'^'^1 
 
 ANTENNA LABORATORY 
 Technical Report No. 58 
 
 I JNo Ll^A«,y 
 UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
 ON INCREASING THE EFFECTIVE APERTURE 
 OF ANTENNAS BY DATA PROCESSING 
 
 by 
 Robert H. MacPhie 
 
 Contract No. AF33(657)-8460 
 Project No. 6278, Task No. 40572 
 
 JULY 1962 
 
 Sponsored by 
 
 AERONAUTICAL SYSTEMS DIVISION 
 
 WRIGHT-PATTERSON AIR FORCE BASE, OHIO 
 
 ELECTRICAL ENGINEERING RESEARCH LABORATORY 
 
 ENGINEERING EXPERIMENT STATION 
 
 UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
Antenna Laboratory 
 Technical Report No. 58 
 
 ON INCREASING THE EFFECTIVE APERTURE 
 OF ANTENNAS BY DATA PROCESSING 
 
 by 
 
 Robert Ho MacPhie 
 
 Contract AF33(657)-8460 
 Project No. 6278^ Task No. 40572 
 
 July 1962 
 
 Sponsored by 
 
 AERONAUTICAL SYSTEMS DIVISION 
 
 WRIGHT-PATTERSON AIR FORCE BASE, OHIO 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
rt 
 
 ACKNOWLEDGEMENT 
 
 The author wishes to acknowledge the many helpful discussions he has 
 had with Professor Deschamps. Some comments of Professor Lo also proved 
 helpful. A special debt of gratitude is owed to Mr. Fred Wise who wrote 
 the programs for the problem which were used on the University of Illinois 
 digital computer, ILLIAC A special program written by Dr. R. L. Carrel 
 was of considerable help in the section on error analysis . 
 
CONTENTS 
 
 Page 
 
 1. Introduction 1 
 
 2. The Antenna as a Filter of Spatial Frequencies 5 
 
 2.1 The Voltage Description 5 
 
 2.2 The Power Description 11 
 
 3= Properties of the Temperature Distribution T(u) and its 
 
 Spatial Frequency Spectrum t(x) 21 
 
 3»1 The Temperature Distribution - T(u) 21 
 
 3.2 The Spatial Frequency Spectrum - t(x) 22 
 
 3.3 Direct Measurement of the Spatial Frequency Spectrum 25 
 
 3.4 Analytic Continuation 27 
 
 4. Fourier Polynomial Approximation 28 
 
 4.1 Fourier Polynomial Approximation to t(x) 28 
 
 4.2 The Least Square Formulation 32 
 
 4.3 Critical Value of N = N 34 
 
 4.4 Case of N Larger than the Critical Value 35 
 
 5. The Properties of the Coefficient Matrix for N > N 37 
 
 o 
 
 5.1 The Gramian as a Measure of Independence of the 
 
 Approximating Functions 37 
 
 5 .2 The Inverse Matrix S 39 
 
 6. Error Analysis for Ill-Conditioned Matrices 43 
 
 7. An Example of the Increased Resolution of an Antenna 
 
 as a Result of the Data Processing 55 
 
 8. Conclusions 57 
 References 58 
 Appendix 6 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/onincreasingeffe58macp 
 
ILLUSTRATIONS 
 
 Figure Number Page 
 
 1 Coordinate System of a Linear Antenna 6 
 
 2 The Relation between u and 9 8 
 
 3 Fourier Transform Pairs for the Voltage 
 Description of the Antenna as a Filter of 
 
 Spatial Frequencies 12 
 
 4 Conventional Antenna System Employing Square 
 
 Law Detection and Low Pass Filtering 13 
 
 5 Compound Interferometer System 18 
 
 6 Fourier Transform Pairs for the Power De- 
 scription of the Antenna as a Filter of 
 
 Spatial Frequencies 20 
 
 7 Cross-correlation Interferometer 26 
 
 8a Partitioning of Temperature Distribution 
 
 Into N Sections 31 
 
 8b Delta Function Representation of Tempera- 
 
 ture Distribution 31 
 
 9 Value of Gramian T as a Function of Aperture 
 
 Size in Wavelengths ~L/\ } for the Case of 
 
 Normalized Effective Aperture A = N/N =2 40 
 
 o 
 
 10 Value of Gramian T as a Function of the 
 Normalized Effective Aperture A with Actual 
 
 Aperture in Wavelengths as a Parameter 41 
 
 11 Spatial Frequency Spectrum of a Uniform 
 Temperature Distribution with the Standard 
 Deviation of the Measurement Error Indicated 
 
 by the Width of the Shaded Area About the Curve 49 
 
 12 Normalized Standard Deviation in the Elements 
 of the Calculated T* 8 / Vector as a Function of 
 the Normalized Standard Deviation in the Measured 
 
 Data for L = \ } N = 8, A = 2 52 
 
 13 Ratio of Calculation Error to Measurement 
 Error as a Function of the Normalized Effective 
 Aperture with the Actual Aperture in Wavelengths 
 
 as a Parameter 53 
 
1. INTRODUCTION 
 
 In recent years the concept of an antenna as a filter of spatial frequen- 
 cies has come to play a central role in antenna theory in general and in 
 Radio Astronomy in particular ' . Given an antenna of finite size, its 
 output as it scans the sky with its more or less narrow main beam and smaller 
 sidelobes will be a smoothed version of the true source distribution of the 
 sky . 
 
 This smoothing property is possessed by all finite antennas . In par- 
 ticular,, if the antenna is linear (e.g., an array whose elements lie on a 
 straight line in space), there is an exact one dimensional Fourier transform 
 relation between the aperture weighting distribution a(x), where x is the aper- 
 ture coordinate in meters, and the field strength pattern A(u), where u = (3 sin 
 (3 is the phase constant in radians per meter and 9 is the polar radiation angle. 
 Consequently a(x) is the spatial frequency spectrum of A(u), and is non-zero 
 only over the finite interval -L/2 < x < L/2 where L is the length of the 
 antenna in meters. This interval, centered at x = 0, is the spatial fre- 
 quency bandwidth of the antenna and indicates that the antenna acts as a 
 
 low-pass filter, i.e., a smoothing device, 
 
 4 
 A similar situation obtains in the case of the planar antenna . The aper- 
 ture coordinates x and y correspond to pattern coordinates u = (3 sin 8 sin ^ and v = 
 (3 sin cos <p and a two dimensional Fourier transform relation exists between 
 the aperture weighting function and the field strength pattern. Most of the 
 spatial frequency literature is concerned with either the linear or planar 
 types of antennas, chiefly because of the above mentioned Fourier relation- 
 ships which permit of exact quantitative analysis of the problem of spatial 
 filtering . Consequently, this report considers only the linear type of antenna 
 but the results can be quite easily extended to the planar case, 
 
 Now since the linear antenna is a low-pass filter of the ideal kind with 
 a response to spatial frequencies lying outside the interval - L/2 £ x < L/2 
 which is identically zero, it has been believed that no information about these 
 higher frequency components is available at the antenna terminals. In theory 
 at least, this is not so. If the sources which excite the receiving antenna 
 are remote and radiate a finite amount of power in the direction of the 
 
5 
 
 antenna, then by the Paley-Wiener Theorem it can be shown that the spatial 
 
 frequency spectrum of such a source distribution is a band-limited analytic 
 function Only the part lying in the interval - L/2 < x < L/2 is passed by 
 the antenna. But this part is not independent of the other due to the ana- 
 lytic nature of the total spectrum . Since this function can be measured 
 for the lower frequencies its values at the higher frequencies could be ob- 
 tained by analytic continuation, Indeed if the function could be measured 
 with arbitrarily high accuracy then an arbitrarily large extrapolation 
 could be made. In effect, one would be calculating the higher spatial 
 frequency components of the source distribution which would excite the antenna 
 aperture if it were larger. The result of course would be an increase in 
 the effective aperture and resolution of the antenna. 
 
 This problem is very similar to that of prediction in the theory of com- 
 
 6 
 muni cation which has been thoroughly investigated by Wiener . More recently 
 
 7 
 Ville ; in a short paper, considered the problem of speech prediction and 
 
 * 
 by assuming that the frequency spectrum of the human voice had a compact support 
 
 arrived at the startling conclusion that by sampling an individual's speech 
 
 over a finite period of time one could predict all future utterances of 
 
 that person, However, it is an open question whether speech is a function 
 
 let alone an analytic one. In addition, Ville showed that for his method of 
 
 extrapolation the solution is very sensitive to small errors in the measured 
 
 8 
 data and only a small extrapolation is practical. A paper by H. Wolter 
 
 on the analytic continuation of optical images has also shown that very 
 
 little extension of the size of the image is possible due to measurement 
 
 inaccuracies, Finally, two very recent papers by D. Slepian, H. J, Landau, 
 
 9 10 
 and Ho 0, Pollak ' , which unfortunately were unknown to the author when 
 
 the work of this report was being done, deal with the problem of taking 
 finite samples of band limited signals and an elegant method of extrapo- 
 lation is indicated but no quantitative results are given. 
 
 Other than in a short note by Y . T , Lo , there seems to be no work of 
 a quantitative nature on the problem of extrapolation as applied specifically 
 to linear antennas. The purpose of this report was to investigate this 
 possibility. In particular an extrapolation of the time-averaged power 
 spectrum of spatial frequencies was considered. The Fourier transform of 
 
 The term support used in distribution theory, means the complement of the 
 open set over which a distribution (the voice's spectrum in this case) is zero, 
 
this function is the temperature brightness distribution of the sky and 
 
 I. 3 ,4,. 11 
 is of principal interest to radio astronomers 
 
 Witnessing the ease of formulation and mathematical advantages of the 
 
 6 
 least square method, as well as its success in the case of Wiener, it was 
 
 decided to do the following; 
 
 1. approximate the spatial frequency spectrum on the finite interval 
 I x I < L by a Fourier polynomial whose coefficients are to be determined; 
 this replaces the exact Fourier transform by a Fourier series of say N 
 terms,, 
 
 2 . by the method of least squares obtain a set of N normal equations 
 for the unknown coefficients* 
 
 Solution of the normal equations of course gives a least square approximation 
 to the spectrum on the interval I x I < L. However, if the period of the ap- 
 proximating function is increased to a value greater than 2L and at the same 
 time the number N is increased, then it is reasonable to suppose that due to 
 the analytic and band- limited nature of the function being approximated, the 
 Fourier polynomial will not only produce a least square fit over the original 
 interval but to a certain extent will continue to approximate the function 
 outside the interval, The amount of effective extrapolation would depend 
 on the degree of the Fourier polynomial and its period. For the special 
 case when N is four times the aperture length L in wavelengths the coefficient 
 matrix of the normal equations becomes an identity matrix. However this case, 
 whose solution can be obtained by inspection, does not correspond to an in- 
 crease in effective aperture, Rather, it leads to the principal solution 
 for the aperture of" length L. 
 
 Unfortunately the extrapolation results were quite disappointing. 
 Whenever the number of terms of the approximating polynomial and its period 
 were increased to a point where the solution of the normal equations would 
 have meant an increase in effective aperture, the equations inevitably became 
 ill-conditioned o The coefficient matrix would always have an inverse whose 
 elements were very large and of alternating sign, As a consequence, small 
 errors in the measured data resulted in much larger relative errors in the 
 calculated Fourier coefficients. In the report an analysis of the effect 
 of such errors is made and for a given accuracy of measurement the amount of 
 
useful extrapolation which one can expect, on the average, is determined. 
 In addition, a simple example of the increase in resolution of an aperture 
 of one wavelength that is achieved by this data processing method is des- 
 cribed . 
 
 Although this problem is not quite the same physically as that of 
 supergaining an antenna the end results are very similar. Thus, although the 
 supergain antenna usually has a very large reactive field associated with it 
 and the present one does not, the performances of both are very sensitive to 
 small errors in the physical parameters of the antennas themselves. The 
 lack 01 highly reactive fields in the present system might be thought of as 
 being replaced by the "highly reactive" coefficient matrix which must be used 
 to process the measurement data. Since this matrix can be specified to an 
 arbitrary number of decimal places one might think that on a suitably large 
 digital computer the accuracy problem could be obviated. However, one still 
 must make some physical measurements which lead to the elements on the right 
 hand side of the set of normal equations. In general these elements cannot 
 be specified to more than say four decimal places and hence the process is 
 still strictly limited by the errors in the physical parameters of the system 
 just as in the supergain case. 
 
 For a demonstration of the sensitivity of supergain antennas to small 
 errors see E. C, Jordan, "Electromagnetic Waves and Radiating Systems,' 
 Prentice-Hall, 1950, pp. 445-450. 
 
2. THE ANTENNA AS A FILTER OF SPATIAL FREQUENCIES 
 
 As was mentioned in the introduction, we shall, for simplicity, consider 
 only the linear antenna by which is meant an antenna whose elements are located 
 on a straight line in space . In this section the various Fourier transform 
 relations between the far field source distribution, the antenna pattern and 
 their corresponding spatial frequency spectra will be formally presented. 
 
