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IEEE TRANSACTIONS ON COMPUTERS, VOL. c-28, NO. 12, DECEMBER
[2] S. H. Fuller, V. R. Lesser, C. G. Bell, and C. H. Kaman, "The effects of emerging
technology and emulation requirements on microprogramming," IEEE Trans.
Comput., voL C-25, pp. 1000- 1009, 1976.
[3] E. F. Moore, "Gedanken-experiments on sequential machines," in Automata
Studies. Princeton, NJ: Princeton University Press, 1956.
[4] S. Dasgupta, "The organization of microprogram stores," Comput. Surveys, vol.
11, pp. 39-65, 1979.
[5] K. Hwang, "Fault-tolerant microprogrammed digital controller design," IEEE
Trans. Industrial Electronics and Control Instrumentation, vol. IECI-23, 1976, pp.
200-206.
[6] A. K. Agrawala and T. G. Rauscher, Foundations of Microprogramming. New
York: Academic, 1976, p. 62.
[7] E. E. Swartzlander, Jr., "Microprogrammed sequential machines," in New Components and Subsystems for Digital Design, Santa Monica, CA: Technology Ser-
[8]
1979
tions and the resulting flow graphs are presented. The structure of
the flow graphs enables one to develop computational algorithms
for sequence length N = 8, 16, 32,*-.
Haar Transform and Rationalized Haar Transform
The HT and RHT of a data sequence {x(n)}T = {X(O)X(1)
X(N - 1)} where N = 2 , n = positive integer and superscript T
denotes transpose can be respectively defined as
I
{Yx(n)}
vice Corp., 1975, pp. I 1 1-1 14.
-, "Microprogrammed control for signal processing," 10th Workshop on
Microprogramming, (IEEE # 77CH1266-6C), 1977, pp. 80-84.
[HO(n)]{x(n)}
(1)
[ifO(n)]{x(n)}.
{YR(n)}=j
[HO(n)] and [Hio(n)] are the HT and RHT matrices of size
(2" x 2n). {Yx(n)} and {YR(n)} are the corresponding transform vectors. Based on [1], [Ho(n)] and [Hlo(n)] can be expressed as
Unified Matrix Treatment of
Discrete Tramforms
V. VLASENKO AND K. R. RAO
Abstract-An unified matrix treatment is developed for some
discrete transforms such as Haar (HT), rationalized Haar (RHT),
rapid (RT), modified Wash-Hadamard (MWHT), HadamardHaar (HHT)r, and rationalized Hadamard-Haar (RHHT)If. For
RT a technique for recovering the data sequence from the trans.
form vector is presented.
Index Terms-Discrete transforms, fast algorithms, Kronecker
products.
[Ho(n)] =
-i
Hl
F
r=n-l s=l
[Is,z] [[P0][MS][B¶
.]]
(D [[P'r][ffl2--r]] I
[H-O(n)] = 2n2r=n-1
f] |,EMIr]
1
(2)
(3)
s=
where r is the iteration number in the fast algorithm, s = number
of blocks in each iteration, and ® is the Kronecker product:
0
[Ilr] =
0
-
0
2r
0
0
INTRODUCTION
An unified matrix theory for various orderings of the Walsh0
Hadamard transform (WHT) has been presented recently [1].
-+
S
This has been also extended to the calsal ordering (WHT)s
4
2r
[2]. Such a matrix treatment is now developed for some other
discrete transforms such as Haar (HT) [3]-[7], rationalized Haar
s= 1
_ I
(3 [H1],
(RHT) [5], [7], rapid (RT) [8], [9], [15], modified WHT (MWHT)
[BS2i
u-ni I [I2--®]
[of
si
s >1
[3], Hadamard-Haar (HHT), [4], [18] and rationalized
Hadamard-Haar (RHHT), [17]. As before [1] this development is [I.] is un.it matrix of size (m x m),
[H1] is the WHT matrix
based on Kronecker products and permutation matrices. HT has
been applied to character recognition, data compression, and
1]
[HI]= [l
image processing [3], [7], [18]. In view of its invariance to cyclic
shift and to small rotations, RT has found applications in alphanumeric and Chinese character recognition [8], [9], [15]. As the
s= 1
| [I2---1 ]® [0 2(1-r- j/2
WHT power spectrum can be computed faster using the MWHT
=
2ir1/
[MS']
[3], [11], [16] and as the data sequence can be recovered faster
s > 1 scale matrix
[I2 n- r]
through the phase and power spectra using the MWHT [16], the
latter has gained some importance. WHT, of course, has been
p,] I[P2n-r--1, 2] S = 1
l [P2n-r, 1
S > 1.
