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[4] A. A. Kishk, “An elliptic dielectric resonator antenna designed for circular polarization with single feed,” Microw. Opt. Technol. Lett., vol. 37, no. 6, pp. 454–456, Jun. 2003. [5] R. K. Mongia, A. Ittipiboon, M. Cuhaci, and D. Roscoe, “Circularly polarized dielectric resonator antenna,” Electron. Lett., vol. 30, no. 17, pp. 1361–1362, Aug. 1994. [6] M. T. K. Tam and R. D. Murch, “Circularly polarized circular sector dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 48, no. 1, pp. 126–128, Jan. 2000. [7] M. Haneishi and H. Takazawa, “Broadband circularly polarized planar array composed of a pair of dielectric resonator antennas,” Electron. Lett., vol. 21, no. 10, pp. 437–438, May 1985. [8] K. K. Pang, H. Y. Lo, K. W. Leung, K. M. Luk, and E. K. N. Yung, “Circularly polarized dielectric resonator antenna subarrays,” Microw. Opt. Technol. Lett., vol. 27, no. 6, pp. 377–379, Dec. 2000. [9] A. A. Kishk, “Performance of planar four-element array of single-fed circularly polarized dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 38, no. 5, pp. 381–384, Sep. 2003. [10] R. Chair, A. A. Kishk, K. F. Lee, and C. E. Smith, “Wideband flipped staired pyramid dielectric resonator antennas,” Electron. Lett., vol. 40, no. 10, pp. 581–582, May 2004. [11] Ansoft Corp., Pittsburgh, PA, Ansoft HFSS 9.2. [12] C. T. Rodenbeck and K. Chang, “Automated pattern measurement for circularly-polarized antennas using the phase-amplitude method,” Microw. J., vol. 47, no. 7, pp. 68–78, Jul. 2004.
Response of Retrodirective Array in the Presence of Multiple Spatially Separated Sources
to date. During the development of retrodirective antenna systems it was noted without theoretical proof that a retrodirective array placed in the presence of multiple transmitters would share its return signal in equal proportion to the power transmitted from each spatially diverse transmitter, [3], this was shown experimentally in, [4]. In this paper a simple generic proof that this is the case and importantly that no other possibility exists is given. II. THEORY OF RETRODIRECTIVE ARRAY RESPONSE As was shown in [5] it is possible to cast the retrodirective array scenario as a multiport S-parameter problem, and in this way systematically include the effects of multiple coupling paths between elements in the system. In [5] this method was combined with the use of the Schwarz inequality to prove that the signal returned to a source, even in the presence of mutual coupling between elements, will be at its maximum only if the original signal has its phase conjugated. The proof was restricted to the case of a single transmitter and no consideration of multiple transmitters each producing unequal power levels, nor their effect on re-radiated power levels was given. We will now extend the previous analysis in order to incorporate these additional complexities and show that no straightforward possibility exists for additional power level control in regard to the signals re-transmitted by the retrodirective array back to the multiple transmitters exciting the system. Fig. 1 shows a generic representation of the multi-transmitter retrodirective array situation. This system can be represented by means of the multiport scattering parameter matrix S:
S00 S01 S02 S10 S11 S12 S20 S21 S22
Vincent F. Fusco Abstract—In this paper, we describe a canonical proof based on the use of multiport scattering parameters which shows that in the presence of multiple coupling paths between elements that is not possible to individually control the power returned to each transmitter in a retrodirective array which is responding to stimulation by multiple transmitters. It is proven that it is not possible to maximize retransmission to only one of the transmission elements at the expense of the others if more than one transmitter is broadcasting, nor is it possible to simultaneously equalize the signal sent to all broadcasting antennas when they each transmit unequal power levels. Index Terms—Multiple sources, retro-directive array, self-phasing array.