 Now just as in communication theory where a specific time signal is 
 described exactly by some complex voltage spectrum and a random signal is 
 usually described statistically by some real average power spectrum (in which 
 all phase information is lost), it is also true in linear antenna theory 
 that the exact description is in terms of the field strength or voltage 
 spectrum, while the time-averaged statistical description is more conven- 
 ient in terms of power,, For brevity these two types of spectra will be 
 called voltage and power spectra and will receive separate treatment. 
 2.1 The Voltage Description 
 
 Let us consider an antenna of length L meters as is shown in Figure 1. 
 If there is a remote point source of unit field strength in the direction 9, 
 then it is well known that except for a constant factor the terminal volt- 
 age due to the source is 
 
 L/2 
 
 trs -un D I t \ J(3x sin -jSx sin j u t ,_ . 
 
 v., (0 ,t) = Re a(x)e r o e r dxe u (1) 
 
 1 O' yj 
 
 -L/2 
 
 Re indicates The real part of. 
 
 x = distance measured in meters along the aperture, 
 
 a(x) = the aperture weighting function and may be complex, 
 
 (3 = propagation constant in radians per meter, 
 
 Bsin0 = progressive phase shift across the aperture in radians per meter which 
 1 o 
 
 causes the main lobe of the antenna pattern to be pointed in the 
 direction, 
 
i 
 
 i 
 
 
 
 
 «^0 / 
 
 X/ INCIDENT 
 /,X PLANE 
 WAVE 
 
 
 APERTURE^ 
 
 
 
 
 i 
 
 f 
 
 I 
 
 
 -L/2 
 
 
 
 L/2 
 
 
 v(u , t; 
 
 Figure 1. Coordinate System of a Linear Antenna 
 
<jJ = angular frequency of the signal in radians per second, 
 t = time in seconds, 
 
 J 
 
 V^n 
 
 If one lets 
 
 U = p sin (2) 
 
 Equation (1) can be rewritten as 
 
 L/2 
 
 v(u Q ,t) = Re \ 
 
 jxu -jxu j" t 
 .(x)e ° e dxe ° (3.) 
 
 ■L/2 
 
 Equation (2) is illustrated graphically in Figure 2. If (3 is thought of 
 
 -A. 
 
 as a propagation vector (3, which indicates the direction of arrival as well 
 as the frequency of the signal, then u is its scalar projection on a line 
 parallel to the x axis and varies from -(3 to (3 for 9 = -ir/2 and 77/2 respec- 
 tively. Note also that the sources at 9 and 77-9 have the same projection 
 and hence are indistinguishable. 
 
 Returning to Equation (3) we see that since the aperture is finite, 
 a(x) = for !xl > L/2 and we can write 
 
 ju t 
 
 v(u ,t) = Re A(u-u ) e ° (4) 
 
 o o 
 
 where the field strength pattern 
 
 00 L/2 
 
 A(u) = a(x)e" JUX dx - J a(x)e" JUX dx (5) 
 
 -00 -L/2 
 
 is the Fourier transform o f a(x), the aperture weighting function. 
 
 Mathematically, the location of the point source in the u domain can 
 be specified by the Dirac delta function, 6(u -u) and the. terminal voltage of 
 the antenna can be written as 
 
PROJECTION OFjQ 
 ONTO U AXIS 
 
 U = /3sin0 
 
 Figure 2. The Relation between u and 
 
P JU, o t 
 
 / 6(u,-u) A(u -u ) du e 
 U 1 1 o 1 
 
 v(u ,t) = R e / 6(u ,-u) A(u -u ) due (6) 
 
 o 
 
 a(u : 
 
 o o 
 
 Re 6(u_-u)* A(u )e ° (7) 
 
 where * is the symbol for convolution. Here we have defined 
 
 A(u) = A(-u) (8) 
 
 Thus the point source or delta function response of the antenna is the reverse 
 of its field strength pattern. In the majority of practical cases the pattern 
 is an even function and the point source response and the pattern are identical. 
 
 Now in general there is not one source but a distribution of sources . 
 The instantaneous field strength at the phase center of the antenna due to 
 a plane wave from the source in the u direction at time t can be written as 
 
 j°° t 
 e(u,t) = Re£(u,t) e ° (9) 
 
 >or C*(u,t 
 
 where the complex phasor C*(u,t) represents the amplitude and phase of the narrow 
 
 band envelope of the carrier at frequency to . The total output voltage of 
 
 the antenna is the integral of all such plane waves weighted by the pattern 
 
 v 
 function A(u -u) . 
 o 
 
 ju t 
 
 v(u ,t) = ReV(u ,t) e ° (10) 
 
 o' o 
 
 P jco t 
 
 !■■> 'Li., I ! 1(U -U) <lu ' J : J I 
 
 Xfi<- 
 
 ■P 
 
 J w t 
 
 Re @ (u .t)* A(u ) e ° 
 
 v--- o 1 o 
 
 (12) 
 
 It has been tacitly assumed that there is an RF filter which allows only a 
 
 narrow band of frequencies centered at to to exist at the antenna terminals . 
 
 o 
 
10 
 
 The limits of +p on the integral indicate that the sources are in the visible 
 
 range . 
 
 Equation (12) shows that the complex amplitude V(u ,t) of the antenna's terminal 
 
 voltage is equal to the convolution of the source distribution and the reverse 
 
 of the antenna pattern. This function has an inverse Fourier transform, 
 
 'V* (x,t), which with the aid of the convolution theorem can be written as 
 
 V (x,t) = e (x,t) a(x) (13) 
 
 where £(x_,t) is the inverse Fourier transform of £, (u,t) and a(x) is the 
 reverse of the aperture weighting function. Since a(x) is identically 
 zero for I x I > L/2 it is clear from Equation (13) that the voltage spec- 
 trum of the output is also zero in these regions. If the modulus of a(x) 
 is greater than zero for all values of x in the aperture then from Equation 
 (13) we define 
 
 6 (x,t) = * < X > t} 7 \x\ < L/2 (14) 
 
 a(x) 
 
 > L/2 
 
 whose Fourier transform is 
 
 ■L/2 
 
 L/2 
 
 e (u , t) = P ^ile' JV dx (15) 
 
 O o o' i) \/ 
 
 This last function is known as the principal solution. It is a smoothed 
 version of the true source distribution with no spatial frequencies greater 
 in absolute value than L/2 . However the frequency components which are 
 present are identical to those of the true source distribution CL (u ,t). 
 From Equation (13) it is clear that the Weighting function ^a(x), in general, 
 distorts the source distribution's spectrum as "seen" at the antenna output 
 in the form of aa (x,t). But, if the antenna aperture is uniformly weighted 
 
11 
 
 with a(x) = 1 for I x I < L/2, then «w (x,t) = L (x,t) in that interval and the 
 output as the antenna scans will be the principal solutionc (u ,t) . If 
 a(x) is not uniform,, the principal solution can still be recovered by measuring 
 V(u ,t) and using Equations (13) and (15). 
 
 Figure 3 is a table of the various pairs of Fourier transforms which have 
 been derived above. The RF carrier factor has been suppressed. 
 2.2 The Power Description 
 
 In many applications it is not the instantaneous signal from a remote 
 source that is of interest but simply its average power. The fields of radio 
 astronomy, radiometry, radio direction finding, and radar are all more or less 
 concerned with measuring the average power radiated or reflected by the remote 
 sources . 
 
 A diagram of the conventional system for measuring such power is shown 
 in Figure 4. The terminal voltage of the antenna is fed into a square law 
 device whose output is 
 
 2 *> V Ju) t 2 
 
 v (u ,t) = [ReF(u ,t)* A(u ) e ] (16) 
 
 o' ^ o o 
 
 V 2 J2u) t 
 = 1/2 Re [£(u ,t) * A(u )] e 
 o o 
 
 + 1/2 Re § (u ,t) .* A(.u ) ■ £ (u ,t) * A " (u ) (17) 
 
 The first term on the right side of Equation (17) is the double frequency 
 component and has a zero average value. The second term is the low frequency 
 component which along contributes to the low pass filter output. Thus, if 
 we assume that the filter response is that of an ideal averager, then the 
 system output after a suitably long integration time approaches 
 
 R(u ) = E \ i Re f (u ,t)* A(u ) • <^*(u ,t)* A*(u ' ) } (18) 
 
 o [2 ^ o o ^ o o J 
 
 ( f( VJ 
 
 where the superscript * indicates the complex conjugate and E 
 the statistical expectation of f(u ). Here it is assumed that time and 
 statistical averaging are equivalent (the sources are ergodic) . Equation 
 (18) can be rewritten as 
 
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 £ H 
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 Ch !=> 
 
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13 
 
 
 J 
 
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 *K/ 
 
 ^APERTURE 
 
 
 ' 
 
 / 
 
 | 
 
 
 L/2 
 
 
 V(u 0t t) 
 
 L/2 
 
 
 
 SQUARE LAW 
 
 
 
 
 DETECTOR 
 
 
 
 
 
 1 
 
 r v 2 ( 
 
 "of) 
 
 
 
 LOW PASS 
 FILTER 
 
 
 
 
 (INTEGRATOR) 
 
 
 
 
 R(u ) = T(u )*P(u ) 
 
 Figure 4. Conventional Antenna System Employing Square Law 
 Detection and Low Pass Filtering 
 
14 
 P P 
 
 R(u o )= i Re J J E pV^^V 5 ] 
 
 -ft -ft V J 
 
 A(u -u ) A(u -u ) du du (19) 
 
 ■P -P 
 
 In a wide range of applications the following statistical properties of the 
 phasor (^ (u,t) obtain: 
 
 If 
 
 then 
 
 Cu,t) = C 1 (u,t) - j ^(u,t) 
 
 (20) 
 
 (21) 
 
 E[S 1 (u,t)J = B^ a (u,t)J = eJ^Cu,!)^^^)/ =0 
 
 and 
 
 2EJ^ i 2 (u,t) = 2Ei£ 2 2 (u,t) = E | i £(u,t) I 2 1 = T(u) (22) 
 
 The real and imaginary parts of Q,(u,t) have zero means and are statistically 
 independent. Their variances are equal to 1/2 T(u), where the real number 
 T(u) is the variance of (£, (u,t) and is a measure of the temperature bright - 
 ness of the source in the direction u. Usually the source in one direction 
 is time-wise independent of the sources in all other directions and so one 
 can write 
 
 EV£( Ul ,t)£* (u 2 ,t)j = T(u x ) 6 Cu 1 -u 2 ) (23) 
 
 Substituting this, result back into Equation (19) gives 
 
 R(u ) = i Re f T(u ) A (u -u, ) A*(u -u, ) du, 
 O 2 J 1 
 
 o 1 o 1 1 
 
 1 T(u )* I A(u ) I 2 (25) 
 
 2 o o 
 
15 
 or if we let 
 
 i I A(u ) I 2 = P(u ) (26) 
 
 2 o o 
 
 then 
 
 V 
 R(u ) = T(u ) *P(u ) (27) 
 
 is the time-averaged response of the antenna as a function of the scan 
 "angle'' u . 
 
 Again we can take the inverse Fourier transform of this result and by 
 the convolution theorem we have 
 
 r(x) = t(x) p(x) (28) 
 
 V 
 where r(x), t(x) and p (x) are the inverse Fourier transforms of R(u ), T(u ) 
 
 and P(u ) respectively. In particular, p(x) is the transform of —I A(u ) I 
 o 2 o 
 
 (Equation 26) , and again by invoking the convolution theorem one obtains the 
 following relation between the power spectrum of spatial frequencies p(x) 
 and the voltage spectrum a(x) for the antenna. 
 
 p(x) = |"t(x)* a*(-x) = | a(-x)* a*(x) (29) 
 
 Since a(x) "= for I x I > — it follows from the properties of the convolution 
 operation that p(x) = for I x I > L. For example^ if a(x) were uniform in 
 the interval I x I < — then p(x) would be triangular in the interval I x I < L. 
 Thus the band-width of power spatial frequencies is twice that for the corres- 
 ponding voltage frequencies . This doubling effect also occurs in the time 
 
 12 
 domain. 
 
 Since p(x) = for I x I > L we know that R(u ) contains no spatial fre- 
 quencies for I x I > L and those for I x I < L will be related to the spatial 
 
16 
 
 frequencies of the brightness distribution T(u) by Equation (28) . Just as 
 
 in the voltage case the principal solution T (u) can be obtained <> Thus we 
 
 o 
 
 define 
 
 t ( X ) = £<*>, I x I < L 
 
 (30) 
 
 = , ! x I > L 
 
 and 
 
 V P V (x) 
 
 T q (u) = , 
 -L 
 
 Indeed the term principal solution was first applied to this type of power 
 
 distribution by Bracewell and Roberts rather than to the instantaneous 
 
 voltage distribution of Equation (15) „ T (u) is a smoothed version of T(u)j 
 
 its frequency components for I x I < L are identical to those of T(u) and 
 
 are equal to zero for I x I > L. 
 
 It would be convenient if the antenna power spectrum p(x) were uniform. 
 
