Pseudo-Random Sequences and Arrays - CiteSeerX

PROCEEDINGS OF THE IEEE, VOL.

[ 541 [SS]

[ 561

[57] [ 581 [59] [60]

64, NO. 12, DECEMBER 1976

1715

rent rotating arcs,” Inst. Elec. Eng. Int. Conference on Gas Discharges, pp. 5 3 4 4 3 8 , 1 9 7 4 . A. E. Guile, “Model of emitting sites and erosion on nonrefractory arc cathodes with relatively thick oxide fii8,”Proc. Inst. Elec. Eng.,vol. 121,pp. 1594-1598, 1974. I. E. Harry, “The measurement of the erosion rate at the electrodes of an arc rotated by a transverse magnetic field,” J. Appl. PhyS., VOI. 40, pp. 265-270, 1969. A. E. Guile and J. G. J. Sloot, “Sudden transitions in velocity of magnetically driven arcs,” in Con5 Rec., IEE Int. Con$ on Gas Discharges, pp. 544-548,1974. K. Taylor, “BritishRailways’ experiencewithpantographsfor high-speed running,” J. Inst. Mech. Eng. (Railway h’visfon), vol. 2, pt. 4 , pp. 349-373, 1971. A. E. Guile, V. W. Adams, W. T. Lord, and K. A. Naylor, “Highcurrent arcs intransverse magneticfields in air at atmospheric pressure,”Roc. Intt. Elec. Eng., vol. 116, pp. 645-652, 1969. A. E. Guile and A. H. Hitchcock, “Time variation in copper cathode erosion rate for long duration arcs,” 1. Phys. D., vol. 8, pp. 427-433, 1975. C. W. Kimblin, “Anode phenomena in vacuum and atmospheric pressure arcs,” IEEE ?Yam. Plasma Science, vol. PS-2, pp. 310319, 1974.

of rotatingarc (611 A. E. Guile and A. H. Hitchcock,“Theeffect velocity on copper cathode erosion,” J. Phys. D, vol. 7, pp. 597606, 1974. [ 621 H. Edels, “The arc discharge, Pt. 2, Interruption and reignition,” F’roc. V I Yugoslav Symposium and Summer School on the Physics o f Ionized Gases (Miljevac by Split, Yugoslavia), pp. 419439, 1972. [ 631 J. F. Perkins, “Dielectric recovery and interruption variability of air arcs at atmospheric pressure,” IEEE i%ms.Power App. Syst., VOl. PAS-93, pp. 281-288, 1974. ( 6 4 1 J. Clark, “Electromagnetic interference shielding,” Electron, No. 5 1 , pp. 80, 85, 87, and 89, May 1974. [ 6 5 ] D. Klapas, “Anode spot transitionphenomena inlow current copper vacuum arcs,” Ph.D. thesis, Liverpool University, 1973. [ 661 T. Takakura, K. Baba, K. Nunogaki. and H. Mitani, “Radiation of plasma noise fromarc discharge,” J. Appl. Phys., vol. 26, pp. 185-189,1955. [ 671 M. I. Skolnik and H.R. Puckett Jr., “Relaxation oscillations and noise from low-current arc discharges,” 1. Appl. Phys., vol. 26, pp. 74-79,1955. [ 6 8 ] G.R. Jordan, B. Bowman, and D. Wakelam,“Electrical and photographic measurements of high-power arcs,” J. Phys. D: vol. 3, pp. 1089-1099, 1970.

Pseudo-Random Sequences and Arrays F. JESSIE MAcWILLIAMS AND NEIL J. A. SLOANE, MEMBER,

IEEE

~brtroct-wienysequencw of h g t h n 2”’ - 1 w b a e aIrtocoak- (281, 1291, Kautz [421, Selmer [601, Zierler [761, [771, and t i o n h n c t i o a L ~ l o r - I / n b . n c ~ h r o n n f o l r ~ ~ , r[44, a C ch. 141. rteallcd~(orpN)~pcacs~ofot~ MNevertheless they are not as widely known as they shouldbe, rsghrta .qasplc# ~ n amp of M n = 2 h - 1 with especially outside of the area of communication theory-see theamepopgtrlnrs~tlybscafoundby~daethon TLb . [62], [66]-and there does not seem to exist a simple, ~ g i ~ a ~ d c r a i p t i o l l o f r m c b a e q a a n c g m d r m y r d t b s[61], i comprehensive account of theirproperties.The reader will n u n y h find such an account in Section I1 of this paper. I. INTRODUCTION Recently, several applications ([371, [381, [431, [591) have SEUDO-RANDOM SEQUENCES (which are also called called for two-dimensional arrays whose two-dimensional autopseudo-noise (PN) sequences, maximal-length shift- correlation function should satisfy p ( 0 , O ) = 1, A i , i ) small for (i, 1) # (b,o). We shall see in Section 111 that it is very register sequences, or m-sequences) are certainbinary easy to use pseudo-random sequences to obtain n l X nz arsequences of length n = 2m - 1 (the construction is given in Section 11). They have many useful properties, one of which is rays with that their periodic autocorrelation functionis given by 1 p(i,j)=--, forO 4 . T h e A f j s a r e 0 , 1 , 0 0 r 0 2The . desired binary arrays aregiven by at/=1ifandonlyif

1 (number of timesti is 1, w or u2;ti+' 255 i s O , w ~ r u ' ; f i + ~ ~ i s1Oo ,r o 2 ; a n d fi+3f

34 =-=255

is 0, 1 or0) 81 255'

(43)

ifandodyif A l j = o

d t j = 1 i f a n d o w i f A,,=u2.

f i r , s) =

If rn > 4 then

A,j=O

b t l = li f m d o n l y i fA t i = l Ct/=1

(42)

u,a ' } exactly once. Then

m-2

=-

(41)

(40)

To test whether these sequences arelinearly independent we

MACWILLIAMS SLOANE: AND

AND ARRAYS

PSEUDO-RANDOM SEQUENCES

map them into GF(4m) using the mappinggiven in Section 11-K.NOW

f(F11,

1,

E-'f(t-l

K ZW 1 ) ,

F 3'f(P

1

1727

kl k2 and minimum distance 2 k l k z - l . The rows of C may be taken to be the binary kt kz-tuples which are rectangular coordinates for the elements 1, a, a', * , a"-' of GF(2k1 ka ). Of course a" = 1. Let Z(i, j ) be the unique integer satisfying

are linearly independent over GF(4) if the equation 00

+a16 +az62 +a363

I ( i , j ) E i (mod n l )

=o

where 6 = t-', ui E GF(4), implies all ui = 0. This means that the equation of least degree satisfied by 6 over GF(4) must have degree 2 4. Most elements of GF(4m) satisfy a minimal equation of degree m (which is the reason for the condition m 2 4). However, the elements 1, 5(4m-1)/3 and [2(4m-1)/3 satisfy anequation ofdegree 1. If m is divisibleby 2 or 3 there are also elements which satisfy equations of degree 2 or 3. ii) Suppose t = (4m - 1)/3.Then z'f(z) = w * ' f ( z ) , z2 ' f ( z ) = w * ' f ( z ) , and z3'f(z) = f ( z ) . The columns of (42) are of one of the types 0 1 ow2

O1ww2 0 w w2 1

or

owwz 1

0 1 ww2

0 1 ww2

-'

T ( O , O ) = ~ ( 1 0 , O= ) 0, 7(5,0) = 0.502

< 14,

r # 5, 10

T(r, s) = 0.3 18 for all other pairs ( r , s) in

therangeO