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Antenna reception of nonisotropic stochastic fields
Antenna Reception of Nonisotropic Stochastic Fields by D.
G. CHILDERS
Department of Electrical Engineering University of Florida, GainesviUe, Florida
ABSTRACT: The response of an antenna with specified bandwidth characteristics to a complex, partially polarized, random process of arbitrary bandwidth is treated from the combined viewpoint of the theory of statistical communications, antenna theory, and the theory of optics. A stochastic spectral representation that accounts for the spatial variations of a random wave process is developed and used to determine the open circuit voltage excited in an antenna via the interaction of the incident process with the frequency varying far field pattern of the antenna. A general formula is derived for the autocorrelation function at the terminals of a receivin# antenna in terms of the coherency matrix of the incident process and the antenna height function. This formulation applies for incoherent stochastic processes. The available power at the terminals of the antenna is related to this autocorrelationfunction. Introduction
Shortly before and since the advent of the first laser there has been an expanding interest in microwave optics. Ko's (1) outstanding paper on the reception of quasi-monochromatic, partially polarized radio waves, among other things, ties together many relationships in optics to those in microwaves. It also provides a brief history and some early and contemporary references of published material relating to the field. Others have followed (2, 3). In this paper we extend the work of Ko (1) to include nonisotropic stochastic processes of arbitrary bandwidth, and consider antennas with bandwidth functions so that the quasi-monochromatic restriction may be removed. We also relate the theories of noise, statistical communications, and information to those of antenna theory and optics. This is accomplished by showing how: 1) a stochastic spectral representation can be developed to represent a random noise field, 2) this representation can be employed in the reception of a partially polarized, nonisotropic random process by antenna elements with specific bandwidths, 3) the auto-correlation function of the received process is related to the coherency matrix of the incident process and the antenna height function, and 4) how the crosscorrelation between received processes
216
Antenna Reception of Nonisotropic Stochastic Fields by different antenna elements is related to the coherency matrix and the antenna height function. A method of interpreting the antenna as a filter is shown and related to the conventional filtering formula of statistical communication theory. A possible way of synthesizing antenna arrays with certain noise reduction properties is suggested via correlation theory. The spectral representation is developed because of the motivation to simultaneously account for frequency and spatial variation in an incident random process. In addition, a method is needed whereby the open circuit voltage of an antenna can be found from the interaction of the spectral representation as a function of the spatial variables of the incident random process with the antenna far field pattern. We may ask: why determine the received open circuit voltage and not the received power? The reception of power masks many of the important parameters of the received process, e.g., correlation between electric field components, coherence, and polarization. However, if we can express the received open circuit voltage of the antenna in terms of the spectral representation which is a function of the spatial parameters, it is then possible to determine the autocorrelation function of the received process as a function of the coherency matrix of the incident process and the antenna height function. The autocorrelation function yields the received power as a function of polarization and other parameters. The crosscorrelation between open circuit voltages of different antenna elements can also be determined and this in turn yields a method that can be used to synthesize antenna arrays with noise reduction properties that are a function of the orientation of the antenna elements.
Spectral Representation of a Partially Polarized, Nonisotropic R a n d o m Process Consider a stochastic representation for a general partially polarized, nonisotropic random field process, (see (4, 5-11)). The representation accounts for the three-dimensional aspects of the noise field. The representation provides an aid in the determination of the coherency matrix of the free space process a n d the correlation function for the received process. In addition, the representation can be used to handle monochromatic or quasimonochromatic signals, provided they are member functions of an ensemble of stochastic processes. In the language of optics, this process is termed incoherent, i.e., the inteusities (or powers) are additive as distinct from complex amplitudes being additive for a coherent process. This is a confusing point since coherent and incoherent also refer, for example, to the degree of correlation (or coherence) between two components of an electric field vector, i.e., two components are often said to be incoherent (uncorrelated) when their crosscorrelation is zero, i.e., when the off diagonal terms of the coherency matrix are zero and said to be coherent (perfectly correlated) if the magnitude of the off diagonal terms of the coherency matrix is unity.
Vol. 282, No. 4, October 1966
217
D. G. Childers The random field is due to any number of sources considered to be noise or signal. Possible sources for such random fields are radar or sonar clutter or reverberation caused by returned echoes from a random collection of moving scatterers in the vicinity of the desired target. In fact, the echo from the target itself may be a random field. Electronic counter measures may also cause such random fields. In the area of radio astronomy, the field may be due to galaxy noise, the sun, radio stars or local interference. The following spectral representation of a homogeneous but nonisotropic random field depicts one of these typical processes. Only electric field components are considered since a parallel derivation can be made for magnetic field components. The one-sided complex representation of the polarized, nonisotropic random field is given by =
= 2
LE+(t)
_
_
e2~II~
"o ~o "o
d~+(f, ¢, O)
(1) where Eo(t) is the 0 component of the electric field vector in spherical coordinates, i.e., the component corresponding to the depression from the z axis (see Fig. 1 for the coordinate system) ; E, (t) is the ~ component of the electric field vector in spherical coordinates; ~(f, ¢, O) is a complex process of orthogonal increments such that one of its properties is
dle(f, 4~, 0)]
E
I
[d~o*(f', ~', 0') d~¢*(f', ~', 0')']
Ld~.(f, ¢p, 0) -Noo(f, 4), O) Nee(f, ~h, 0)]
J
~ ( f - f ' ) ~(~ - ~') df dr' dU d~'
(2)
N®o(f, oh, O) N,¢(f, ¢, O)
where E denotes the expected value of the quantity within the brackets, the * denotes complex conjugates; ~2 and iY are solid angles as functions of and ~b 0; N(f, 4~, 0) is the average power spatio-spectral density of the random process expressed in watts per unit solid angle per unit area per cycle per second; sin 0 d¢ de is the infinitesimal solid angle upon which the noise is incident. Thus, N(f, ¢h, O) sin 0 de d~ df is the power per unit area incident upon the antenna from the direction (0, ~) within an infinitesimal solid angle of cone d~. In the language of the radio astronomer, N(f, 8, ~) is the spectral brightness of the received radiation. Because the representation depends upon ensemble averages and not upon time averages the process may be of arbitrary bandwidth and thus is not limited to quasi-monochromatic processes (2, 6). Equation 1 is the spectral stochastic representation and a function of the
218
Journal of The Franklin Institute
Antenna Reception of Nonisotropic Stochastic Fields Z
t /
I
\\
"x \
,,
///
t. ', /
%
e# Ea
v
-
'\l " ¢ /
x ×
Fro. Spherical coordinate system.
FIG. 2. Dipole configuration and a section of a plane w a v e incident upon t h e dipole configuration.
spatial parameters. Equation 2 is one of the important properties of this representation that allows us to relate this representation to existing and past analytical forms as well as to physical processes themselves. More will be said about N ( f , ¢, 0) and its relation to the coherency (correlation) matrix. It is difficult to give a physical interpretation to Eq. 1. However, its usefulness as well as a physical interpretation is given in the next section. The representation chosen for/~ in Eq. 1, known as a stochastic spectral representation, is the incident (or transmitted) electric field vector with both 0 and ¢ components. However, we express these components as a continuous superposition of harmonic functions modulated by the stochastic weighting factor d~(f, 8, ¢), (5, 6). This representation is applicable for any probability density function that describes the behavior of the electric field. Note that Eq. 1 is analogous to the Fourier transform used for a deterministic process which cannot be used for random processes. Note also that Eq. 1 is similar to a RiemannStieltjes integral although it is properly a stochastic integral. Thus Eq. 1 is simply a stochastic spectral representation for a random electric field process.
Reception of the Complex, Partially Polarized, Nonisotropic Stochastic R a n d o m Process We process chastic matrix excited
show how the stochastic spectral, representation of a random wave previously developed can be used to determine the open circuit stovoltage excited in a receiving antenna. We then derive the coherency for the received process and show how the cross-correlation between antenna voltages can be determined.
