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Digitized by the Internet Archive 
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ANTENNA LABORATORY 
 Technical Report No. 49 
 
 EVALUATION OF CROSS-CORRELATION METHODS 
 IN THE UTILIZATION OF ANTENNA SYSTEMS 
 
 by 
 Robert H, MacPhie 
 
 25 January 1961 
 
 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 
 
ENGINEERING LIBKARY 
 
 ACKNOWLEDGMENT 
 
 The author is very grateful for the many helpful suggestions from 
 Professor Deschamps. The discussions with Professor Lo^ Mr. Craig Allen 
 and other members of the Antenna Laboratory were also of considerable' help, 
 
 
ABSTRACT 
 
 The theory of the cross-correlation of signals of two or more antennas to 
 produce antenna product patterns is reviewed and presented in a generalized 
 form. The problem of distortion of amplitude modulation by the multiplicative 
 processes is considered and a system to remove this distortion by filtering 
 is proposed. An analysis is made of the correlation system output when in the 
 presence of a primary desired signal there is a secondary interfering signal 
 incident on the antenna system from a different direction. It is shown that 
 the rejection of this interference does not depend entirely on the product 
 pattern itself. Indeed^ if the interfering signals amplitude is relatively 
 low, the interference in the system output is determined mainly by a low-level 
 interference pattern which is essentially the sum of the factor patterns of 
 the individual antennas rather than their product. Some ostensibly good 
 product patterns have associated with them very poor low-level interference 
 patterns. The analysis considers the generalization to more than one inter- 
 fering signal and more than one pattern multiplication as well as the effect 
 of time averaging (filtering) which is sometimes possible, for example, in 
 the case of Radio Astronomy. 
 
 Use is made of the negative sidelobes of the product pattern to suppress 
 the integrated effect of average background noise. It is shown that for any 
 arbitrary noise background with the signal known to be absent the time average 
 of the total contribution of this noise in the system output can be reduced to 
 zero. Then when the signal is transmitted its time averaged effect alone will 
 be observed in the system output. It is assumed that the noise from any given 
 direction is a quasi-stationary random process which is statistically independent 
 
of the noise from any other direction. 
 
 A discussion of the noise limitation on the formation of arbitrary 
 patterns from a two element array by means of multiplicative operations is 
 presented and it is shown that any background noise will generally cause a 
 shift in the apparent direction of arrival of the signal as 'seen" by the 
 two element product pattern. The formation of an arbitrary pattern from 
 this basic two element pattern with its error due to the shifting action 
 of the noise does nothing to reduce this error. Rather^ it only serves to 
 emphasize the apparent direction of arrival of the signal and if the noise 
 is relatively large the error between the apparent direction and the true 
 direction can be considerable. 
 
CONTENTS 
 
 Page 
 
 1, I rr reduction 1 
 
 2, The Multiplication of Antenna Patterns bv Signal Cross-Correlation 3 
 
 2„1 Two Antenna Case 3 
 
 2„2 Demodulation and Multiplicative Distortion 5 
 2„3 N Antenna Case and Removal of Amplitude Modulation Distcr- 
 
 ^ion by filtering 6 
 
 3, Analysis of the Effect of More Than One Plane Wave Incident en 
 
 the Antenna System 13 
 
 3 1 0;+put of An Additive System with Two Plane Waves Incident 13 
 3„2 Output of a Two Antenna Multiplicative System with Two 
 
 : nndpp' Plane Waves 14 
 3 3 Output of a Two Antenna Multiplicative System with R 
 
 Interfering Signals 22 
 
 3.4 '•.■*" erf erence in the Case of More Than One Multiplication 24 
 
 4c Suppr-? -" <~>n ft T he Effect of Average Background Noise by Use 
 
 of the Negative Sidelobes of the Product Pattern 28 
 
 5„ The NToise Lim ition on the Formation of Arbitrary Patterns 
 from a Two Element Interferometer by Means of Multiplicative 
 Operations 36 
 
 6, Cone 1 us i on' 42 
 
 Reference? 44 
 
ILLUSTRATIONS 
 
 Figure 
 Number 
 
 Page 
 1. Mul tipli ca T j en of Two Antenna Patterns 4 
 
 2. Frequency Spectra of Signals at Successive Stages of the 
 Process of Removing Undesirable Modulation on the Signals 
 
 to be Cross-Correlated. 8 
 
 3. System for Removing Modulation from All but the Last 
 Product Signal for the Case of Multiplication of N 
 
 Antenna Patterns 10 
 
 4. Comparison of Product Pattern and Low-Level Interference 
 
 Pattern for the Compound Interferometer 19 
 
 5. System for the Suppression of the Average Value of the Output 
 
 of a Compound Interferometer Due to an Arbitrary Distribution of 
 Background Noise 34 
 
 6„ Phaser Representation of the Signal Plus Noise Output from a 
 
 Two Element Multiplicative Array. 40 
 
 7. <a) Interferometer Pattern of Signal Alone, (b) Interferometer 
 Pattern of Noise Alone, (c) Interferometer Pattern of Signal 
 Plus Noise^ (d) Synthesized Pattern of Signal Plus Noise 42 
 

 INTRODUCTION 
 
 For some time cross-correlation techniques have been used to obtain receiving 
 antenna patterns which are effectively the product of the patterns of two or 
 
 more different antennas. Practical examples of such systems are the Mill's 
 
 1 2 
 
 Cross and the Compound Interferometer both used in Radio Astronomy. In the 
 
 former, two long, mutually perpendicular antennas of lengths L and L in the 
 form of a cross, have their signals cross-correlated to produce a power pattern, 
 with a pencil beam, roughly equal to the voltage pattern of an antenna whose 
 area is L L , The Compound Interferometer places a uniformly weighted aperture 
 beside a simple interferometer and the cross-correlated output combines the 
 sharpness of the interferometer pattern with the directivity of the aperture. 
 If both are of length $ (hence the system is of length 2i ) , the product 
 power pattern is equal to the voltage pattern of a uniform aperture of length 
 4i. 
 
 In this report the product pattern for the general case of two antennas 
 separated by a distance i and with complex patterns is presented Then the 
 further generalization to the multiplication of N antenna patterns is made, 
 and a system to accomplish this is proposed. In this connection the problem of 
 demodulation is considered. The multiplicative operations cause distortion of 
 the modulating signal and it is shown that this multiplicative distortion can 
 be eliminated (in theory) by filtering. 
 
 The problem of rejecting interfering signals from arbitrary directions 
 other than that of the desired signal is more complicated than in the case of 
 additive arrays. It will be shown that the product pattern alone does not 
 describe fully the rejection of interference. Another pattern, which in general 
 
depends on the relative directions of the primary and interfering signals, has 
 to be introduced. In the case of a single cross-correlation the pattern is 
 essentially the sum of the factor patterns of the two antennas rather than 
 their product It will be shown that some good product patterns have associated 
 with them rather poor low-level interference patterns „ More generally, if the 
 signals of N antennas are cross-correlated, in the system output there are N-l 
 distinct interference patterns all of which must be analysed when interference 
 rejection is required. 
 
 The product pattern, unlike the power pattern of additive antenna systems, 
 has negative sidelobes and in Section 4 it is shown that, they can be used to 
 eliminate the effect of background noise in the time-averaged output of the 
 correlation system. The proposed scheme will work only when it is known a 
 priori when the signal is present and when it is absent,, In addition the 
 noise from any direction must be quasi-stationary and statistically inde- 
 pendent of noise coming from any other direction„ 
 
 Finally an analysis of the noise limitation on the formation of arbitrary 
 patterns from a two element array by means of multiplicative operations is 
 presented. It is shown that errors introduced by the noise into the two element 
 product pattern cannot be reduced in the formation from i t of patterns of 
 ostensibly much greater resolution. 
 
2. THE MULTIPLICATION OF ANTENNA PATTERNS 
 BY SIGNAL CROSS -CORRELATION 
 
 2,1 Two Antenna Case 
 
 A diagram of the multiplication system for two antennas is shown in Figure 
 1. The signal from antenna A is 
 
 A(9,t) = Re A(0)e Jwt volts (1) 
 
 and that from antenna B is 
 
 B(6,t) = Re B(e)e" jf3isin6 e Jwt volts (2) 
 
 = Re B (Q) e JCOt volts (2a) 
 
 where 9 = angle which the propagation vector of an incident plane wave makes 
 with the normal to the line joining the two antennas, 
 
 t = time in seconds, 
 
 u> = angular frequency of the incident plane wave in radians per second, 
 
 i = spacing between the antennas in meters, 
 
 P = propagation constant in radians per meter, 
 
 Re indicates 'the real part of". 
 The pattern coefficients A(Q) and B (0) are in general complex. Note that the 
 effect of the spacing i between the antennas is the phase factor e " K 
 which is absorbed into B.(0)„ 
 
 Now as is shown in Figure 1, the signal from antenna A enters a frequency 
 shifter (a nonlinear device) together with a local signal at frequency oj . 
 The output is filtered so as to pass only the difference frequency (oj = to - co ) 
 
LOCAL 
 
 PUMP 
 
 SIGNAL 
 
 (J 
 
 ANTENNA A 
 
 9 
 
 ANTENNA B 
 
 ¥ 
 
 W 
 
 i 
 
 > x 
 
 A(0,<U) 
 
 B^(0,W) 
 
 FREQUENCY 
 SHIFTER 
 
 AND 
 FILTER 
 
 A(0,o/) 
 
 It 
 
 MIXER 
 
 W. 
 
 i 
 
 Pp<0,W |f <) 
 
 SYNCHRONOUS 
 DETECTOR 
 
 I 
 
 R,(0,f) 
 
 '' '• Multiplication of Two Antenna Patt 
 
 erna 
 

 signal which is simply the signal from A shifted in frequency to co . Then this 
 signal and that from B enter a mixer which is a second nonlinear device 
 whose output contains among other components a signal at frequency to . By means 
 of a synchronous detector the amplitude of this signal can be obtained and it 
 is given by 
 
 p (e, i) = Re A(e) B*(e) o> 
 
 ft 
 
 where indicates the complex conjugate quantity. In trigonometric form, Equa- 
 tion (3) can be written as 
 
 p (e, i) = :A(e) B(e>l cos[pisine + a (e> - p(e)l (3a) 
 
 where a(G) = Arg A(Q) and (3(9) = Arg B(Q). Since P (Q,4) is the product of two 
 voltage patterns it varies as the power of the incoming waves and consequently 
 any comparison with additive antenna systems patterns will be with the power 
 patterns of those systems. We note incidentally from Equation (3) that if 
 / = and A(Q) = B(Q), then the product pattern degenerates to an additive 
 systems power pattern given by 
 
 P o (0) = |A(e)^ 2 (4) 
 
 I 
 
 2.2 Demodulation and Multiplicative Distortion 
 
 I — . 
 
 If the incoming plane wave is modulated, with amplitude modulation for 
 example, then the output of the system will be 
 
 P 2 (0, F(t),l) =[14 mF(t)] 2 Re A(e) B*(e) (5) 
 
 = [1 + 2mF(t) + m F(t) ] Re A(e) B*(0) (5a) 
 
where m = percent modulation, (0 < m < 1), 
 F(t) = modulating signal. 
 
