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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