6 and performing the same essential functions necessary for a directed RF link. .... each other at time t = 0, we list the received, IF, and trans- mitted frequencies ...
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. XX, NO. Y, MONTH 1999
100
Frequency Stability in Adaptive Retrodirective Arrays
Keywords | Phase Conjugating, phase ltering, frequency stability, adaptive beam forming.
φRETURN
R ef er en ce
Abstract | The frequency stability of communications or remote sensing systems using phase conjugation as a means of retrodirective signaling is developed under the in uence of constant rectilinear motion. The results indicate that for dynamic mobile platforms such as aircraft, helicopters, and ground vehicles the retrodirective properties of phase conjugating arrays is not guaranteed due to a possible instability of the frequency. It is shown that the frequency transmitted by each platform has a dependence on the open loop gains of the transceivers on each platform in the link. Fast and accurate reconstruction of the signal's phase requires a high open loop gain, however, too high a gain will cause instability and eventual loss of lock. Design tradeo s are discussed.
Pl an e
Leo D. DiDomenico and Gabriel M. Rebeiz
φREF Ln
φINPUT
φOUTPUT ∆φ ∆φ
∆φ ∆φ
Phase Conjugation Cell
Fig. 1: Each phase processor modi es its input RF phase that the total return signal will retro-re ect and be N recent years many subsystems have being designed such sent back to its source. Iwith directed RF beam forming capabilities for both military and civilian applications, including: remote sensing, {f − f , φ − φ } {f , φ } secondary surveillance radar, and communications. In addition, these systems are being located on mobile platforms that are fast movers such as land vehicles, aircraft, spacecraft, and permanent satellites. These designs use phase shifters in order to physically con ne the antenna's beam so as to subtend a narrow angle and also perform tracking. However, another technique exists. Developed in 1955 [1]6 GHz 6 GHz [6] and performing the same essential functions necessary for a directed RF link. It is called phase conjugation and has been used by many in the optics eld for adaptive re12 GHz construction of highly distorted optical signals [7]. Phase conjugation at microwave and millimeter wave frequencies {f , φ } may be implemented using single-side-band mixers. In order to use phase conjugation in highly dynamic mobile systems it is necessary to understand the frequency stability Fig. 2: An example of a phase conjugation cell that transof electromagnetically coupled systems in relative motion. forms the input RF energy to its phase conjugate. This is the subject of this paper. The idea is based on the arrangement shown in Fig. 1. an input RF signal is the requirement for constructive adHere an unmodulated electromagnetic wave is traveling to- dition of a signal back at the reference plane. In addition, ward an antenna array. The distance to the nth array el- so long as this phase conjugation is done to within the ement is Ln and the wavelength is . The phase at the same additive constant in phase at each of the antenna reference plane is Ref = 0. It is easy to see that the round elements, then retro-directivity will be achieved. A consetrip phase or return phase is Return = 2(2Ln =) + . quence that follows is that phase modulating all the array If we wish to have constructive interference at the refer- elements at the same time by a coding angle of =2 raence plane then we may set the round trip phase equal dians will not impact the retro-directivity property of the to zero. Therefore, the phase at the output of a pro- array. Of course, other modulation types are also possible. cessing cell is the negative of the input phase, that is, Output = , Input = ,2Ln=. Hence, conjugation of II. Generating a Conjugated Signal Various methods for producing phase conjugation and Both authors are with the Radiation Laboratory, EECS Dept., Uni- full duplexed communications are possible . Fig. 2 shows a versity of Michigan at Ann Arbor. Mr. DiDomenico is also with the US Army Research Labs. simple phase conjugation cell [8]. The cell consists of an anI. Introduction
L
RF
RF
RF
LO
LO
LO
RF
{f RF 2 ,φ LO1 + φ LO 2 − φ RF 2 }
{f RF1,φRF1}
(0 , fTA0)
(t0 , fRB0)
(t0 , fRA0)
(0 , fTB0)
A
B
(t0 , fTA1)
{f LO1 , φ LO1 }
{f LO 2 ,φ LO 2 }
A
B (t1 , fRA1)
{f IF ,φ LO1 − φ RF1 }
PLL
(t1 , fRB1)
{f IF , φ LO1 − φ RF 2 }
(t0 , fTB1)
(t1 , fTA2)
(t2 , fRB2)
(t2 , fRA2)
(t1 , fTB2)
A
Base Band Message Phase
Fig. 3: A phase conjugation cell that uses an IF signal.