 Then r(x) would be equal to t (x) (except possibly for a constant factor) 
 
 o 
 
 and the antenna output R(u ) would be the desired principal solution. Al- 
 
 o 
 
 though in the voltage case this can be accomplished with little difficulty 
 by letting a(x) = 1 for I x I < — , it is impossible in the power case when 
 a single antenna and a square-law detection are used. No aperture function 
 a(x) exists which leads to a uniform power spectrum p(x) = 1, through the 
 
 convolution operation of Equation (29) „ However, if the terminal voltages 
 
 13 
 of two antennas are cross-correlated it can be shown that the power pattern 
 
 that results is 
 
 P 2 (u) = ^ Re A(u) B*(u) e jiu (32) 
 
 where A(u) is the pattern of the first antenna, 
 
 B(u) is the pattern of the second 
 
 jiu 
 and e is an interferometer-type pattern resulting from the spacing of I 
 
17 
 
 meters between the antennas . If we let 
 
 B i (u) = B(u) e jiU (33) 
 
 Equation (32) can be written as 
 
 P 2 (u) = i Re A(u) B i *(u) 
 
 V 1 v V * 
 P 2 (u) =-Re A(u) B (u) 
 
 (34) 
 
 The associated spatial frequency spectrum is 
 
 P 2 (x) = j [a(x)* b (x) + a*(x)* t^ (x) ] (35) 
 
 Now if the correlation system is in the form of a Compound Interfero- 
 
 14 v 
 
 meter, then the spectrum p„(x) will be uniform and the system output will 
 
 be the principal solution T (u ) . A diagram of this system is shown in 
 
 ° ° L 
 
 Figure 5. The two antennas are a uniformly weighted aperture of length — 
 
 L l 
 
 meters adjacent to a simple interferometer also of length — meters . Thus 
 
 we let 
 
 a(x) =2, -— < x < 
 
 < 36 > 
 = otherwise, 
 
 b (x) = 26(x) + 26(x-|) (37) 
 
 Substituting these functions into Equation (35) gives 
 
 p (x) = 1 I x I < L 
 
 (38) 
 = otherwise 
 
18 
 
 UNIFORMLY WEIGHTED 
 APERTURE^ 
 
 L/2 
 
 % 
 
 SIMPLE INTERFEROMETER 
 L/2 
 
 CORRELATOR 
 
 1 
 
 T (u )*T(u )* Si " LU » 
 
 Lu o 
 
 J 
 
 Figure 5. Compound Interferometer System 
 
19 
 and 
 
 p( u)= L^f^ (39) 
 
 2 Lu 
 
 the correlator output is 
 
 sin Lu 
 
 R(u ) = T (u ).= T(u )* L — (40) 
 
 o o o o Lu 
 
 A table of the various Fourier transform pairs for the power description is 
 given in Figure 6. Like all Fourier transform relations these results are 
 linear (in average power in this case) . This very fortunate property is 
 due to the assumption that sources in different directions are statistically 
 independent in time. 
 
 Because of the rather startling fact that an antenna is a perfect low- 
 pass filter of both the voltage and power spatial frequencies of a source 
 distribution it has generally been concluded that the principal solution is 
 the most one can hope for and that no information about the higher spatial 
 frequencies is available at the terminals of the antenna. In the next 
 chapter it will be shown that this is not the case. 
 
20 
 
 
 
 
 
 
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21 
 
 3. PROPERTIES OF THE TEMPERATURE DISTRIBUTION T(u) AND ITS 
 SPATIAL FREQUENCY SPECTRUM t(x) 
 
 In this chapter mathematical models of the temperature distribution 
 T(u) and of its Fourier transform t(x) will be derived from some general 
 assumptions about the physical nature of the sources whose radio waves 
 are intercepted by the receiving antenna. Indeed, in the remainder of this 
 report we will be concerned only with the time-averaged functions T(u) and 
 t(x) rather than with their instantaneous counterparts 0(u,t) and e(x,t). 
 This is partly for convenience and partly for historical reasons since most 
 of the early work dealt with the former functions . 
 3.1 The Temperature Distribution - T(u) 
 
 It will be recalled (Chapter 2,1) that the variable u is related to 
 the physical direction by Equation (2) 
 
 u = p sin 
 
 Thus T(u) is the projection on a line parallel to the antenna axis of the 
 actual brightness distribution T (9) and hence is a distorted version of it. 
 There is no difficulty, however, in correcting for this distortion. Given 
 
 T(u), there are two values of 9 corresponding to say u = u , = sin u /p 
 
 ii * 
 
 and =77-0 and hence 
 
 T A (9 l /) + T A (e x * ) = T(u 1 ) (41a) 
 
 If the antenna is located on a ground plane, then the source at and image 
 at 77 - are always identical and we get a unique relation between a source 
 in the upper half space and its projection, i.e., 
 
 T A (0) = T A (sin _1 p = \ T<u) (41b > 
 
 Any mathematical model of a source distribution must necessarily be 
 rather general and incomplete because if one knew exactly the properties 
 of the temperature- distribution it would not be necessary to ever build 
 
22 
 
 devices to measure them. In a wide range of application the sources possess 
 the following properties: 
 
 1. they are stationary (with respect to their time statistics), 
 
 2. they are remote from the receiving antenna and fixed in space for 
 the duration of the time of observation^ 
 
 3. a source in one direction is statistically independent (in the 
 time domain) of a source in any other direction. 
 
 Indeed these three properties have been used to derive Equation (27) for the 
 time-averaged response of the antenna. The function T(u) is a measure of 
 the temperature brightness of the remote sources and is the expected value 
 of the absolute square of the field phasor £ (u,t) 
 
 {i £(u,t)i 2 j 
 
 T(u) = E | I C (u,t) I J (42) 
 
 Sometimes it is written as, kT(u) where k is Boltzmann's constant and hence 
 is a measure of the energy being radiated by the sources toward the antenna 
 system. Consequently, T(u) is non-negative since it is assumed that the 
 average energy flow from the sources cannot be negative. It has also been 
 assumed that the sources are remote. This means that they are in the so- 
 called visible range of the antenna which requires the angle 9 to be real. 
 
 I I * 
 As a result the sources all lie in the range for which I u I < (3 . Outside 
 
 this range the source distribution is identically zero. Indeed the word 
 "distribution" is a better description of the sources in space than the word 
 
 function' . This is because in some applications it is convenient to re- 
 present remote sources as "point" sources whose mathematical description is 
 in terms of delta "functions" which of course are distributions. In summary 
 the model of the temperature distribution T(u) has the following properties: 
 
 1. T(u) is non-negative and real, 
 
 2, T(u) = for I u I > (3 ., (compact support) 
 3.2 The Spatial Frequency Spectrum - t(x) 
 
 The inverse Fourier transform of T(u) is by definition 
 
 oo 
 
 5 I- 
 
 t(x) = — \ T(u) e u du (43) 
 
 The source distribution T(u) has a compact support. 
 
23 
 But since T(u) - for I u I > (3 we can write 
 
 (3 
 
 i 
 
 t(x) = ^r J T(u) e JUX du (44) 
 
 This is the inverse Fourier transform of a real function and hence possesses 
 complex symmetry,, i.e., if 
 
 t(x) = t 1 (x) + j t 2 (x) (45) 
 
 then 
 
 t 1 (x) = t 1 (-x), t 2 (x) = -t 2 (-x] 
 
 (46) 
 
 t(x) = t*(-x) (47) 
 
 Physically, t(x) is a function which depends on the field at one point 
 x , with the field at another point x , where x = x - x . In complex nota- 
 tion it is given by half of the expected value of the Hermitian product of the 
 field phasors at the two points 
 
 i r 
 
 ; X ) = -e ; e(x^t) e (x 2 , 
 
 t(x)=-E<' e( X;L ,t) e (x 2 ,t)| (48) 
 
 This function is therefore measurable, at least for some finite values of 
 its arguments Since it is the Fourier transform of a distribution which 
 
 is identically zero for I u I > (3 (compact support), it follows from the 
 
 5 
 Paley-Wiener theorem that it is a band-limited analytic function and never 
 
 becomes identically zero over any finite interval of x. 
 
 However, t(x) must satisfy the energy condition (Parseval's theorem) 
 
 A°° f 3 
 
 J I t(x) I 2 dx = J I T(u) ! 2 du 
 
 (49) 
 
 By band limited function we mean one whose Fourier transform has a compact 
 support . 
 
24 
 
 and since it is assumed that the sources radiate a finite amount of energy 
 towards the antenna the integral on the right exists. Consequently, in 
 order that the improper integral of I t(x) I converge we must have 
 
 l+£ 
 
 lim lt(x)l 2 = or (-) ] (50) 
 
 " x 
 x »°o 
 
 where 0(— ) means "of order at most — " and 6 > » 
 
 15 
 The function t(x) is related to the complex degree of coherence between 
 
 the field at the two points x and x which is given by 
 
 1 2 
 
 E^e ( X;L ,t) e r (x 2 ,t-T)J 
 e[I e ( X]L ,t) I 2 J 1/2 E [l e (x~t) I 2 J 
 
 Y 12 (T) " -7T-- " ,2,1/2 r, I ; „,2, 1/2 (51) 
 
 This function also depends on the time delay t. When t = the numerator 
 reduces to 2t(x) and in the case of remote sources the denominator reduces 
 to 2t(o). Hence we can write 
 
 \ 2 <°> - Hiy (52) 
 
 which is the normalized spatial frequency spectrum and is valid when all 
 sources are remote. 
 
 To summarize, the spatial frequency spectrum t(x) has the following 
 properties : 
 
 * 
 
 1. t(x) = t (-x), (complex symmetry) 
 
 2. t(x) = 1/2 e{ 6 (jc,,t) e (x t)l with x -x = x, (measurability) 
 
 3. t(x) is analytic, 
 
 4. t(x) is band-limited, 
 
 5. lim I t(x) I 2 = 0[(-) 1 ^ 6 ], e > (finite energy) 
 I x I > °o 
 
25 
 
 In the next part it will be shown that t(x), or rather a finite section 
 
 of it, can be measured directly by means of a correlation interferometer. 
 
 3.3 Direct Measurement of the Spatial Frequency Spectrum 
 
 In Chapter 2,2 it was shown that by analyzing the output of an antenna 
 
 of length L one could deduce the spatial frequency spectrum of the source 
 
 distribution that was passed by the antenna „ This spectrum, t (x) ? is a 
 
 truncated version of the true spectrum, being equal to t(x) for I x I < L 
 
 and identically zero for I x I > L„ It is also possible to measure directly 
 
 t (x) by means of one of the simplest of antennas . 
 o 
 
 Consider a two-element interferometer in which the terminal voltages 
 of the two isotropic elements are cross-correlated as is shown in Figure 7. 
 The phase shifter in the feed line from the right-hand element can be used 
 for scanning but if it is set at zero and if the element spacing is x meters, 
 then the system output due to a source distribution T(u) is 
 
 ^(x) =^ J T(u) cos ux du (53) 
 
 The limits +_ (3 on the integral are due to our assuming that all sources are 
 in the visible range . 
 
 If the phase shifter is set at 7T/2 radians, the output is 
 
 p 
 
 :x > - w S T<u) 
 
 t 2 (X) = W *' T(u) Sin UX dU (54) 
 
 thus we can write 
 
 t(x) = t 1 (x) + j t 2 (x) = ^ f T(u) e JUX du (55) 
 
 -P 
 
26 
 
 X 
 
 + 
 X 
 
27 
 
 t(x) = F l < 
 
 {t<u>] 
 
 the inverse Fourier transform of T(u) . 
 
 In practice the interferometer element spacing x cannot exceed some maximum 
 
 value, say L meters . Hence the function which we measure is 
 
 t (x) = t(x) for ! x I < L 
 
 (57) 
 =. for ! x I > L 
 
 The value of t (x) for negative x has been deduced from the complex symmetry 
 property of t (x), i.e,, t (-x) = t (x) . Thus, by varying the element 
 spacing of a correlation interferometer we can measure directly the complex 
 spatial frequency power spectrum of the distance sources on the interval 
 I x I <_ L . 
 3.4 Analytic Continuation 
 
 Perhaps the most important property of t(x) is its analyticity. It is 
 well known that if an analytic function is specified exactly over any finite 
 
 region its value at any point outside the region can be deduced by the pro- 
 
 1 fi 
 cess of analytic continuation = Since t(x) is analytic for all values of x 
 
 and its values for I x I < L can be measured, it is theoretically possible to 
 
 calculate t(x) for any x, This would yield the complete spatial frequency 
 
 spectrum t(x) for -°o < x < °o and the true temperature brightness distribution 
 
 T(u) could be obtained by taking its Fourier transform. 
 
 Of course this. assumes that t(x) can be measured exactly. For such an 
 idealized situation it is interesting to note that the size oi the measure- 
 ment interval is of no importance. As long as t(x) is measured with no 
 error an aperture of one wavelength is "equivalent" to one of a thousand 
 wavelengths'. However, error is unavoidable and in its presence the amount 
 of meaningful extrapolation of t(x) will be shown, later in this report, 
 to be very limited. In practice, therefore, the large aperture is still 
 preferable to the small aperture. 
 
