr
clip0
9
256
clip1
9
256
86
36
36
86
175
72
clip2
9
256
88
309
79
clip3
9
256
93
161
81
clip4
9
256
105
85
61
9sym
9
420
272
102
102
rnd10
10
871
398
200
195
rd84_1
8
121
137
77
56
rd84_2
8
128
85
85
23
rd84_4
8
162
143
56
56
298
149.3
128.6
76.1
13.8
49.02
average
improvement compared to RM (%)
236
Table 2. The number of nonzero coefficients in
the spectrum of fixed-polarity RM and hybrid
transforms.
Extensions to the generalized Reed-Muller and ReedMuller Haar transforms for multiple-valued functions are
possible and will be the subject of a future work.
Bench
in
#1
FRM
FRMH
FHybr
8. References
clip0
9
256
86
35
35
clip1
9
256
82
174
60
clip2
9
256
78
307
68
clip3
9
256
81
152
78
clip4
9
256
65
73
47
9sym
9
420
135
90
90
rnd10
10
871
309
176
174
rd84_1
8
121
40
77
40
rd84_2
8
128
12
85
12
rd84_4
8
162
64
54
44
298
95.2
122.3
64.8
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average
improvement compared to FRM
-28.5%
32 %
Table 3. Number of 4-variable functions
requiring t products.
t
0
1
2
3
4
5
6
7
8
9
10
PRM
1
16
120
560
1820
4368
8008
11440
12870
11440
8008
Hybr
1
33
287
1347
4121
8926
14015
15744
12312
6417
2026
11
12
13
14
15
16
av.
4368
1820
560
120
16
1
8.00
301
6.71
FRM
1
81
836
3496
8878
17884
20152
11600
2336
240
32
FRMH
1
31
322
1658
5114
10746
14962
14890
10262
4978
1906
FHybr
1
81
1324
6080
14686
20580
17576
4488
696
24
530
122
10
2
2
5.50
6.52
4.96
7. Conclusions
In this paper the hybrid binary and ternary RM-RMH
transforms as well as fixed-polarity binary RM-RMH
transforms are proposed. We give also an example of
application of these transforms to the reduction of the
number of nonzero spectral coefficients. The experiments
show that the proposed techniques on average outperform
both the RM and fixed-polarity RM expressions.
237