B
(t2 , fTA3)
(t3 , fRB4)
(t2 , fRA4)
(t2 , fTB3)
B
A
tenna for transmitting and receiving a signal, and a mixer driven from an LO at twice the RF frequency. Note that the mixer's output frequency (typically called IF) is therefore the same as its input frequency and that the mixer takes the di erence of the phases of the RF and LO signals. Unfortunately, this method does not allow for easy full duplex communications since it is hard to process the phase at such high IF frequencies. As a result only retransmission of the original signal is practical, making this approach useful for transponders but not for full-duplex communications. Fig. 3 shows a phase conjugation cell that uses an intermediate frequency. A BPSK modulated signal is received at a frequency and phase of ffRF 1 ; RF 1 g and down converted to an IF via the rst mixer with a local oscillator at frequency fLO1 . The received phase will change as function of time due to motion of the platforms communicating and the BPSK signaling. Hence, the signal's phase has two components to its power spectral density (PSD): messagephase and geometry-phase. The message-phase is associated with the digital BPSK type signal and the geometryphase is associated with the platform dynamics. However, it is possible through data encoding to separate the PSD of the message-phase and geometry-phase in the frequency domain. The geometry-phase can then be used to create a regenerated and conjugated IF signal. This regenerated IF signal is then remodulated with a new message during the up conversion process by superimposing the message on LO2 . The use of a low IF frequency increases the system complexity but allows access to the phase at a frequency where phase ltering may be performed. A digital Costas or squaring type phase-lock-loop (PLL) can be used to do the phase ltering and ensure that a regenerated carrier with zero phase error can be obtained. An especially attractive ltering method is based digital PLL technology and is called a serial digital phase lter (SDPF) [9]-[12]. The SDPF is a type of PLL that does not require a voltage controlled oscillator and performs the phase-lock function with only logical operations, as opposed to oating point calculations. Hence, the SDPF can operate real time on the conjugated and down-converted signal. The details of
U=V/2
U=V/2
Fig. 4: Platforms A and B are shown moving towards each other in a series of time snap-shots from the point of view of an observer in the center of mass frame of reference. The arrows indicate the direction of signal ow. phase ltering will not be given here, however, we note that the SDPF can be easily extended to any higher order PLL for improved signal tracking. Also, the step response in frequency of the SDPF can be made to approximate an analog PLL response with some phase noise [12]. Hence, a linear model may be used for analysis. III. The Physics of Frequency Stability
Consider two phase conjugation cells which are on two separate and moving platforms: A and B. Let A move relative to B with a constant velocity that is directed along the center line of the phase conjugation cells (see the time sequence of Fig. 4). Each phase conjugation cell is composed of two mixers and a PLL. Further assume that A and B may be con gured for approximate phase conjugation operation, i.e. o set frequencies for transmit or receive are possible. The block diagram of the phase conjugation cells are shown fully in Fig. 5. The goal is to analyze the transmission frequency from A to B and vice versa. Initially let us assume that A and B are at rest relative to each other. As a consequence the frequencies in the combined system are stable. The RF transmit and receive frequencies are therefore a constant, fa0 for A and fb0 for B. Next, assume that A and B continue transmitting, but also abruptly start moving at a velocity v=2 towards each other. Hence, the total velocity of each platform is v relative to the other. Therefore, A's signal is received by B at a di erent frequency fRB0 , the di erence being due to the velocity-induced doppler frequency. This is a step function. In addition, since all motion is relative, B's signal is received by A at a di erent frequency fRA0 , the di erence again being due to the velocity induced doppler frequency.