 The problem is to find a suitable method for performing the extrapo- ' 
 lation. In Chapter 4 a method which takes into account the Fourier and 
 band-limited aspects of t(x) will be described, 
 
28 
 
 FOURIER POLYNOMIAL APPROXIMATION 
 
 Having obtained by measurement the real empirical curves t (x) and 
 t (x) tor -L _<__ x -_ L, it is necessary to express them analytically by 
 means of mathematical functions, Since t(x) = t (x) + jt (x) is known 
 to be the Fourier transform of the unknown source distribution T(u), it is 
 natural to approach the problem with the methods of Fourier analysis. We 
 will use a Fourier polynomial to obtain a least square approximation to 
 t(x) on the interval -L < x < L. Then by increasing both the degree N and 
 the period L of the polynomial we will attempt to increase the least square 
 fit on the interval to the point where it cont inues to approximate the 
 function t(x) for points just outside the interval. This continuation is 
 predicated on the fact that t(x) is analytic, 
 4,1 Fourier Polynomial Approximation to t(x) 
 
 By definition we have 
 
 (x) - A- J T(u) e- 
 
 (58) 
 
 If the visible range -(3 ^_ u <_ (3 is divided into N equal regions } the kth of 
 which is centered at 
 
 then we can write 
 
 N/2-1 
 
 27T 
 
 k -N/2 
 
 $ 
 
 W) f 
 
 t(x) - S -^ T(u) e JUX du (59) 
 
 k & 
 
 k N 
 
 We have for convenience chosen N even 
 
N, 2-1 
 t(x) = 2 
 
 (3/N 
 
 ju, x 
 
 1 P ^ J vx _> 
 
 - J T(v + u k ) e J dv e 
 
 fc=-N/2 _p/ N 
 
 29 
 
 (60) 
 
 N/2-1 
 
 2 T k CN) (x) e JU k X 
 k^-N/2 
 
 (61) 
 
 where 
 
 T k (N) ( x) 1 
 
 (3/N 
 
 jvx 
 
 - j T(v + u k ) e^ dv 
 
 (62) 
 
 -(3/N 
 
 If we expand the exponential in the integral as a power series in (jvx), 
 the integral becomes 
 
 k 
 
 -(3/N 
 
 JVX 
 
 (vx)' 
 
 dv (63) 
 
 The leading term is clearly 2(3/N times the average value of T[v + u ] on the 
 interval I v I < (3/N and if we assume that T(u) is continuous, then 
 
 lim 
 
 N >°o 2(3 
 
 P/N 
 
 h S — V 
 
 -(3/N 
 
 N_ 
 2(3 J 
 
 (3/N 
 
 Tjv + u, ] dv 
 k 
 
 (64) 
 
 ■(3/N 
 
 T(v + u R ) 
 
30 
 Thus we let 
 
 (N) 6 N/2 ~ 1 (N) JU k X 
 k=-N/2 
 
 which is a Fourier polynomial of N terms with a period L meters, given by 
 
 L = g- = NX (66) 
 
 o (3/N 
 
 where X = 271/(3 is the wavelength of the signal in meters „ The real, constant 
 
 (N) 
 coefficient T, is the approximation to the average value of T(u) on the kth 
 
 interval » The superscript N indicates the degree of the approximating poly- 
 nomial o Figure 8a is a graph of a "typical" source distribution which has 
 
 been partitioned into 10 sections (N = 10) „ In Figure 8b is shown a distri- 
 
 1 2B 
 bution of 10 delta functions the kth of which is located at u, = (k + — )— {■; 
 
 on, k 2 N 
 
 and has a magnitude T k which is the approximation to the average value 
 of T(u) on the kth interval. The distribution can be given mathematically 
 by the formula 
 
 N/2-1 
 
 T 
 
 k 
 
 |u - Uj 
 
 T (N) (u) = S T W 6lu - u, ( (67) 
 
 k=-N/2 
 
 Except for the constant factor — ^ the Fourier transform of T (U) is the approxi- 
 
 (N) 
 
 mating Fourier polynomial t (x) . We see that the approximation of t(x) by 
 
 (N) 
 
 t (x) has the effect of quantizing the distribution T(u) into the discrete 
 
 (N) (N) 
 
 set of delta functions T (u) „ We also note that the bandwidth of t (x) 
 
 (N) I i 
 
 is roughly the same as that of t (x), i a e., T (u) = for I u I > (3 • 
 
 (N) 
 Equation (66) indicates that t (x) v s period L , and its degree N, 
 
 are linearly related and in the limit, L \ °o a s N 4-°°= But in such a 
 
 o ' ^ 
 
 situation 
 
31 
 
 Figure 8a 
 
 Partitioning of Temperature Distribution 
 Into N Sections 
 
 (N) 
 
 T(u) 
 
 .(N) 
 
 -P 
 
 Figure 8b. Delta Function Representation of 
 Temperature Distribution 
 
32 
 
 N >«"" k= _^ A , ^ ,,, 
 
 277 J 
 
 ■P 
 
 where we have let 
 
 T k = T t( k+ |> Au] > T(u) 
 
 u fe - (k + i) Au > u (69) 
 
 and Au = 2(3/N ^ du 
 
 Thus by formally going to the limit as N approaches infinity the approximating 
 polynomial approaches the actual Fourier transform of the source distribution 
 T(u). 
 
 Of course, in practice this is impossible since only a finite number of 
 coefficients can be determined . However for a given aperture it is reasonable 
 to believe that by making N (and hence L ) suitably large, the polynomial that 
 results will, to some extent, continue to approximate t(x) for I x I > L and 
 hence increase the effective aperture of the antenna, 
 
 4,2 The Least Square Formulation 
 
 (N) 
 
 In the usual manner we require, when we approximate t(x) by t (x), 
 
 that the average of the squared modulus of the error be a minimum: 
 
 / I t(x) - t 
 
 00, v i 2 
 
 (x) I dx = min (70) 
 
 / 
 
 I t(x) . fi_ *V T «*> e JV ' 2 <* = *» ™ 
 
 ™ k=-N/2 k 
 
33 
 
 Then we require that 
 
 
 8A 
 
 (N ,= for k = 
 
 N 
 
 2 
 
 (72) 
 
 This yields N normal equations which in matrix form can be written rs 
 
 S _N _N S _N N 
 2' 2 2'~"2 + 
 
 S N , N S N , N 
 -V^~2 -2 +1 ^J fl 
 
 _N N 
 2>~2 H 
 
 2 ^2 
 
 (N) 
 
 ,(N) 
 
 1(N) 
 
 5-1 
 
 -I L 2 J 
 
 ,(N) 
 
 -N 
 
 2 
 
 (N) 
 
 0_N 
 
 2~ 
 
 r (N) 
 
 !Li 
 
 2 
 
 (73) 
 
 S T (N) = T (N) 
 o 
 
 (74) 
 
 (N) (N) 
 where S is an N x N square matrix, and T and T are N dimensional 
 
 column vectors; the matrix element in the kth row and i th column is 
 
 ki 
 
 sin (u - u ) L 
 
 ( \- V L 
 
 sin (k - f) -2| L 
 
 (75) 
 
34 
 
 (N) 
 
 The kth element of T is given by 
 
 L 
 
 „ 00 TO M , x " JU k X 
 T 0k =2? ■» * (X> e 
 
 (76) 
 
 ip VV 
 
 where T (u ) is the value of the principal solution sampled at u = 
 
 (k + 1/2) 2 (3/N. 
 
 4.3 Critical Value of N = N 
 
 S) 
 
 The formula for the T elements is quite similar to that for the 
 o 
 
 coefficients of an orthogonal Fourier series expansion of the complex func- 
 tion t(x) on the interval -L < x < L. Indeed if the number of terms N of 
 
 (N) 
 
 our polynomial t (x) is four times the length of the aperture in wavelengths 
 
 (N = N = 4L/\), then the matrix elements are given by 
 
 where 
 
 sin (k - i) V .. 
 S ki = (k - i) 71 = 6 ki (?7) 
 
 6^ = 1 if k = I 
 = if k ^ i 
 
 is the Kronecker delta » 
 
 In this case the approximating polynomial is the usual complex Fourier 
 series whose individual functions are orthogonal on the measurement interval 
 -L <. x < L, The matrix S has become an identity matrix I and from Equation 
 (74) one can write 
 
 See Equation (31). Chapter 2,2. 
 
35 
 
 (N ) (N ) ._ M M 
 
 \ ° =V ° f o rNo =iian dk = -§, . ...f-1 (78) 
 
 
 (N ) 7TN p -jux 
 
 T k ° = -=§> .1 * (x) e 
 
 -L 
 
 which of course is 
 
 ( 
 
 N ) 
 
 TIN 
 
 T 
 
 o 
 
 = ~£ T < u > 
 
 k 
 
 
 2(3 o k 
 
 (80) 
 
 where T (u ) is the principal solution sampled at u = u . 
 ok k 
 
 It is easy to show for this critical case of N = N = 4L/X. that the 
 
 o 
 
 spacing between sample points u is that which is specified by the sampling 
 theorem . The elements of the solution vector To are, except for a 
 constant factor, the samples of the principal solution in the visible range. 
 Using the sampling theorem one can write 
 
 T <») - H ° 2 T<V Sl ", (LU ' ™\ N . 4LA (81) 
 
 ° " ™o fe-N /2 k Lu - KIT ' 
 o 
 
 N /2-1 
 
 T (u) ~ ° S T (u ) Sin / LU " k77) , N = 4L A (82) 
 
 ° ~ k=-N /2 ° k LU " W 
 o 
 
 Of course this does not give the principal solution exactly (since there are 
 
 only N terms) but since T (u) ~ for I u I > (3 it will usually be a very 
 
 good approximation to T (u), especially for large N . 
 o o 
 
 4.4 Case of N Larger than the Critical Value 
 
 If N, the degree of the approximating polynomial, is now increased 
 
 beyond N to say 2 N which incidentally also doubles the period L of 
 o o o 
 
36 
 
 the polynomial, then the least squares fit to the function t(x) will clearly 
 be improved on the measurement interval and, 'hopefully, we expect it to 
 continue to fit the curve for values of x which lie just outside the inter- 
 
 Val * (2N Q ) 
 
 Physically, we can think of the 2N coefficients, T , as estimates 
 
 of the average value of T(u) on the corresponding intervals of u. Since 
 
 there are twice as many distinct intervals as before the effective resolution 
 
 has been doubled. However the solution is not identical to that of the 
 
 principal solution of an antenna which is twice as large. In the latter case 
 
 the 2N = 4(2L/\) estimates are obtained by fitting the curve over its 
 
 entire length of 4L whereas in the former the fitting is done over the 
 
 original length of 2L, The result is a better fit on this interval but a 
 
 worse fit on the [ L, 2L ] interval. 
 
 From the practical viewpoint the increase of N beyond N causes the 
 
 o 
 
 coefficient matrix to revert to a general form and as a result the coefficients 
 
 (N) 
 
 T must be obtained by matrix inversion. 
 
 T (N) = S -\ (N) (83) 
 
 o 
 
 (N) 
 From Equation (76) we see that when N is increased the elements of T 
 
 are obtained by merely taking more closely spaced samples of the principal 
 
 solution T (u) in the visible range. But according to the sampling theorem, 
 o 
 
 for N > N these samples are no longer completely independent . This lack of 
 o 
 
 independence is indeed the reason why the coefficient matrix S has reverted 
 
 to its general form and in the next section it will be shown that as N is 
 
 increased beyond N ,S becomes more and more ill-conditioned as a result of this 
 o 
 
 reduction of independence. 
 
37 
 
 5. THE PROPERTIES OF THE COEFFICIENT MATRIX FOR N > N 
 
 o 
 
 In this chapter it will be shown when N and hence L are increased 
 
 (N) 
 
 beyond their critical values that not only do the elements of the T 
 
 vector lose some of their independence but that the individual functions, 
 
 JUlrX 
 
 e , of the approximating polynomial rapidly depart from their mutual 
 
 orthogonality which occurs for N = N » A measure of the linear dependence 
 
 18 
 
 of these functions is the Gram determinant or Gramian which quickly re- 
 duces to a very small value when N exceeds N . In this particular case 
 
 o 
 
 the Gramian is the determinant of S and it is well known that matrices 
 
 19 
 with very small determinants are usually ill-conditioned . As a result 
 
 S possesses an inverse, S } whose elements are large in magnitude and 
 
 (N) 
 of alternating sign. This means that the solution vector T involves 
 
 the differences of large and usually nearly equal numbers and hence is 
 
 very sensitive to errors . 
 
 5.1 The Gramian as a Measure of Independence of the Approximating Functions 
 
 In a manner which is similar to that used for measuring the independence 
 
 18 
 of vectors one can calculate the Gramian or Gram determinant associated with 
 
 the approximating functions ana the interval -L < x < L. 
 
