II B HAR.Y 
 
 OF THE 
 U N IVER.SITY 
 Of 1LLI NOIS 
 
 621 365 
 
 Ue655-te 
 
 no. 40-49 
 
 cop. 2 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/onsynthesisofstr44mitt 
 
ANTENNA LABORATORY 
 
 Technical Report No. 44 
 
 ON THE SYNTHESIS OF STRIP SOURCES 
 
 by 
 Raj Mittra 
 
 4 December 1959 
 
 Contract AF33(616)-6079 
 Project No. 9-(13-6278) Task 40572 
 
 Sponsored by: 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
'lr 
 
 11 
 
 ACKNOWLEDGEMENTS 
 
 The author wishes to acknowledge the Wright Air Development Center 
 for sponsoring the research reported here through Contract AF33 (616)-6079. 
 It is also a pleasure to thank several people in the Antenna Laboratory 
 of the University of Illinois for helpful discussions. 
 
Ill 
 CONTENTS 
 
 Page 
 
 Acknowledgements ii 
 
 1. Introduction 1 
 
 2. The Lagrangian Interpolation Method 3 
 Conclusions 17 
 References 18 
 
1. INTRODUCTION 
 
 2 
 It is well known that the space factor of the radiation pattern due 
 
 to an infinite strip source lying in the y = plane between x = + a and 
 
 with edges parallel to the z-axis, can be expressed in the form 
 
 p* f 
 
 S(u) = — \ ex P (ipu) g(p) dp (1) 
 
 Here u = *-— cos p = — £> = -— X = wavelength 
 
 The problem of synthesis can be stated as in the following: given a 
 
 S(u) for either a set of values for u or for continuous values of u in 
 
 a range, find g(p) using Equation (1). 
 
 The range of u is usually divided up into the visible and invisible 
 
 range of the spectrum. The visible range, from the definition of u is 
 
 obviously given by u < >—- , and S(u) for u > *-— is associated with stored 
 
 3 
 energy in the system. The properties of S(u) have been discussed by Taylor. 
 
 4 5 
 
 Woodward and Woodward and Lawson have considered the synthesis problem 
 
 when S(u) is assigned at a finite number of points in the visible range. 
 Knudsen pointed out the relation between this problem and the sampling 
 theorem of bandwidth limited functions. Yen discussed the problem of 
 synthesis when the assigned points are distributed arbitrarily in the 
 visible range. Yen s paper is the most general work on the subject available 
 so far. 
 
 The purpose of the paper is to show, in the first instance that the 
 expression developed by Yen is what is obtained by using Lagrangian inter- 
 polation through the assigned values of u. Once this is realized a more 
 general type of interpolation becomes possible. The approach to be used 
 
2 
 here is therefore different from that of the sampling theorem idea used by 
 Yen. A proof based on lines entirely different from Yen s is also derived 
 for the form of the source function which is associated with minimum energy 
 storage in the system. 
 
 The second part of the paper deals with the problem of synthesizing 
 g(p) when S(u) is known for a range of u rather than at a set of points. 
 The range would ordinarily be the visible range. When S(u) is synthesized 
 by interpolating through a finite number of points there is little control 
 over the value of S(u) for the intermediate u's. One would, however, like 
 in many cases, for the pattern to behave in a specified manner in the entire 
 visible range of u, allowing of course, a certain degree of tolerance in 
 the design. This problem is dealt with in the second part of the paper. 
 The solution to this problem is derived by directly solving the integral 
 equation (1). 
 
2. THE LAGRANGIAN INTERPOLATION" METHOD 
 
 The synthesis problem for S(u) will be considered as the problem of 
 interpolating it through an assigned number of points, by the method of 
 Lagrangian Interpolation. It will be shown that for a particular case, 
 the solution obtained is the same as given by the sampling theorem approach. 
 It will then be pointed out that the interpolation may be done in terms of 
 type of functions more general than those used in the sampling theorem 
 approach. 
 
 Consider first of all the problem of interpolating a function f (u) 
 
 which has the value f (u ) at u = u , p = -r -r-1. ...0 1 2 ...n. Apply- 
 
 P P 7 7 > 
 
 ing Lagrange's Interpolation formula, one has 
 
 n ( U -u) n ( U -u) 
 
 r. , ^ *, x (-r) m „, x -(r-1) m 
 
 f(u) = f(u ) = — ■ — -, * + f(u , , . )„ -. 
 