 2 2 
 Hence there is square law distortion (m F(t) ) which can be significant for 
 
 large m. This problem also occurs in additive systems which employ square law 
 
 detection. Gray* has shown for the simple case of sinusoidal modulation that 
 
 for less than 10$ square law distortion m must be less than 40$ which is not 
 
 too severe a restriction. However, if N antenna patterns are multiplied together 
 
 (as will be shown in the next section) the output is proportional to 
 
 P N (G, F(t)) = [1 + m F(t)] N P (0) (6) 
 
 As N increases the restriction rapidly becomes intolerable; for N = 6 and 10$ 
 
 2 2 
 distortion from only the m F(t) term in the binomial expansion, m must be 
 
 kept less than 8$, In the following section a system for multiplying N antenna 
 
 patterns which removes this multiplicative distortion is proposed. 
 
 2,3 N Antenna Case and Removal of Amplitude Modulation Distortion by Filtering 
 
 If there were N antennas the spectrum of the signal at the terminals of 
 each would be that of the carrier and its modulation while the amplitude and 
 phase of the signal would be determined by the antenna's pattern and its loca- 
 tion. Now since any one of the N antennas signals is modulated in the same way 
 as any other, nothing would be lost if the modulation were removed from the 
 carrier ol - • 1 1 but one of these signals. Not only will there be no loss of 
 information bill L1 Will be shown that this actually eliminates the multiplicative 
 distortion brouL'M aboul by the repeated cross-correlation of the antenna 
 
 '•8 . 
 
 1 Applied Electronics, Wiley, pp. 748-740 
 
Let the N antennas have terminal voltages given by A (0, t,F(t)), 
 
 1 
 
 A (0,t,F(t)) A (9,t,F(t)) where 
 
 k N 
 
 A, (G,t,F(t)) = [1 + mF(t)] Re A (0) e "J Cpi k sin0-ct) 
 £ k k 
 
 = [1 + mF(t)] Re A (0) e'^ 1 (7) 
 
 k 
 
 For simplicity we consider the case of colinear antennas whose patterns are 
 
 functions of alone. The more general case of non-colinear antennas with 
 
 "two-dimensional" patterns (Q,§) is of no greater difficulty as far as the 
 
 multiplicative operations are concerned In Figure 2(a) are shown the spectra 
 
 of A (0, t, F( t)) and of the local signal at frequency co . The separation of 
 
 1 
 the local signals frequency co from the carrier's at to is A co and is chosen 
 
 soas to be slightly larger than the spectral width of one of the carrier's 
 
 sidebands. Hence, the output of the first frequency shifter (FS) will contain 
 
 the difference frequency at A co along with the modulation spectrum of the signal 
 
 This difference frequency can be held constant by an automatic frequency control 
 
 device (AFC). The low frequency output spectrum is shown in Figure 2(b). A 
 
 high Q filter tuned to frequency A co will remove the unwanted modulation on the 
 
 new "carrier" leaving a single frequency signal whose amplitude varies as the 
 
 antenna pattern A. (0), (see Figure 2(c)). The frequency shifting operation 
 
 1 
 to a much lower frequency is made necessary by the difficulty of removing the 
 
 modulation from an RF carrier by direct filtration. 
 
 A second frequency shifter (FS) ? this time with a local signal at frequency 
 
 U = co - 2 A co is used to return the unmodulated "carrier" at A co to the RF 
 
SIGNALS INTO FIRST FREQUENCY SHIFTER 
 
 RADIO 
 
 (a) 
 
 FIRST PUMP 
 
 FREQUENCY . ... n/ ,. Q . 
 FREQUENCIES 4 ♦ A^.FM.e) 
 
 nrv . 
 
 J. cj ^^ 
 
 (b) 
 
 .1 LOW FREQ. OUTPUT FROM FIRST FREQ. SH 
 
 ♦ A{(A£J,F(6)),e) = A{«J,F(a»,G) SHIFTER DOWN 
 y<"~"^ /^N IN FREQUENCY TO AW 
 
 FTER 
 
 |A^(AW,e) /MODULATION SPECTRUM F(U)\ 
 \REMOVED BY HIGH Q FILTER/ 
 
 A6J 
 
 (d) 
 
 i SIGNALS INTO SECOND FREQ. SHIFTER 
 | A^(A6),9) 
 
 1 
 
 At) 
 
 2ND PUMP 
 FREQUENCY 
 
 (e) 
 
 T 2 2 Ad) 
 
 SIGNALS OUT OF 2ND FREQ. SHIFTER 
 
 A {| (6J-3A^J,e) 
 
 SUPPRESSED 
 CARRIER 
 
 A.(6J-A6J,0) 
 ■li 
 
 ii SIGNALS INTO FIRST MIXER 
 
 (f) 
 
 CJ-3AO) Gi^ GO-ACJ O) 
 
 t A < 
 
 (g) 
 
 W-3A6) 
 OW MIXER 
 
 2 
 
 a., (6),F(6)),e 
 
 CJ-ACJ 0) 
 
 P 2 AlA^I' |P 2 
 
 i/ir\17ir\ 
 
 A<0 2AOJ 3A6J 
 
 (h) 
 
 "P (ACJ.9) /MODULATION SPECTRUM F«J) 
 
 V REMOVED BY HIGH Q FILTER; 
 
 Figure 2. 
 
 FREQUENCY, RAD./SEC. 
 PROCESS CAN BE CONTINUED BY STARTING AT STAGE (d) ABOVE AND 
 USING A. {CJJ{Q),Q) IN THE NEXT MIXER OPERATION 
 
 BCy Spectra of Signals at Successive Stages of the Process of 
 Bmovlng f.'ndosi rable Modulation on the Signals to be Cross-Correlated, 
 
range. The FS output will be a new carrier at frequency oo with sidebands at 
 
 oo - A oo and oo + A co. At this point it is necessary to suppress the carrier 
 
 (there is no technical difficulty in doing this), Then the two sidebands whose 
 
 amplitudes are proportional to A (Q) are mixed with the modulated signal from 
 
 1 
 antenna two, A (0, t,F(t)) as is shown in Figure 2(f). The low frequency 
 
 2 
 difference output spectra are shown in Figure 2(g). Again an AFC device is 
 
 used to hold this output constant in frequency. A filtertuned to A co passes 
 
 the product signal given by 
 
 P 2 (9,t, i 1 ,i 2 ) = Re A i (0) A i *(0) e 
 
 j Aw t 
 
 (8) 
 
 and which has all its modulation removed. 
 
 The process can be continued by returning this signal to RF as was done 
 in stages 2(d) and 2(e) (see Figure 2). Then by mixing with A ,(0, t,F(t)) and 
 filtering one obtains 
 
 P 3 (0,t,i 1 ,i 2 ,i 3 ) = Re A i (0) A i (0) A jf (0) e 
 
 -I *-i O 
 
 j A oo t 
 
 (9) 
 
 which again has all the modulation removed. A diagram of the system is shown 
 in Figure 3. The process is continued until the (N - 2)ND unmodulated pro- 
 duct is formed, i.e., 
 
 N-2 
 
 j A w t 
 
 P N _ 1 (0,t,i i ,...i N _ 1 ) = Re 
 
 A A (0) A *(0) 
 4 x 2q-l 2q 
 
 A (0) e^ 
 
 N-l 
 
 if N is even and > 3, and 
 
 (10) 
 
 Vl^^l'-W =R 
 
 N-l 
 
 n a ( Q ) a *(0) 
 
 4 2q-l 2q 
 
 j A oo t 
 
 if N is odd and > 2 r (in the above formula i is zero), Then, instead of 
 
c 
 bo 
 
 •H 
 CO 
 
 O 
 
 3 
 T3 
 O 
 
 0, 
 
 W 
 
 3 
 
 J2 
 
 3 
 
 •H 
 
 ■o 
 
 r ( 
 
 
 
 a 
 
 a 
 
 •H 
 
 
 ■^ 
 
 bfl 
 
 H 
 
 e 
 
 3 
 
 •H 
 
 3 
 
 > 
 
 
 
 
 <H 
 
 e 
 
 o 
 
 0) 
 
 
 ad 
 
 
 
 
 a 
 
 h 
 
 Ed 
 
 
 
 U 
 
 >H 
 
 
 
 0) 
 
 a 
 
 d 
 
 u 
 
 +J 
 
 M 
 
 
 ■ 
 
 Fi 
 
 >) 
 
 
 
 03 
 
 <M 
 
 3 
 ho 
 
 •H 
 
11 
 
 shifting this signal back up to RF, we mix it directly with the modulated signal 
 
 from the last antenna A (0, t,F(t)), The modulated RF difference frequency is 
 
 N 
 if the mixer output signal at this frequency is fed into a synchronous detec- 
 tor along with a reference signal also at co taken from the local source, then 
 the detector output will be 
 
 N 
 2 
 
 ^(0, F(t) £ 2? l 3 ,...l }i ) = [1 + m F(t)] Re U A < 0) A 
 
 2q-l 
 
 (0) A, *(Q) 
 
 2q 
 
 (11) 
 
 if N is even and > 1 and 
 
 P N (0, F(t), i 2 ,i 3 , ...,1 N ) = [1 + m F(t)] Re 
 
 f N-l 
 
 q 2i A i (e) A % <°> 
 
 4 X 2q-1 *2q 
 
 A iN (8) 
 
 if N is odd and > 2. 
 
 The output has no multiplicative distortion and consequently m need not be 
 
 limited to say, less than 40$ as with a square-law detector. The practical 
 
 success of the system depends on the filtering efficiency which in turn depends 
 
 on the modulation spectrum. The noise problem should be alleviated by the 
 
 filtering but the extra multiplicative operations will clearly degrade the signal 
 
 to noise ratio. Finally, mention should be made of amplification which probably 
 
 will be necessary if N is large. The most convenient place for an amplifier 
 
 'would be immediately after the filter that removes the modulation from the 
 
 pseudo-carrier at A co. A narrow band low noise parametric amplifier could be 
 
 ised. 
 
 It should be noted that the pattern P ( 0, F . __ . , i_, . . , ,i ) is proportional 
 
 N (t) 2 ' N 
 
 ;o the Nth power of the electric field strength of the incoming plane wave, 
 tence the equivalent "field strength pattern" (voltage) is 
 
 T> t Q V t +■ \ 
 
 \ — ID (d T?f + \ 
 
 e » 
 
 1 
 ,N 
 
 l 1 CM 
 
12 
 
 and the equivalent power pattern is 
 
 2 
 
 p N p (e,F(t)i 2 , ....i N ) = P N ( Q ,F(t),i 2 , ... i N ) N (is) 
 
 Any comparison with conventional additive patterns should be made on this 
 basis. In the rest of this report only the power pattern will be used and the 
 superscript will be omitted. 
 