 We define the Hermitian scalar product 
 
 ;ju k x -ju 3 
 e e 
 
 ^ t . * f ^ 1 f JUkX " JU ^ h 
 <e (x) e (x)> = Trr | e e dx 
 
 sin (u - u ) L 
 
 —7 ri — (84) 
 
 (u k - u,) L 
 
 and interpret it as the cosine of the ."angle" in Hilbert space between the 
 functions e^x) = e JUkX and e i °° = ^^ ° The Gramian T is the determi- 
 nant of the following matrix of such scalar products. 
 
r = det 
 
 < e (x) e (x) > < e (x) e (x) > 
 
 -N -N -N -N , 
 
 — — _ —4-1 
 
 2 2 2 IT 
 
 < e (x) e (x) > <e (x) e (x)> . 
 
 -N . -N -N . -N _ 
 
 2+ 1 2 2 +1 2 +1 
 
 • * ' " ' ' <6 N_ (X) S N (X)> 
 I" 1 2" 1 
 
 38 
 
 (85) 
 
 But in this case we see from Equations (75) and (84) that 
 
 r = det S 
 
 (86) 
 
 It will be recalled that for N = N the matrix S is an identity matrix and 
 
 o 
 
 T= det I = 1 
 
 (87) 
 
 Consequently the approximating functions are all mutually orthogonal since 
 the cosine of the "angle" between any distinct two of them is zero. 
 
 Now as N increases beyond N to say 2N = 8L/\, the coefficient matrix 
 
 o o 
 
 becomes 
 
 S = 
 
 sin 77/2 
 77/2 
 
 sin 77/2 
 77/2 
 
 sin 77/2 
 
 77/2 
 
 sin 77/2 
 77/2 
 
 sin 377/2 
 377/2 
 
 sin 77/2 
 77/2 
 
 sin 377/2 
 377/2 
 
 (88) 
 
39 
 
 20 
 
 For L = \/2 it will be truncated to N - 4 rows and columns; for L = \ there 
 will be 8 and so on. It is of interest to note that it is a striped matrix 
 since not only are all the elements of the main diagonal equal but all ele- 
 ments of any off-diagonal are also equal, This is known as a Toeplitz f orrrT " . 
 
 The gram determinants, as functions of the original measurement inter- 
 vals in wavelengths L/\, are plotted in Figure 9. Note that there is a very 
 rapid falling off as L increases. From the practical viewpoint this means 
 that it will be easier to extrapolate by 100 percent from a small aperture 
 than from a larger one. In Figure 10 the value ot the Gramian T is plotted 
 
 as a function of the normalized effective aperture A = N/N = L /L with the 
 
 o o 
 
 original measurement aperture L/\ as a parameter. It can be seen that the 
 
 Gramian rapidly falls from 1 at A = 1 (where N = N ) to very small values 
 
 o 
 
 as A increases . Again the value of T for a given A> 1 is much smaller for 
 large L than for small . 
 
 Since the smallness of T is a measure of the lack of independence of the 
 functions e (x) on the interval, we see that any significant extrapolation 
 that is performed with this method must be done with functions which are 
 strongly dependent . 
 
 5.2 The Inverse Matrix S 
 
 Matrices whose determinants are very small usually possess inverses 
 whose elements are extremely large, The coefficient matrix S is no exception 
 and for A» 1, S is very ill-conditioned, i.e., the elements of S are 
 astronomical. For example if L = X and N = 8, A= 2 and the inverse of 
 S is 
 
 s" 1 . 
 
 93. 
 
 -329. 648 
 
 1,205. -2,423 
 
 4,947 
 
 S is symmetric 
 
 s 'i - C 
 
 Jk kj 
 
 -867. 834 
 
 3,296. -3,213, 
 
 ■6,808. 6,710 
 
 9,468. -9,423, 
 
 9,468, 
 
 -574. 
 
 265. 
 
 -66. 
 
 2,242. 
 
 -1,050. 
 
 265. 
 
 -4,732. 
 
 2,242. 
 
 -574. 
 
 6,710. 
 
 -3,213. 
 
 834. 
 
 -6,808. 
 
 3,296. 
 
 -867. 
 
 4,947. 
 
 -2,423. 
 
 648. 
 
 
 1,205. 
 
 -329. 
 
 (89) 
 
 93. 
 
40 
 
 
 .C <M 
 
 
 +-> 
 
 : 
 
 bO II 
 
 
 c 
 
 
 C 
 
 
 rH iz; 
 
 
 <1> X 
 
 
 > 55 
 
 
 aS 
 
 
 & II 
 
 
 c<! 
 
 
 ■H 
 
 
 0) 
 
 
 Ph 
 
 
 N 3 
 
 
 •H 4-> 
 
 
 w ^ 
 
 
 a) 
 
 x< 
 
 a 
 
 S 
 
 3 
 
 lO _1 
 
 -p 
 ^ > 
 -H 
 
 
 a, p 
 
 if) 
 
 < " 
 
 
 
 
 <H 
 
 CD 
 
 H 
 
 ■z. 
 
 C 
 
 LlI 
 
 O TJ 
 
 _1 
 LiJ 
 
 •H CD 
 P N 
 
 > 
 
 Cl rH 
 
 < 
 
 3 aS 
 
 ^ 
 
 fa S 
 
 u 
 
 
 rt o 
 
 ■z. 
 
 fc 
 
 
 co 
 
 in 
 
 aS <H 
 
 ' UJ 
 
 Ph ° 
 
 tr 
 
 
 
 z> 
 
 el en 
 
 
 aJ a! 
 
 •rH O 
 
 s 
 
 UJ 
 
 ca 
 
 Q_ 
 
 Ph .£ 
 
 < 
 
 o -P 
 
 
 <H ^1 
 
 
 
 
 
 «H 
 
 
 
 
 
 3 * 
 
 
 rH ^ 
 
 
 ai \ 
 
 
 > hJ 
 
 NVIlAlVdO 
 
41 
 
 10 u 
 
 
 
 
 
 
 io- 1 - 
 
 < 
 
 ' 1 \ 
 
 
 
 
 io- 2 - 
 
 
 
 
 
 <f> 
 
 T \ 
 
 
 
 
 fe 10-3- 
 
 I 
 
 
 
 
 DETERMINANT 
 5 O 
 
 ; 
 
 
 
 M 
 
 O 
 
 V 
 
 
 L. 1 
 X"4 
 
 
 u |0- 6 - 
 
 - 1 \ 
 
 
 
 
 < 
 
 1 1Q - 7 - 
 
 CD 
 
 <> 1 
 
 > I 
 
 
 
 
 I0" 8 - 
 
 • I 1 
 
 
 
 
 io- 9 - 
 
 >" 2 . 
 
 X 2 
 
 
 
 io- 10 
 
 1 
 
 1 
 
 1 1 
 
 1 
 
 NORMALIZED EFFECTIVE APERTURE Kj=A 
 
 l\lo 
 
 Figure 10. Value of Gramian T as a Function of the Normalized Effective 
 Aperture A with Actual Aperture in Wavelengths as a Parameter 
 
42 
 
 The smallest element (in magnitude ) is 66.00. ,. and the largest is 9,468.5.. 
 This is none too good, but if N is then increased to 12 and L/\ is held fixed 
 
 at 1, then A = 3 and the smallest and largest (in magnitude) elements of the 
 
 5 10 
 
 inverse matrix in this case are 4.89 x 10 and 3.39 x 10 respectively. 
 
 However the size of the elements alone does not necessarily imply that 
 
 the process is an ill-conditioned one. Of equal importance is the fact 
 
 that the elements of S alternate in sign as can be seen in Equation (89) . 
 
 Because of this the solution vector 
 
 T W) = S"V N) (90) 
 
 o 
 
 involves the small differences of very large numbers. To preserve accuracy 
 
 it will be necessary to specify the elements of both S and of T to a 
 
 -1 ° 
 large number of decimal places. This can easily be done for S since the 
 
 elements of S are expressed by the exact formula of Equation (75). In prac- 
 tice one can specify them to say twelve decimal places and then invert S 
 
 on a high speed digital computer. The process of inversion will involve round 
 
 at 
 (N) 
 
 off errors but in most cases the elements of S can be calculated to at 
 
 least eight or nine decimal digits of accuracy. However the vector T 
 is obtained from measurements of t(x) which can in most cases be performed 
 with a maximum precision and accuracy of only about three or four decimal 
 places . 
 
 The weak point of the process is therefore the sensitivity of the solu- 
 
 (N) (N) 
 
 tion vector T to small errors in the measurement vector T . In the 
 
 o 
 
 next section a statistical analysis of the effect of measurement error is 
 presented. 
 
43 
 6. ERROR ANALYSIS FOR ILL-CONDITIONED MATRICES 
 
 — (N) 
 
 Let the data vector T in the matrix equation 
 o 
 
 be written as 
 
 S 7 (N) = ? (N) (91) 
 
 o 
 
 T (N) - T (N) ♦ 6 T (N) (92) 
 
 o o o 
 
 (N) — (N) (N) 
 
 where T is the exact value of T and 6 T is the unavoidable error 
 o o o 
 
 due to measurement . Then we can write 
 
 M N) + 5T (N) ] =T (N) + 6T (I 
 
 L J ° ° 
 
 S +6 T Vi ' i + 6 T ^ (93) 
 
 L 
 
 where 
 
 6 T (N) - S" 1 6 T (N) (94) 
 
 o 
 
 (N) 
 is the error in the calculated vector whose true value is T . Here we 
 
 have assumed that we can obtain an inverse matrix S which is exact. In 
 
 practice, of course, the process of inverting S will involve round-off 
 
 errors. However, the elements of S can be specified to an arbitrarily large 
 
 number of decimal places and if the inversion is done by a large digital 
 
 * -1 
 
 computer with a capacity of 10 or 12 decimal digits, then the error in S 
 
 -(N) 
 is negligible compared to the error in T which is due to measurement and 
 
 o 
 
 which probably is no less than one part in ten thousand or 4 decimal digits 
 
 accuracy. 
 
 — (N) 
 
 The T vector is not itself a measured quantity being obtained from 
 
 the measured function t(x) by means of Equation (76) of Chapter 4.2. We 
 can write the measured function as 
 
 t(x) = t(x) + 8 t(x) (95) 
 
 The University of Illinois Digital Computer, ILLIAC, was used in all numerical 
 work. 
 
44 
 
 where 6 t(x) is the unavoidable error in the measurement of the true spec- 
 trum t(x). For simplicity, let us assume that the source distribution 
 T(u) is uniform with T(u) = T , a positive real constant. In this case 
 the measured function is 
 
 t(x) = 2(3 T S1 ** P X + 8 t(x) (96) 
 
 A px 
 
 where 2(3 T — ° is the spatial frequency spectrum of T(u) = T . 
 P x ( N) A 
 
 The elements of the true T vector can be calculated by means of 
 
 o 
 
 the above mentioned equation. They are given (for N even) by 
 
 > {[i + <* ♦ 1> f] pl] + si{ [i - (k + i) |] pi}] 
 
 T ok } = V Sl ;|! "* ' "' -' pL ' Sl - • ' ' - '■ • 
 
 where 
 
 y 
 Si(y) = \ ^^-^ dx 
 
 P sin x 
 J X 
 
 is the sine integral. 
 
 Now it is reasonable to assume that the error function, 8 t(x), in 
 the measurement of t(x) has statistically independent real and imaginary 
 parts whose means are zero. Thus if t(x) = t (x) + jt (x), 
 
 E < 6 t GO o t (x)j = (98) 
 
 | 6 t 1 (x)J = E J 6 t 2 (x)J = (99) 
 
 We let the autocorrelation function for both 8 t (x) and 6 t (x) be given 
 
 1 2i 
 
 by 
 
 R g GO = E 
 
 ( 5 W 6 W] = E { 6t 2 (X l } 6t 2 (X 2^ (1 ° 0) 
 
45 
 
 If one makes the usual assumption that the error power spectrum is flat 
 
 2 
 (white noise) with a power per unit bandwidth of N /2 and a bandwidth of M 
 
 o 
 
 radians per meter, then the autocorrelation function of 6 t (x) and 6 t (x) 
 is 
 
 M N 2 
 
 o sin If* 
 
 6 277 Mx 
 
 j 2 (101) 
 
 t sin Mx 
 
 2 Mx 
 
 where x = x - x . In what follows it will be assumed that the error spec- 
 trum is much broader than the signal spectrum, M » (3 . 
 