 ~ r ^ , (u -u ) -(r-1) II _ ,v(u -u j 
 (-r) m -r -(r-1) m -(r-1) 
 
 (2) 
 
 n. ,(u -u) 
 
 + -----+ f(u ) 
 
 IT (n) (u m- U n ) 
 
 where 
 
 n 
 
 n (o) (U m- U) = n (U m~ u) = ( U - r - u)(U -(r-l)- u) — (u o' U) - - (u n" U) > 
 VHy m=-r 
 
 m+p 
 
 with the factor (u -u) missing. A simple substitution of u = u shows that 
 p m 
 
 the formula in (2) gives the assigned value f(u ). 
 
 One can easily verify the following rearrangement of (2) for f(u), 
 
 u-u 
 
 f( U ) = s f(u ) n . (i - — — r> (3) 
 
 p=-r p m=-r(p) u m "U p 
 where the subscript (p) in the product implies that the factor corresponding 
 
4 
 to m = p is missing. 
 
 Now consider the particular case when the spacing of assigned points is 
 a unit distance in terms of u, and u = 0, i.e u = p, and p is an integer. 
 Furthermore let both n and r tend to infinity. Then if f(u) is assigned 
 for all values of u in the range - oo < u < oo at unit distances, then using 
 (3), it can be interpolated as 
 
 f(u) = E f(u) n (1 - -H_E), m ± 
 p=-oo p m=-=-= m ' 
 
 m=0 
 
 (4a) 
 
 = e f(u) n i - < u -p> 2 
 
 p=-oo p m=l 2 
 
 L m J 
 
 sin TTx 
 Using the well known infinite product expansion of — — , which is 
 
 7Tx ' ' 
 
 sin ^ = ff (1 _ ±_ ) 
 
 TTx m=l 2 
 
 m 
 
 one can rewrite (4) as 
 
 f(u) = E f(u ) Si " \ (U - P ^ (4b) 
 
 p=- ^ p 7T (u-p) 
 
 Equation (4b) is the expansion usually derived using the sampling theorem 
 
 of band-limited functions. 
 
 Using the Lagrangian interpolation method used above it is fairly 
 
 simple to extend (4b) for the case of arbitrarily distributed assigned u's. 
 
 Suppose, for instance, q of the assigned points do not coincide with the 
 
 unit points. Then the assigned points can be divided into two groups. Let 
 
 u , for n = 1, 2, . . . q be the only non-unit sample points and let the rest 
 
be the unit ones, i.e., u = v , an integer when, p ^ 1, 2, ...q. Then 
 using the general interpolation formula (3), when r and n — »-eo, one can 
 write 
 
 f ( U ) = s " f (v ) n , ji - - i n /l 
 
 p=-co p ■■ - (p) I v -v / m=l u -v 
 
 m=-oo \ m p / y m p / 
 
 (5) 
 
 q co// / u-u^ \ q / u-u \ 
 
 pfei P m =-co ^ »„-« p y» d <p) ^ v u P J 
 
 The double primes on the summation £ and on the product n j.mply.that the 
 terras corresponding to p or m = 1, 2, . . .q are excluded. One can see that 
 the right hand side of (5) has been deliberately broken up into two parts, 
 involving the unit and non-unit sample points. This is done for convenience, 
 as will be clear in what follows. 
 
 00 // U "V 
 
 In the infinite product Ft . v (1 - ), of the first term in the 
 
 (p) um-v » 
 m=-co * p 
 
 r.h.s. of (5) there are missing q factors corresponding to q non-integers, 
 
 u = u , u , . . . u . These factors have been replaced by the second product 
 1' 2' q 
 
 00 // 
 
 containing q factors. But IT can be written as 
 
 m=-co 
 
 But 
 
 f co u-V d 
 
 n (p) « - Hh> 
 
 m=-oo m p 
 
 q u-v 
 
 n (i £) 
 
 r=l v v -v 
 
 r p 
 
 u-v co (u-v ) 2 \ sin 7T(u-v ) 
 