Consider the excitation process between a receiving antenna and the exciting
Vol. 282, No. 4, October 1065
219
D. G. Childers incident random field. Assume that a random plane wave from a very distant source is falling upon the receiving antenna and that the energy loss due to scattering is negligible. The random plane wave is in the far field of the receiving antenna and the electric and magnetic field vectors lie in the plane normal to the direction of propagation of the wave-front. The radial field components are negligible. Due to the action of the incident wave an open circuit voltage is excited in the antenna. Following Burgess (12), Sinclair (13), Tai (14), Ko (1, 15, 15), and Copeland (17), the open circuit voltage is determined by the dot product of the complex vector effective antenna height function with the incident electric field vector. However, we extend the definition of the complex vector effective height of the antenna to include a frequency variation. The previous authors did not include this variation for a variety of reasons. Because of the spectral representation for the random process, it is desirable also to include a frequency variation (or bandwidth limitation) for the antenna height function. Thus, the far field pattern is now a vector function (the vector antenna height function) of the spatial and frequency variables. The fact that the excited antenna open circuit voltage is due to the above mentioned dot product is justified by applying the Reciprocity Theorem. The differential stochastic open circuit voltage excited in the antenna is the result of the dot product of the differential electric field vector of the incident random process with the complex vector (frequency) effective antenna height function (in the far field) of the antenna. Then, if the appropriate integrations are performed the stochastic open circuit antenna voltage is obtained. Mathematically, the above steps are expressed as follows: Let f~(f, @, 0) be the complex vector effective antenna height function and let v(t) be the stochastic open circuit voltage excited in the antenna, then
dr(f, ~, O) = dE(f, 4, o).h(f, ¢, O)
(3)
such that
v(t) = 2
=
2
fU'f e2~/~ dr(f, 0, O) f~.f~fo~ dE(f, @, O).[z(f, @, o) ~0
~0 ~ 0
"0
~0 "0
e 2~'/*
(4)
is the spectral representation for the received process for the general far field excitation case.
Autocorrelation a n d Crosscorrelation F u n c t i o n s f o r Received
Partially Polarized, Nonisotropic Random Process We now determine how the crosscorrelation function between an open circuit voltage excited in one antenna and another open circuit voltage excited in another antenna is related to the coherency matrix of the incident process
220
Journal of The Franklin Institute
Antenna Reception of Nonisotropic StochasticFields and the two different antenna height functions. Consider first the coherency matrix, which is a correlation matrix between the components of the electric field vector, see (1, 2, 5, 18), where each has additional references. For the random process incident upon the antenna, i.e. propagating toward the antenna, the coherency matrix is defined as
Eo(t) I
Rob(r) = E
( E o * ( t - r)
E+*(t - r))
\E+(t)l
=[Roo(.) R0+(.)] LR+o(,) R++(r)
.,::
INooNoo]
=4 f f f "o "o'o
e""
[N+o
N++JsinOdOd~df.
(5)
The coherency matrix for the incident process as seen by the receiving antenna in the far field is given by transposing the matrices in Eq. 5; i.e., T
Ra(r) = E
(Eo*(t - r)
E**(t - r)
[\E+ (0 / = [R0o(r)
I
R+0(r)]
[R0+(~) R++(~) 2, ~o
•
~
-o -o
[Nee
[No,
X°'1sin 0 dO d4~ dr.
(6)
N,+J
Ko (1) has given this necessity for distinguishing between the coherency matrices very elegantly. We repeat part of his reasoning here for completeness. The coherency matrix yields information concerning the degree and type of polarization (among other things). However, we distinguish two types of polarization, right handed and left handed. Right handed polarization (as defined by radio physicists and engineers but opposite to that in optics) is defined as occurring if the rotation of the electric field vector and the direction of propagation is such as to form a right handed screw. If a left handed screw is formed then the polarization is said to be left handed. If we imagine ourselves at the origin, then a wave propagating away from us with a certain coherency matrix (a certain polarization, say, right handed polarization), then the same
Vol. 282, No. 4, O ~ t o ~ ~
221
D. G. Childers
matrix will represent a left handed polarized wave for one propagating towards US.
Transposing of the matrix involved interchanging No, with N,o. But since it can be shown that R,0(r) = R*o,(r) and No, -- N*o,, then according to Ko (1) this results in the change of signs in the phase angle of the off diagonal terms only. This change in a sign simply changes the sense in which the polarization ellipse is described. Equations 5 or 6 necessitates further interpretation. We now show that N (f, 0, ~b) sin 0 d~ dO df introduced previously is the infinitesimal spatio-spectral density in watts per unit area for the random process. The N terms in the matrices represent the auto- and cross-spectral density terms in watts per unit frequency bandwidth per unit solid angle per unit area incident upon the antenna from the direction (0, 4). Thus, these equations account for spatial variations in the process, i.e., the coherency matrix is expressed in terms of the power spatio-spectral density matrix. It should be emphasized that Eqs. 5 and 6 appear at this time to be useful only in conjunction with an antenna receiving element. This is due to the solid angle interpretation for the power spatio-spectral density. That is, the correlation functions (or more properly the spatio-spectral densities) are quantities looking into a cone that expands outward into space with its apex at the origin (or antenna). A u t o c o r r e l a t i o n of V ( t )
We calculate the autocorrelation function for the excited open circuit antenna voltage and determine its relationship to the coherency matrix of the incident random process and the antenna height function. This is given as R ~ ( r ) = E['v(t)v*(t - r)J
=4
= 4
ffffff ffffff fff
e2"II', e~'"(I-f'lE[h. d~]Ed~*, lz*]
d~, dSo*
[No, No,]/h,*\
e','r~ (hoh,) [N,o
tt ,)sinO N**J\h, /
d$od$**](ho*I d$,d~**J\h**/ dSd~df
(7)
where the relationships of the second section are employed to obtain the last step: This equation has many interesting interpretations. First, we examine the power available from the antenna. This power depends upon the radiation resistance (not discussed here). However, the power available from the antenna is directly proportional to R,, (0). This in turn means that the power available
222
Journal of The Franklin Institute
Antenna Reception of Nonisotropic Stochastic Fields is proportional to the integrand of Eq. 7 which is
hoho*Noo q- hoh~*No~ -q- ho*h~N,o q- h,h~*N,~. 1
(s)
Next we define a coherency matrix (actually the power spatio-spectral density matrix) for the receiving antenna by (in the frequency domain with spatial variations)
(ho*h~*) = h~
(9)
Lh~ho*
h~h**J
This definition agrees with Ko's (1) and can be interpreted in terms of the transmitted electric field vectors, as shown by Ko. Equation 8 is obtained by Trace [[hJEN] r]
(10)
where [h] is the 2 X 2 matrix of Eq. 9. Thus, the available power from the antenna is proportional to the trace of the product of the coherency matrix of the receiving antenna with the coherency matrix of the incident process as seen by the receiving antenna. A second interpretation is possible when we note that if the two electric field components of the random process are uncorrelated, then the off diagonal (the cross-spectral density and thus, cross-correlation) terms of Eq. 7 are zero. Also if Noo and N~, are equal, then the process is said to be completely unpolarized and
R~,(r) = 4 f f f
e~'~s" [hoho*Noo + h~h**N,~] sin 0 d~ d4~ dr.
(11)
This is a generalization of a result previously obtained by Childers and Reed (19) for two arbitrarily oriented dipoles in a noise field. The method of derivation is different (though related) from that employed here. Some detailed examples appear in (19) for dipole receiving elements. A third interpretation of Eq. 7 is to n o t e the close analogy with filtering theory. The integrand can be written as
[h][N][h] t
(12)
where I-hi is the (1 × 2) matrix for the antenna height function as a function of frequency and spatial coordinates, IN] is the (2 X 2) matrix of the average power spatio-spectral density expressed in watts per unit solid angle per unit area per cycle per sec, and [-hi* is the complex conjugate and transpose of I-hi. 1 This is directly comparable with the numerator of Eq. (15) of Ko (1).