13 
 
 3. ANALYSIS OF THE EFFECT OF MORE THAN ONE PLANE 
 WAVE INCIDENT ON THE ANTENNA SYSTEM 
 
 In the preceding analysis, only a single plane wave incident at the arbitrary 
 angle 9 was considered. In this section it will be shown that when two plane 
 waves are incident at angles 9 and Q , the output of the synchronous detector 
 of the product system is in important respects different from the output of the 
 conventional square law detector of additive systems. We let one of the plane 
 waves of unit amplitude at an angle of incidence 9 be the desired signal. The 
 other, of complex amplitude Se \ and angle Q , is either due to a discrete, 
 remote, noise source or a secondary component of the desired signal which 
 arises when there is more than one transmission path from the source to the 
 receiver (multipath propagation). 
 3.1 Output of An Additive System wit h Two Plane Waves Incident 
 
 In an additive system the received terminal voltage before detection is a 
 linear superposition of the two signals 
 
 C(t;l,9;Se , %) = C(t;l,9) + C(t;Se ,S ) 
 
 = Re]c(9) + Se'^1 CCQ^ \ e jwt (14) 
 
 The usual detector > a square law device, has a low frequency output proportional 
 to 
 
 . t , 2 2 2 .t 
 
 Ca^iSeW^ = !c(9)^ + S C(9j + 2S Re C(0; C(Q) e" j5i (1! 
 l ■ ' i i 1 
 
 )r alternatively 
 
 3 
 
 Welsby and Tucker have obtained some similar results in which they compare 
 
 multiplicative systems with additive systems employing linear rather than 
 square law detection. 
 
14 
 
 I i ^l I 2 I I 2 2 1 I 2 , 
 
 |cu,e;Se J , e 1 )| = |c(e)| + s |c(e 1 )| + 2s|c(e) c(e 1 )|cos l 1 -Ke)+4(e 1 ) 
 
 (15a) 
 where £{Q) = ARG C(9) . 
 
 o 
 
 If the pattern is normalized with the primary signal incident at = i 
 
 the beam angle of the pattern, Equation (13a) can be written as 
 
 c(se J , 1 e 1 '> 
 
 = 1 + s 
 
 0(9^ 
 
 + 2s|c(e i )|cos(l 1 -t(0°)+^(e i )^ (16) 
 
 C(G 1 ) 
 
 where c(9 ) = , The last two terms of Equation (16) represent interference 
 
 and its rejection is seen to depend on c(0 ) alone. 
 
 3.2 Output of a Two Antenna Multiplicative System with Two Incident Plane Waves 
 
 The output of the synchronous detector in the two antenna multiplicative 
 case and with two incident plane waves is naturally more involved than that of 
 the additive system. It is of the form 
 
 P 2 a,9;Se j ^ 1 9 1 ;i) = ReJA(e)B i *(e) + S 2 Af e^B^^ 
 
 + s(A(9)B i *(9 1 ) e ' l + A(0 l )B i *(9) e * 
 
 (17) 
 
 or alternatively (and with some rearranging) 
 
 ,h 
 
 P (l,e;SB , 9,;i) = A(6) B(Q) cos r : p/sinn-:a(0.'-(3<0> 
 
 & 1 
 
 s a(Oj > B(e 1 )|cos lysine +a(e 1 )-p^e 1 )] 
 
 A(9) B(9)|cofl p;sinG-ta(0)-|i(9)] ■ 
 
 A(Q i | 
 
 
 a(e 1 )-p(e) 
 
 l I n6 a(6 I -p(6) 
 
 B(9j ) 
 
 bo) 
 
 cos| pislnSj -i a(0)-p(e,)] 
 
 i i no n(f)J |ii n ' 
 
 (17a) 
 
15 
 
 Now as in the additive case, if the pattern is normalized with the pri- 
 mary signal incident at 6 = 0° Equation (15a) becomes 
 
 P 9 (Se J , 1 Q t) = 1 + s 2 ia(9.)b(e i ) 
 
 cos[ pisine Td(e )-p(e )] 
 ~" c M[a(0° > ^(M o 5 TT " ~~ 
 
 + S ', ia(6, ) 
 
 cos(e i ^a(e 1 )-p(o°)) 
 
 II cos(a(0°)-p(0°)) 
 
 + ib(e,) 
 
 i J- 
 
 cos(pisin0 1 -^ 1 ^a(O°)-p(e i )) 
 
 cos(a(0O)-(3(0°)) 
 
 (18) 
 
 J 
 
 A(e 1 ) 
 
 B(e 1 ) 
 
 where 0.(0^ = ^^J . b 'V = j^ • 
 
 Comparing Equations (16) and (18) we note that the basic difference is in the 
 
 last terms which is the major interference term when S <* 1. In the additive 
 
 case (Equation (16)) the rejection of interference depends on the voltage 
 
 pattern c(0 ): the worst case occurs when £ = C(O)-C(0.,) and the interference 
 1 11 
 
 term is 2S c(0 )| , In the multiplicative case (Equation (18.)) the low-level 
 interference is proportional, not to the product pattern itself but to the sum 
 
 of the normalized factor patterns | a(0 )| and lb(0 )| each of which is multi- 
 
 1 i 1 
 
 plied by a cosine term. Consequently the rejection of this first order ( - s) 
 interference by a multiplicative system cannot be determined by the product 
 pattern at all. It will be convenient to define the pattern associated with S 
 as the low-level interference pattern since it predominates when S is small and 
 & is negligible Hence we let 
 
 ,ar0 )| cos(£ -q(0 1 )-(3(O°)) * IbO,)! cos(p.fsin0 -S.'crtcft-pfe)) 
 
 cos(a(0°) - (3(0°)) 
 
 This expression is clearly equal to or less than 
 
 ja(0 i )i -» jb(0 1 ) 
 
 (19) 
 
 but the 
 
 2cos[ a(0°)-p(0°)] 
 exact nature of the pattern is quite complicated for this general case. Now 
 
16 
 
 for any given system, i is fixed and the pattern is a function of the two 
 variables G-, and £ „ Likewise for any given Q there will be an £ which 
 
 ma 
 
 ximizes the magnitude of L(Q ,§ ,i). To determine this we differentiate with 
 
 respect to £ and equate the derivative to zero. 
 
 8 L<Q i ,£ v O 1 
 
 -laCe^l sin[^ 1 +a(e i )-p(0°)]+|b(e i )| sin(pisine i -S 1 +a(0 o )-p(6 1 )) 
 
 cos(a(0°)-p(0°)) 
 
 = 
 
 (20) 
 
 from which we obtain the relation 
 
 la(0 )lsin(a(0 )-p(O°))-ib(0 )lsin (pfsin G.+a(O°)-p(0 )) 
 
 i- e i ,i; |" ra(e 1 )!cos(a(e 1 )-p(o°)T+ib(e 1 )lcos(piiin e 1 +a(o°>-p(e 1 )) 
 
 -p<e r 
 
 (21) 
 
 Now there are two values of £,(Q,i) (in the range [ 0, 27T]) that satisfy the above 
 equation, namely § (6,i) and £ (0 ,i) + 77. Substituting these values of £ 
 into the expression for L(0 , 1, -O gives the worst possible patterns L(0 £ ,i ) 
 and L(0 |.+ 7T, i), If the patterns a(0 ) and b(0 ) are real, Equation (21) 
 reduces to 
 
 € j (0 I) ■ tan 
 
 b(9 ) sin (P i sin ) 
 a(0 ) + b(0 ) cos (P 1 sin Gj) 
 
 (21a) 
 
 and the expr< lor the worst patterns are 
 
 '.'V V " 
 
 a(9 1 )+b(9 1 ) cos(Pisin Oj) ! 2 + | b(Q ] ) si n (Pi sin ) . ' 
 
 a^Oj) + b(0 ) cosfPisin 9, ) 
 
 (22) 
 
 cos (1 (9j , i )) 
 
 
17 
 
 ue^ l I+ ir,i) 
 
 a(8,)tb(eJ cos(Pisin ) I 2 + J b(0 n ) sin(Pi sin n K 2 
 
 1 _1_ 1_ L _ 1 1 J 
 
 a(0 ) + b(0 ) cSsTPisin ) 
 
 (22a) 
 
 cosf£ (0 I) + TT) 
 
 But cos (I (0 , I ) + 77) = -cos (| (0 i)) and hence 
 
 L(e x , Ij + ir,i) = -L(0 r e p I) 
 
 (23) 
 
 Subsequently we will consider only 
 
 L o (0 r i) = L(0 r e r i) 
 
 (24) 
 
 knowing that the alternative solution L(0 £, + 77 ? f ) is simply its negative. 
 If i = there is a further reduction to 
 
 L (0 ) = - [a(0 ) + b(0 )] 
 o 1 2 1 1 
 
 (22b) 
 
 Equation (22) can be put into a quite symmetric form by letting 
 
 M(0 
 
 , I) = a(0 ) + b(0 ) cos(P I sin ) 
 
 (25) 
 
 and 
 
 N(0 r i) = b(0 ) sin(P I sin ) 
 
 (26) 
 
 Then Equation (22) becomes 
 
 W U -2 
 
 M(e x , i) 2 + n(0 i) 2 
 
 cos 
 
 tan 
 
 .1 N( 6p i> 
 M(0 £) 
 
 (27) 
 
18 
 
 We can therefore conclude that some apparently good product patterns 
 might have quite unacceptable low level interference patterns. As an example 
 of this we consider the Compound Interferometer mentioned in the introduction. 
 It consists of a uniformly weighted aperture of length i, adjacent to a simple 
 interferometer also of length i. A diagram of the system is shown in Figure 4, 
 The normalized factor patterns are 
 
 sin 
 
 P | sin 9 \ 
 
 a(e 1 ) 
 
 . _v 2 
 
 P i sin 0, 
 
 b(9 1 ) 
 
 = cos 
 
 "'P i sin 9. 
 
 Hence the product pattern is proportional to 
 
 P I sin 0. 
 
 sm 
 
 p 2 ( 8i , D---PT 
 
 P i sin A 
 
 sin 9 n 
 
 cos 
 
 V 
 
 cos (P I sin ) 
 
 sin 
 
 /'P £ sin fl 
 
 p 2 ,e 1 - i "' -tn 
 
 P i sin 9, 
 
 sin 9, 
 
 where x = — - — 
 
 sin(4x) 
 
 4x ' 
 
 (28) 
 
 But the worsl l ->w level interference pattern is 
 
 L (x) 
 
 o 2 
 
 x 2 , , „ v 2 
 
 cos x cos 2x f (cos x sin 2x) 
 
 sin x 
 
 + cos x cos 2x 
 
 
 , s i n x 
 . / • cos x cos 2x 
 
 tan ' - 
 
 c:ds x sin 2x 
 
 (28a) 
 
19 
 
 O) 
 
 C/5 
 
 V 
 
 ( 
 
 \ 
 
 
 r 
 
 \ 
 
 
 V 
 
 CT 
 
 < 
 
 I- 
 O 
 
 Q 
 O 
 
 cr 
 
 Q. 
 
 UJ 
 O 
 
 z. 
 