 In the appendix it is shown that the expected value of each element of 
 
 (N) 
 the error vector 6 T is zero, 
 o ' 
 
 [« C] ■ ° 
 
 -N N 
 for k =_...-- i (102) 
 
 (N) (N) 
 and the covariance of the elements 6 T- and 6 T . is 
 
 f 6 t (n) 
 
 L 0i 
 
 0i 0j 
 
 U ~ 3) N \ 
 
 2 
 
 We see that 77L °"t,./M is the variance ol the individual elements of the error 
 
 With this statistical information we can now derive expressions for the 
 
 — (N) 
 mean and variance of the error in the solution vector T 
 
 P" 
 
 - e s x 8 t^ n) £ o < io,n 
 
 The average error is zero. To calculate the variance we write 
 
 E r 6 T (N)T 6 T (N >7 . E I p6 lf\ T S-h T< N) ] (105) 
 
E i 6 T 
 
 (N) 
 
 r N/2-1 N/2-1 
 
 E J S S-J § T ( ^ } 2 S" 1 6 T (N) 
 i=-N/2 ki 0i j=-N/2 k J ° J 
 
 46 
 
 where the superscript denotes the transpose of the vector, A typical 
 element of the resulting sum is 
 
 (106) 
 
 N/2-1 N/2-1 
 
 s s 
 
 i=-N/2 j=-N/2 
 
 kl- kj 
 
 s:l s: 1 E 6 T (I ! ) 6 T (N) 
 
 of 
 
 oj 
 
 where S is the kj element of the inverse matrix S = From Equation (103) 
 we have 
 
 6 T 
 
 (N)' 
 
 N/2-1 N/2-1 77LO^ sin a_j)M^ 
 
 s s s. „ s. 
 
 i=-N/2 j=-N/2 
 
 ki kj M 
 
 U J; N \ 
 
 (107) 
 
 But since E 
 
 ( 6 r} = 
 
 (N) 
 we can write down the variance of 6 T as 
 
 k 
 
 
 (108) 
 
 8t 
 
 (N) 
 
 A e 2 
 
 kN t 
 
 (109) 
 
 This equation shows by how much the error in measurement is amplified to 
 
 give the error in the calculated data. There is an A, >T for each element 
 
 (N) ^ 
 
 of the 6 T vector although only half are distinct due to the symmetry 
 
 of the matrix . The total variance of the calculated error vector is the 
 
 sum of the individual variances of its elements. 
 
17 
 
 N/2-1 - N/2-1 
 
 a 6T (N) " \ "t (111) 
 
 Geometrically we can think of the positive square root of the variance as 
 
 the radius of a sphere of uncertainty centered at the tip of the true so- 
 
 (N) 
 lution vector T 
 
 Parenthetically, we note from Equation (108) that the coefficients A 
 
 depend inversely on the bandwidth M of the error power spectrum. This 
 
 apparent anomaly is due to our defining the variance of the measurement 
 
 2 
 error as 0" which can also be written as 
 
 M N 2 
 
 (T = - (112) 
 
 t 7T 
 
 and Equation (108) becomes 
 
 N/2-1 N/2-1 sin[a-j)%7 ^ 9 
 
 r ^= 2 2 S" 1 S" 1 N / L N J (113) 
 
 6 T < N) i=-N/2 j=-N/2 W * (i-J)^^ 
 
 in which it can be seen that the calculated variance is proportional to the 
 
 2 
 power per unit bandwidth N of the measurement error function. However, 
 o 
 
 Equation (108) is useful in showing how the calculation process acts as a 
 
 filter which suppresses the high frequency components of the error spectrum. 
 
 For example, if two measurements of t(x) are made, one of which oscillates 
 
 2 
 rapidly about the correct value with variance 0" and the other oscillates 
 
 much more slowly about the correct value with the same variance, then the 
 
 error in the calculated data will be much less for the first measurement 
 
 Since the A in Equation (109) is not identical the actual shape of the 
 uncertainty volume will be ellipsoidal » 
 
48 
 
 than for the second . For a given measurement variance the calculated vari- 
 ance is inversely proportional to the bandwidth of the measurement error 
 power spectrum „ 
 
 Now it is important to know by how much the effective aperture can be 
 increased (by increasing N) before the radius of the sphere of uncertainty 
 exceeds, far example, one percent of the length of the true solitLon vector. Clearly 
 this depends on the relative magnitudes of the signal and error functions, 
 t(x) and 6t(x)^ as well as the width of 6t(x)'s power spectrum. In Figure 11 
 is shown a graph of t(x) for the case of a uniform temperature distribution 
 with a shaded area extending 0" units above and below the curve „ This indi- 
 cates the size of the standard deviation which as a fraction of the value of 
 t(x) at x = is 
 
 / ° t 
 ° '- 7^- (H4) 
 
 t 2pT A 
 
 For larger values of I x I it is always greater than this . From Equation (97) 
 
 (N) 
 
 we can calculate the value of the elements of the true data vector T and 
 
 o 
 
 then 
 
 T (N > = S' 1 T W) (115) 
 
 (N) 
 
 is the value of the correct T vector. Then we let 
 
 / a o T (N) , , 
 
 (J = , x (116) 
 
 8T (N) I T (N) I 
 
 -(N) 
 
 be the normalized standard deviation of the calculated vector T about its 
 
 true lengthy i.e., 0" i s the radius of the sphere of uncertainty as a 
 
 6T (N) 
 
 fraction of the length of the solution vector T „ As mentioned previously 
 
 the calculated variance depends on the bandwidth of the measured error 
 
 power spectrum M which we assumed was much larger than (3° In all numerical 
 
 calculations it was assumed that 
 
 M = 10 (3 (117) 
 
49 
 
 
 
 
 
 
 T3 
 
 
 +-) 
 
 CD 
 
 
 fl 
 
 
 
 £3 
 
 cd 
 
 
 ■H 
 
 O 
 
 
 ^H 
 
 ■ H 
 
 
 +J 
 
 -d 
 
 
 01 
 
 a 
 
 
 
 
 
 p 
 
 ?h 
 
 
 CD 
 
 o 
 
 
 Ih 
 
 r M 
 
 
 3 
 
 ?H 
 
 
 -t-> 
 
 H 
 
 
 CO 
 
 
 CD 
 
 U 
 
 +-> 
 
 > 
 
 CD 
 
 CI 
 
 Sh 
 
 a 
 
 cu 
 
 3 
 
 S 
 
 E 
 
 o 
 
 0) 
 
 fu 
 
 
 H 
 
 Sh 
 
 CD 
 
 
 3 
 
 £ 
 
 e 
 
 co 
 
 +-> 
 
 fn 
 
 cd 
 
 
 O 
 
 tt) 
 
 +-> 
 
 tH 
 
 S 
 
 3 
 
 
 
 O 
 
 fl 
 
 CD 
 
 -O 
 
 ^ 
 
 -P 
 
 < 
 
 cd 
 
 
 cd 
 
 
 «H 
 
 CD 
 
 <h 
 
 O 
 
 Sh 
 
 O 
 
 CI 
 
 < 
 
 S 
 
 o 
 
 -d 
 
 3 
 
 
 
 
 Sh 
 
 -p 
 
 -a 
 
 -p 
 
 vi 
 
 cd 
 
 O 
 
 •rH 
 
 xi 
 
 CD 
 
 > 
 
 CO 
 
 a 
 
 0) 
 
 
 co 
 
 Q 
 
 
 
 xj 
 
 >> 
 
 T3 
 
 +-> 
 
 CJ 
 
 U 
 
 
 CI 
 
 cd 
 
 HH 
 
 CD 
 
 -a 
 
 o 
 
 3 
 
 c 
 
 
 a* 
 
 cd 
 
 X3 
 
 CD 
 
 -p 
 
 
 Sh 
 
 CO 
 
 T3 
 
 c*h 
 
 
 ■rH 
 
 
 
 
 Ss 
 
 ■H 
 
 £1 
 
 
 OS 
 
 -P 
 
 Q) 
 
 •H 
 
 
 X! 
 
 -P 
 
 -C 
 
 
 cd 
 
 +-> 
 
 
 a 
 
 
 >, 
 
 CO 
 
 5 
 
 XJ 
 
50 
 
 Admittedly this is rather arbitrary and in an actual application the value 
 
 of M might be considerably dif f erent ■> However, if some estimate of the 
 
 actual value of M is made^ then one can use Equation (108) to calculate 
 
 2 
 corrected value of the variance 0" , . ° For example, if the estimate of 
 
 2 &T 
 
 Mwas20 (3, then since 0" ,. is inversely proportional to M (Equation 108) 
 
 5t ( } 
 the actual variance is one half of that which is calculated for M = 10 (3 . 
 
 In general, to obtain the actual variance corresponding to a value of M = 
 
 W(3 one can use the following simple formula 
 
 8t 
 
 (N-) 
 
 = A° o- 2 
 w 6t (n) 
 
 M=W(3 M=10(3 
 
 (118) 
 
 and 
 
 8'T 
 
 (N) 
 
 M=10P 
 
 is the variance obtained in all of the following calculations. 
 
 As a typical example the case of L = X. and N = 8 will be considered 
 
 in detail. The inverse of the coefficient matrix is given by Equation (89). 
 
 (8) 
 
 The variance of the kth element of St is 
 
 2 T 3 
 
 3 -1 _i sin «"J)f 2 
 
 2 S 1 S 1 - Cr 2 
 
 k£ kj 
 
 U-J): 
 
 (119) 
 
 But from Equation (114) we substitute for v to obtain 
 
 2 277 3 3 -1 1 Sin (i -J } f 2 2 
 
 a 2 , x = fJL p L s S s * s" 1 - o- 2 t 2 
 
 ,(8) 5 K „ ~ .. , k£ °k.i v ?7 t A 
 
 8t 
 
 *=-4 j=-4 M « (i-j)§ 
 
 (120) 
 
 This can be evaluated for the 8 values of k and for k = 1 one obtaii 
 
51 
 
 ^(8,= ¥»••«•" >< 2T / 
 
 '2 2 
 39,061.0 0" T 
 t A 
 
 (121) 
 
 (8) 
 By means of Equation (97) and (115) the value of T can be obtained and 
 
 the normalized standard deviation of the solution error about its true 
 
 value is 
 
 a 
 
 / 5T (8) 197.8 a /T 
 
 a - = = 99,8 0" ' (122) 
 
 5t (8) - T (8) 1.9862 T A t UW 
 
 This normalized error line is plotted in Figure 12 along with those of the 
 other 7 elements of the vector. Note that only half are distinct. The 
 graph shows that if the normalized standard deviation of the measurement error 
 0" is .01 the normalized standard deviation of the calculated error varies 
 from .156 to as much as 1.955. Conversely if one wants, on the average, a 
 certain accuracy in all of the calculated results the measurement accuracy 
 must be about 200 times greater than the calculated accuracy. An error 
 line for the vector T as a whole is also shown in Figure 12. It is the 
 
 heavy line with a slope of 110.5. It indicates that if one requires that 
 
 -(8) 
 the normalized radius of the sphere of uncertainty of the calculated T 
 
 vector be less than one percent then the measured data must have a normalized 
 
 standard deviation of ( ,01/110.5) 100 = .00903 percent. Thus we see that 
 
 even for the seemingly modest increase in effective aperture of from one to 
 
 two wavelengths (A= 2) the method of processing the data is such that a 
 
 one hundred fold increase in relative error is to be expected. 
 
 This ratio of calculated error to measurement error is plotted in 
 
 Figure 13 as a function of the effective aperture /\. with h/k } the original 
 
 aperture size in wavelengths, as a parameter. It can be seen that not only 
 
 does 0" /6 increase with increasing A. but it increases as the original 
 6t / 
 
52 
 
 
 
 
 A=2, N=8, 
 
 L=X, M = IO/3 
 
 
 2.0 
 
 - 
 
 
 
 CO 
 
 
 
 
 
 — JC 
 
 
 
 
 
 \- 
 
 
 
 
 
 2 V s 
 
 
 
 
 
 1 {2 
 
 
 
 
 
 H Z 
 
 
 
 
 
 < LJ 
 
 
 
 
 
 > 2 
 
 1.5 
 
 
 
 
 IJ UJ 
 
 
 
 
 / ^k=-l,0 
 
 UJ 
 
 
 
 
 
 1 1 
 
 
 
 
 
 Q »- 
 
 
 
 
 
 Z <-> 
 
 
 
 
 / ' ^ 
 
 2 > 
 
 CO £ 
 
 
 
 
 / c V 8>=|,0 ' 5a t'---->^ 
 
 1.0 
 
 — " 
 
 
 
 Q "?" 
 
 
 
 
 
 UJ 
 
 
 
 
 
 ALIZ 
 ATED 
 
 
 
 
 k= -2,1— ~>^ / ^ ^^^ 
 
 2 _i 
 
 
 
 
 
 O o 
 
 
 
 
 ^^^/^^k =3,2 
 
 Z _l 
 
 < 
 
 0.5 
 
 
 
 
 o 
 
 
 
 
 — i- r 
 
 1 1 1 
 
 .002 .004 .006 .008 .01 .012 
 
 NORMALIZED STANDARD DEVIATION IN MEASURED DATA a}' 
 
 Figure 12. Normalized Standard Deviation in the Elements of the Calculated 
 T^ 8 ) Vector as a Function of the Normalized Standard Deviation 
 in the Measured Data for L = \, N = 8, A=2. 
 
