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BLOCK-DIAGONAL STRUCTURE OF
WALSH-HADAMARD/DISCRETE COSINE
TRANSFORM
where G\m) is the dyadic conversion function (Gray code conversion function). The [£(#)] is another N x N, bit-reversed
permutation matrix. The entity of [B(N)] is given as
Indexing term: Transforms
The /(-matrix, the conversion matrix for Walsh-Hadamard/
discrete cosine transform, is known for its efficient blockdiagonal structure. This associates with the even/odd
structure of the transform kernels. In the letter we present a
direct matrix derivation by using the intrinsic properties of
the discrete cosine transform and the Walsh-Hadamard
transform.
Introduction: Hein and Ahmed have shown that the discrete
cosine transform (DCT) can be implemented via the WalshHadamard transform (WHT) through a conversion matrix
which has a block-diagonal structure.1 Kwak and Rao even
derived a C-matrix transform by trial-and-error to keep the
block structure in integer form.2 However, Jones, Hein and
Knauer have shown conceptually that a conversion matrix
from one even/odd transform to another requires only multiplication by a sparse matrix,3 yet there is no formal derivation
for the block-diagonal structure of this conversion matrix, the
/1-matrix. Recently, Hsu and Wu proposed a fast computation
algorithm for the discrete Hartley transform via the Hadamard transform4 and derived the block-diagonal structure in
recursive matrix operation.5 It is our aim in this letter to
provide a direct proof of Hsu and Wu's derivation for the
block-diagonal structure of the ,4-matrix, by using the intrinsic properties of the DCT and the WHT.
Conversion matrix: A-matrix: The DCT is derived through the
WHT by means of a conversion matrix, the /1-matrix. It was
proposed by Hein and Ahmed that
{CT(N)} = IA(N)11WT(N)]{X(N)}/N
= (DCT(N)-\IWT(N)']T/N
(2)
In eqn. 1 {CT(Nj} is the N x 1 column vector which denotes
the DCT-transformed data of size N. [DCT{N)] and [WT{N)]
are the N x N transform matrices of the DCT and the
sequency-order WHT, respectively. The superscript ^ \ stands
for the row ordering of the underlined matrix in bit-reversed
order, and the superscript T stands for matrix transpose. The
/1-matrix is orthonormal and has block-diagonal structure of
the form
0
1
L °
for n = bit reversal of m
= 0,
otherwise
Since
[WT{N)Y =
(7)
then we substitute eqns. 4-6 into eqn. 7 to obtain
IWT(N)Y
= [D(Ny][WH(Ny]
"N
where
[Y(A0] =
(9)
The permuted DCT matrix [Y(iV)] is the rearrangement of
[DCT(iV)] such that the rows of the [DCT(iV)] are ordered in
bit-reversed form while the columns are ordered in dyadic
one. In other words, the input data are permuted in dyadic
ordering while the outputs are in bit-reversed order. Then, the
symmetrical property of the entities of this matrix can be
shown and expressed in the following recursive form:
\_B(N/2)
Intrinsic properties of WHT and DCT: It is known that the
natural order (Hadamard order) WHT can be generated from
the lower-order matrices by
(3)
-WH(N/2)\
Eqn. 10 means that the N x N permuted matrix of [£>CT(AT)],
[F(iV)], can be decomposed into an N/2 x N/2 decompositable submatrix [Y(iV/2)] and a nondecompositable submatrix
[fi(N/2)] recursively. This is owing to the even/odd structure
of the DCT. In the second iteration, the [Y{N/2)'] can be
decomposed into N/4 x N/4 submatrices [Y(N/4y\ and
[B(AT/4)], respectively. After iteration of (log2 N — 1) we
obtain
-[! -A-
[WH{2)-\
Proof of block-diagonal structure of WHT/DCT: Using the
intrinsic properties given in eqns. 3 and 10, we can rewrite
eqn. 2 as
I \Y{N/2)
N\_B(N/2)
Y(N/2)JWH(N/2)
-B(N/2)j_WH(N/2)
[WT{N)] =
(4)
(5)
IWT(N)Y =
(6)
where [D(N)~\ is an N x N permutation matrix. The entity of
[D(A0] is given as
n, m = 0, ..., N - 1
= 0, otherwise
ELECTRONICS LETTERS 8th October 1987
Vol. 23
No. 21
WH(N/2)1
-WH{N/2)\
\_V2Y{N/2)WH{N/2)
~N \_B(N/2WH(N/2) - WH(N/2))
Y(N/2)(WH(N/2) - ^H(iV/2))"|
2B{N/2)WH{N/2)\
The properties of the sequency-order WHT matrix [WT{N)~\
and the natural-order WHT matrix [WH(N)~] are given a s 7 8
for n = G{m)
(10)
-B(N/2)j
More interestingly, a similar property is also exhibited by the
discrete Hartley transform.49
where A(N/2) is the conversion matrix of dimension
N/2 x JV/2 and R(N/2) is the remainder matrix of the same
dimension.6 This feature results in substantial savings in the
number of multiplications and additions.