 Ul 
 
 cr 
 ui 
 Li- 
 ce: 
 
 UJ 
 
 UJ 
 > Z 
 
 uj or 
 
 5 
 
 o 
 
 . t= 
 
 4 
 
 ) 
 
 0) 
 as 
 
 a, 
 
 0) 
 
 u 
 
 c 
 a> 
 
 i« 
 a> 
 
 0) 
 
 CD 
 
 > 
 
 M 
 
 o +j 
 
 •c 
 
 a 
 
 a) 0) 
 
 C w 
 CD -u 
 
 •o 
 
 c 
 
 3 
 
 o 
 o, 
 s 
 o 
 o 
 
 CD 
 
 o o 
 en <h 
 
 •r-l 
 
 h 
 
 IS 
 
 s 
 
 O 
 
 O 
 
 0) 
 
 3 
 CUD 
 
 •H 
 
20 
 The two patterns are plotted in Figure 4 as functions of x. It should be 
 
 noted that this graph shows only the normalized patterns (each with its max- 
 imum equal to unity). In any particular case, the relative magnitudes of the 
 
 two patterns depend on the value of S. For example, suppose that S = 0.25 
 
 5 
 and corresponds to x = — 77. The output signal to interference ratio for 
 1 8 
 
 the worst case would be 
 
 R(0 1 S) = — 
 
 S P 2 (9 1 , 1) + 2S L o (0 p I) 
 
 (.25) (.127) + 2(.25) .395 
 
 s = .25 
 ~ 577/8 
 
 00794 + .1974 
 
 (29) 
 
 = 4.88 
 
 It can be seen that the low-level interference pattern dominates in this case 
 
 because of its high sidelobes which are only 3 db down at x = +77, + 277 ( +377 etc. 
 
 2 
 
 Note that in some cases S P (0 , SL ) < and hence the worst case of 
 
 interference would occur when 2S L (0,, 1) were negative also. If L (0,, i) 
 
 o 1' o 1 ' 
 
 is not negative as in the above example, we would select the alternate worst 
 
 pattern -L (0, , 1 ) = L (0 , , £ _ + 77 t ) (see Equation (23)). The worst possible 
 o 1 o 1' I 
 
 signal to interference ratio for the system, without regard to sign, is therefore 
 
 R <G., i, S 
 
 f> 1 ' ' 
 
 )l = 
 
 S I P.O., 1)1 + 2S|L (0 i ) 
 2 1' o 1 ' 
 
 (30) 
 
 By way of comparison, the worst case for an additive system would be 
 
 R (6 , S)| ■ 
 
 8 2 |c(9 1 )| 2 + 2810(8] » 
 
 (31) 
 
21 
 In the latter equation it is seen that high directivity of c(0 ) also means 
 a large signal to interference ratio while the high "directivity'' of P (0 , I ) 
 in the former equation can be negated by L (0 S.) to give a low net signal 
 to interference ratio. 
 
 It is important to realize^ however, that this situation obtains only 
 when the interfering signal is at its worst possible phase with respect to 
 the desired signal. Usually the interference phase £ is a time varying quantity 
 with the probability of any one value of £ (t) in the interval [0, 27T] being the 
 same as any other. Hence, the average of the low-level interference when taken 
 over a suitably long period of time approaches zero, i.e.. 
 
 1 im 
 T -r 
 
 A(0 X ) 
 Me)" 
 
 cos (P I sin + |(t) + a(Q ) - (3(e)) 
 ~cos (f3 £ "sin + cl(6) - (3(e) 
 
 (32) 
 
 P«9 , ) . cos ((3 i sin 0, - £(t) + a(0) - (3(0 )) 
 
 1 
 
 ft©; 
 
 1 r___ _ 
 
 "cos Wl sin + o-(0) _r T(0)) 
 
 dt = 
 
 In practice this means that a low pass filter would be used to reduce the 
 low-level interference signal in the system output. This filtering process 
 is quite feasible in Radio Astronomy (hence the usefulness of Compound Inter- 
 ferometer and indeed of the Mill's Cross), since only the steady state flux 
 of energy from each point on the celestial sphere is required. For instan- 
 taneous reception of modulated signals or for any application where a lengthy 
 time averaging is not practicable, the low-level interference pattern must 
 be considered. 
 
22 
 
 3.3 Output of a Two Antenna Multiplicative System with R Interfering Signals 
 
 If in addition to the desired signal of unit amplitude incident at the 
 
 th 
 angle 0, there are R secondary interfering signals^ the q of which has 
 
 amplitude S e and direction , the output of the square law detector 
 
 q q' 
 
 of the additive system will be 
 
 j£. 
 
 j&, 
 
 6(1, S ie , i; ...; S R e , R ) 
 
 R # -jb 
 
 = |'C(6)| + 2Re fo'e) 2 S C (0 )e q 
 
 q-1 q q 
 
 r r : 
 
 . Z SsSc(0)c(0)e 
 
 -, n q J" q r 
 
 q=l r=l 
 
 6 -I ) 
 
 (33) 
 
 The analogous output from the cross-correlation system is 
 
 P 2 (l, 0, S x e e, . 
 
 ji f * R R 
 
 • • s o e R , D ^) = Re i A (e) B , (e) + s s s s 
 
 R ' R £ q r 
 
 q=l r=l 
 
 £(e > - l(e„) 
 
 A(9 ) B . (0 ) e q 
 
 q f r 
 
 R # -jl(9 ) 
 
 r + A(0) 2 S B. (0 ) e 
 r i r 
 r=l 
 
 t-j€ e i 
 
 + B. (0) Z S A(0 ) e 
 
 ^ , q q 
 
 q=l 
 
 (34) 
 
 A comparison oJ » h<;se equations with those for a single interfering signal 
 shows thai th< ten again has the low-level interference pattern 
 
 van by Equation N9). The rejection of Interference Li 
 
23 
 
 S . <1 for all q will depend primarily on this pattern which, as was shown 
 
 q 
 
 in Section 3.2, can be quite poor even when the product pattern itself is 
 good. The step from R discrete interfering signals to a continuous back- 
 ground of interference is quite straightforward, and the rejection of low- 
 level interference in this case also depends on the low-level interference 
 pattern. 
 
 A major hindrance in the design of a satisfactory interference pattern 
 is the interferometer factor cos (P i sin 0) due to the separation i of the 
 two antennas Its oscillations between +1 and -1 cause high sidelobes, 
 especially if bi0 ) has high sidelobes also. 
 
 Indeed, it is quite easy to show that it is impossible to get a product 
 pattern and a low-level interference pattern that both possess low sidelobes 
 if one of the factor patterns does not have them. 
 
 If I is quite large the cosine factor in the noise term 
 
 cos[P I sin + a(e ) - P(9 )] 
 
 s lace^ b( 0l )' Ho^To") -(3(o o )i 
 
 from Equation (18) will oscillate rapidly between -1 and +1 as varies 
 Hence, for a number of values of this noise term will be given by 
 
 !a(0 ) b(0 )1 
 
 S _ i y. sHa(0 ) b(0 )| 
 
 cos[a(0°) - P(0°)] L X 
 
 and if b(0 ) has high sidelobes for these values of the noise term will be 
 
 large. It can be reduced only by making ia(0 )| small, much smaller th 
 
 an 
 
 Ibce^ 
 
 However, the low-level interference noise term is given by 
 
24 
 
 cos (I + a(0 ) _ (3(0°)) cos (Pi sin - | * a(0°) - (3(0 )) 
 
 s ; a(0 ) __ _i 1 + |b(0 )| L_-i_ _ .J. 
 
 1 cos[a(0°) -P(0°)] cos;a(0 ) -(3(0°)] 
 
 As £ varies this noise component varies also. It can be shown that the max- 
 imum value is at least 
 
 S \ ibO.M - la(0 )| r 
 
 ___^_ i i__i_ S - [b(0)| - |a(0 n )| ' 
 
 cos[a(0°) - (3(0°)] [ 1 l 
 
 To minimize this component we therefore must make ia(0 )\ srf (b(0 , ) ■ , 
 and this is not what is required for the suppression of the other noise compon- 
 ent. It follows that one or the other of the noise components can be minimized 
 but not both. 
 
 3.4 Interference in the Case of More Tnan One Multiplication 
 
 The case of the multiplication of N antenna patterns with R interfering 
 signals is obviously an involved situation. For N even the product pattern is 
 
 N/2 . jPl sin R j| 
 
 V 1 °< S l *, °! - *> - " A 2q -l (9)e * = V \q-l (9 j ) 
 
 q=l v J=l 
 
 (35) 
 
 e jPi 2q-l Sin 9 j 
 
 -jPi .me r -jft -^ 2q .i Bin e k > 
 
 A 2q (6) e ^IS k e A 2q (0 k ) e 
 
 Bj way Oi b N i 4, R ■ 1, and let the patterns be real. Even with 
 
 the expression that results when put into trigonometric 
 
25 
 
 P (1, 0: Se , . i i i ) = A (0)A(0)A(9)A (6)cos<Pl sine-Pi sinG+Pl sine) 
 
 1 * "7, 
 
 cos iPi sine -Pi sin0+Pi' sin0) 
 2 3 4 
 
 A (0 ) 
 2 1 
 
 /A (0 ) 
 
 S ! -r- 7 -r— cos(Pi sin0-Pi sin© 
 A ( e ) 2 3 
 
 1 1 
 
 + Pi 4 sin0 + l 1 ) + a~7q7- cos(Pi 2 sin0 1 - Pi 3 sin0 4- 0| sine - ^) 
 
 A (0 ) 
 
 t A / Q x ~ cos (Pi sin0 - Pi sin0 1 + Pi sine 4 ^) 
 
 W \ 
 
 r-7r T - cos (Pi sine - Pi sine + Pi sine 1 - (L) 
 A (0) 2 3 41" 
 
 4 
 
 A (0 ) A (0 ) 
 A (0) A 2 (0T- C ° S(Pi 2 5ine i " Pi 3 Sin6 + Pi 4 Sin9) 
 
 J. u 
 
 W A 3 (0 1 ) 
 
 -nenrier cos((3i 2 sine - Pi 3 slne i + pi 4 sin9 - 2 V 
 
 J- o 
 
 VV A (6 )* 
 
 a (0TA-T0T cos((3 V inG " Pi 3 sine + p V in V 
 
 1 4 
 
 W A 3 ,0 1 ) 
 
 1~(e) A (0) cos(Pi 2 sin 0i - Pi 3 sin0 1 + Pi^in©) 
 
 ~K^Q) A 4 (0) cos (Pi ^^ - Pi 3 sin t Pi 4 sin ei f 2^) 
 
 VV A 4 ' e i ) I 
 
 ^ 3 T0TA 4 l0T~' COS(,3i 2 Sin0 " P V ine ! + Pl 4 Sine i , | 
 
26 
 
 [A (0 ) A (0 ) A (0 ) 
 \m 1(8) v(e> —tfV-i " Pi 3 sine i " p V ine - V 
 
 A (G ) A (9 ) A (B ) 
 V(8) 1(8) A*<9) c ° s(! V in9 - P V ln9 l - P V in6 l ' V 
 
 A (0 ) A (8 ) A (0 ) 
 
 \ (9) 1(9) A* (9) c -«V ine i - ? V in6 * P V ln9 l " V 
 12 4 
 
 A (9 ) A (6 ) A (8 ) 
 
 \ C 9 ) A (6) A*(9) <"»«V 1 '* 1 " P's'^! + "V 1 " 8 ! 
 2 3 4 
 
 ,! 
 