Vol. 282, No. 4, October 1966
223
D. G. Childers Equation 12 is interpreted as the power spatio-spectral density function seen at the output of the antenna height function expressed in watts per unit solid angle per cycle per see. This is a generalization of the one-dimensional result
(13)
I h(f) I~ N ( f )
where h(f) is the weighting function of a filter and N ( f ) is the power spectral density at the input. It is also a variation (or an extension, depending upon one's viewpoint) of a result for four pole networks discussed by Barakat (5) and O'Neill (3). Thus, Eq. 12 might be interpreted as a filtering equation for antenna height functions that vary with frequency.
Crosscorrelation F u n c t i o n We consider now the crosseorrelation function between two open circuit voltages excited in two different antennas by the same incident far field random process. Let vx(t) be the open circuit voltage excited in the reference antenna at the origin with antenna height function ha(f, 0, ¢) and v~(t) be the open circuit voltage in the second antenna located at distance d from the origin along the z-axis (see Fig. 2). This antenna has a height function denoted by h~(f, O, ¢). v~(t) and v~(t) are given by Eq. 4. However, in order to calculate the crosscorrelation function between v~ and v~, we need some reference with respect to incident plane waves. Thus, if a plane wave is arriving at an angle (0, ~b), then
vl (t) = 2 f f f
e2"u* hi" d~
(14)
v,(t) = 2 f f f
exp [2mf (t + d_ c cos 0)J ~.d~
(15)
and
where (d/c) cos 0 is the time it takes for a plane wave of the random process to travel from h~ to hi. Thus, the spatiotemporal crosscorrelation between v~(t) and v2(t) as a function of the time separation r and the spatial separation d Rn(r, d) -- EEv,(t)v~*(t -- 4
--
z)-I
/// exp [2rif (r -- -' cos 0)] (hi,0 hi,c) C
sin 0 dO dep df
INto
~24
(16)
N**J\h,,**/
Journal of The Franklin Institute
Antenna Receptionof NonisotropicStochasticFileds and R,,(r, d) = E[v~(t)vl*(t - r ) ] = 4 f f f e x p [ 2 ~ r i f ( r 4 - d- cos 0)] (h,.eh,.a) c
sin 0 dOdch df.
X
(17)
The latter integrand (with the exception of the e2T;/* term) is the complex conjugate of the former integrand as it should be. It is assumed that the far field spatio-spectra as seen by the two receiving antennas separated by d are the same. Some examples for which the crosscorrelation function is determined explicitly for h t a n d h~ as dipoles are computed in (19), where it is shown that for certain dipole orientations it is possible to make the isotropic unpolarized noise crosscorrelation function always zero for all r. Thus, it becomes possible to synthesize arrays with zero noise crosscorrelation but still detect an incident signal. It is also possible to detect polarization by means of such arrays (see Cohen (20), who gives further references). Others discuss crosscorrelation as applied to antenna systems (16); correlation intefferometers, (21, 22, 23). (These references do not touch on the same method used here, but do discuss related material).
Signal Reception Characteristics The results of the previous sections can be applied to monochromatic or quasi-monochromatic signal reception provided we allow delta functions. Thus, Eqs. 16 and 17 can be used to obtain the correlation function of a received signal process that is stochastic or random. As an example consider the situation when the random signal process is a plane wave incident at an angle (00, ~b0) and has a power spectral density S(f) which is separable from the spatial variables, then we can replace the N matrix with
Soo(f) S~o(f)
Sos(f)S~(f)
l_L_ ~(O -
0o)~(4~ - ¢o)
(18)
sin00
where 1/sin 00 is a normalizing factor and
2
[Soo(f) -4- S**(f)] df = Roo(O) A- R**(O)
(19)
is the total intensity of the received signal, and the Roe and R ~ are ½ the real parts of elements in Eq. 6.
vow. ~82. ~o. 4, O ~ t o ~ ~
225
D. G. Childers TABLE I R
Type of Polarization
S
Randompolarized °[: i] [ 1 '] Right hand circular
½
Left hand circular
-i
1
i
1
-~
½
Odir io
i] [ 10~] -½i
1
½i
1J
1
[i i]
[: 00]
o
The normalized coherency matrix of Eq. 6 is often used to specify the polarization of the incident process; i.e.,
1
-Roo(O)
Re,(0)]
=R
Roo(O) + R~,(O)
(20)
R~o(O) R ~ , ( 0 ) J
is tabulated in Table I for various types of polarization. 2 (See (18, 3, 15)).
Monochromatic Signals W e n o w c o n s i d e r m o n o c h r o m a t i c s i g n a l s so t h a t S ( f )
= c ~ ( f - J'0), w h e r e
c is a constant proportional to power per cycle per sac. If we let [S-] represent the normalized power spectral density for monochromatic, plane wave signals, then we can express the type of polarization in terms of [-S]. This is also shown in Table I, where Eq. 19 is applied to determine the normalized total intensity. z
I: Jx
(a)
y
'~
(b)
×
(c)
= -s,°. -V
FIG. 3. Antenna height functions for three dipole arrangements. 2 The notation J is sometimes used instead of R to denote the normalized coherency matrix.
226
Journalof The FranklinInstitute
Antenna Reception of Nonisotropic Stochastic Fields T.~
II s
Crosscorrelation functions, Rt~(r, d), for monochromatic plane wave signals incident at an angle (00, ~o) for the dipole arrangement of Fig. 3(a). Type of Polarization
R~(~, d)
Random (unpolarized)
sin~Oo exp
Right hand circular
sin20o exp
Left hand circular
sin~Oo exp
Plane, 0 direction
2 sin20o exp
Plane, ¢ direction
0
00)]
[~,,0(~ cOO~
TXBLE I I I s
Crosscorrelation functions, R,~(r, d), for monochromatic plane wave signals incident at an angle (0o, ¢o) for the dipole arrangement of Fig. 3(b). Type of Polarization
RI~(r, d)
R a n d o m (unpolarized)
~o~0~o~o0+ ~o~oo~ex, [~,0 (~ ~ ~o~00)]
Right h a n d circular
Eco~2 00 sin 2 ¢0 + cos 2 ~,o] exp
Left h a n d circular
~co~00~in~o0+ ~o~o0~cup[~,o (~ ~ ~o~,0)]
Plane, 0 direction
[(~)]
27fifo ~-- c co~ 00
[(~)]
2 c o s ~00sin~boexp 27fifo
v - - c- cos
00
Plane, 4, direction
3 The received power is proportional to ½ the real part evaluated at variou~ r.
VoL ~ , So. 4, O c t o ~ , ~
227
D. G. Childers TABLE IV a
Crosscorrelation functions, R~( r, d) for monochromatic plane wave signals incident at an angle (0o, ~o) for the dipole arrangement of Fig. 3(c).
Type of Polarization
Rl~(r, d)
Random (unpolarized)
Right hand circular
- - [ s i n 80 cos O0 cos ~bo-~i }sinOosin~o]exp[27rifo(r--!