101
This is also a step function. The receivers on each platf f LOB1 LOA2 form mix the received signal down to the PLL's IF input frequencies fIFA1 or fIFB1 by using local oscillators characterized by frequency fLOA1 and fLOB1 for platforms A fTAn fRBn and B, respectively. It is assumed that fLOA1 > fb0 and fLOB1 > fa0 so that phase conjugation can take place at f f IFB1 both platforms. The PLL's on each platform now receive IFA2 a signal with a step change in frequency and respond with PLL PLL a changing output frequency centered at the PLL's output IF frequencies of fIFA2 and fIFB2 . Hence, in general the f f IFB2 fTBn output response of the PLL will be a signal with a time fRAn IFA1 varying frequency. This time varying frequency is up converted to RF by mixing with local oscillators characterized by frequencies fLOA2 and fLOB2 where fLOA2 fIFA2 and fLOB2 fIFB2 . This entire procedure is repeated again as the signal is received by the complimentary platf f LOB2 LOA1 form, with another velocity-induced doppler frequency being added. The signals bounce back and forth between A and B generating a sequence of frequency responses corre- Fig. 5: Diagram of signal ow in a microwave phase conjusponding to the time steps of each path traversal. Let a gation link. This simple model assumes no sources of phase subscript of T stand for transmit, a subscript of R stand noise. for receive, and a subscript of d stand for doppler. If the platforms are at rest relative to each other and have been fTA1 (t) = fLOA2 + fLOA1 , fb0 1 + v wA (t , 0 ) c at rest for a long time, then we have: = fsysA , fb0 A (t , 0 ) (9) fLOB2 + (fLOB1 , fa0 ) = fb0 (1) v w (t , ) f ( t ) = f + f , f 1 + TB 1 LOB 2 LOB 1 a 0 0 and c B fLOA2 + (fLOA1 , fb0 ) = fa0 : (2) = fsysB , fa0B (t , 0 ) (10) It is therefore possible to de ne the \system frequencies", Where k is given by v wk (t , n ) n t n+1 fsysA = fLOA1 + fLOA2 (3) c k (t , n ) = 11 + + vc wk ( n+1 , n ) t n+1 (11) and fsysB = fLOB1 + fLOB2 (4) and k = A; B . Note that this factor accounts for the nite bandwidth of the closed loop response of the PLL. The where fsysA = fsysB = (fa0 + fb0 ) under ideal zero velocity PLL's time response to a step in frequency is w(t , ). conditions. Let us also de ne w(t , n ) as the normalized This function is zero for all time t < and is a time varying time response of the PLL to a step change in the input function after until the next input is received at the next frequency at time n . Note that n is the time it takes to path traversal of the signal. Then a new step response is traverse path number n. Both w(t) and n will be deter- manifest due to the next doppler-induced frequency-step. mined latter. After the platforms start moving relative to Continuing, each other at time t = 0, we list the received, IF, and transt = 1 mitted frequencies explicitly in order to arrive at a general form of the governing equations by induction. fRA1 (t) = fTB1 (t) + fdB1 = fTB1 (t) 1 + vc (12) t=0 fTA0 (t) = fa0 (5) fRB1 (t) = fTA1 (t) + fdA1 = fTA1 (t) 1 + vc (13) fTB0 (t) = fb0 (6) v w (t , ) f ( t ) = f , f ( t ) 1 + TA2 sysA TB 1 1 t = 0 c A = fsysA , fsysB A (t , 1 ) fRA0 (t) = fTB0 (t) + fdB0 +fa0B ( 1 , 0 )A (t , 1 ) (14) v = fb0 1 + c (7)
fRB0 (t) = fTB0 (t) + fdA0 = fa0 1 + vc
(8) 102
fTB2 (t) = fsysB , fTA1 (t) 1 + vc wB (t , 1 ) = fsysB , fsysA B (t , 1 ) +fb0 A ( 1 , 0 )B (t , 1 ) (15)
We may rewrite these equations but with platform labels instead of by time. Platform A
fTA0 (t) = fa0 fTA1 (t) = fsysA ,fb0 A (t , 0 ) fTA2 (t) = fsysA ,fsysB A (t , 1 ) +fa0B ( 1 , 0 )A (t , 1 ) fTA3 = fsysA ,fsysA B (t , 2 ) +fsysB A ( 2 , 1 )B (t , 2 ) ,fb0 B ( 1 , 0 )A ( 2 , 1 )B (t , 2 ) .. .
Fig. 6: Diagram of the relative positions of the platforms A and B as a function of bounce number. be the total distance between the platforms at time t = 0. Then path number zero will have a length of R0 = R,R0 . But R0 = ut0 and t0 = R0 =c. Therefore,
(16)
Platform B
c
(24)
For the next path traversal we have that R1 = R0 , R1 , where R1 = ut1 and t1 = (R0 + R1 )=c. Where u = v=2. Hence,
fTB0 (t) = fb0 fTB1 (t) = fsysB ,fa0B (t , 0 ) fTB2 (t) = fsysB ,fsysAB (t , 1 ) +fb0A ( 1 , 0 )B (t , 1 ) fTB3 (t) = fsysB ,fsysB A (t , 2 ) +fsysAB ( 2 , 1 )A (t , 2 ) ,fa0A ( 1 , 0 )B ( 2 , 1 )A (t , 2 ) .. .