 , A (0 ) A (0 ) A (0 ) A (0 ) 
 -4 I 1 1 2 1 3 1 4 1 ,n. . _ 
 
 Tl) A o (0) A (9) A f8) " cos(Pi 2 Slne i 
 12 3 4 
 
 Pi sin0 
 
 + Pi sine, ) 
 
 4 1 I 
 
 (36) 
 
 2 3 
 
 There are no* 3 distinct interference patterns associated with S, S and S 
 
 4 
 in addition to the product pattern associated with S . It follows a fortiori 
 
 that care must be exercised in the design of product patterns involving severa] 
 
 multiplicative operations if interference is to be rejected. 
 
 It is important to note that even in the case where time averaging is 
 
 2 
 possible, there are still terms in the S interference pattern that are not 
 
 functions of £i t ) and hence cannot be averaged out. The terms of the S and 
 
 S patterns are all functions of £.(t) and can be removed. If 6 is and 
 
 the patterns ire normal ized, the time-averaged output reduces to 
 
 iyi, o° s, e t v i 3 , i> 4 ) = i + s 2 [■ 1 ce 1 ) a. 2 (e 1 ) coscPi^mo^ 
 
 B) coB(Pl 4 «in6 1 ) • :i 2 U) \ ) '•/ ,J i ) cos (Pi si no - Pi sine J 
 
27 
 
 * (6,) MM cos (-(3| sine, + Pi .sine.) 
 3141 31 41 
 
 + S 4 a ce n ) a (6. ) a O-.) a (0 ) cos (Pi sine - Pi sin0 * Pi sine. ) (37) 
 
 in which we see that the S interference pattern is a sum of 4 cross products, 
 e.g., a (0 ) a (0 ) cos(Pi sin - Pi sin ), of the original patterns 
 
 ^ J. *J3 1 ^ X *j X 
 
 modified as usual by a cosine term. A useful design would minimize this 
 pattern in the sidelobe region as well as the product pattern itself. 
 
 We note also that if N is odd the random phase angles from the signal as 
 well as the noise are present in all output terms and consequently time 
 averages of the product pattern systems for N odd all go to zero. Hence, 
 only an even number of antenna patterns can be cross-correlated in say a 
 Radio Astronomy application. 
 
28 
 
 4. SUPPRESSION OF THE EFFECT OF AVERAGE BACKGROUND NOISE BY 
 USE OF THE NEGATIVE SIDELOBES OF THE PRODUCT PATTERN 
 
 An interesting and useful feature of product patterns is their negative 
 sidelobes. Thus they contrast with the power patterns of additive antennas 
 which are non-negative functions of x = Pi sin 8./2„ It will be shown that 
 for an arbitrary distribution of background noise in the absence of the signal 
 the pattern can be adjusted so as to reduce to zero the time-averaged response 
 to this noise in the system output. We assume that the noise coming from any 
 direction is a narrow band quasi-stationary random process, statistically 
 independent of the noise coming from any other direction. If its average 
 power density is given by S (x, t) (which is generally a slowly varying unkno\ 
 function), then the time-averaged response of the additive antenna system 
 to this noise is 
 
 I (x , t) = 
 o' 
 
 S 2 (x, t)|C (x - x)! 2 dx (38) 
 
 7 o 
 
 2 
 
 where . C(x) " is the power pattern of the antenna with x corresponding to the 
 
 o 
 
 / (3£ s i n 9 \ 2 
 main lobe direction I x = — . S (x } t) is clearly non-negative and 
 
 the integral will be positive for any noise background that is not identically 
 
 zero for all x. 
 
 For the produd pattern case, the time-averaged output is 
 
 I (x , t) = 
 
 J 
 
 r 
 
 S 2 (x, t) P o (x - x) dx (39) 
 
 
 and since P, (x) can tak'.- on negative values it should be possible to select 
 
 iny given time t the integral equation 
 
29 
 
 r 
 
 S 2 (x, t ) P (x - x) dx = (40) 
 
 ' o 2 o 
 
 To make Equation (40; hold for all t it is clear that P (x) must become a 
 
 slowly varying function of time in order to compensate for the variation 
 
 2 
 
 in S (x, t). Thus Equation (40) can be generalized to 
 
 r 
 
 S 2 (x, t) P (x - x, t) dx = (40a) 
 
 J 2 o 
 
 In addition, as the beam angle scans it will be necessary, in general, for the 
 pattern to change if the output due to the noise is to remain at zero. 
 
 In this derivation it is assumed that it is known a priori when the signal 
 is present and when it is absent. During the periods when it is absent the 
 antenna pattern is adjusted to eliminate the effect of the slowly varying 
 average background noise. Then when the signal is transmitted its effect 
 alone will be observed in the system's output, A slowly varying background 
 noise density will gradually change this ideal condition and so there should 
 be some prearranged sequence of time intervals in which the signal is turned 
 off and the receiver pattern modified to compensate for this gradual change 
 of background noise density. 
 
 To illustrate, let us consider the Compound Interferometer with the uni- 
 formly weighted aperture replaced by a linear array of m * 1 elements with 
 
 real and symmetric weighting (a = a where r indicates the r element to 
 
 r -r' 
 
 the right from the center of the array and -r indicates the r element to 
 the lef t; : The patterns of the two antennas are 
 
 m/2 
 
 A(x - x ) = 2_ a cos[2r(x - x )] (41) 
 
 o r=0 r *• o 
 
 Dl 
 
B(x - x ) = cos[ m(x - x )] 
 o o 
 
 30 
 
 (42) 
 
 R« ■ n Pi sin 
 
 Pi sin 9 o 
 
 where x = - — — ■ — — , x = ■ 
 
 2 ' o 2 
 
 and G is the beam angle of the patterns, 
 o 
 
 Now from Equation (3a) it follows that 
 
 m/2 
 
 P (x - x ) = 2 a cosi r(x - x )] cos[m(x - x )] cos[2m(x - x )] 
 2 o r o o o 
 
 r=0 
 
 (43) 
 
 m/2 
 
 m/2 
 
 = 2af(x-x)=2 af(x-x) 
 rr o rro 
 
 r=0 r=0 
 
 Substituting this into the expression for the output due to the arbitrary 
 background noise density (Equation (39)) we get 
 
 I(x , t) = 
 
 o' 
 
 m 2 
 = 2 
 r=0 
 
 2 m/2 
 S (x, t) 2 a f (x - x) d: 
 rro 
 r=0 
 
 S (x, t) f (x - x) dx 
 r o 
 
 (44) 
 
 and from Equation i40) we require that 
 
 m 2 
 2 a 
 , ' 
 
 2 m/2 
 
 S x t) f fx - x) dx = 2 a q (x , t) = 
 r o r r o' 
 
 r=0 
 
 (45) 
 
 where q (x t ) 
 r o 
 
 S (x, t) f (x - x ) dx 
 r o 
 
 (46) 
 
 To get the coil I , q (x , l) it is necessary to install switches in the 
 
 r o 
 
 trar. from the various array elements. If all switches but the 
 
 the center element' I Lne were open then the output from the system 
 
 i 
 
31 
 
 I (x , t) = a q (x , t) (47) 
 
 o o o o o 
 
 and if during the measurement starting at, say, t , a is set equal to unity, 
 we get 
 
 q (x t ) = I (x t ) (47a) 
 
 o <r o o o o 
 
 Likewise opening; all but the lines from the a elements and setting a equal 
 ft. r r 
 
 to unity we get 
 
 q (x , t ) = I (x , t ) (48) 
 
 r o' o r o o 
 
 In this way all of the coefficients can be obtained. 
 
 We can think of the q (x , t)s as components of an — + 1 dimensional noise 
 
 r o- 2 
 
 vector Q'x t 1 whose scalar product with the weighting vector A('x , t ) also 
 o o o 
 
 f _ a. i dimensions, is required to be zero by Equation (45) „ It is clear 
 
 that there is no unique vector A(x , t ) that will satisfy this condition since 
 
 o o 
 
 any vector will do which lies in the hyperplane to which Q<x t ) is perpen- 
 
 o o 
 
 dicular Out of this infinity of vectors that satisfy Equation (45) it would 
 
 be best to select one which gave the maximum response to the signal when it 
 
 was incident at the beam angle 6 . This means that we pick Aix t ) such that 
 
 o o o 
 
 m/2 
 P (0) = 2 a (x , t ) = MAX (49) 
 
 2 r=0 
 
 However it is obvious that some limit on the size of the vectors A('x , t ) 
 
 o o 
 
 themselves must be established because if any vector is doubled in size then 
 the sum of its components will also be doubled, Hence we seek out from all 
 
32 
 
 vectors of a given length lying in the hyperplane the one whose sum of its 
 components is the largest. We set this length (without loss of generality) 
 equal to unity and we obtain the condition equation 
 
 m/2 
 
 2 a (x , t ) = 1 (50) 
 
 r=0 r ° ° 
 
 Note that there are — + 1 coefficients a (x , t ) and still only 3 
 
 2 r o o 
 
 simultaneous equations. There is thus a possible — - 2 additional conditions 
 that could be satisfied, e.g. 1) first null occurs at 9 - A, 2) first 
 sidelobe has maximum at = TT , etc. 
 
 However, let us consider the five-element case, m = 4 in which there 
 are just three possible conditions; the three equations are 
 
 a (x , t ) + a, (x , t ) 4 a (x , t ) = MAX 
 
 o o' o loo 2 o 7 o 
 
 c (x , t ) a fx , t ) + c, (x , t ) a (x , t ) f c (x , t ) a <x , t ) - 
 oooooo lo'oloo 2 o o 2 o o 
 
 2 2 2 
 
 a (x , t ) + a, (x , t)+a„(x.t)=l 
 o o' o loo 2o'o 
 
 (51) 
 
 The last two equations represent a plane and a sphere, respectively, in three 
 dimensional space. Their intersection is a circle of unit radius centered at 
 the origin and a certain point (or points) on it satisfy the first equation. 
 To get this point I arrange' s method of undetermined multipliers can be used 
 and the following s< I -,\ equal ions results. 
 
 L + c (x , t ) X (x , t ) + 2a (x , t ) X (x , t ) » 
 
 -/ o o o O o o 
 
33 
 
 1 .. c ix , t i X (x , t ) + 2a, (x . t ) \ (x t ) = 
 loo ooo loo lo o 
 
 (52) 
 
 1 - c (x , t ) v 'X , t ) + 2a (x , t ) X (x , t ) = 
 2 " o o ooo 2 o o loo 
 
 c (x , t ) a <x , t ) + c, (x , t ) a n (x , t ) + c„ (x t ) a (x t ) = 
 
 ooo o o o 1 o' o 1 o' o 2 o 7 o 2 o o 
 
 2 2 2 
 
 a ix t ) + a (x , t ) + a„ (x , t ) = 1 
 
 o o' o 1 o' o 2 o' o 
 
 A digital computer could be used to solve these equations for the weighting 
 
 coefficients a ( x s t ) a (x , t) and a (x , t ) Then instructions from 
 ooolo'o 2 o o 
 
 the computer could be fed back to each of the lines coming from the array 
 
 elements where a variable attenuator would be adjusted to give the correct 
 
 weighting. A diagram of the system is shown in Figure 5. Note that the 
 
 computer also sends phase shift instructions (for scanning) and switching 
 
 instructions i for obtaining the q (x , t ) coefficients) The conditions 
 
 r o o 
 
 fed into the computer are, of course, those that are mentioned above and 
 which give rise to Equations (52) which the computer must solve. 
 