co80o)]
Left hand circular
--[-sin O0 cos 8o cos ~0 -- i ½sin Oo sin ~0J exp 2~rifo "r---c
cos .0)]
Plane, 0 direction Plane, ~bdirection
0
It is worth noting that Table I can be easily determined for monochromatic signals, but is not so easily determined for quasi-monochromatic signals or broadband noise. In fact, this is a research project under current investigation. We axe now in a position to determine the received signal correlation functions provided we specify the antenna height functions. Let hi and h2 be dipoles in three common arrangements, as shown in Fig. 3, where we assume the dipole bandwidths to be either infinite or at least constant over the bandwidth of the received signal. If we now substitute the results of Table I for ES] into Eq. 18 and in turn use Eq. 16, we m a y determine the crosscorrelation function for monochromatic, plane wave signals of various polarizations incident at an angle (00, ¢~) for the three dipole configurations shown in Fig. 3. The results of these calculations are given in Tables II, III, and IV. All of the cross correlation functions in Tables I I - I V are periodic with argument 2~rfoEr - (d/c) cos 8o3, where fo is the frequency of the plane wave, Oo is part of the incident angle, and d is the separation of the dipoles along the z-axis. One half of the sum of the right hand circular and left hand circular terms is equal to the unpolarized term as it should be. In Table II the crosscorrelation or received power is independent of the ~b component of polarization since the dipoles are not excited by this term. The maximum is achieved when r = (d/c) cos 00 and 80 = ~r/2 or the plane wave is parallel to the x-z plane. In Table I I I we examine and interpret the results for plane polarization. When a plane wave is incident at an angle (0o, ¢o) and we say it is polarized
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Journal of The Franklin Institute
Antenna Reception of Nonisotropic Stochastic Fields
in the 0 direction we mean that there is only one electric field component and this component is the 0 component shown in Fig. 2. If we now imagine the plane in Fig. 2 moved so that ¢0 = ~r/2 and then shifted until 00 = 0, we obtain a plane wave propagating down the z-axis parallel to the x-y plane with the electric field vector aligned with the y-axis. This gives the maximum received power provided we delay the signal received at h~ by an amount d/c. Thus, ~he result is reasonable physically. A similar interpretation holds for the other crosscorrelation functions. In Table IV we can maximize the received signal for random polarization or plane polarization (0 direction) if 00 = 7r/4 and ¢o = 0 (if we neglect the minus sign which is simply a 180 ° phase shift). The maximization for right or left circular polarized waves is not as easily interpreted. The dipole configuration of Table IV is interesting since this configuration yields zero crosscorrelation for all r if the incident noise is isotropic and unpolarized (19). The correlation functions shown in Tables II-IV give the antenna directivity pattern for dipoles arranged in the configurations shown and for which their outputs are multiplied (one being delayed with respect to the other). By varying r we can achieve electronic beaming of the dipole configuration so as to maximize the output. This maximization by beaming is achieved, naturally, when we have r = d/c cos 0o or the directivity pattern is aligned in the direction of the incoming signal. Some related work in this area by Ko (24, 25), gives the noise temperature concepts employed in radio astronomy and elsewhere to the reception of partially polarized fields. This work is related to Ko's and is, in fact, strongly motivated by his results; however, many of the considerations are different and thus, have led to new results. Discussion
As mentioned earlier some examples for the noise case are given in (19) but they do not employ the antenna height function directly but use the spectral representation. The calculations are rather tedious and lengthy, but they illustrate the usefulness of the theory. The signal calculations of the previous section show how the theory can be used for monochromatic, polarized, plane W ayes.
We have shown how a spectral representation for a random spatio-temporal process can be used to determine the response of an antenna to this incident process. The spectral representation does not include the restriction to quasimonochromatic waves. The representation is valid for arbitrary bandwidth. The antenna height function is also assumed to have some bandwidth associated with it. The auto- and crosscorrelation functions between open circuit voltages
Vol. 282, No. 4, October 1966
229
D. G. Childers excited in different antennas by the same far field process are related to the coherency matrix of the incident process and the antenna height functions. It is also shown how the antenna can be viewed from the aspect of filtering theory. This combination of communication, antenna, and optical theory appears extremely useful in that we demonstrate how some of the ideas and analytical tools of noise, communication, and information theory can be brought to bear upon antenna theory. Much work remains for the future. Antenna gain functions and the radiation resistance for non-monochromatic waves must be defined. A possible definition of the gain function is (following Ko ( 1) ), G(f, 0, ~) =
(21)
f ~ f
J Ill 47r JJJ
dg df
However, the radiation resistance is not so easily defined for non-monochromatic input currents and frequency varying antenna height functions. Antenna directivity patterns are determined here for correlation arrays with certain noise reduction properties. However, the case of quasi-monochromatic signals must be worked out. Further work must be done to relate the coherency matrix of the incident process with the polarization detectability of the antenna. And nonisotropic (and possibly non-homogeneous) processes must be used to synthesize antenna correlation arrays. The list is long. References (1) H. C. Ko, "On the Reception of Quasi-Monochromatic, Partially Polarized Radio Waves," PIRE, Vol. 50, pp. 1950-1957, Sept. 1962. (2) M. J. Beran, and G. B. Parrent, "Theory of Partial Coherence," Englewood Cliffs, N. J. Prentice-Hall, Inc., 1964. (3) E. L. O'Neill, "Introduction to Statistical Optics," Reading, Mass., Addison-Wesley Pub. Co., 1963. (4) D. G. Childers, "A Covariance (Coherency) Matrix for Back Scattered and Spatial Electromagnetic Noise," Canadian Jour. Phys., Vol. 43, pp. 1099-1114, June, 1965. (5) R. Barakat, "Stochastic Generalization of the Green-Wolf Complex Scalar Potential of Electromagnetic Theory," J. Opt. Soc. Am., Vol. 53, pp. 252-255, Feb. 1963. (6) R. Barakat, "Theory of the Coherency Matrix for Light of Arbitrary Spectral Bandwidth," J. Opt. Soc. Am., Vol. 53, pp. 317-323, Mar. 1963. (7) J. L. Doob, "Stochastic Processes," New York, John Wiley and Sons, Inc., 1953. (8) H. Cramer, "A Contribution to the Theory of Stochastic Processes," Proc. Sea Berkeley Syrup. on Mathematical Statistics and Probability, Univ. of Calif. Press, 1951. (9) A. M. Yaglom, "An Introduction to the Theory of Stationary Random Functions," Englewood Cliffs, N. J., Prentice-Hall, Inc., 1962. (10) A. M. Yaglom, "Some Classes of Random Fields in n-dimensional Space, Related to Stationary Random Processes with a Rational Spectrum," Theory of Probability and Its Applications, Vol. 2, pp. 273-320, 1957.
230
Journal of The FranklinInstitute
Antenna Reception of Nonisotropic Stochastic Fields
(11) S. Stein, and D. E. Johansen, " A Theory of Antenna Performance in Scatter-Type Reception," I R E Trans. Antennas and Propagation, Vol. AP-9, No. 3, pp. 304-311, May 1961. (12) R. E. Burgess, "Aerial Characteristics," Wireless Eng., Vol. 21, pp. 145-160, Apr. 1944. (13) G. Sinclair, "The Transmission and Reception of Elliptically Polarized Waves," P I R E , Vol. 38, pp. 148-151, Feb. 1950. (14) C. T. Tai, "On the Definition of the Effective Aperture of Antennas," I E E E Trans. on Antennas and Propagation, Vol. AP-9, pp. 224-225, March 1961. (15) H. C. Ko, "The Use of the Statistical Matrix and the Stokes Vector in Formulating the Effective Aperture of Antennas," I E E E Trans. on Antennas and Propagation, Vol. AP-11, No. 6, pp. 581-582, Nov. 1961. (16) H. C. Ko, "Microwave Scanning Antennas," Vol. 1 : Apertures, Edited by R. C. Hansen, New York, Academic Press, pp. 263-337, 1964. (17) J. R. Copeland, "Radar Target Classification by Polarization Properties," P I R E , Vol. 4~% pp. 1290-1296, July 1960. (18) M. Born and E. Wolf, "Principles of Optics," New York, Pergamon Press, 1959. (19) D. G. Childers, and I. S. Reed, "Space-Time Cross-Correlation Functions for Antenna Array Elements in a Noise Field," I E E E Trans. on Infor. Theory, Vol. IT-11, No. 2, pp. 182-190, Apr., 1965. (20) M. H. Cohen, "Radio Astronomy Polarization Measurements," P I R E , Vol. 46, pp. 172183, Jan. 1958. (21) R. H. MacPhie, "On the Mapping by a Cross-Correlation Antenna System of Partially Coherent Radio Sources," I E E E Trans. on Antennas and Propagation, ¥ol. AP-12, No. 1, pp. 118-124, Jan., 1964. (22) W. G. Jaeckle, "Space-Frequency Equivalence for Correlation Arrays and Linear Additive Arrays," URSI, 1964. (23) C. J. Drane and G. B. Parrent, "On the Mapping of Extended Sources with Nonlinear Correlation Antennas," I E E E Trans. on Antennas and Propagation, Vol. AP-10, No. 2, pp. 126-131, Mar. 1962. (24) H. C. Ko, "Antenna Temperature and the Temperature of Electromagnetic Radiation," I E E E Trans. on Antennas and Propagations, Vol. AP-12, No. 1, pp. 126-127, Jan., 1964. (25) H. C. Ko., "Radio-Telescope Antenna Parameters," I E E E Trans. on Military Electronics, Vol. MIL-8, Nos. 3 and 4, pp. 225-232, July-Oct. 1964.