utilized in various aspects of digital signal and image processing
[3], [4], [10], [12]-[14], [18]. (HHT)T which serves as a compromise
] is the "perfect shuffle" matrix, which can be expressed for
between WHT and HT has also been applied to image processing [P'
N= 8 as
[18]. Detailed development of the unified matrix treatment of
these discrete transforms is not presented here as the techniques
1 0 0 0 0 0 0 0
are similar to those of the (WHT)h [1]. Only the matrix formula0 10 0 000
0 0 00 1 00 0
[P4,2] = 0 O 00 00 1 0
[P2,4] = [P4,2]TManuscript received March 10, 1978; revised May 11, 1979.
0 1 00 0 00 0
V. Vlasenko was a visiting Professor at the Department of Electrical Engineering,
University of Texas at Arlington, Arlington, TX 76019 under the International
0 O 01 0 00 0
Research and Exchanges Board Scholarship during September 1977-June 1978. He
0 O 00 0 10 0
is now with the Odessa Polytechnic Institute, Odessa, USSR.
K. R. Rao is with the University of Texas at Arlington, Arlington, TX 76019.
.0 0 00 0 0 0 1
0018-9340/79/1200-0934$00.75 @) 1979 IEEE
935
IEEE TRANSACTIONS ON COMPUTERS, VOL. c-28, NO. 12, DECEMBER 1979
XM(l
0
2
3
4
5
6
7
(a)
X(i)
yR
0
0
2
2
3
3
4
4
5
5
6
6
7
7
(b)
Fig. 1. Signal flow graphs for (a) HT and (b) RHT.
Signal flow graphs for the fast computation of (1)-based on (2) and
(3) are represented in Fig. 1(a) and (b), respectively. The notation
and structure used in the flow graphs are the same as those in [1].
the recurrence relation [3]
[H1(k)] [11(k)]
[H(k + 1)] = [2k2I2k
15
k = 0, 1, 2,
.
(5)
Modified Walsh-Hadamard Transform
[H(O)] = 1
The idea of "perfect shuffle" for developing the fast algorithmic
computation of WHT has been shown to yield good results [1],
[2]. The signal flow graph for fast computation of (WHT)a [2] Analytical form of [H(k + 1)] can be expressed as
based on [1] is shown in Fig. 2. This technique can be applied to
1 0
MWHT for developing its fast algorithm.
H
=
[H(n)]
2
is
The MWHT of {x(n)} [3]
r=n- 1
{F(n)} = N [H(n)]{x(n)}
2r
(4)
.
1: ([P2] (9
s'l
I
) f-
where [H(n)] is the (2" x 2n) MWHT matrix, which is defined by Signal flow graph of (6) is shown in Fig. 3 for N = 8.
(6)
936
IEEE
X( i )
0
natural orderr
o
TRANSACTIONS ON COMPUTERS, VOL. c-28, NO. 12, DECEMBER 1979
Walsh
(sequency)
order
ca-sal order
0 wal (O,t)
2 cal(I,t)
2
4 -cal(2,t)
3
6 cal(3,t)
--
4
7 sol(4,t)
5
5 scl(3,t)
62
3 sal(2,t)
7
sol(l,t)
r-
Xr=O
|
r=2-
-Permutation stage
Fig 2. Signal flow graph for calsal ordeered Walsh-Hadamard transform (WHT),..
x i1
o
F(i )
0O
5,
2
2
2
2
3
3
-
4
4
2l
5
5
6
6
--
7
2
7
Fig. 3. Signal flow graph for mo3dified Walsh-Hadamard transform.
Hadamard-Haar Transform and Rationalized
Hadamard-Haar Transform
The Ith order Hadamard-Haar transform (HHT), is defined as
[4], [18]
*[[12.-k- l] (0 [H1]][P2, 2.-k-
{L,(n)}
where
[l ([I2k] () [[P23--kl, 2]
[RHH,(n)] =
=
I
[HHI(n)]{x(n)}
(7)
[HH,(n)] [Go(l)] ® [HO(n 1)] and [Go(l)] is the (2' x 2')
=
Il]
2r
[I2 ]II- r=n-l-l
fI zE
[IS2]j
s=l
-
(WHT)h matrix. The rationalized version of (HHT), and its inverse can be expressed as [17]
{L,(n)} = [RHH,(n)]{x(n)}
(8)
{x(n)} [RHH,(n)]f[P,(n)]{L,(n)}
=
®
is
and
the
dia[RHH,(n)] [Go(l)] [fio(n 1)]
[P,(n)]
(3 [[P"r][BW2"-,]
| ]-
(10)
Signal flow graphs of (9) and (10) are represented in Figs. 4 and 5,
respectively (N = 8, 1 = 1).