 (6) 
 
 m=-co <P> (1 " v^F ) = m5l K 1 
 
 r * 
 
 m p 
 
 l 
 
 since v s are integers. 
 P 
 
Using this manipulation in (5) one can obtain 
 
 J n (i e_)l 
 
 <x> // sin 7T(u-v ) ) m=l u -v f 
 
 S(u) =2 S(V ) —. JL L. E_ZJ 
 
 P= -oo p 7T(u-V p ) q 
 
 n (i £_> 
 
 m=l v -v 
 
 m p 
 
 co^ u-u q u-u 
 
 n (l- p > n v (l- p ) 
 
 q sin 7r(u-u ) m= i " v -u ' -; ( p ) u-u 
 
 + 2 S (u ) v ■ P m ^ m=1 ^_P_ ( 7 ) 
 
 p=l p 7T(u-u ) oo u-u 
 
 p n (i H> m t p 
 
 m=-oo m 
 m^O 
 
 It can be verified that if u = u , p = 1, 2,...q are also integers then 
 (7) simply reduces to (4b). 
 
 There is a definite reason why (7) wag chosen to be written in the 
 particular form used in that equation. Up until now all the efforts were 
 directed towards obtaining an expression by which S(u), which is specified 
 at a set of values for u, either unit of non-unit, could be interpolated. 
 However that is only a part of the task. The main object is to obtain the 
 function g(p) when S(u) is known. Stated in terms of the mathematical 
 language, it would mean that S(u) is to be interpolated in such a form that 
 
 oo 
 
 2 ^ / S(u)e" ipU du = h(p) (8) 
 
 > -oo 
 
 where 
 
 h(p) = g(p) for -1 < p < 1 
 
 = for I p| > 1 
 
 Equation (8) is of course obtained by regarding S(u) as the Fourier 
 
 transform of " ^ . and taking the inverse transform of S(u) thed 
 7T ' & 
 
yields h(p). Hence one has to impose the following two conditions on the 
 interpolated form of S(u), viz. 
 
 (a) That the Fourier transform integral of S(u) i.e the l.h.s. 
 of (8) exist. 
 
 (b) That h(p), the transform of S(u) satisfy the condition, 
 
 h(p) = 0, for p > 1. 
 As may be easily shown, the Fourier inverse transform of any 
 function F. which is of the form 
 
 , order of Q(u) > order of P(u) (9) 
 
 •77- (u-u/ ) Q(u) ' 
 
 where P(u) and Q(,u) are polynomials, has a Fourier transform say, f(p), 
 
 where f(p) = for |p| ? 1. 
 
 One can verify that each term in the summation (7) is a function of 
 
 the type (8), and hence satisfies the required condition that its transform 
 
 is of the type f(p). It is also quickly verified that since the order of 
 
 the polynomials in the numerator is equal to the denominator, (each term 
 
 sin(u-u ) 
 
 behaves as K ■" -, (k = constant] as jui — ^oo j The expression in (5) 
 p u-u \k " 
 
 P " 
 
 was rewritten in the form of (7), in order that the form of S(u) be of the 
 
 desired type. 
 
 The expression given in (7) is not the only one which can be used to 
 
 sin tt(u-u ) 
 
 express Sfu) . Instead of using the factor ————_-— J- to modify the second 
 
 IT (U-U ) 
 
 , . sin itu _, 
 
 summation in (5), it is possible to use the factor — — — — - — ' The expression 
 7 sin iru 
 
 P 
 
 for S(u) then reads 
 
u-u u-u 
 
 p 
 
 q n' /(1 -Hr> & (p) d - 
 
 v- sin mi m=l m u n m-1 py 
 
 S(u) = S, + S flzi. ■ m ~P m^j^l __V U P ( 10 ) 
 
 1 p=l Sin TfU ox, 
 
 p n (i - «) 
 
 m ^ 
 
 where S is the first series in the right hand side of Equation (7). With 
 this choice of representation for S(u) the expression in (10) resembles 
 (as can be shown by simple manipulation), the one given by Yen The 
 method of approach in this paper is quite different and somewhat more 
 general . 
 
 It is easily seen that more general interpolation using Lagrange s 
 formula is possible by replacing the variable u by ^(u) , u by ^(u ) and 
 
 v by Ti(u ) at least when the number of points specified are finite. It 
 
 m m 
 
 is necessary though that the function "n(u) be so chosen as to make S(u) 
 satisfy (8) . 
 