Vol. 282, No. 4, October 1966
231
G. CHILDERS
Department of Electrical Engineering University of Florida, GainesviUe, Florida
ABSTRACT: The response of an antenna with specified bandwidth characteristics to a complex, partially polarized, random process of arbitrary bandwidth is treated from the combined viewpoint of the theory of statistical communications, antenna theory, and the theory of optics. A stochastic spectral representation that accounts for the spatial variations of a random wave process is developed and used to determine the open circuit voltage excited in an antenna via the interaction of the incident process with the frequency varying far field pattern of the antenna. A general formula is derived for the autocorrelation function at the terminals of a receivin# antenna in terms of the coherency matrix of the incident process and the antenna height function. This formulation applies for incoherent stochastic processes. The available power at the terminals of the antenna is related to this autocorrelationfunction. Introduction
Shortly before and since the advent of the first laser there has been an expanding interest in microwave optics. Ko's (1) outstanding paper on the reception of quasi-monochromatic, partially polarized radio waves, among other things, ties together many relationships in optics to those in microwaves. It also provides a brief history and some early and contemporary references of published material relating to the field. Others have followed (2, 3). In this paper we extend the work of Ko (1) to include nonisotropic stochastic processes of arbitrary bandwidth, and consider antennas with bandwidth functions so that the quasi-monochromatic restriction may be removed. We also relate the theories of noise, statistical communications, and information to those of antenna theory and optics. This is accomplished by showing how: 1) a stochastic spectral representation can be developed to represent a random noise field, 2) this representation can be employed in the reception of a partially polarized, nonisotropic random process by antenna elements with specific bandwidths, 3) the auto-correlation function of the received process is related to the coherency matrix of the incident process and the antenna height function, and 4) how the crosscorrelation between received processes
216
Antenna Reception of Nonisotropic Stochastic Fields by different antenna elements is related to the coherency matrix and the antenna height function. A method of interpreting the antenna as a filter is shown and related to the conventional filtering formula of statistical communication theory. A possible way of synthesizing antenna arrays with certain noise reduction properties is suggested via correlation theory. The spectral representation is developed because of the motivation to simultaneously account for frequency and spatial variation in an incident random process. In addition, a method is needed whereby the open circuit voltage of an antenna can be found from the interaction of the spectral representation as a function of the spatial variables of the incident random process with the antenna far field pattern. We may ask: why determine the received open circuit voltage and not the received power? The reception of power masks many of the important parameters of the received process, e.g., correlation between electric field components, coherence, and polarization. However, if we can express the received open circuit voltage of the antenna in terms of the spectral representation which is a function of the spatial parameters, it is then possible to determine the autocorrelation function of the received process as a function of the coherency matrix of the incident process and the antenna height function. The autocorrelation function yields the received power as a function of polarization and other parameters. The crosscorrelation between open circuit voltages of different antenna elements can also be determined and this in turn yields a method that can be used to synthesize antenna arrays with noise reduction properties that are a function of the orientation of the antenna elements.
Spectral Representation of a Partially Polarized, Nonisotropic R a n d o m Process Consider a stochastic representation for a general partially polarized, nonisotropic random field process, (see (4, 5-11)). The representation accounts for the three-dimensional aspects of the noise field. The representation provides an aid in the determination of the coherency matrix of the free space process a n d the correlation function for the received process. In addition, the representation can be used to handle monochromatic or quasimonochromatic signals, provided they are member functions of an ensemble of stochastic processes. In the language of optics, this process is termed incoherent, i.e., the inteusities (or powers) are additive as distinct from complex amplitudes being additive for a coherent process. This is a confusing point since coherent and incoherent also refer, for example, to the degree of correlation (or coherence) between two components of an electric field vector, i.e., two components are often said to be incoherent (uncorrelated) when their crosscorrelation is zero, i.e., when the off diagonal terms of the coherency matrix are zero and said to be coherent (perfectly correlated) if the magnitude of the off diagonal terms of the coherency matrix is unity.
Vol. 282, No. 4, October 1966
217
D. G. Childers The random field is due to any number of sources considered to be noise or signal. Possible sources for such random fields are radar or sonar clutter or reverberation caused by returned echoes from a random collection of moving scatterers in the vicinity of the desired target. In fact, the echo from the target itself may be a random field. Electronic counter measures may also cause such random fields. In the area of radio astronomy, the field may be due to galaxy noise, the sun, radio stars or local interference. The following spectral representation of a homogeneous but nonisotropic random field depicts one of these typical processes. Only electric field components are considered since a parallel derivation can be made for magnetic field components. The one-sided complex representation of the polarized, nonisotropic random field is given by =
= 2
LE+(t)
_
_
e2~II~
"o ~o "o
d~+(f, ¢, O)
(1) where Eo(t) is the 0 component of the electric field vector in spherical coordinates, i.e., the component corresponding to the depression from the z axis (see Fig. 1 for the coordinate system) ; E, (t) is the ~ component of the electric field vector in spherical coordinates; ~(f, ¢, O) is a complex process of orthogonal increments such that one of its properties is
dle(f, 4~, 0)]
E
I
[d~o*(f', ~', 0') d~¢*(f', ~', 0')']
Ld~.(f, ¢p, 0) -Noo(f, 4), O) Nee(f, ~h, 0)]
J
~ ( f - f ' ) ~(~ - ~') df dr' dU d~'
(2)
N®o(f, oh, O) N,¢(f, ¢, O)
where E denotes the expected value of the quantity within the brackets, the * denotes complex conjugates; ~2 and iY are solid angles as functions of and ~b 0; N(f, 4~, 0) is the average power spatio-spectral density of the random process expressed in watts per unit solid angle per unit area per cycle per second; sin 0 d¢ de is the infinitesimal solid angle upon which the noise is incident. Thus, N(f, ¢h, O) sin 0 de d~ df is the power per unit area incident upon the antenna from the direction (0, ~) within an infinitesimal solid angle of cone d~. In the language of the radio astronomer, N(f, 8, ~) is the spectral brightness of the received radiation. Because the representation depends upon ensemble averages and not upon time averages the process may be of arbitrary bandwidth and thus is not limited to quasi-monochromatic processes (2, 6). Equation 1 is the spectral stochastic representation and a function of the
218
Journal of The Franklin Institute
Antenna Reception of Nonisotropic Stochastic Fields Z
t /
I
\\
"x \
,,
///
t. ', /
%
e# Ea
v
-
'\l " ¢ /
x ×
Fro. Spherical coordinate system.
FIG. 2. Dipole configuration and a section of a plane w a v e incident upon t h e dipole configuration.
spatial parameters. Equation 2 is one of the important properties of this representation that allows us to relate this representation to existing and past analytical forms as well as to physical processes themselves. More will be said about N ( f , ¢, 0) and its relation to the coherency (correlation) matrix. It is difficult to give a physical interpretation to Eq. 1. However, its usefulness as well as a physical interpretation is given in the next section. The representation chosen for/~ in Eq. 1, known as a stochastic spectral representation, is the incident (or transmitted) electric field vector with both 0 and ¢ components. However, we express these components as a continuous superposition of harmonic functions modulated by the stochastic weighting factor d~(f, 8, ¢), (5, 6). This representation is applicable for any probability density function that describes the behavior of the electric field. Note that Eq. 1 is analogous to the Fourier transform used for a deterministic process which cannot be used for random processes. Note also that Eq. 1 is similar to a RiemannStieltjes integral although it is properly a stochastic integral. Thus Eq. 1 is simply a stochastic spectral representation for a random electric field process.
Reception of the Complex, Partially Polarized, Nonisotropic Stochastic R a n d o m Process We process chastic matrix excited
show how the stochastic spectral, representation of a random wave previously developed can be used to determine the open circuit stovoltage excited in a receiving antenna. We then derive the coherency for the received process and show how the cross-correlation between antenna voltages can be determined.