=
where
gonal matrix whose diagonal elements are negative integer powers
of 2. Matrices [HH,(n)] and [RHH,(n)] can be expressed as
[HH,(n)] = y
l {[I2s] ® [[P2-5-k1, 2]
[[I2n-k-1] 0) [H1]][P2,2-0-1l])
0
I[[P2][M
_ Z['r
[B=n(-]}JS
{[IS21
(3[Pr=[s]B2lr} ]
(
Rapid Transform
Although RT is not an orthogonal transform it has some interesting properties (apart from the computational simplicity), such
as invariance to circular shift, to reflection of the data sequence
and to the slight rotation and inclination of a two-dimensional
pattern [8], [9]. Also there is some correspondence between periodicity and null subspace in the pattern and transform domains.
These properties have led to its application in character recognition [8], [9], [15]. The algorithm for RT [8] can be described by
X(r +) = X(r) + X)(N/2)
N= O, 1 2 * 2
=y(rp)
12, DECEMBER 1979
IEEE TRANSACTIONS ON COMPUTERS, VOL. c-28, NO.
937
[P4,I [P2,4]
X(i)
0'
2
3 '
4'
5 '
6 '
Fig. 4. Signal flow graph for
XCi)
[P2,2 ]
(HHT)j, N = 8.
[ ]P4 ] [.2,4]
[P -2
;p
Li
)
A
2
3
4
5
5
6
Fig. 5. Signal flow graph for
XS+(N2) = X -
and powers of odd elements along the main diagonal are constructed from
(Nl2)
=|P1) |
p=0
1, 2,*--, N -1
(RHHT)j, N = 8.
(11)
K
0
[R(n)]= r =nfI 1 {[Mr(n)][I2n--]
...
...
kp,o
L
k(N,2) - 1,0
nn
(-1)k
[M,(n)] =
r
1
(13)
(13)
( I1 (xt2) - I.
0
(14)
-I
k(N12)- 1,n- 1
A1/2
(15)
X
The matrix of states (15) can be used for computing the inverse
RT. Signal flow graph for RT is shown in Fig. 6(a). Structure. of
systems for RT and its inverse is shown in Fig. 6(b).
0
(_l1to.r
ko,n
Kl,n-lI
kp,r
where
1
nl+ I <<°0.
The number of elements kp,+ 1 is Nk = N/2 log2 N, which can be
arranged in a rectangular matrix (N/2 x n) as
(12)
(3 [Hl][P2, 2X-1]}
np+l
)o
P,7 I 1
CONCLUSIONS
In this paper the "perfect shuffle" method [1] was applied to
several orthogonal linear and nonlinear transforms, which are
widely used for digital signal processing. Analytical formulas
which facilitate the computation of transform coefficients and the
practical organization of signal processing algorithms were ob-
938
IEEE TRANSACTIONS ON COMPUTERS, VOL. c-28, NO. 12, DECEMBER 1979
X{;
)I[diog
[diog
[Hl]
0
2
Y
4
5
6
Fr-O~~~~~4
k1r-
(a'I)
{X ..}
ln-l}
*
-r--r
-1-T
Compute
(I
l
-I
pI
pute
I1
I
y
I
DecisionI
{KP
{Kpo}
l
{Kp2}
{Kpn--1
Construction of Matrix
[kPr]
(b
)
Fig. 6. (a) Signal flow graph for the one >dimensional RT, N = 8. (b) Implementation of RT a nd its inverse.
tained. A technique for recovery of input data for the RT was also
presented.
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[3] N. Ahmed and K. R. Rao, Orthogonal Transformsfor Digital Signal Processing.
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[4] S. Welding, G. Gagneux, and G. Stamon, "A set of invariants within the power
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N. Ahmed, P. J. Milne, and S. G. Harris, "Electrocardiographic data compression via orthogonal transforms," IEEE Trans. Biomed. Eng., vol. BME-22, pp.
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[15] M. A. Narasimhan and K. R. Rao, "Printed alphanumeric character recognition
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[16] J. Ninan and V. U. Reddy, "BIFORE phase spectrum and the modified Hadamard transform," IEEE Trans. Acoust. Speech Signal Processing, vol. ASSP-23,
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[17] K. R. Rao, A. Jalali, and P. Yip, "Rationalized Hadamard-Haar transform," in
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[18] K. R. Rao, M. A. Narasimhan, and K. Revuluri, "Image data processing by
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