R0 = 1 +R u
R1 = 1 +R v 2c
1 , 2vc 1 + 2vc
(25)
1 , 2vc n 1 + 2vc :
(26)
and in general we may write for n 0,
Rn = 1 +R v 2c
In addition, the total travel time n is then given by n X
(17)
n = 1c Rk k"=0 # 1 , 2vc n+1 R = v 1, 1+ v 2c
From this we see that the transmit frequencies of platform (27) A depend on the system response of platform B and vice versa. In addition, by multiplying by the appropriate factor of and adding we may derive a set of coupled time and the time di erence n+1 , n is dependent, constrained di erence equations. The govern 1 , 2vc n+1 1 R ing set of equations is therefore: (28) n = c 1 + v 1 + 2vc 2c fTA(n+1) (t) + fTB(n) (t)A (t , n ) = fsysA (18) fTB(n+1) (t) + fTA(n) (t)B (t , n ) = fsysB (19) where it has been assumed that the delay time in processing the signal in each platform's PLL is small compared to the fLOA1 + fLOA2 = fsysA (20) travel time associated with each path, so that the PLL fLOB1 + fLOB2 = fsysB (21) processing is assumed to be instantaneous. fTA(0) = fa0 (22) IV. Infinite Bandwidth PLL Response fTB(0) = fb0 (23) If we assume that both platforms are identical except In equations 11, and 18-19 the round trip starts at n and for the system frequencies, initial transmit frequencies, and nishes with the next step change in frequency due to the that the PLL's on each platform have in nite bandwidth return trip doppler at n+1 . These are functional equations then that can be made into nite di erence equations by only = A (t , n ) = B (t , n ) = = 1 + vc (29) looking at a particular time instance. This will make it easier to establish stability properties. Finally, an expression for n is required that takes into so that account the motion of the two platforms, Fig. 6. Let R w(t , n ) = 1: (30) 103
Thus is a constant that depends only on the relative velocity of platforms A and B, and is not a function of time. Equations 18-19 become constant coecient coupled di erence equations. Writing the time step explicitly we have: fTA (n + 1) + fTB (n) = fsysA (31) fTB (n + 1) + fTA(n) = fsysB (32) By taking the one sided Z-transform to account for the initial conditions in equations 22-23 we also obtain the equivalent form in the z-domain. ,1
z FTA (z )z ,1 + FTB (z ) = f1sysB (33) + z ,1 + fTB (0) z ,1 + f (0) FTB (z )z ,1 + FTA (z ) = f1sysA (34) TA + z ,1 Where FTA (z ) and FTB (z ) are the Z-transforms of fTA (n) and fTB (n) respectively. We note that there are two equations with two unknowns, the functions FTA (z ) and FTB (z ). Solving for FTA (z ):
Fig. 7: Pole plot of the step frequency response over the z-plane for a system of phase conjugation cells.