 The problem of calculating g(p) from S(u) is straightforward The 
 expression for g(p) which is simple to derive, is 
 
 ^ gfp) = g S(.n) < ipn (ID 
 
 IT n--x> 
 
 S(n) is obtained by substituting u = n in the expression (5) or (7). It 
 
 should be pointed out that it was not necessary to recast (5) into (7) 
 
 except to show that the inverse of S(u) is zero outside the range jpj< 7T 
 
 and perhaps to put it in a form more suitable for computation. 
 
 Minimum Supergain Criterion;- 
 
 As pointed out by Yen, for increased efficiency one should set the 
 
 criterion 
 
 P = j | g(p) ] 2 P = mm 
 
 " -17 
 
for a given number of specified values of S(u). The Supergain ratio of 
 v as defined by Taylor, is 
 
 2f \ U(P)J 2 dp 
 
 (12) 
 
 r-2a/\ 
 (J^ 2 I S(u)| 
 
 ^2a/X 
 
 It is seen that for different g(p)'s which have the same visible spectrum 
 
 the one with the minimum "Y has the lowest supergain ratio. 
 
 From (11) it is seen that the problem is to minimize SlS(n)j , under 
 
 the conditions imposed on S(u), that it have the required values at 
 
 n = v , V , . ...V (integer values) and u = u , u , „ „ . u (non-integer 
 l 7 2 r 1 2 q 
 
 values)^ where, r + q - N, say ? is finite. 
 
 This problem has been attacked by Yen, by the method of Lagrangian 
 mpltipliers. An alternative method^ which uses the concept of a vector 
 space is to be described here Consider first of all ? an analogous problem 
 of constructing a column vector U ^hich satisfies the equations 
 
 a • U = X , n = 1, 2, ... N (13) 
 
 n n' 
 
 where, 
 
 a = known row-vectors 
 n 
 
 X = given numbers 
 n 
 
 and the dot indicates the operation of a product of the matrices,, 
 
 it is clear that if the dimension (number of elements in the row-vector' 
 
 of a is greater than N, the equivalent matrix equation does not have an 
 
 n 
 
10 
 
 unique solution for U. Now consider imposing an extra condition on U 
 which requires it to satisfy 
 
 Magnitude of U = min. (14) 
 
 When the above constraint is added an unique solution is indeed 
 
 obtained as shown by the following argument. 
 
 Suppose the dimension of a s is R, where R > M. i.e. 
 
 n ' ' ' 
 
 a = [ a , , a; . . . a .... a 1 (15) 
 
 n L nl' n2' nM' nR J 
 
 then the vector U is also R dimensional in general. Now consider the vectors 
 
 a , a . ... a as N linearly independent vectors in the R-dimensional space. 
 
 Let a set of additional vectors b , b .... b be so chosen that they are 
 
 1 2 R— N 
 
 orthogonal to all the vectors a , i.e., 
 
 t= n' ' 
 
 a • b = n = 1, 2, ... N, s = 1, 2, ... R-N (16) 
 
 n s ' ' 
 
 It is possible to express any R-dimensional vectors in terms of the a' s 
 and b' s which may be used as the basis So ? let. 
 
 N N-R 
 
 U = 2 C a + Z d b , (17) 
 
 n=l n n s=d s s 
 
 Then the coefficients C are determined from the set of following equations 
 
 n 
 
 which are derived with the use of (16) ? 
 
 N 
 
 X=U-5 = 2 n C5.a, r = 1, 2, ... N (18) 
 
 r r n=l n n r 7 
 
 The task at hand therefore is to determine the coefficients d g under the 
 constraint (14) . 
 
11 
 
 Since all the vectors b 1 ie in a space orthogonal to the a 's it is 
 
 s n 
 
 clear that the magnitude of U will be minimum only and only if, 
 
 d g = 0, 8=1,... N-R (19) 
 
 The vector U is therefore uniquely determined in terms of a ' s through 
 
 n 
 
 the use of (18), which does have a unique solution. 
 
 In order to apply this method to the problem of constructing S(u), 
 one has to extend his ideas to an infinite dimension function space. This 
 is explained in the following. 
 