I. INTRODUCTION Retrodirective or self phasing antenna arrays can provide beam steering action automatically through the process of local phase conjugation of a spatially sampled incident wavefront, [1]. A retrodirective array continuously adapts its phase response in order to track a remote signal source without a-priori knowledge of its position. In so doing this type of array can compensate for wavefront aberrations caused by inhomogeneous effects in the propagation path and array misalignment. The means for producing local phase conjugation of the sampled incoming wavefront as suggested by Pon [2], i.e., by heterodyne mixing of the incoming wavefont sample at radian frequency ! , with a local oscillator running at 2! , still remains the principal approach used Manuscript received August 30, 2005. This work was supported by the N. Ireland Department of Education and Learning SPUR2 SoCaM programme. The author is with the Institute of Electronics Communications and IT, Queens University of Belfast, Queens Island, Belfast BT3 9DT, N. Ireland (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAP.2006.872652
S0N S1N S2N
111111 111111 111111
.. .
.. .
S=
Sk0
111
SN 0
111
111
(1)
SkN
111
.. .
.. .
111
SNN
111
Consider the right-hand side of Fig. 1, the retro-directive array section. Using the notation in [5] let the transmitted signal for this portion be represented as c = (0; 0; 0; 1 1 1 0; ck+1 ; . . . ; cN ):
Note that c0 . . . ck represent the k transmitters in the system, since these do not take part in retro-directive re-transmission, their value is set to zero. Thus, the total power re-transmitted by the retro-directive portion is
N j
Assume now that Soo . . . Skk = 0. Thus
C i j2 :
(2)
i=k+1 transmitters co . . . . . . ck
[0; 0; . . . ; 0; ck+1 ; . . . ; cN ]
are all matched, thus
0 S01 S02 S10 0 S12 S20 S21 0
111 111
111
S0N S1N S2N : .. .
.. .
.. .
SN 0
111
111
111
SNN
Hence, the power received at the transmitting terminals is equal to
N
i=k+1
0018-926X/$20.00 © 2006 IEEE
Ci Si0
2
N
+
i=k+1
Ci S11
2
N
+ 111
i=k+1
2
Ci Sik :
(3)
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006
Fig. 1.
Generic multitransmitter retrodirective array.
Now by taking each term in (3) and applying the Schwarz inequality in the form given by (4)
1
01
Ci Sij3
2
1
jC i j 2
1
jSij j2
(4)
01 01 3 , and j is an arbitrary constant, Here, equality occurs when ci = j Sij we establish from (3) and (4) that
Total power sent from RDA +
Total power returned to T=X1
+
Total power returned to T=Xk
Total power sent from RDA
Total power sent from RDA
=
+
N i=k+1 Ci Si0 + N i=k+1 jCi j2 2 N i=k+1 Ci Sik : N i=k+1 jCi j2
+
N i=k+1 Ci Si1 N i=k+1 jCi j2
bk =
2 +
3 i k + 1: ci = 0 Si30 = 1 Si31 = 1 1 1 = k Sik
iv)
v)
a = [ 0 ; 1 ; . . . ; k ; 0; 0; 0; . . . ; 0]:
b0 =
111
111
bl = (5)
where T=Xk is the kth transmitting source. If a source mismatch is present its effect is to reduce the amount of available transmit power within the system, and all other aspects of functionality remain unaffected. Two observations follow from application of Schwarz’s inequality. j S i0 j 2 ; i) Each term in (5) is less than jSi1 j2 ; . . . ; jSik j2 , respectively. ii) Each term in (5) takes on a maximum value when iii)
antennas, i.e., make them all have the same receive signal amplitude, since we cannot discriminate on which path energy is arriving and weight it accordingly. This situation arises because these paths cannot be individually processed as signals collected from these paths are collectively gathered at the receiver terminals of the retro-directive array, see 3 it (8)–(11). In general since Si30 6= Si31 6= . . . . . . 6= Sik would therefore be necessary in order to keep ci the same that 0 6= 1 6= 1 1 1 1 1 1 6= k . Thus the signal returned to each transmitter cannot be optimally equalised but instead the signal strength returned by each retro-directive element will vary according to the sum of the individual signals arriving at the RDA terminals prior to retransmission. Therefore signals are returned in proportion to the energy transmitted from each source, once again this consistent with experimental observation, e.g., [4]. If we let the k transmitters be matched, i.e., S00 = S11 = . . . . . . . . . = Skk = 0, and since b = a S and in addition we let each transmitter send a different power level then we can write (7)