Consider the excitation process between a receiving antenna and the exciting
Vol. 282, No. 4, October 1065
219
D. G. Childers incident random field. Assume that a random plane wave from a very distant source is falling upon the receiving antenna and that the energy loss due to scattering is negligible. The random plane wave is in the far field of the receiving antenna and the electric and magnetic field vectors lie in the plane normal to the direction of propagation of the wave-front. The radial field components are negligible. Due to the action of the incident wave an open circuit voltage is excited in the antenna. Following Burgess (12), Sinclair (13), Tai (14), Ko (1, 15, 15), and Copeland (17), the open circuit voltage is determined by the dot product of the complex vector effective antenna height function with the incident electric field vector. However, we extend the definition of the complex vector effective height of the antenna to include a frequency variation. The previous authors did not include this variation for a variety of reasons. Because of the spectral representation for the random process, it is desirable also to include a frequency variation (or bandwidth limitation) for the antenna height function. Thus, the far field pattern is now a vector function (the vector antenna height function) of the spatial and frequency variables. The fact that the excited antenna open circuit voltage is due to the above mentioned dot product is justified by applying the Reciprocity Theorem. The differential stochastic open circuit voltage excited in the antenna is the result of the dot product of the differential electric field vector of the incident random process with the complex vector (frequency) effective antenna height function (in the far field) of the antenna. Then, if the appropriate integrations are performed the stochastic open circuit antenna voltage is obtained. Mathematically, the above steps are expressed as follows: Let f~(f, @, 0) be the complex vector effective antenna height function and let v(t) be the stochastic open circuit voltage excited in the antenna, then
dr(f, ~, O) = dE(f, 4, o).h(f, ¢, O)
(3)
such that
v(t) = 2
=
2
fU'f e2~/~ dr(f, 0, O) f~.f~fo~ dE(f, @, O).[z(f, @, o) ~0
~0 ~ 0
"0
~0 "0
e 2~'/*
(4)
is the spectral representation for the received process for the general far field excitation case.
Autocorrelation a n d Crosscorrelation F u n c t i o n s f o r Received
Partially Polarized, Nonisotropic Random Process We now determine how the crosscorrelation function between an open circuit voltage excited in one antenna and another open circuit voltage excited in another antenna is related to the coherency matrix of the incident process
220
Journal of The Franklin Institute
Antenna Reception of Nonisotropic StochasticFields and the two different antenna height functions. Consider first the coherency matrix, which is a correlation matrix between the components of the electric field vector, see (1, 2, 5, 18), where each has additional references. For the random process incident upon the antenna, i.e. propagating toward the antenna, the coherency matrix is defined as
Eo(t) I
Rob(r) = E
( E o * ( t - r)
E+*(t - r))
\E+(t)l
=[Roo(.) R0+(.)] LR+o(,) R++(r)
.,::
INooNoo]
=4 f f f "o "o'o
e""
[N+o
N++JsinOdOd~df.
(5)
The coherency matrix for the incident process as seen by the receiving antenna in the far field is given by transposing the matrices in Eq. 5; i.e., T
Ra(r) = E
(Eo*(t - r)
E**(t - r)
[\E+ (0 / = [R0o(r)
I
R+0(r)]
[R0+(~) R++(~) 2, ~o
•
~
-o -o
[Nee
[No,
X°'1sin 0 dO d4~ dr.
(6)
N,+J
Ko (1) has given this necessity for distinguishing between the coherency matrices very elegantly. We repeat part of his reasoning here for completeness. The coherency matrix yields information concerning the degree and type of polarization (among other things). However, we distinguish two types of polarization, right handed and left handed. Right handed polarization (as defined by radio physicists and engineers but opposite to that in optics) is defined as occurring if the rotation of the electric field vector and the direction of propagation is such as to form a right handed screw. If a left handed screw is formed then the polarization is said to be left handed. If we imagine ourselves at the origin, then a wave propagating away from us with a certain coherency matrix (a certain polarization, say, right handed polarization), then the same
Vol. 282, No. 4, O ~ t o ~ ~
221
D. G. Childers
matrix will represent a left handed polarized wave for one propagating towards US.
Transposing of the matrix involved interchanging No, with N,o. But since it can be shown that R,0(r) = R*o,(r) and No, -- N*o,, then according to Ko (1) this results in the change of signs in the phase angle of the off diagonal terms only. This change in a sign simply changes the sense in which the polarization ellipse is described. Equations 5 or 6 necessitates further interpretation. We now show that N (f, 0, ~b) sin 0 d~ dO df introduced previously is the infinitesimal spatio-spectral density in watts per unit area for the random process. The N terms in the matrices represent the auto- and cross-spectral density terms in watts per unit frequency bandwidth per unit solid angle per unit area incident upon the antenna from the direction (0, 4). Thus, these equations account for spatial variations in the process, i.e., the coherency matrix is expressed in terms of the power spatio-spectral density matrix. It should be emphasized that Eqs. 5 and 6 appear at this time to be useful only in conjunction with an antenna receiving element. This is due to the solid angle interpretation for the power spatio-spectral density. That is, the correlation functions (or more properly the spatio-spectral densities) are quantities looking into a cone that expands outward into space with its apex at the origin (or antenna). A u t o c o r r e l a t i o n of V ( t )
We calculate the autocorrelation function for the excited open circuit antenna voltage and determine its relationship to the coherency matrix of the incident random process and the antenna height function. This is given as R ~ ( r ) = E['v(t)v*(t - r)J
=4
= 4
ffffff ffffff fff
e2"II', e~'"(I-f'lE[h. d~]Ed~*, lz*]
d~, dSo*
[No, No,]/h,*\
e','r~ (hoh,) [N,o
tt ,)sinO N**J\h, /
d$od$**](ho*I d$,d~**J\h**/ dSd~df
(7)
where the relationships of the second section are employed to obtain the last step: This equation has many interesting interpretations. First, we examine the power available from the antenna. This power depends upon the radiation resistance (not discussed here). However, the power available from the antenna is directly proportional to R,, (0). This in turn means that the power available
222
Journal of The Franklin Institute
Antenna Reception of Nonisotropic Stochastic Fields is proportional to the integrand of Eq. 7 which is
hoho*Noo q- hoh~*No~ -q- ho*h~N,o q- h,h~*N,~. 1
(s)
Next we define a coherency matrix (actually the power spatio-spectral density matrix) for the receiving antenna by (in the frequency domain with spatial variations)
(ho*h~*) = h~
(9)
Lh~ho*
h~h**J
This definition agrees with Ko's (1) and can be interpreted in terms of the transmitted electric field vectors, as shown by Ko. Equation 8 is obtained by Trace [[hJEN] r]
(10)
where [h] is the 2 X 2 matrix of Eq. 9. Thus, the available power from the antenna is proportional to the trace of the product of the coherency matrix of the receiving antenna with the coherency matrix of the incident process as seen by the receiving antenna. A second interpretation is possible when we note that if the two electric field components of the random process are uncorrelated, then the off diagonal (the cross-spectral density and thus, cross-correlation) terms of Eq. 7 are zero. Also if Noo and N~, are equal, then the process is said to be completely unpolarized and
R~,(r) = 4 f f f
e~'~s" [hoho*Noo + h~h**N,~] sin 0 d~ d4~ dr.
(11)
This is a generalization of a result previously obtained by Childers and Reed (19) for two arbitrarily oriented dipoles in a noise field. The method of derivation is different (though related) from that employed here. Some detailed examples appear in (19) for dipole receiving elements. A third interpretation of Eq. 7 is to n o t e the close analogy with filtering theory. The integrand can be written as
[h][N][h] t
(12)
where I-hi is the (1 × 2) matrix for the antenna height function as a function of frequency and spatial coordinates, IN] is the (2 X 2) matrix of the average power spatio-spectral density expressed in watts per unit solid angle per unit area per cycle per sec, and [-hi* is the complex conjugate and transpose of I-hi. 1 This is directly comparable with the numerator of Eq. (15) of Ko (1).