 A final word of caution should be mentioned Although in theory the 
 equations can always be solved to reduce the noise output to zero, some noise 
 backgrounds can be eliminated only by resorting to a supergain condition For 
 example, a not uncommon case is that of a single point source of noise which 
 can be eliminated only if the pattern always has a null in the direction of 
 the point source This presents no difficulty until the beam angle x itself 
 approaches the angle of the point source. This requires the null to approach 
 the beam angle, and since the pattern is an analytic function the response 
 of the pattern to the signal incident at the beam angle will approach zero. 
 
V 
 
 3 
 
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 CO 
 bfl 
 cd 
 
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 x: 
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 <H 
 
 O 
 
 c 
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 W 
 
 
 
 a 
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 3 
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 (1) 
 XI 
 
 0J 
 
 in 
 
 •H 
 
 o 
 
 c 
 
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 c 
 
 3 
 
 o 
 
 bD 
 
 o 
 
 CQ 
 <H 
 
 o 
 
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 cd (-< 
 
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 en co 
 
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 3 
 be 
 
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 a m 
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 3£ 
 
35 
 
 To maintain the response at, say, unit amplitude it will be necessary to 
 increase the magnitude of the weighting coefficients a (x , t ) which means 
 going to a supergain condition. Furthermore, if the beam angle and the angle 
 of the point source of noise coincide it will be necessary to split the beam to 
 satisfy Equation (45). This means that it is impossible to eliminate the noise 
 and not the signal if both are incident as plane waves at the same angle. How- 
 ever, there is usually more than one source of noise and, in theory at least, 
 the signal from a point source which may or may not coincide with one of the 
 noise sources can be retained and the noise rejected. 
 
36 
 
 5. THE NOISE LIMITATION ON THE FORMATION OF ARBITRARY PATTERNS 
 FROM A TWO ELEMENT INTERFEROMETER BY 
 MEANS OF MULTIPLICATIVE OPERATIONS 
 
 5,6 
 Several writers •' have discussed the possibility of forming arbitrary 
 
 antenna patterns by multiplicative operations on the cross-correlated output 
 
 from a single pair of elements spaced a distance I apart^ i.e., from a simple 
 
 interferometer „ Patterns of arbitrary sharpness of the main lobe could be 
 
 synthesized and used to advantage in direction-finding applications. However^ 
 
 no account was taken of noise in these schemes^ and it will be shown below 
 
 that noise severely limits their usefulness. 
 
 From Equation (3a) the pattern of the interferometer is 
 
 P (6, i) = cos (Pi sin 8) 
 
 (58) 
 
 It is raised to various powers from say to M and a linear combination of 
 these terms each weighted by a factor q is formed 
 
 M k 
 
 Q (x) = 2 q cos x 
 k=0 
 
 (59) 
 
 where x = Pi sin 9 But 
 
 k 
 cos X 
 
 k-1 
 
 k 
 
 1 l k 
 cos[ (k - 2r)x] -4 — 
 
 „ k k 
 
 2 la 
 
 (60) 
 
 if k is even, and 
 
 k 
 COS X 
 
 k-1 
 
 2 | ') 
 
 r, 
 
 cos[ (k - 2r) x] 
 
 (60a) 
 
37 
 
 if k is odd and where 
 
 is the binomial coefficient. Hence 
 
 M 
 Q (x) = 2 w cos(kx) 
 k=0 
 
 where the w s are linear combinations of the q.'s. By a proper choice of the 
 k K 
 
 q :'s any arbitrary pattern of a 2M + 1 element array (M even) can be simu- 
 lated. If M is odd the array has 2M elements. 
 
 The assumption is made that there is no noise. In particular, it is 
 assumed that only a single plane wave ; the signal ; is incident on the antennas, 
 In the following it will be shown that if the signal is received in an 
 arbitrary background of noise there will generally be a shift in the apparent 
 direction of the signal as "seen" by the basic pattern cos (Pi sin 0) and the 
 process of forming a "better" pattern by multiplicative operations cannot 
 remove this error. 
 
 Let the signal of say unit average power be incident on the system from 
 
 direction x = Pi sin 9 and let the noise have an average power density given 
 
 2 
 
 by S (x) . Then the time-averaged output from the low-pass filter of the 
 
 correlation system is 
 
 I(x) = 
 
 o 
 
 ;5( X;l - x) + S^( Xl ), 
 
 cos(x - x ) dx 
 oil 
 
 cos(x - x) + 
 o 
 
 S (x, ) cos x dx 
 1 11 
 
 cos X 
 
 S (x ) sin x dx 
 
 sin x 
 
 (61) 
 
 where x is the beam angle of the interferometer pattern. We see that the 
 
38 
 
 terms in the square brackets are the Fourier coef f icients, a and b , of the 
 series expansion of S(x) given by 
 
 where 
 
 S(x) = Z a cos kx + 2 b sin k x 
 
 k=o k k=i k 
 
 cos(kx) cos(jx) dx = 6 
 
 sin(kx) sin(jx) dx = 6, , 
 
 kj 
 
 sin(kx) cos(jx) dx = , 
 
 and 
 
 5 kj =l, k-J 
 
 = 0, k £ j 
 
 Therefore Equation (60) can be expanded as 
 
 I < x ) = cos(x - x) + a cos x + b sin x 
 o o 1 o i o 
 
 = cosCx - x) + c cos(x - x ) 
 o l o c 
 
 where 
 
 2 2 . 2 . -1 
 c, = a, + b, , x = tan 
 
 1 1 1 c 
 
 The arbitrary nod ie background is "seen" by the pattern as a point source at 
 
39 
 
 ang 
 
 le x = Pi sin with average power c . 
 c c 1 
 
 Equation (63) can be reduced finally to 
 
 I(x ) = [ cos x + c cos x 1 cos x + [ sin x + c sin x ■] sin x 
 o 1 c J o 1 c o 
 
 d, cos(x - x ) 
 1 o d 
 
 (66) 
 
 where 
 
 2 ,2 
 
 [cos x + c cos x + [sin x + c, sin x J 
 L l c J L 1 c J 
 
 x = tan 
 d 
 
 -1 
 
 sin x + c sin x 
 1 c_ 
 
 cox x + c cos X 
 1 c. 
 
 (67) 
 
 The relations are best shown by the phasor diagram of Figure 6. The output is 
 
 jx 
 the scalar product of the unit scanning phasor e and the phasor sum e ' + 
 
 J x ^ J x h 
 c d 
 
 c e = de There is a magnitude error and an angular error in the esti- 
 mated value of the signal. However^ if x = x or x -77 the angular error is 
 
 ze 
 
 ro; the magnitude error is maximized and is - c. . Contrariwise if x = 
 
 1 c 
 
 x - cos (c ,/2) the amplitude of d is unity and no magnitude error occurs. 
 
 The maximum error in angle for a given c < 1 is (x -x) =- sin c 
 
 1 d max 1 
 
 and occurs at x = x + (77/2 + sin e, ) . If c ' • 1 the error can be as much 
 c 11 — 
 
 as 77, 
 
 The basic patterns due to the signal alone ; the noise alone and the signal 
 plus noise are shown in Figures 7a, 7b ? and 7c respectively. In Figure 7d a 
 pattern Q„(x - x ) synthesized from the basic signal plus noise pattern. 
 
 MOd or- r , 
 
 d cos(x - x ) is shown. The error (x, - x ) occurring in the basic pattern 
 1 o d d 
 
 can in no way be reduced by this synthesizing of Q (x - x ) since the process 
 
 Mod 
 
40 
 
 0) 
 
 
 
 a 
 
 SI. 
 
 in o 
 
 o 
 
 c 
 
 3 Z 
 
 M 
 
 
 
 H 
 
 ^< 
 
 —i 
 
 0. in 
 
 o 
 
 < 
 
 3 
 
 x-a 
 
 
 — 1 r-l 
 
 (U 
 
 CO CU 
 
 c 
 
 05 
 
 a 
 
 bO 
 
 •^1 
 
 60 r-l 
 
 i-( 
 
 
 
 ■H ffl 
 
 CO 
 
 z 
 
 CO C 
 
 bo 
 
 <M 
 
 VI 
 
 <H -H 
 
 
 
 
 
 O CO 
 
 c 
 
 c 
 
 C <H 
 
 u 
 
 u 
 
 U O 
 
 0) 
 
 4) 
 
 0) 
 
 4-> 
 
 •P 
 
 P c 
 
 ■p 
 
 ■»-> 
 
 P u 
 
 ed 
 
 0) 
 
 3) CO 
 
 0, 
 
 £ 
 
 a, *> 
 p 
 
 k >-• 
 
 t< 
 
 in at 
 
 1 0) 
 
 0) 
 
 a) a, 
 
 •p 
 
 ■u 
 
 p 
 
 0) 
 
 0) 
 
 d) T3 
 
 s 
 
 G 
 
 E a» 
 
 ■s. o 
 
 
 
 « 
 
 V ^ 
 
 l* 
 
 U -H 
 
 -^ 0> 
 
 CO 
 
 0) in 
 
 <H 
 
 <*-{ 
 
 <H 0) 
 
 u 
 
 ^ 
 
 fc £ 
 
 <D 
 
 0) 
 
 1) P 
 
 +) 
 
 P 
 
 P c 
 
 —^ a 
 
 G 
 
 c >> 
 
 J w 
 
 i—i 
 
 HH CO 
 
 d A u Q 
 
 D 
 
 O 
 
 "O 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \\ 
 
 
 
 
 
 tr 
 
 
 
 
 
 
 M, 
 
 
 
 
 UJ 
 CO 
 
 
 
 CO 
 
 < 
 
 1 
 1 
 
 
 
 
 
 
 
 
 
 
 z 
 
 I 
 
 w 1 
 
 X *J 
 ^1 
 
 
 
 
 
 a: 
 
 \ 
 
 
 
 
 0" 
 
 / / 
 
 
 
 ^ 
 
 _l 
 
 z 
 
 CO 
 CO 
 
 O 
 
 CO 
 
 < 
 
 X 
 0. 
 
 \ 
 
 1 
 
 \ 
 
 \ \ 
 
 
 
 AV 
 
 
 
 
 
 
 
 \ 
 
 \ \ 
 
 
 
 / \ 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 ' / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 ^/ 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ — 
 
 
 
 
 
 — \ 
 
 X 
 
 \ 
 \ 
 
 z / 
 
 
 
 \ 
 
 
 
 "O 
 
 
 
 \ 2 
 
 Z CO / 
 
 
 
 
 V 
 
 
 
 
 
 *— *• 
 
 \ 
 
 < <\/ 
 
 
 
 
 
 \ 
 
 
 
 
 X\ 
 
 
 1 7\. 
 