 Let S(u) be expressed as 
 
 S( U ) = ZA ?in T(u-n) 
 n (u-n) 
 
 Let the specified values of SCu) for u = v, ; v~, . „. V and 
 u = u , u , ... u , r + q = N„ The following equations are therefore 
 obtainable^ 
 
 sin 7T(v -n) 
 
 S(v ) = 2 A — ? p = 1, 2, ... r 
 
 p n (v -n) ' ' 
 
 (20) 
 
 sin 7T(u -n) 
 
 S(u ) = 23 A — -. v t = 1 , 2 , 
 
 t n (u -n) 
 
 The problem at hand is to solve for the A 's satisfying (20) and the 
 ^ n 
 
 condition 
 
 Si A I 2 =Z:|s(n)| 2 = min. 
 
 n 
 
 Consider for the moment the special case of specified values of S(v ) and 
 S(u ) which are real. The case of complex S(u) can also be handled with 
 a simple extension of the idea developed below. The N equations in (20) 
 
 WTVERS/TY OF ILlfNOIf 
 
12 
 
 nay be written as 
 
 x a = \ ' A a = 1, 2, ... N (21) 
 
 Where X s are the specified values of S i e S(v ) S(u ) etc a 's 
 a P ' t "' a 
 
 are the row vectors representing the rows of the matrix Equation (20) and 
 A is a column vector with elements A , A , etc. 
 
 J- <~i 
 
 In view of the previous discussion, one can immediately write using 
 (17) and (19) 
 
 N 
 
 A = Z C "a (22) 
 
 a=l a a v ' 
 
 Where C s are to be determined from the set of N equations 
 a 
 
 N 
 
 X = S. C a • a (23) 
 
 r a=l a a r 
 
 Eqjations (22) and (23) determine A and hence S(u) uniquely. 
 
 Since the main problem is to obtain g(p) the following step helps to 
 express the aperture distribution in a suitable form 
 
 When (22) is translated into function space it is seen that the space 
 factor S(u) is a superposition of the functions T (u) which have, as the 
 
 coefficients of their expansion in terms of the function , the 
 
 u-n ' 
 
 elements of a „ The function S(u) is obtained by weighting the T (u) s with 
 a a 
 
 the corresponding coefficients C , and summing the individual terms. 
 
 Now, take T (u), a = 1, 2, . . „ r, which can be written explicitly as 
 
 } a 7 7 7 7 
 
13 
 
 T (u) = 2 a Sin(u - n)7r 
 a an u-n 
 
 sin 11 (v -n) ' 
 
 y, a sin 7T(u-n) 
 
 (v -n) (u-n) 
 
 n a 
 
 (24a) 
 
 sin 7T(u-v. ) 
 
 = IT 
 
 (u-v ) 
 a 
 
 since v is an integer, 
 a 
 
 T (u) also admits an integral representation 
 a 
 
 V U) = 2i / G • e P dp . (24b) 
 
 Next consider T (u) , for a = r +1, 2, . . . N. One then has, for say, T 
 
 a ' ' r+l> 
 
 rp / X V Sil1 ^( U ~ n ) 
 
 T r +1 (U) = fr a (r+l)n u^n" 
 
 sin 7T( Ul -n) sin TT(u-n) ,__. 
 
 = 2 7 r o - (25) 
 
 n (u n -n) (u-n) 
 
 T (u) may also be represented in the integral form. 
 
 .77 
 
 7T / " 1U 1 P ipu 
 
 T , (aa) = 77 / e e dp (26) 
 
 r+1 2 
 
 -IT 
 
 -iu lP 
 
 which may be verified by first expanding e as 
 
 -iu p sin 7T (u, -n) 
 
 1 = _ A a — -1 (27) 
 
 77 n (Uj^-n) 
 
 and then integrating the right hand side of (26) after replacing e 
 
 -iu p 
 1 
 
14 
 by its series in expansion (27) . 
 
 From (24) and (27), it follows that g(p), for minimum supergain criterion 
 has to be of the form 
 
 n -iv p q -iu p 
 
 g(p) = nil P n e n + t =l & t e (B n and 6 t (28) 
 
 are constants) 
 
 Equation (28) is identical with the result obtained by Yen by using an 
 
 entirely different approach. 
 