The output signals at terminals of each of the respective transmitters in the system can now be written as
Total power returned to T=X0
2
vi)
1353
(6)
As stated in [5] for the case of a single transmitter since transmission/retransmission paths are reciprocal then Sik = Ski 3 = Ski 3 . Thus the signal retransmitted and consequently Sik from the (k + 1)th to N th terminals should be the conjugate of the corresponding received signal received by each retro-directive element in order to maximize the signal returned to each of the transmitter source terminals. This is in accordance with all previously reported observations on this class of array, e.g., [1] and , [3]. From (6) we can establish the condition for the maximum signal return to each of the k transmitters in a generalized retro-directive array. We can see that it is not possible to maximize retransmission to only the j th transmission element if more than one transmitter is broadcasting, since all S parameter combinations relevant to the number of transmitters turned on will exist. Similarly it does not appear possible to systematically equalize the signal sent to all k simultaneously broadcasting
k
j =1 k01 j =0 l01 j =0
j S j 0
(8)
j Sjk
(9)
j Sjl +
k
j Sjp ; j =l+1
1
l k 0 1:
(10)
Equations (8) and (10) represent the retro-directive power received at each transmit element terminal including the effect of transmitter to transmitter and retransmit to retransmit element coupling. For the re-transmit (RDA) elements (11) holds
bm =
k
j =0
j Sjm ; k + 1 kn N:
(11)
Therefore the signals presented at the output terminals of each of the elements in the retro-directive system are defined. The effect of mutual coupling between ports is at both transmitter and retransmit portions of the array to modify the absolute magnitude and phase of the transmitted and subsequently the retrodirected signals, with the effect being most profound for nearest neighbor coupling. However for rectangular patch antennas at 0.5 spacing E-plane coupling is 022 dB while H-plane coupling is 034 dB, [6]. Consequently the assertion of significant retransmitted signal angular and/or power aberration by this means in a real retrodirective array is unlikely. III. CONCLUSION In this paper, we have shown that the presence of mutual coupling in an unequal power multiple transmitter retrodirective situation that signals will be returned to each transmitter pro-rata the level at which each transmitter stimulates the retrodirective array. In addition it was shown that it is not possible to maximize retransmission to only one transmission element if more than one transmitter is broadcasting, nor is to systematically return equal power signals to all k simultaneously broadcasting antennas since individual retrodirective array elements cannot discriminate along which individual path energy has arrived.
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The observations in this paper allow formal constraints to be placed on the functionality that can be realistically obtained from a retrodirective array which is expected to operate in the presence of multiple signal excitation. REFERENCES [1] V. F. Fusco and S. Karode, “Self-phasing antenna array techniques for mobile communications applications,” Electron. Commun. Eng. J., pp. 279–286, Dec. 1999. [2] C. Y. Pon, “Retrodirective array using heterodyne technique,” IEEE Trans. Antennas Propag., vol. AP-12, no. 1, pp. 176–180, Jan. 1964. [3] R. C. Chernoff, “Large retrodirective arrays for space applications,” IEEE Trans. Antennas Propag., vol. 27, no. 4, pp. 489–496, Jul. 1979. [4] S. Karode and V. F. Fusco, “Multiple target tracking using retrodirective antenna arrays,” in Proc. Inst. Elect. Eng. Antennas and Propagation Conf., York, U.K., Mar. 1999, pp. 178–181. [5] A. F. Kay, “Comments on “self-phasing array antennas” and electronically adaptive antenna systems,” IEEE Trans. Antennas Propag., vol. AP-12, no. 6, pp. 792–793, Nov. 1964. [6] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-30, no. 6, pp. 1191–1196, Nov. 1982.