53 
 
 1000 
 
 500 
 
 *200 
 
 en 
 o 
 
 ol 100 
 
 LU 
 
 50 
 
 20 
 
 10 
 
 
 
 
 
 
 
 
 i 
 
 / , 
 
 / 
 
 1 
 
 
 
 r 
 
 ■) 
 
 L 
 
 _ 1 
 
 "2 
 
 A-, i 
 
 / X "4 
 
 /L_ 1 
 /X" 8 
 
 i 
 
 1 i 
 
 / 
 
 / 
 
 / 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 / 
 
 
 
 M = IO/3 
 
 i 
 
 
 
 
 
 
 >i 
 
 
 
 
 
 
 1.5 2 
 
 NORMALIZED 
 
 EFFECTIVE APERTURE 
 
 6 
 = A 
 
 10 
 
 Figure 13. Ratio of Calculation Error to Measurement Error as a Function of 
 the Normalized Effective Aperture with the Actual Aperture in 
 Wavelengths as a Parameter 
 
54 
 
 aperture^ size increases. For example, if 4 is fixed at 2 the value of 
 increases from 1*225 for L/\ = 1/4, to 4,41 for LA = 1/2, 
 
 and to 110-5 for LA = 1. On the other hand, if one can accept a value of 
 of 100, then the normalized effective aperture will range from 
 
 aperture size : 
 
 '*<»>/< incl 
 
 and to IK 
 
 o' lo' 
 6t (N) / * 
 
 6=6 for L/\ = 1/8 to about 1,3 for L/\ = 4, 
 
 It is quite evident that such a process is of negligible practical 
 value since for moderately large antennas (L/\ > 10) the amount of useful 
 increase in effective aperture would be very small . For very small an- 
 tennas (L/X. < 1/8) the process does cause a significant increase in effec- 
 tive aperture and it is only in this area, i.e., very low frequency systems 
 where the apertures are necessarily small in terms of wavelengths, that 
 there might be a use for the process . 
 
55 
 
 7. AN EXAMPLE OF THE INCREASED RESOLUTION OF AN ANTENNA 
 AS A RESULT OF THE DATA PROCESSING 
 
 Although it has been shown that the proposed process is too sensitive 
 
 to errors in the measured data to be of much practical value, we will (for 
 
 academic purposes) consider briefly the increased resolution that is obtained 
 
 in the case of L = X. and A= N/N - 2. As shown in Chapter 6 the normalized 
 
 o 
 
 average solution error for this case is roughly one hundred times larger than 
 the normalized average error in the measured data. Consequently, for the 
 results that follow it has been assumed that the measurement accuracy was 
 about one part in ten thousand. This leaves an average error in the calcu- 
 lated data of about one percent. This corresponds to the usual accuracy of 
 the graphical representation of radiation patterns in which form the results 
 will be given. 
 
 Thus in Figure 14 the dotted curve is the principal solution T (u) for 
 
 a unit point source on the u = direction. The solid curve is that which 
 
 (8) 
 
 results when the elements of the solution vector T, are taken as the esti- 
 mates of the average value of T(u) at intervals u = 2P/8 and a smooth curve 
 is passed through these 8 points . We note that the main lobe of the latter 
 pattern is about one half the width of the principal solution's main lobe. 
 In Figure 15 are plotted the corresponding curves for the case of a point 
 source located at u = P//\/2 . We note that the curve T (u)has a main lobe 
 
 which is only about one third as wide as that of T (u) but has much larger 
 
 o 
 
 side lobes . Comparing the two cases of point sources at u = and u = P/V2 
 we see that T (u) is translation invariant, i.e., except for a translation 
 
 to the right of P///2 units its shape is unchanged. T (u) is not transla- 
 
 (8) 
 
 tion invariant, however; T (u) for the point source at u = is not just 
 
 a shifted version of T (u) for the same source at u = P/V~2~- Had the error 
 sensitivity not vitiated the process to begin with this would have been another 
 difficulty in any practical application. 
 
 Finaly, we consider the case of two point sources of equal strength 
 located at u = -P/4 and u = P/4 respectively. In Figure 16 is shown the 
 
 principal solution T (u) for the one wavelength aperture together with the 
 
 (8) ° 
 
 A = 2 solution T (u). Although the two sources are far from being resolved 
 
 by the single lobed principal solution, the A = 2 solution has two distinct 
 peaks which show quite closely the locations oi the two sources whose true 
 locations are represented by delta functions. 
 
56 
 
 Figure 14. Principal Solution Pattern T (u) and A= 2 Pattern T (u) 
 for a One Wavelength Aperture and a Point Source at u = 
 
 
 - 
 
 T (8) (u)^^ 
 
 \ \ 
 
 \ \ 
 
 
 V-^^7 
 
 !-^A< ; 
 
 M / / 
 
 / \ / / 
 / \ / / 
 
 h A / 
 
 
 — ► 
 
 -$ 
 
 o / 
 
 k 
 
 V2 
 
 Figure 15. Principal Solution Pattern T (u) and A= 2 Pattern T (u) 
 
 for a One Wavelength Aperture and a Point Source at u = pA/2 
 
 ( o\ 
 
 Figure 16. Principal Solution Pattern T (u) and A= 2 Pattern T (u) 
 for a One Wavelength Aperture and Two Equal Point Sources 
 at u = -[3/4 and u = (3/4 Respectively . 
 
57 
 8. CONCLUSIONS 
 
 In this report it has been established that although a finite antenna 
 acts as a perfect low pass filter of the spatial frequency spectrum of re- 
 mote sources, it is theoretically possible to deduce the entire frequency 
 spectrum from the output of the antenna. To do this we noted that since 
 the spectrum is a band-limited analytic function, its values for the fre- 
 quencies outside the pass band could be obtained by a process. of analytic 
 continuation of the function from within the band where it could be measured. 
 
 However., in the presence of measurement error it has been shown that 
 very little meaningful extrapolation is possible, at least with the method 
 
 described in this report . The general lack of success of this and several 
 
 7 8 9 
 other proposed methods ' would indicate that although the analytic func- 
 tion t(x) on the interval I x I < L contains an infinite amount of infor- 
 mation (if it could be measured exactly), in practice it yields a negligible 
 amount of information about its values outside the interval and it can be 
 adequately described within the interval by N = 4L/\ numbers. This, inci- 
 dentally, is the same number that is required by the Shannon Sampling Theorem 
 but the result is obtained here by a different approach. 
 
 It has also been shown that the process of expanding a function on a 
 finite interval by an orthogonal Fourier series is not only very convenient 
 (since S = I) but it is the only one of any practical value since the ex- 
 pansion of the non-orthogonal series is accompanied by the extreme error 
 sensitivity described in Chapters 5 and 6 of this report , 
 
 Finally, we note the similarity between this problem and that of a 
 supergain antenna and assert that although the data processing approach does 
 not necessitate having antennas with relatively large reactive fields, it 
 does have in common with supergain antennas an- extreme sensitivity to errors 
 in the physical parameters of the system. Since these errors cannot be made 
 arbitrarily small, the data processing system is inherently as unstable as 
 the conventional supergain system. 
 
58 
 
 REFERENCES 
 
 1. Bracewell, R.N., and Roberts, J.A,, "Aerial Smoothing in Radio Astro- 
 nomy", Austral. J. of Phys,, Vol . 7, pp , 615-640, December, 1954. 
 
 2. Booker, H.G., and Clemmow, P,C, "The Concept of an Angular Spectrum 
 of Plane Waves and its Relation to that of Polar Diagram and Aperture 
 Distribution", Proc, IEE, Vol . 97, Pt. 3, pp. 11-17, January, 1950. 
 
 3. Bracewell, R.N,, "interf erometry and the Spectral Sensitivity Island 
 Diagram", Trans, IRE, PGAP, Vol, AP-9, No. 1, pp. 59-67, January, 1961. 
 
 4. Bracewell, R.N,, "Two Dimensional Aerial Smoothing in Radio Astronomy," 
 Austral. J. of Phys,, Vol, 9, pp . 297-314, September, 1956. 
 
 5. Paley, R,E.A,C, and Wiener, N«, "Fourier Transforms in the Complex 
 Domain", Am. Math. Soc, Colloq . Pub., Vol „ 19, 1934. 
 
 6. Wiener, N., "Extrapolation, Interpolation, and Smoothing of Stationary 
 Time Series, Wiley, 1949. 
 
 7. Ville, J, A., "Sur le Prolongement des Signaux a Spectre Borne," Cables 
 et Transmissions, Vol. 10, No. 1, pp , 44-52, 1956. 
 
 8. Wolter, H,, "On Basic Analogies and Principal Differences between Opti- 
 cal and Electronic Information", Progress in Optics, Vol I, pp. 157-209, 
 Emil Wolf, Editor, North-Holland Publishing Co., Amsterdam, 1961. 
 
 9. Slepian, D., and Pollak, H.O., "Prolate Spheroidal Wave Functions, 
 Fourier Analysis and Uncertainty, i", Bell System Technical Journal, 
 Vol. 40, No. 1, pp, 43-63, January 1961. 
 
 10. Landau, H.J., and Pollak, H,0,, "Prolate Spheroidal Wave Functions, 
 Fourier Analysis and Uncertainty, II", Bell System Technical Journal, 
 Vol, 40, No. 1, pp. 65-85, January, 1961, 
 
 11. Lo, Y.T., "On the Theoretical Limitation of a Radio Telescope in De- 
 termining the Sky Temperature Distribution", J. Applied Phys., Vol. 32, 
 No. 10, pp. 2052-2054, October, 1961. 
 
 12. MacPhie, R.H., "Evaluation of Cross-Correlation Methods in the Utiliza- 
 tion of Antenna Systems", Technical Report No. 49, January, 1961, An- 
 tenna Laboratory, Electrical Engineering Research Laboratory, University 
 of Illinois, Urbana, Illinois, 
 
 13. Covington, A.E., and Broten, N.W., "An Interferometer for Radio Astro- 
 nomy with a Single Lobed Radiation Pattern", Trans. IRE, PGAP, Vol. 
 AP-5, No, 3, pp, 247-255, July, 1957, 
 
59 
 
 14. Shannon, C,E,, Communication in the Presence of Noise , Proc. IRE, 
 Vol, 37, No, 1, pp. 10-21, January 1949 . 
 
 15. Courant and Hilbert, "Methods of Mathematical Physics," Vol, 1, p. 62, 
 Interscience, 1953, 
 
 16. Hildebrand, F .B ,, "introduction to Numerical Analysis/' p, 439, McGraw- 
 Hill, 1956 = 
 
 17. Calderon, A., Spitzer, F„, and Widom, H „ , "inversion of Toeplitz 
 Matrices", Illinois J. of Math,, Vol . 3, p. 490, 1959. 
 
60 
 APPENDIX 
 
 (N) 
 STATISTICAL PROPERTIES OF THE ERROR VECTOR &T ' 
 
 The kth element of the data error vector is given by 
 
 *™ -J I W cos t (k + s>tf x i ] + 6t 2 (x i ) sin t (k + i )5 i x i ] 
 
 where 
 
 6t(x) = 6t (x) + j 5 t 2 (x) (A-2) 
 
 is a function representing the difference between the measured and the true 
 value of t(x). We assume that 
 
 E 
 
 < 6t 1 (x) ' = E < 5 t 2 (x)i = E j 6t 1 (x 
 
 1> 5t 2 (X 2 ) 
 
 = (A-3) 
 
 The error has zero average and its real and imaginary parts are statistically 
 independent . The autocorrelatic 
 to be the same and are given by 
 
 independent . The autocorrelation functions of St (x) and 6t (x) are assumed 
 
 Cr 2 sin Mx 
 
 B < VV 5t i (x 2>{ ■ E i W 6t 2 <x 2 ) f ■ -i~-a^- (A - 4) 
 
 r, , x t sin Mx , . _ . 
 
 V X >=— "Mx" (A " 5) 
 
 where x = x -x and M » P„ The error spectrum is flat and much broader than 
 
 the spectrum of t(x) =, Since the autocorrelation function is a function of x, 
 
 the difference of the two sample points x and x , the measurement error pos- 
 sesses the stationary property. 
 
 For convenience we have suppressed the constant factor 7rN/(3 which would make 
 Equation (Al) similar to Equation (76) (page 34). 
 