 
 
 
 
 
 \ 
 
 
 
 1 
 
 \ X ^«- — ► 
 
 to Q- / ' 
 
 
 
 
 
 
 
 ^ 
 
 Xy 
 
 w\S\ 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 \ \ "°v 
 
 
 
 
 
 
 
 
 
 
 
 
 •-* 
 
 G 
 
 
 ct) 
 
 V 
 
 
 C 
 
 B 
 
 
 hfi 
 
 0) 
 
 
 -H 
 
 r-H 
 
 
 w 
 
 W 
 
 
 CO 
 
 O 
 
 
 JG 
 
 p 
 
 ^ 
 
 
 <H 
 
 m 
 
 
 O 
 
 a 
 
 
 c 
 
 
 
 
 
 
 h 
 
 >. 
 
 ■H 
 
 m 
 
 c« 
 
 *-> 
 
 
 ^ 
 
 fit 
 
 *-> 
 
 U 
 
 P 
 
 D 
 
 < 
 
 C 
 
 D. 
 
 
 CO 
 
 ♦^ 
 
 <x> 
 
 
 3 
 
 > 
 
 t< 
 
 
 *-> 
 
 a 
 
 <u 
 
 Cfl 
 
 <D 
 
 Ul 
 
 U 
 
 IX 
 
 •H 
 
 H 
 
 
 O 
 
 r-< 
 
 u 
 
 ^. 
 
 D. 
 
 
 
 
 •H 
 
 '/i 
 
 U5 
 
 •(-> 
 
 Tj 
 
 g 
 
 rH 
 
 x: 
 
 .-1 
 
 3 
 
 E 
 
 a 
 
 fl 
 
 'D 
 
 
 
 Cb 
 
 
 
 ^ 
 
 
 
 3, 
 
 
 
41 
 
 only defines more sharply the value of x for the benefit of the observer. 
 The synthesized pattern is limited not by its own shape and sharpness (in 
 theory it can be made to approach a delta) but by the basic interferometer 
 pattern to which any combination of point source plus background noise 
 appears as another equivalent point source. This is all that the synthesized 
 pattern will ever "'see". If there is a fairly low signal to noise ratio it 
 can be concluded that a method of pattern syntheses such as this can lead 
 to large errors in the estimated value of the angle of incidence of the 
 signal . 
 
42 
 6. CONCLUSIONS 
 
 It has been shown that the cross-correlation of antenna voltages to 
 obtain product patterns gives rise to two major problems: 
 
 1. Multiplicative distortion of amplitude modulation, 
 
 2. Interference patterns which differ from the original product pattern 
 and which can have high sidelobes even when the product pattern itself has 
 low sidelobes. 
 
 The first problem can in theory be overcome by a rather complicated 
 system which requires frequency shifting and filtering. An analysis of the 
 second problem shows that if instantaneous reception of, say, modulated signals 
 is required, special care must be exercised in the design of both product 
 and interference patterns if an adequate rejection of interference is to be 
 achieved. For systems in which the time average of the incoming signals is 
 all that is required the problem simplifies considerably but if N, the number 
 of patterns to be multiplied, is large it is still quite involved. 
 
 It has also been shown that product patterns possess a distinct advantage 
 over conventional additive power patterns in that their negative sidelobes can 
 be used to suppress the integrated effect of any arbitrary distribution of 
 background noise. 
 
 Finally, the artificial formation of arbitrarily sharp antenna patterns 
 by multiplicative operations on the time-averaged output of a two element 
 interferometer system has been shown to be of limited value in the presence 
 of ,i background noise distribution because the noise causes a shift in the 
 apparent direction of arrival of the signal. This shift is not reduced by 
 
 formation '<( i more directive" pattern whose effect is therefore nothing 
 
43 
 
 more than to emphasize the apparent direction of arrival of the signal as 
 indicated by the interferometer pattern. If the noise is relatively strong 
 the error in the apparent direction of arrival can be quite large. 
 
44 
 REFERENCES 
 
 1. Mills, B. Y. and Little, A. G., "A High-Resolution Aerial System of a 
 New Type." Austral. J. Phys. , Vol. 6, pp. 272-278, September, 1953. 
 
 2. Covington, A. E. and Broten, N. W "An Interferometer for Radio 
 Astronomy with a Single Lobed Radiation Pattern," IRE Trans, PGAP, 
 AP-5, No. 3, pp. 247-255, July, 1957. 
 
 3. Welsby, V, G. and Tucker, D. G. , "Multiplicative Receiving Arrays," 
 J, Brit, IRE, 19, pp. 369-382, June, 1959 
 
 4. Berman, A. and Clay, C. S., "Theory of Time-Averaged Product Arrays," 
 J. Acoustical Soc. America , 29, No. 7, pp. 805-812, August, 1957. 
 
 5. Drane, C. J,, Jr., "Phase Modulated Antennas," presented at IRE-URSI 
 Joint Fall Meeting, 1959, San Diego, California. 
 
 6. Brown, J. L M Jr. and Rowlands, R. 0., "Design of Directional Arrays,' 
 J. Acoustical Soc. America , 31, No. 12, pp. 1638-1643, December, 1959. 
 
 7. Dwight, H. B., Tables of Integrals and Other Mathematical Data , 
 The Macmillan Company, p.l. 
 
ANTENNA LABORATORY 
 TECHNICAL REPORTS AND MEMORANDA ISSUED 
 
 "Synthesis of Aperture Antennas, " Technical Report No : 1, C.T.A, Johnk, 
 October. 1954. * 
 
 A Synthesis Method for Broad-band Antenna Impedance Matching Networks," 
 Technical Report No. 2, Nicholas Yaru, 1 February 1955 
 
 The Asymmetrically Excited Spherical Antenna," Technical Report No, 3, 
 Robert C Hansen. 30 April 1955,* 
 
 "Analysis of an Airborne Homing System," T ech nical Report No, 4, Paul E„ 
 
 Mayes r 1 June 1955 ( CONFIDENTIAL) . 
 
 Coupling of Antenna Elements to a Circular Surface Waveguide, Technical 
 Report No 5, H. E King and R. H. DuHamel , 30 June 1955.* 
 
 Axial ly Excited Surface Wave Antennas," Technical Report No. 7, D, E. Royal, 
 10 October 1955 * 
 
 Homing Antennas for the F-86F Aircraft f 450-2500mc) , " Technical Report No. 8, 
 P.E t Mayes R.F, Hyneman, and R.C. Becker, 20 February 1957, (CONFIDENTIAL). 
 
 Ground Screen Pattern Range," Technical Memorandum No 1, Roger R. Trapp, 
 10 July 1955 * ~~~ ~~~ 
 
 Conj rag t AF33 ( 61 6 i ^3220 
 
 Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J. 
 Scott and R H DuHamel, 16 April 1956. 
 
 An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report 
 No. 10, D G Berry and J B, Kreer, 10 May 1956. 
 
 "A Technique for Controlling the Radiation from Dielectric Rod Waveguides," 
 Tech_nical_Repot No. 11, J W. Duncan and R = H DuHamel, 15 July 1956.* 
 
 Directional Characteristics of a U-Shaped Slot Antenna, ' Technical Report 
 No^_12, Richard C Becker, 30 September 1956 „** 
 
 Impedance of Ferrite Loop Antennas," Technical Report No 13, V, H, Rumsey 
 and W. L Weeks. 15 October 1956 
 
 "Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report 
 No^_l_4, Richard F Hyneman, 20 December 1956. 
 
Distributed Coupling to Surface Wave Antennas," Technical Report No 15. 
 
 jj^ ' — — — * 
 
 Ralph Richard Hodges, Jr , 5 January 1957. 
 
 The Characteristic Impedance of the Fin Antenna of Infinite Length, Technical 
 Report No 16 Robert 1 Carrel, 15 January 1957 
 
 On the Estimation of Ferrite Loop Antenna Impedance, Technical Re port No. 17, 
 Walter L. Weeks, 10 April 1957.* 
 
 A. Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source 
 Antenna fechnic .1 Report No. 18, Walter L. Weeks, 20 April 1957, 
 
 Broadband logarithmically Periodic Antenna Structures. Technical Report No . 19 
 R K DuHamel and D E. Isbell, 1 May 1957. 
 
 Frequency Independent Antennas," Tec hnical Report No 20 V. H Rumsey, 25 
 
 October 1957 
 
 The Equiangular Spiral Antenna," Techni cal Report No, 21, J D, Dyson, 15 
 September 195? 
 
 Experimental Investigation of the Conical Spiral Antenna Technical Report 
 Vo 22. R L. Carrel 25 May 1957.** ' """ 
 
 Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical 
 Report \ T o. 23 E ' Scott, 10 August 1957. 
 
 Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," 
 rechni .. | i 24 J. W. Duncan and R. H DuHamel, August 1957. 
 
 The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary 
 Cross Section, hnical Repo rt No. 25, Robert L Carrel, August 1957. 
 
 -Backed Slot Antennas, ' Technical R eport No. 26_, R. J. Tector, 30 
 October 1957 
 
 Coupled Waveguide Excitation of Traveling Wave Slot Antennas/ Techni cal 
 Re] [0 27 W I Weeks, 1 December 1957. 
 
 Pti Red ingular Waveguide Partially Filled with Dielectric," 
 
 28 W I Weeks, 20 December 1957 
 
 p< r lint Length of Biconical Structures of Arbitrary 
 Cross rechnli i] Rep ort No. 29 , J. D Dyson. 10 January 1958 
 
 Lcally Periodic Antenna Structur Technical Report No. 30, 
 D ^58 
 
 ( ' pular Slots,' technical Reporl ^'< 31, N. J. 
 ; , L958 
 
 tation oi i Surface Wave on a Dielectric Cylinder/' 
 I w Du J5 May L058 
 
"A Unidirectional Equiangular Spiral Antenna, " Technical Report No. 33, J. D. 
 Dyson, 10 July 1958 
 
 Dielectric Coated Spheroidal Radiators," Technical Report No. 34^ W. L Weeks, 
 12 September 1958. 
 
 "A Theoretical Study of the Equiangular Spiral Antenna," Technical Report 
 No. 35, P E Mast, 12 September 1958. 
 
 Contract AF33 '616) -6079 
 
 Use of Coupled Waveguides in a Traveling Wave Scanning Antenna,' Technical 
 ILERELLJ^JL 6 -.' R H MacPhie, 30 April 1959. 
 
 On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and 
 Infinite Ranges, Technical Report No. 37, Raj Mittra, 15 May 1959. 
 
 "Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical 
 Repojr£_No ^_38 , W, L, Weeks, 5 June 1959. 
 
 Log Periodic Dipole Arrays," Technical Report No. 39, D E Isbell, 1 June 1959. 
 
 A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical 
 Report No^ 40. 3. Dasgupta and Y. T„ Lo, 17 July 1959 
 
 The Radiation Pattern of a Dipole on a Finite Dielectric Sheet/" T echnical 
 ReportNq 41 KG Balmain, 1 August 1959 
 
 The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem 
 in a Waveguide. Technical Report No. 42, Raj Mittra, 1 October 1959. 
 