subscriber by steering the main beam [4]. These algorithms will be called “signal tracking” in the following. Clearly, these systems do not filter out the interference signals, so decreasing the system performance compared to optimal beamforming. In this paper, a novel approach is proposed that is “half-way” between optimal beamforming and signal tracking beamforming. The basic idea is to estimate the spatial distribution of the interference signals using classical spectral estimation, and the direction of arrival (DOA) of the desired signal by means of a noise subspace method, taking advantage of the code division multiple access (CDMA) processing gain. The weights of the array are then calculated according to the estimated DOA of the desired signal and of the estimated spatial distribution of the interference signals. It will be shown that the proposed approach reaches better performance compared to tracking beamforming, and avoids some problems affecting the optimal beamforming. II. THE PROPOSED BEAMFORMING AND COMPARISON TO OPTIMAL BEAMFORMING AND TRACKING BEAMFORMING A. The Signal Model
A Beamforming Algorithm for Adaptive Antennas Operating in Crowded CDMA Signal Environment Marco Donald Migliore Abstract—A novel beamforming algorithm for adaptive antennas operating in a crowded signal code division multiple access environment is presented. The novelty of the algorithm consists in the use of both classical and modern spectral estimation algorithms. In particular, the estimation of the direction of arrival of the desired signal is performed using the signal covariance matrix, while the distribution of the interference signals is obtained by means of the spectrogram. This approach permits satisfactory performances even when the number of signals is much larger than the number of elements of the array. Index Terms—Adaptive antennas, code division multiple access (CDMA) systems, wireless communication.
I. INTRODUCTON Adaptive antennas are, at least theoretically, able to dramatically increase the capacity of wireless communication systems [1]. In fact, the use of optimal beamforming techniques [2], [3] enables us to maximize the signal of the desired subscriber and to filter out the signals of the undesired subscribers by introducing nulls (or minima) of the antenna pattern in the directions of the interference signals. However, in real scenarios the optimal beamforming suffers from some drawbacks. In particular, the number of signals must be smaller than the number of elements of the adaptive antenna [2], otherwise the performance of the antenna is not guaranteed since the antenna might filter out the desired signal. This restricts the practical application of optimal beamforming in wireless communication systems. In order to overcome this problem it is possible to adopt beamforming algorithms which maximize the signal of the desired Manuscript received April 8, 2005; revised November 10, 2005. The author is with the DAEIMI, Università di Cassino, 03043 Cassino, Italy (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAP.2006.872645
As a preliminary step, let us introduce the scenario wherein the adaptive antenna operates. We consider a CDMA wireless network with K subscribers. The base station is equipped with a uniform linear array (ULA) of M equispaced identical elements (Fig. 1). The array receives the signals from the K subscribers located in the far field zone of the array. We suppose all the signals to be uncorrelated. The received signal at the ith element of the array is
()=
xi t
K
k=1
() ( )
sk t f uk ej (
(t)+(i01)8(u )) + n (t) i
(1)
wherein sk is the amplitude of the k th signal, k is the phase of the k th ; signal, uk is the DOA of the k th signal in the u-space (u being the angle drawn in Fig. 1), f u is the voltage at the output of the element when a plane wave having unit amplitude and zero phase u =; is the impinges on the element from direction u; u is the interelement distance, and ni t is the noise at wavelength, the ith element of the array. The noise is assumed to be a zero-mean stationary random process with variance 2 , statistically independent from sensor to sensor and independent from the signals. In the text we will suppose that u1 is the DOA of the desired signal. The received signals can be written in vector form as
= cos
()
8( ) = 2 1 ()
1
x(t) = As(t) + n(t) wherein x(t) = [x1 (t); . . . ; xM (t)]T s(t) = s1(t)ej (t) ; . . . ; sK (t)ej (t) T n(t) = [n1(t); . . . ; nM (t)]T A = [a(u1 ); . . . ; a(uK )] a(u) = f (u) 1; ej8(u) ; . . . ; ej(M 01)8(u) T
(2)
is the steering vector, and the superscript T denotes transposition. Fi+ t , wherein is the nally, the output of the antenna is y t vector of the array weights and the superscript denotes the transpose conjugate. For further discussion it is useful to recall that, given the observation vector , the covariance matrix is defined as [2]
( ) = w x( ) +
x
R = E fx(t)x+(t)g
wherein E is the expectation operator.
0018-926X/$20.00 © 2006 IEEE
w
(3)