61 
 (N) 
 
 AVERAGE VALUE OF 6t 
 
 ok 
 
 The average or expected value of the error vector's kth element is 
 
 cos [ (k + -) — x 
 
 + E ^t 2 ( Xl )j sin r (k + |)2£ Xi j ^ 
 
 (A-6) 
 
 But since the average values of St (x) and 6t (x) are zero (Equation A-3) 
 we have 
 
 [<\ 
 
 k - - f, . ' . .. ■ , f - 1 (4-7) 
 
 (N) 
 
 The average value of the error vector §T is zero. 
 
 o 
 
 CO VARIANCE OF THE ERROR VECTOR 
 
 We now will determine the covariance of the elements St , and St . 
 
 ok oi 
 
 of the error in the data vector 
 
 ( 6T ok } *«] = E )J ^W C ° S W 6t 2 (x i ) Sin Vl ] dX l ' 
 " I ^ 6 t 1 (x 2 ) cos l^ 2 + 5 t 2 (x 2 ) sin l^ 2 ] dx 2 } 
 
 1 2(3 1 23 
 
 where k ± - (k + -)-£ and ^ = (i + -)-£ 
 
 The above can be written as 
 
62 
 
 {«*<]•]] B ( 6 w 6 w) 
 
 E ] 5t_,/ 8t /( = / ;i E ^ 5t,(xJ 6t,(x„)f cos k x cos £ x 
 
 + E { 5t (x ) 5t (x ) [ cos k x sin I x 
 
 + E < 5t 1 (x 1 ) 6t l (X 2M Sln k l X l C ° S ^1 X 2 
 
 (A-9) 
 
 { 6t 2 (x l 
 
 ) 6t 2 (x 2 ) sin k Xj sin I x 2 dx dx 2 
 
 Due to the statistical independence of 6t (x) and St (x) the two middle terms 
 of the integrand are zero and by Equations (A-4) and (A-5) we have 
 
 L L 
 
 JJ 
 
 1 ! 5T ok } 6T oi } \ : | V X > COS k i X ! COS V 2 dX l dX 2 
 
 L L 
 
 JJ- 
 
 + I I Rr(x) sin k x sin i x dx dx (A-10) 
 
 After much algebraic manipulation and by noting that Rr(x) = R,(-x), the 
 above pair of double integrals can be reduced to 
 
 L (N) on) cos [(k r'i )L] + x J 4 
 
 W? *™] - l — (577751 — J v- 
 
 E ) 6t:;/ 6t_/( = L- ~ J R R (x) [sin | x - sin k x] dx 
 
 sin (k -i,)L 
 
 W l J 
 
 o* 
 
 + L _ — rr I Rc(x) [cos i x + cos k xj dx (A-ll) 
 
63 
 Then by substituting for Rg(x) from Equation (A-5) we get 
 
 L „ 2 
 
 Mx 
 
 1 -i 1 )L / 2 Wh 
 
 sin i n x - sin kxj dx 
 
 Mx L 1 1 
 
 sin [<k-i >L] /- L or 2 ,, 
 
 (k-l\)L ^"^ tcos i x x , cos V ] 
 
 cos [ (k -i,)L]+l 0" 2 
 
 ° L - 2^-i^L - :, ', 
 
 + L 
 
 k[ c it (M ^i )L ]- c i[< 
 
 -C |(M-k 1 )L S +C. j (M+k 1 )L ( ] 
 
 sin [(k-i )L] O 2 1 -, 
 
 ". <> (M+i 1 )L j + S. (M-i 1 )LV 
 
 2(k -i )L 2M 
 
 4- S j (M+k 1 )L | + S < (M-k^ l| ] 
 
 (A-13) 
 
 where 
 
 /cos y 
 
 is the cosine integral* Now if M » $, then M » i = (i+1/2) 2^/N and 
 M » k = (k+1/2) 2|3/N. The above expression then simplifies to 
 
 Alt Li rr 2 
 
 I ok oi J (k-i)ifi 2 
 
 77 
 
 2M 2 
 
 77L Cr 2 sinl(k _ n ML 
 
 t N \ . 
 
 (A-14) 
 
 M <->^ 
 
ANTENNA LABORATORY 
 TECHNICAL REPORTS AND MEMORANDA ISSUED 
 
 Contract AF33(616)-310 
 
 "Synthesis of Aperture Antennas," Tec hn ical 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,* AD 61049. 
 
 "The Assymmetrically Excited Spherical Antenna," Technical Report No, 3 , 
 Robert C, Hansen, 30 April 1955.* 
 
 "Analysis of an Airborne Homing System," Technical Report Ho. 4 , Paul E. 
 Mayes, 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.* 
 
 "Axially Excited Surface Wave Antennas," Technical Report No. 7 , D. E. Royal, 
 10 October 1955.* 
 
 "Homing Antennas for the F-86F Aircraft (450-2500 mc)V ' Technical Report No. 8, 
 P: E, 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 o* 
 
 Contract AF33(616)-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," T echnical Report 
 No. 10 , Do Go Berry and J, B. Kreer, 10 May 1956. AD 98615 
 
 "A Technique for Controlling the Radiation from Dielectric Rod Waveguides," 
 Technical Report 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 AD 119780 
 
 "Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report 
 No, 14, Richard F, Hyneman, 20 December 1956. 
 
"Distributed Coupling to Surface Wave Antennas >-," Te chnical Report No 15, 
 Ralph Richard Hodges Jr , 5 January 1957 
 
 "The Characteristic Impedance of the Fin Antenna of Infinite Length," Tec hn ical 
 Report No 16, Robert L Carrel, 15 January 1957, * 
 
 "On the Estimation of Ferrite Loop Antenna Impedance," Technical R eport No, 1 7, 
 Walter L Weeks. 10 April 1957 * AD 143989 
 
 "A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source 
 Antenna," Technical Report No_ 18, Walter L Weeks, 20 April 1957, 
 
 "Broadband Logarithmically Periodic Antenna Structures," Technical Repo rt No., 1 9, 
 R t H, DuHamel and D, E Isbell, 1 May 1957 AD 140734 
 
 "Frequency Independent Antennas," Technical Report No, 20, V, H. Rumsey, 25 
 October 1957 ~~~ 
 
 "The Equiangular Spiral Antenna," Technical Report No, 21, J. D, Dyson, 15 
 September 1957. AD 145019 
 
 "Experimental Investigation of the Conical Spiral Antenna," Technical Report 
 No, 22, R, L, Carrel, 25 May 1957,** AD 144021 
 
 "Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical 
 
 Report No 23, E J, Scott, 10 August 1957, 
 
 "Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," 
 Technic al Report No _ 24, J W Duncan and R, H, DuHamel, August 1957, 
 
 "The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary 
 Cross Section," Tec hnical Repoi , t_jJo I _25, Robert L, Carrel, August 1957. 
 
 "Cavity-Backed Slot Antennas," Technical Report No. 26, R, J, rector, 30 
 October 1957 
 
 "Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical 
 Report No__27 i W L Weeks. 1 December 1957 
 
 "phase Velocities in Rectangular Waveguide Partially Filled with Dielectric," 
 Technical Report No , 28, W= L Weeks, 20 December 1957 
 
 "Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrary 
 Cross Section," T echnical Report No 29, J D Dyson, 10 January 1958. 
 
 "Non-Planar Logarithmically Periodic Antenna Structure," Technical Rep ort No 30, 
 D E Isbell, 20 February 1958 AD 156203 
 
 "Electromagnetic Fields in Rectangular Slots," Technical Report No, 31, N, J, 
 Kuhn and P E, Mast, 10 March 1958 
 
"The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder, 
 Technical Report No 32 J W Duncan, 25 May 1958 
 
 "A Unidirectional Equiangular Spiral Antenna" Technical Report No 33 , 
 J D Dyson 10 July 1958 AD 201138 
 
 "Dielectric Coated Spheriodal Radiators," Technical Report No. 34 , W. L. 
 Weeks, 12 September 1958 AD 204547 
 
 "A Theoretical Study of the Equiangular Spiral Antenna," Technical Report 
 No 35, P E Mast, 12 September 1958 AD 204548 
 
 Contract AF33(616)-6079 
 
 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," Technical 
 Report No 36, R H MacPhie, 30 April 1959 AD 215558 
 
 "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," Techn ic al 
 Report No 38 , W L Weeks, 5 June 1959 
 
 "Log Periodic Dipole Arrays," Technical Repo rt No 39 ,, D E Isbell, 1 June 1959, 
 AD 220651 ~~~ 
 
 "A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical 
 Report No_ 40. S Dasgupta and Y T Lo„ 17 July 1959 
 
 "The Radiation Pattern of a Dipole on a Finite Dielectric Sheet/' Technical 
 Report No 41 „ K G Balmain,, 1 August 1959 
 
 "The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem 
 in a Waveguide," Te c h nic a 1 Re po r t No 42, Raj Mittra, 1 October 1959, 
 
 "impedance Properties of Complementary Multiterminal Planar Structures , V 
 Technical Rep ort 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," 
 Technica 1 Report ' No_ 45 _ C H Tang and Y T Lo, 11 March 1960 
 
 "New Circularly Polarized Frequency Independent Antennas with Conical Beam or 
 Omnidirectional Patterns," Technical Report No 46, J D Dyson and P E. Mayes, 
 20 June 1960 AD 241321 
 
 "Logarithmically Periodic Resonant-V Arrays," Technical Report No 47, P, E. Mayes 
 and R L Carrel, 15 July 1960 AD 246302 
 
"A Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror, Technical 
 Report No 48 , Y T Lo June 1960 
 
 "Evaluation of Cross-Correlation Methods in the Utilization of Antenna Systems," 
 Technical Repor t No 49 . R H MacPhie, 25 January 1961 
 
 "Synthesis of Antenna Product Patterns Obtained from a Single Array," Technical 
 Report No 50 R H MacPhie, 25 January 1961 
 
 "On the Solution of a Class of Dual Integral Equations/' Technical Report No 51 , 
 R Mittra 1 October 1961 AD 264557 
 
 "Analysis and Design of the Log-Periodic Dipole Antenna/' Technical Report No 52 , 
 Robert L Carrel 1 October 1961* AD 264558 
 
 "A Study of the Non-Uniform Convergence of the Inverse of a Doubly- Infinite 
 Matrix Associated with a Boundary Value Problem in a Waveguide, ** Technical Report 
 No, 53 s R ; Mittra, 1 October 1961, AD 264556 
 
 * Copies available for a three-week loan period, 
 ** Copies no longer available . 
 
AF33;657 )-8460 
 
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 Stanford, California 
 
 Ramo-Wooldridge Corporation 
 Attn: Librarian (Antenna Lab) 
 Conoga Park, California 
 
 Stanford Research Institute 
 Attn: Librarian (Antenna Lab) 
 Menlo Park, California 
 
 Rand Corporation 
 
 Attn: Librarian (Antenna Lab) 
 
 1700 Main Street 
 
 Santa Monica, California 
 
 Rantec Corporation 
 Attn: Librarian (Antenna Lab) 
 23999 Ventura Blvd. 
 Calabasas, California 
 
 Raytheon Electronics Corporation 
 Attn: Librarian (Antenna Lab) 
 1089 Washington Street 
 Newton, Massachusetts 
 
 Republic Aviation Corporation 
 Applied Research & Development 
 
 Division 
 Attn: Librarian (Antenna Lab) 
 Farmingdale, New York 
 
 Sylvania Electronic System 
 Attn: Librarian (M/F Antenna & 
 
 Microwave Lab) 
 100 First Street 
 Waltham 54, Massachusetts 
 
 Sylvania Electronic System 
 Attn: Librarian (Antenna Lab) 
 P. 0. Box 188 
 Mountain View, California 
 
 Technical Research Group 
 
 Attn: Librarian (Antenna Section) 
 
 2 Aerial Way 
 
 Syosset, New York 
 
 Ling Temco Aircraft Corporation 
 Temco Aircraft Division 
 Attn: Librarian (Antenna Lab) 
 Garland, Texas 
 
 Sanders Associates 
 
 Attn: Librarian (Antenna Lab) 
 
 95 Canal Street 
 
 Nashua, New Hampshire 
 
 Texas Instruments, Inc. 
 Attn: Librarian (Antenna Lab) 
 6000 Lemmon Ave . 
 Dallas 9, Texas 
 
 Southwest Research Institute 
 Attn: Librarian (Antenna Lab) 
 8500 Culebra Road 
 San Antonio, Texas 
 
 A. So Thomas, Inc. 
 Attn: Librarian (Antenna Lab) 
 355 Providence Highway 
 Westwood. Massachusetts 
 
New Mexico State University Aeronautical Systems Division 
 
 Head Antenna Department Attn: ASAD - Library 
 
 Physical Science Laboratory Wright-Patterson Air Force Base 
 
 University Park, New Mexico Ohio 
 
 Bell Telephone Laboratories, Inc. National Bureau of Standards 
 
 Whippany Laboratory Department of Commerce 
 
 Whippany, New Jersey Attn: Dr. A. G. McNish 
 
 Attn: Technical Reports Librarian Washington 25, D. C. 
 Room 2A-165 
 
 Robert C. Hansen 
 Aerospace Corporation 
 Box 95085 
 Los Angeles 45, California 
 
 Dr. Richard C. Becker 
 10829 Berkshire 
 Westchester, Illinois 
 
 Dr. Harry Letaw, Jr. 
 
 Raytheon Company 
 
 Surface Radar and Navigation 
 
 Operations 
 State Road West 
 Wayland, Massachusetts 
 
 Dr. Frank Fu Fang 
 
 IBM Research Laboratory 
 
 Poughkeepsie, New York 
 
 Mr. Dwight Isbell 
 1422 11th West 
 Seattle 99, Washington 
 
 Dr. Robert L. Carrel 
 Collins Radio Corporation 
 Antenna Section 
 Dallas, Texas 
 
 Dr. A. K. Chatterjee 
 
 Vice Principal & Head of the Department 
 
 of Research 
 Birla Institute of Technology 
 P. 0. Mesra 
 District-Ranchi (Bihar) India