Vol. 282, No. 4, October 1966
223
D. G. Childers Equation 12 is interpreted as the power spatio-spectral density function seen at the output of the antenna height function expressed in watts per unit solid angle per cycle per see. This is a generalization of the one-dimensional result
(13)
I h(f) I~ N ( f )
where h(f) is the weighting function of a filter and N ( f ) is the power spectral density at the input. It is also a variation (or an extension, depending upon one's viewpoint) of a result for four pole networks discussed by Barakat (5) and O'Neill (3). Thus, Eq. 12 might be interpreted as a filtering equation for antenna height functions that vary with frequency.
Crosscorrelation F u n c t i o n We consider now the crosseorrelation function between two open circuit voltages excited in two different antennas by the same incident far field random process. Let vx(t) be the open circuit voltage excited in the reference antenna at the origin with antenna height function ha(f, 0, ¢) and v~(t) be the open circuit voltage in the second antenna located at distance d from the origin along the z-axis (see Fig. 2). This antenna has a height function denoted by h~(f, O, ¢). v~(t) and v~(t) are given by Eq. 4. However, in order to calculate the crosscorrelation function between v~ and v~, we need some reference with respect to incident plane waves. Thus, if a plane wave is arriving at an angle (0, ~b), then
vl (t) = 2 f f f
e2"u* hi" d~
(14)
v,(t) = 2 f f f
exp [2mf (t + d_ c cos 0)J ~.d~
(15)
and
where (d/c) cos 0 is the time it takes for a plane wave of the random process to travel from h~ to hi. Thus, the spatiotemporal crosscorrelation between v~(t) and v2(t) as a function of the time separation r and the spatial separation d Rn(r, d) -- EEv,(t)v~*(t -- 4
--
z)-I
/// exp [2rif (r -- -' cos 0)] (hi,0 hi,c) C
sin 0 dO dep df
INto
~24
(16)
N**J\h,,**/
Journal of The Franklin Institute
Antenna Receptionof NonisotropicStochasticFileds and R,,(r, d) = E[v~(t)vl*(t - r ) ] = 4 f f f e x p [ 2 ~ r i f ( r 4 - d- cos 0)] (h,.eh,.a) c
sin 0 dOdch df.
X
(17)
The latter integrand (with the exception of the e2T;/* term) is the complex conjugate of the former integrand as it should be. It is assumed that the far field spatio-spectra as seen by the two receiving antennas separated by d are the same. Some examples for which the crosscorrelation function is determined explicitly for h t a n d h~ as dipoles are computed in (19), where it is shown that for certain dipole orientations it is possible to make the isotropic unpolarized noise crosscorrelation function always zero for all r. Thus, it becomes possible to synthesize arrays with zero noise crosscorrelation but still detect an incident signal. It is also possible to detect polarization by means of such arrays (see Cohen (20), who gives further references). Others discuss crosscorrelation as applied to antenna systems (16); correlation intefferometers, (21, 22, 23). (These references do not touch on the same method used here, but do discuss related material).
Signal Reception Characteristics The results of the previous sections can be applied to monochromatic or quasi-monochromatic signal reception provided we allow delta functions. Thus, Eqs. 16 and 17 can be used to obtain the correlation function of a received signal process that is stochastic or random. As an example consider the situation when the random signal process is a plane wave incident at an angle (00, ~b0) and has a power spectral density S(f) which is separable from the spatial variables, then we can replace the N matrix with
Soo(f) S~o(f)
Sos(f)S~(f)
l_L_ ~(O -
0o)~(4~ - ¢o)
(18)
sin00
where 1/sin 00 is a normalizing factor and
2
[Soo(f) -4- S**(f)] df = Roo(O) A- R**(O)
(19)
is the total intensity of the received signal, and the Roe and R ~ are ½ the real parts of elements in Eq. 6.
vow. ~82. ~o. 4, O ~ t o ~ ~
225
D. G. Childers TABLE I R
Type of Polarization
S
Randompolarized °[: i] [ 1 '] Right hand circular
½
Left hand circular
-i
1
i
1
-~
½
Odir io
i] [ 10~] -½i
1
½i
1J
1
[i i]
[: 00]
o
The normalized coherency matrix of Eq. 6 is often used to specify the polarization of the incident process; i.e.,
1
-Roo(O)
Re,(0)]
=R
Roo(O) + R~,(O)
(20)
R~o(O) R ~ , ( 0 ) J
is tabulated in Table I for various types of polarization. 2 (See (18, 3, 15)).
Monochromatic Signals W e n o w c o n s i d e r m o n o c h r o m a t i c s i g n a l s so t h a t S ( f )
= c ~ ( f - J'0), w h e r e
c is a constant proportional to power per cycle per sac. If we let [S-] represent the normalized power spectral density for monochromatic, plane wave signals, then we can express the type of polarization in terms of [-S]. This is also shown in Table I, where Eq. 19 is applied to determine the normalized total intensity. z
I: Jx
(a)
y
'~
(b)
×
(c)
= -s,°. -V
FIG. 3. Antenna height functions for three dipole arrangements. 2 The notation J is sometimes used instead of R to denote the normalized coherency matrix.
226
Journalof The FranklinInstitute
Antenna Reception of Nonisotropic Stochastic Fields T.~
II s
Crosscorrelation functions, Rt~(r, d), for monochromatic plane wave signals incident at an angle (00, ~o) for the dipole arrangement of Fig. 3(a). Type of Polarization
R~(~, d)
Random (unpolarized)
sin~Oo exp
Right hand circular
sin20o exp
Left hand circular
sin~Oo exp
Plane, 0 direction
2 sin20o exp
Plane, ¢ direction
0
00)]
[~,,0(~ cOO~
TXBLE I I I s
Crosscorrelation functions, R,~(r, d), for monochromatic plane wave signals incident at an angle (0o, ¢o) for the dipole arrangement of Fig. 3(b). Type of Polarization
RI~(r, d)
R a n d o m (unpolarized)
~o~0~o~o0+ ~o~oo~ex, [~,0 (~ ~ ~o~00)]
Right h a n d circular
Eco~2 00 sin 2 ¢0 + cos 2 ~,o] exp
Left h a n d circular
~co~00~in~o0+ ~o~o0~cup[~,o (~ ~ ~o~,0)]
Plane, 0 direction
[(~)]
27fifo ~-- c co~ 00
[(~)]
2 c o s ~00sin~boexp 27fifo
v - - c- cos
00
Plane, 4, direction
3 The received power is proportional to ½ the real part evaluated at variou~ r.
VoL ~ , So. 4, O c t o ~ , ~
227
D. G. Childers TABLE IV a
Crosscorrelation functions, R~( r, d) for monochromatic plane wave signals incident at an angle (0o, ~o) for the dipole arrangement of Fig. 3(c).
Type of Polarization
Rl~(r, d)
Random (unpolarized)
Right hand circular
- - [ s i n 80 cos O0 cos ~bo-~i }sinOosin~o]exp[27rifo(r--!