 "impedance Properties of Complementary Mul titerminal Planar Structures " 
 Te chjrica.1 JRepor t No __43_, G, A, Deschamps, 11 November 1959 
 
 On the Synthesis of Strip Sources," Technical Report No 44 Raj Mittra, 
 4 December 1959 
 
 Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides," 
 Technical Report No^ 45. C H Tang and Y. T, Lo, 11 March I960, 
 
 Ve* Circularly Polarized Frequency Independent Antennas With Conical Beam or 
 Omnidirectional Patterns," Tec hnica l Report No, 46, J.D Dyson and P.E. Mayes, 
 20 June 1960 
 
 Logarithmically Periodic Resonant-V Arrays,'" Technical Report No, 47, P.E. 
 Waves and R L Carrel, 15 July 1960. 
 
 * Copies available for a three week loan period 
 Copies no longer available. 
 
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 Seal t : <■ 3, WashJ rig I on 
 
 ITT Laboratories 
 Attn;, L„ DeRosa 
 
 M F Contract AF33( 616)-5120 
 500 Washington Avenue 
 Nutley 10, New Jersey 
 
 ITT Laboratories 
 
 A Div. of Int, Tel. & Tel, Corp, 
 Attn: G B S. Giffin, ECM Lab, 
 3700 E. Pontiac Street 
 Fort Wayne, Indiana 
 
 Jansky and Bailey, Inc 
 Engineering Building 
 
 Attn: Mr. D. C, Ports 
 1339 Wisconsin Avenue, N.W. 
 Washington, D.C 
 
 Jasik Laboratories, Inc, 
 100 Shames Drive 
 Westbury, New York 
 
 John Hopkins University 
 Radiation Laboratory 
 Attm Librarian 
 
 M/F Contract AF33(616)-68 
 1315 St. Paul Street 
 Baltimore 2, Maryland 
 
 Applied Physics Laboratory 
 Johns Hopkins University 
 8621 Georgia Avenue 
 Silver Spring, Maryland 
 
 Lincoln Laboratories 
 Attn.. Document Room 
 
 M/F Contract AF19(122)-458 
 Massachusetts Institute of Technology 
 P.O. Box 73 
 Lexington 73, Massachusetts 
 
AF 33(616)-6079 
 
 Litton Industries, Inc. 
 Maryland Division 
 
 Attn Engineering Library 
 
 M F Contract AF33( 600)-37292 
 4900 Calvert Road 
 College Park, Maryland 
 
 University of Michigan 
 Aeronautical Research Center 
 Attn: Dr, K, Seigel 
 
 M/F Contract AF30( 602)-1853 
 Willow Run Airport 
 Ypsilanti, Michigan 
 
 Lockheed Aircraft Corporation 
 Attn: C, D, Johnson 
 
 M/F Contract NOa(s) 55-172 
 P.O. Box 55 
 Burbank, California 
 
 Microwave Radiation Co,, Inc, 
 Attn: Dr, M, J. Ehrlich 
 
 M F Contract AF33(616'>-6528 
 19223 S. Hamilton Street 
 Gardena, California 
 
 Lockheed Missiles & Space Division 
 
 (Attn: E, A, Blasi 
 M/F Contract AF33( 600) -28692 & 
 AF33(616)-6022 
 Department 58-15 
 Plant 1, Building 130 
 Sunnyvale, California 
 
 The Martin Company 
 
 Attn; W, A, Kee, Chief Librarian 
 
 M/F Contract AF33(600)-37705 
 Library & Document Section 
 Baltimore 3, Maryland 
 
 Ennis Kuhlman 
 
 McDonnell Aircraft 
 
 P.O. Box 516 
 
 Lambert Municipal Airport 
 
 St, Louis 21, Missouri 
 
 Melpar, Inc . 
 
 Attn, Technical Library 
 
 M/F Contract AF19(604)-4988 
 Antenna Laboratory 
 3000 Arlington Blvd. 
 Falls Church, Virginia 
 
 Melville Laboratories 
 Walt Whitman Road 
 Melville, Long Island, 
 New York 
 
 Motorola, Inc , 
 
 Attn: R, C, Huntington 
 8201 E, McDowell Road 
 Phoenix, Arizona 
 
 Physical Science Lab. 
 
 Attn. R. Dressel 
 New Mexico College of A and MA 
 State College, New Mexico 
 
 North American Aviation, Inc , 
 
 Attn: J. D Leonard, Eng . Dept . 
 M/F Contract NOa(s) 54-323 
 4300 E, Fifth Avenue 
 Columbus, Ohio 
 
 North American Aviation, Inc, 
 Attn: H, A. Storms 
 
 M/F Contract AF33( 600)- 36599 
 Department 56 
 International Airport 
 Los Angeles 45, California 
 
 Northrop Aircraft, Inc. 
 
 Attn: Northrop Library, Dept. 2135 
 M F Contract AF33 (600)-27679 
 Hawthorne, California 
 
 Dr, R. E, Beam 
 Microwave Laboratory 
 Northwestern University 
 Evanston, Illinois 
 
AF 33' 616'. -6079 
 
 Ohio State University Research 
 Foundation 
 
 Attn: Dr, T, C, Tice 
 
 M F Contract AF33(616)-6211 
 1314 Kinnear Road 
 Columbus 8, Ohio 
 
 University of Oklahoma Res, Inst. 
 Attn Prof, C, L, Farrar 
 
 M F Contract AF33( 616)-5490 
 Norman, Oklahoma 
 
 Dr. D. E. Royal 
 
 Ramo-Wooldridge, a division of Thompson 
 
 Ramo Wooldridge Inc, 
 
 8433 Fallbrook Avenue 
 
 Canoga Park, California 
 
 Rand Corporation 
 Attn: Librarian 
 
 M/F Contract AF18(600)-1600 
 1700 Main Street 
 Santa Monica, California 
 
 Philco Corporation 
 
 Government and Industrial Division 
 Attn Dr, Koehler 
 
 M. F Contract AF33(616)-5325 
 4700 Wissachickon Avenue 
 Philadelphia 44, Pennsylvania 
 
 Prof, A A, Oliner 
 Microwave Research Institute 
 Polytechnic Institute of Brooklyn 
 55 Johnson Street - Third Floor 
 Brooklyn, New York 
 
 Radiation, Inc , 
 Technical Library Section 
 Attn Antenna Department 
 
 M F Contract AF33(600)-36705 
 Melbourne, Florida 
 
 Radio Corporation of America 
 RCA Laboratories Division 
 Attn Librarian 
 
 M F Contract AF33( 616)-3920 
 Princeton, New Jersey 
 
 Radioplane Company 
 
 M/F Contract AF33( 600) -23893 
 
 Van Nuys, California 
 
 Ramo-Wooldridge, a division of 
 Thompson Ramo Wooldridge, Inc. 
 
 Attn technical Information Services 
 8433 Fallbrook Avenue 
 P 0. Box 1006 
 Canoga Park California 
 
 Rantec Corporation 
 
 Attn: R. Krausz 
 M/F Contract AF19( 604)-3467 
 Calabasas, California 
 
 Raytheon Manufacturing Corp „ 
 Attn; Dr, R, Borts 
 
 M/F Contract AF33(604)-15634 
 Wayland, Massachusetts 
 
 Republic Aviation Corporation 
 Attn: Engineering Library 
 
 M/F Contract AF33( 600)-34752 
 Farmingdale 
 Long Island, New York 
 
 Republic Aviation Corporation 
 Guided Missiles Division 
 Attn: J. Shea 
 
 M/F Contract AF33(616)-5925 
 223 Jericho Turnpike 
 Mineola, Long Island, New York 
 
 Sanders Associates, Inc. 
 95 Canal Street 
 
 Attn: Technical Library 
 Nashua, New Hampshire 
 
 Smyth Research Associates 
 
 Attn.; J. B. Smyth 
 3555 Aero Court 
 San Diego 11, California 
 
 Space Technology Labs, Inc, 
 
 Attn; Dr., R. C Hansen 
 P.O. Box 95001 
 Los Angeles 45, California 
 M/F Contract AF04( 647 ) -361 
 
AF 33(616^-6079 
 
 Sperrv Gyroscope Company 
 Attn B, Berkowitz 
 
 M. F Contract AF33( 600)-28107 
 Great Neck 
 Long Island, New York 
 
 Stanford Electronics Laboratory 
 Attn. Applied Electronics Lab. 
 Document Library 
 Stanford Iniversity 
 Stanford, California 
 
 Stanford Research Institute 
 
 Attn; Mary Lou Fields, Acquisitions 
 Documents Center 
 Menlo Park, California 
 
 Stanford Research Institute 
 Aircraft Radiation Systems Lab, 
 Attn D, Scheuch 
 
 M F Contract AF33(616)-5584 
 Menlo Park, California 
 
 Sylvania Electric Products, Inc. 
 Electronic Defense Laboratory 
 M/F Contract DA 36-039-SC-75012 
 P.0, Box 205 
 Mountain View, California 
 
 Mr, Roger Battie 
 
 Supervisor, Technical Liaison 
 
 Sylvania Electric Products, Inc, 
 
 Electronic Systems Division 
 
 P.O. Box 188 
 
 Mountain View, California 
 
 Sylvania Electric Products, Inc , 
 Electric Systems Division 
 Attn- C. Faflick 
 
 M/F Contract AF33(038)-21250 
 100 First Street 
 Waltham 54, Massachusetts 
 
 Technical Research Group 
 M/F Contract AF33< 61 6 ) -6093 
 2 Aerial Way 
 Syosset, New York 
 
 Temco Aircraft Corporation 
 Attn: G, Cramer 
 
 M/F Contract AF33( 600) -36145 
 Garland, Texas 
 
 Electrical Engineering Res, Lab, 
 University of Texas 
 Box 8026, University Station 
 Austin, Texas 
 
 A, S 5 Thomas, Inc„ 
 M/F Contract AF04C 645;-30 
 161 Devonshire Street 
 Boston 10, Massachusetts 
 
 Westinghouse Electric Corporation 
 Air Arm Division 
 Attn: P, D, Newhcuser 
 
 Development Engineering 
 M/F Contract AF33(600)-27852 
 Friendship Airport 
 Baltimore, Maryland 
 
 Professor Morris Kline 
 
 Institute of Mathematical Sciences 
 
 New York University 
 
 25 Waverly Place 
 
 New York 3, New York 
 
 Dr , S. Dasgupta 
 
 Government Engineering College 
 
 Jabalpur, M.P, 
 
 India 
 
 Dr , Richard C. Becker 
 10829 Berkshire 
 Westchester, Illinois 
 
 Tamar Electronics, Inc, 
 Attn LB McMurren 
 2045 W Rosecrans Avenue 
 Gardena, California 
 
 The Engineering Library 
 Princeton University 
 Princeton, New Jersey 
 
AF 33(616)-6079 
 
 Dr. B, Chatterjee 
 Communication Engineering Dept , 
 Indian Institute of Technology 
 Kharagpur (S.E. Rly,) 
 India 
 
 Sperry Phoenix Company 
 Attn: Technical Librarian 
 P.O. Box 2529 
 21111 North 19th Avenue 
 Phoenix, Arizonia 
 
 Dr. Harry Letaw, Jr„ 
 
 Raytheon Company 
 
 Surface Radar and Navigation Operations 
 
 State Road West 
 
 Wayland, Massachusetts