 Synthesis for Continuous Specification 
 
 When the function S(u) is specified over a continuous range of u 1 
 
 I I ^ 2a 
 usually the entire visible range lu| <• — <r- } the problem becomes one of 
 
 solving the integral equation, 
 
 S(u) = P| / g(p) e ipU dp , - |* < u < £ . 
 
 J -IT 
 
 Transforming the variable of integration from p to p , where 
 
 p = -If cos p (29) 
 
 one obtains 
 
 n/ n a I "i^u cos p' / N / ( n v 
 
 S(u) = pa e h(p ) dp (30JI 
 
 where^ 
 
 h(p ) = g(-7T cos p / ) sin p y (31) 
 
 This equation can be solved if S(u) is expansible in a Neumann series* as 
 
 * For details of Neuman Series^ see Watson, Theory of Bessel Functions > 
 Cambridge University Press, 
 
15 
 
 follows 
 
 oo 
 
 S(U) = n£) C n^n (77u) < 32 > 
 
 in the appropriate range in which S(u) is specified . 
 
 The solution is constructed on the basis of the well known integral 
 
 representation of J (x) , viz., 
 
 n 
 
 J 
 
 7T 
 r 
 
 ix cos a , ,„„,, 
 
 e cos na da (33) 
 
 Using (33) one readily derives the series representation for the 
 unknown h(p ) in (29), as 
 
 1 °° I 
 
 h(P ^ = Sa¥ nS) Cn 1 ° OS nP (34) 
 
 The unknown g(p) in the original integral equation is then obtained 
 using (30), (31) and (34). The formal solution of the synthesis problem 
 
 for the continuous specification of S(u) is therefore complete. 
 
 i 
 An alternative approach consists of expanding (u) 2 S(u) in a series 
 
 of Bessel functions in the appropriate range as, 
 
 ! oo 
 
 (u) 2 " S(u) = S. D J ,(u) (35) 
 
 n=0 n n+^ 
 
 which again is a Neumann series expansion. Now the Fourier transform of 
 
 -n-ii^ is g iven °y 
 
 CD 
 
 y~ 2 J t (y) e" lxy dy = P (x) i" n ^iF I x I < 1 
 n+J n 
 
 /-oo 
 
 = I x | > 1 
 
1 . d n 
 
 77 
 
 -1 
 
 Hence g(p) is given by 
 
 CO 
 
 -co 
 
 Substituting (35) in (38) and using (36) f we obtain 
 
 <TI 1 ^ „ . -n 
 
 16 
 
 P (x) = Legendre polynomial = ' — - (x 2 -l) n . (36) 
 
 n _ n. " 
 
 2 n. dx 
 
 Also S(u) can be written as 
 
 r 1 
 
 S(u)=(3a g(p / ) e iP?rU dp/ p' = g (37) 
 
 xv 1 / , v -ipu 
 
 g(p) = -^ag I S(u) e du p| < 77 (38) 
 
 g(p) ^2 'pi n5) D n * P n (p) | P | * * (39 > 
 
 Equation (39) provides us with an alternative representation of g(p) and 
 one may choose either this equation or (34) for synthesizing the strip 
 source. 
 
17 
 CONCLUS ION 
 
 In this paper we have presented a means for synthesizing a strip 
 source when its space factor is known, either at a number of sample points 
 or over the entire visible range. It is believed that the methods 
 presented here have not been discussed elsewhere in connection with the 
 same problem. 
 
18 
 REFERENCES 
 
 1. Knudsen, H.L., Sartryck ur Teknisk Tidskr., Vol. 82, pp. 1-8, Dec. 1952. 
 
 2. Riblet, H.J., Note on the Maximum Directivity of an Antenna, Proc . IRE 
 Vol. 36, pp. 620-623, May 1948. 
 
 3. Taylor, T.T., Trans IRE Vol. AP-3, pp. 16-28, Jan. 1955. 
 
 4. Woodward, P.M., J.I.E.E . Vol. 93, pt . 3a, pp. 1554-1558, 1946. 
 
 5. Woodward, P.M. and Lawson, J.D., J.I.E.E. Vol. 95, pt. 3, pp. 363-370, Sept. 1948 
 
 6. Yen, J.L., Trans IRE , Vol. AP-5, pp. 40-46, Jan. 1957.