co80o)]
Left hand circular
--[-sin O0 cos 8o cos ~0 -- i ½sin Oo sin ~0J exp 2~rifo "r---c
cos .0)]
Plane, 0 direction Plane, ~bdirection
0
It is worth noting that Table I can be easily determined for monochromatic signals, but is not so easily determined for quasi-monochromatic signals or broadband noise. In fact, this is a research project under current investigation. We axe now in a position to determine the received signal correlation functions provided we specify the antenna height functions. Let hi and h2 be dipoles in three common arrangements, as shown in Fig. 3, where we assume the dipole bandwidths to be either infinite or at least constant over the bandwidth of the received signal. If we now substitute the results of Table I for ES] into Eq. 18 and in turn use Eq. 16, we m a y determine the crosscorrelation function for monochromatic, plane wave signals of various polarizations incident at an angle (00, ¢~) for the three dipole configurations shown in Fig. 3. The results of these calculations are given in Tables II, III, and IV. All of the cross correlation functions in Tables I I - I V are periodic with argument 2~rfoEr - (d/c) cos 8o3, where fo is the frequency of the plane wave, Oo is part of the incident angle, and d is the separation of the dipoles along the z-axis. One half of the sum of the right hand circular and left hand circular terms is equal to the unpolarized term as it should be. In Table II the crosscorrelation or received power is independent of the ~b component of polarization since the dipoles are not excited by this term. The maximum is achieved when r = (d/c) cos 00 and 80 = ~r/2 or the plane wave is parallel to the x-z plane. In Table I I I we examine and interpret the results for plane polarization. When a plane wave is incident at an angle (0o, ¢o) and we say it is polarized
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in the 0 direction we mean that there is only one electric field component and this component is the 0 component shown in Fig. 2. If we now imagine the plane in Fig. 2 moved so that ¢0 = ~r/2 and then shifted until 00 = 0, we obtain a plane wave propagating down the z-axis parallel to the x-y plane with the electric field vector aligned with the y-axis. This gives the maximum received power provided we delay the signal received at h~ by an amount d/c. Thus, ~he result is reasonable physically. A similar interpretation holds for the other crosscorrelation functions. In Table IV we can maximize the received signal for random polarization or plane polarization (0 direction) if 00 = 7r/4 and ¢o = 0 (if we neglect the minus sign which is simply a 180 ° phase shift). The maximization for right or left circular polarized waves is not as easily interpreted. The dipole configuration of Table IV is interesting since this configuration yields zero crosscorrelation for all r if the incident noise is isotropic and unpolarized (19). The correlation functions shown in Tables II-IV give the antenna directivity pattern for dipoles arranged in the configurations shown and for which their outputs are multiplied (one being delayed with respect to the other). By varying r we can achieve electronic beaming of the dipole configuration so as to maximize the output. This maximization by beaming is achieved, naturally, when we have r = d/c cos 0o or the directivity pattern is aligned in the direction of the incoming signal. Some related work in this area by Ko (24, 25), gives the noise temperature concepts employed in radio astronomy and elsewhere to the reception of partially polarized fields. This work is related to Ko's and is, in fact, strongly motivated by his results; however, many of the considerations are different and thus, have led to new results. Discussion
As mentioned earlier some examples for the noise case are given in (19) but they do not employ the antenna height function directly but use the spectral representation. The calculations are rather tedious and lengthy, but they illustrate the usefulness of the theory. The signal calculations of the previous section show how the theory can be used for monochromatic, polarized, plane W ayes.
We have shown how a spectral representation for a random spatio-temporal process can be used to determine the response of an antenna to this incident process. The spectral representation does not include the restriction to quasimonochromatic waves. The representation is valid for arbitrary bandwidth. The antenna height function is also assumed to have some bandwidth associated with it. The auto- and crosscorrelation functions between open circuit voltages
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D. G. Childers excited in different antennas by the same far field process are related to the coherency matrix of the incident process and the antenna height functions. It is also shown how the antenna can be viewed from the aspect of filtering theory. This combination of communication, antenna, and optical theory appears extremely useful in that we demonstrate how some of the ideas and analytical tools of noise, communication, and information theory can be brought to bear upon antenna theory. Much work remains for the future. Antenna gain functions and the radiation resistance for non-monochromatic waves must be defined. A possible definition of the gain function is (following Ko ( 1) ), G(f, 0, ~) =
(21)
f ~ f
J Ill 47r JJJ
dg df
However, the radiation resistance is not so easily defined for non-monochromatic input currents and frequency varying antenna height functions. Antenna directivity patterns are determined here for correlation arrays with certain noise reduction properties. However, the case of quasi-monochromatic signals must be worked out. Further work must be done to relate the coherency matrix of the incident process with the polarization detectability of the antenna. And nonisotropic (and possibly non-homogeneous) processes must be used to synthesize antenna correlation arrays. The list is long. References (1) H. C. Ko, "On the Reception of Quasi-Monochromatic, Partially Polarized Radio Waves," PIRE, Vol. 50, pp. 1950-1957, Sept. 1962. (2) M. J. Beran, and G. B. Parrent, "Theory of Partial Coherence," Englewood Cliffs, N. J. Prentice-Hall, Inc., 1964. (3) E. L. O'Neill, "Introduction to Statistical Optics," Reading, Mass., Addison-Wesley Pub. Co., 1963. (4) D. G. Childers, "A Covariance (Coherency) Matrix for Back Scattered and Spatial Electromagnetic Noise," Canadian Jour. Phys., Vol. 43, pp. 1099-1114, June, 1965. (5) R. Barakat, "Stochastic Generalization of the Green-Wolf Complex Scalar Potential of Electromagnetic Theory," J. Opt. Soc. Am., Vol. 53, pp. 252-255, Feb. 1963. (6) R. Barakat, "Theory of the Coherency Matrix for Light of Arbitrary Spectral Bandwidth," J. Opt. Soc. Am., Vol. 53, pp. 317-323, Mar. 1963. (7) J. L. Doob, "Stochastic Processes," New York, John Wiley and Sons, Inc., 1953. (8) H. Cramer, "A Contribution to the Theory of Stochastic Processes," Proc. Sea Berkeley Syrup. on Mathematical Statistics and Probability, Univ. of Calif. Press, 1951. (9) A. M. Yaglom, "An Introduction to the Theory of Stationary Random Functions," Englewood Cliffs, N. J., Prentice-Hall, Inc., 1962. (10) A. M. Yaglom, "Some Classes of Random Fields in n-dimensional Space, Related to Stationary Random Processes with a Rational Spectrum," Theory of Probability and Its Applications, Vol. 2, pp. 273-320, 1957.
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(11) S. Stein, and D. E. Johansen, " A Theory of Antenna Performance in Scatter-Type Reception," I R E Trans. Antennas and Propagation, Vol. AP-9, No. 3, pp. 304-311, May 1961. (12) R. E. Burgess, "Aerial Characteristics," Wireless Eng., Vol. 21, pp. 145-160, Apr. 1944. (13) G. Sinclair, "The Transmission and Reception of Elliptically Polarized Waves," P I R E , Vol. 38, pp. 148-151, Feb. 1950. (14) C. T. Tai, "On the Definition of the Effective Aperture of Antennas," I E E E Trans. on Antennas and Propagation, Vol. AP-9, pp. 224-225, March 1961. (15) H. C. Ko, "The Use of the Statistical Matrix and the Stokes Vector in Formulating the Effective Aperture of Antennas," I E E E Trans. on Antennas and Propagation, Vol. AP-11, No. 6, pp. 581-582, Nov. 1961. (16) H. C. Ko, "Microwave Scanning Antennas," Vol. 1 : Apertures, Edited by R. C. Hansen, New York, Academic Press, pp. 263-337, 1964. (17) J. R. Copeland, "Radar Target Classification by Polarization Properties," P I R E , Vol. 4~% pp. 1290-1296, July 1960. (18) M. Born and E. Wolf, "Principles of Optics," New York, Pergamon Press, 1959. (19) D. G. Childers, and I. S. Reed, "Space-Time Cross-Correlation Functions for Antenna Array Elements in a Noise Field," I E E E Trans. on Infor. Theory, Vol. IT-11, No. 2, pp. 182-190, Apr., 1965. (20) M. H. Cohen, "Radio Astronomy Polarization Measurements," P I R E , Vol. 46, pp. 172183, Jan. 1958. (21) R. H. MacPhie, "On the Mapping by a Cross-Correlation Antenna System of Partially Coherent Radio Sources," I E E E Trans. on Antennas and Propagation, ¥ol. AP-12, No. 1, pp. 118-124, Jan., 1964. (22) W. G. Jaeckle, "Space-Frequency Equivalence for Correlation Arrays and Linear Additive Arrays," URSI, 1964. (23) C. J. Drane and G. B. Parrent, "On the Mapping of Extended Sources with Nonlinear Correlation Antennas," I E E E Trans. on Antennas and Propagation, Vol. AP-10, No. 2, pp. 126-131, Mar. 1962. (24) H. C. Ko, "Antenna Temperature and the Temperature of Electromagnetic Radiation," I E E E Trans. on Antennas and Propagations, Vol. AP-12, No. 1, pp. 126-127, Jan., 1964. (25) H. C. Ko., "Radio-Telescope Antenna Parameters," I E E E Trans. on Military Electronics, Vol. MIL-8, Nos. 3 and 4, pp. 225-232, July-Oct. 1964.
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