+ i)(n+1)
(8)
Substituting eqn. 8 into eqn. 2, we obtain
(1)
where
= 1,
2_rY(N/2)WH(N/2)
N|
0
0
"
B(N/2)WH{N/2)
and finally
01(2)] = [»7f(2)]|T(2)] =
So the fact that the /1-matrix has a block-diagonal structure
for any length of power 2 is now proved. Following the same
1123
derivation, it is also easily shown that the block-diagonal
structure is also exhibited by the Walsh-Hadamard
transform/discrete Hartley transform.5
Conclusion: A bare-bones summary of the proof is that the
intrinsic properties of the DCT and Hadamard-order WHT
result in the block-diagonal structure. An interesting benefit of
this method is that a similar structure can also be derived for
the conversion between the natural-order WHT and the
unitary transforms which possess the prescribed intrinsic
properties.
CHAU-YUN HSU
JA-LING WU
voltage becomes, and the induced voltage causes severe
damage to equipment, repeaters, cables and personnel.
According to the calculation of induced voltage on a 700 km
length of coiled cable, a few hundred kilovolts will be induced
by a sudden change of 1-6 A of current, the nominal power
feed current for an optical submarine repeater.
3rd September 1987
Department of Electrical Engineering
Tatung Institute of Technology
40 Chung-Shan North Road
3rd Sec. Taipei, 10451
Taiwan, ROC
h=3 m
References
1
HEIN, D., and AHMED, N.: 'On a real-time Walsh-Hadamard/cosine
transform image processor', IEEE Trans., 1978, EMC-20, pp.
453-457
2
KWAK, H. s., SRINIVASAN, R., and RAO, K. R.: 'C-matrix transform',
ibid., 1983, ASSP-31, pp. 1304-1307
3
JONES, H. w., HEIN, D. N., and KNAUER, s. a : 'The Karhunen-Loeve,
discrete cosine, and related transforms obtained via Hadamard
transform'. Conf. on int. telemeter., Los Angeles, California, 1978
pp. 87-98
4
CHAU-YUN HSU and JA-LING WU.: 'The Walsh-Hadamard/discrete
Hartley transform', Int. J. Electron., to be published
5
6
7
8
9
CHAU-YUN HSU and JA-LING WU: 'Fast computation of discrete
Hartley transform via Walsh-Hadamard transform', Electron.
Lett., 1987, 23, pp. 466-468
ELLIOTT, D. F., and RAO, K. R.: 'Fast transforms: algorithms,
analysis and applications' (Academic, New York, 1982)
AHMED, N., and RAO, K. R.: 'Orthogonal transforms for digital
signal processing' (Springer-Verlag, New York, 1975)
BEAUCHAMP, K. G.: 'Applications of Walsh and related functions'
(Academic Press, 1984)
HOU, H. s.: 'The fast Hartley transform algorithm', IEEE Trans
1987, C-36, pp. 147-156
REDUCTION OF HIGH SURGE VOLTAGE
ON COILED OPTICAL FIBRE SUBMARINE
CABLE
optical submarine
repeater
|95U1|
Fig. 1 Dimensions of coil winding up 700 km length of OS submarine
cable
This letter proposes some useful methods to reduce the
induced voltage, and their effects have been examined experimentally. The methods are categorised in two principles; the
former is based on the addition of capacitance to cables and
the latter on the reduction of inductance of coiled cable.
Fig. 2 Measured waveform of induced surge voltage on test cable 1 in
air
Io = 2mA
Indexing terms: Optical fibres, Cables
The induced high surge voltage caused by a sudden change
of feed current on the long coiled optical fibre submarine
cable has been measured. The reduction methods of the
induced voltage have been examined and their effects are
demonstrated.
Optical fibre submarine cable, especially deep-water cable, is
so small in diameter that a very long cable can be wound up
in the cable tank at the cable depot and on board ship, and
consequently it makes a big coil with extremely large inductance. For example, as shown in Fig. 1, a coil, consisting of
27 000 turns (N) of 22-5 mm in cable diameter (dj), 3 m in
height (/i) and 13m/3-6m in outer/inner diameter (Do/D.) is
formed by winding up a 700 km length of OS cable,1 which is
the nominal length in a cable tank at a depot and on a ship.
The inductance is estimated to be about 4000 H. Furthermore,
the capacitance of the deep-water cable is small because of the
lack of an outer conductive layer. For these circumstances a
very high surge voltage will be induced if current on the cable
is changed abruptly by some kind of disconnection of the
power line. The longer the cable, the higher the induced
1124
Fig. 2 shows the measured waveform of induced surge
voltage caused by a sudden cut of 2 mA of current / 0 on a
53-8 km length of deep-water cable (test cable 1) coiled in 6 m
diameter of cable pan, and its peak voltage Ep is 36-3 V. The
dimensions of the coil are as follows: Do = 6m, £), = 2-7m,
ft =1-13 m and N = 3942 turns. The surge voltage was
recorded by wave memory. According to the result, a peak
voltage of 29 kV is estimated to be generated by an abrupt
change of 1-6 A in current. The experimental results to reduce
the induced voltage are as follows:
(i) Addition of capacitance to cables is expected to reduce the
induced voltage because of filtering high-frequency components of the induced voltage. To confirm the effect, cable 1
was immersed into earthed water. Fig. 3 shows the experimental result of induced voltage caused by a current change of
2 mA, and the peak voltage is 6-5 V. Comparing to the experimental result shown in Fig. 2, a suppression factor of induced
voltage (SFIV) of 18% is obtained. On the other hand, to cope
with the cables that cannot be immersed into water, which is
usual in a cable depot, a conductive layer composed of copper
sheet 1 mm thick and 1 m wide was set on the top layer of test
cable 2 along its circumference and was earthed. The length
ELECTRONICS LETTERS 8th October 1987
Vol. 23 No. 21
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