than zero and > 1. This results in two of the poles of the step response being outside the unit circle. These poles are unstable and cause the transmit frequency to diverge from the value of the rst term. When the platforms are moving away from each other then the relative velocity is less than zero and < 1. The poles are then inside the unit circle so that unconditional stability is achieved. f (0) FTA (z ) = (z , 1)(zTA, )(z + ) z 3 Note that it is possible to avoid any divergence issues by forcing the coecients of n and (,)n in equation 37 to f , f (0) , f (0) sysA TB TA 2 be identically zero. This results in two equations in two z + (z , 1)(z , )(z + ) unknowns, the system frequencies fsysA and fsysB . The f (0) , f TB sysB + (z , 1)(z , )(z + ) z (35) result is fsysA = fa0 + fb0 (38) which may be inverted for all n 0 by contour integration f = f + f (39) sysB b0 a0 along a path C that encloses all the poles in Fig. 7. Hence, I and makes physical sense when viewed in light of Fig. 5. 1 fTA (n) = 2j FTA (z )z n,1dz (36) The method of changing the system frequencies at each C time step to force convergence would not be easy since The result is knowledge of the other platform's velocity and initial transmit frequency would have to somehow be determined. fTA (n) = Even thought the above results were derived for an in fsysA , fsysB nite bandwidth PLL, it is interesting to see just how many (1+ )(1 , ) path traversals would have to be performed before there are ( f + f )(1 + ) , ( f + f ) any observable problems with the frequency stability. Conb 0 a 0 sysA sysB n + (,) sider a situation where the two platforms approach each 2(1 + ) other at a relative velocity of v = 25 , 100 m/s. In the case ( f , f )(1 , ) + ( f , f ) , b0 a0 2(1 , ) sysA sysB n (37) of in nite open-loop gain the time response as a function of bounce number is independent of range, although the A similar equation also exists for the transmitted frequen- time for a given number of bounces will vary with range. cies of platform B and may be written by exchanging all This will be shown not to be the case for nite gain openreferences for platform A with the corresponding references loop responses. It is further assumed that both platforms for B. Equation 37 contains three terms: the rst term is a have the same initial transmit frequency (relative to each of constant that depends only on the system frequencies and the platform's own reference frames) and that the system the relative velocity of the two platforms and is indepen- frequencies for both platforms are identical. The transmit dent of path number n. The next two terms are functions frequency is assumed to be 5.99 GHz. We may use equaof the step number n and oscillate back and forth about the tion 37 to calculate the response for even and odd path constant frequency term. When n is even the frequency is traversal numbers. Where the even and odd response physat a local maximum and when n is odd it is at a local mini- ically means that on every transmission of a new frequency mum. The stability of the frequency is then easily analyzed the frequency changes value. It is much easier to plot the in terms of the poles at z = ; ,, Fig. 7. When the plat- even and odd frequency response since they are not highly forms approach each other the relative velocity is greater oscillatory functions. Fig. 8 shows the results for the di104
2500
100 m/s
2000
2000
-100 m/s
fn- fao1500
-75 m/s
1000 fn- fao
25 m/s
0
25 m/s
(Hz)
- 1000
50 m/s 75 m/s
- 2000
100 m/s
10 2
10 3
(Hz)
10 4
10 5
10 6
10 7
1000
-50 m/s
500
-25 m/s
10 8
10 2
10 3
Bounce Number, n
10 4
10 5
10 6
10 7
10 8
Bounce Number, n
Fig. 8: Frequency response for approaching platforms, parameterized by velocity. Even and odd responses plotted. Fig. 9: Frequency response for separating platforms parameterized by velocity. Even and odd responses plotted. verging case and Fig. 9 shows the result for the converging case. Both gures are based on writing equation 37 for even and odd values of bounce number n. Hence, the transmit frequency oscillates between the values speci ed by these even and odd path index equations.
+
+
+
+
fTA (n)jeven = fsysA , fsysB + f , fsysA , fsysB n (40) a0 (1 , )(1 + ) (1 + )(1 , ) Fig. 10: Finite bandwidth calculations based on second order PLL. fTA (n)jodd = fsysA , fsysB + ,f + fsysB , fsysA n(41) b0 (1 , )(1 + ) order. Let us restrict the analysis to second order systems (1 + )(1 , ) -
This frequency oscillation between the even and odd index values causes a swapping of frequencies between the moving platforms on every pass of the signal. Note that when v = 100 m/s the one-way doppler frequency is 2 KHz, and that the average frequency in both the converging and diverging graphs is 1 KHz. In the case of approaching platforms the doppler frequency is negative, and in the case of diverging platforms it is positive. This is the opposite of the sense of the doppler shift normally expected for closing or diverging platforms and is due to the di erencing process associated with phase conjugation. It takes on the order of 5 106 bounces before the oscillations die out and convergence or divergence is obtained for platforms separating or closing at +100 m/s or ,100 m/s, respectivly. This is shown when fa0 = fb0 = 5:99 GHz. At 1 Km range with a closing speed of 100 m=s the instability at N 2 , 3 106 will take place about 8.1 s after the platforms start moving as indicated by equation 27. At this time, the platforms are about 200 m from each other. When the distance is 10 Km this time increases to 81 s and the distance is about 2 Km apart. This is a relatively long time and suggests that by designing an appropriate PLL frequency response, it might be possible to stabilize the frequency response of the phase conjugation process.
and mathematically model the electronics via an analog model. As noted before it is possible to construct a SDPF that is similar in response to an analog PLL but with added phase noise. Consider the PLL in Fig. 10. Its response is characterized by [11] (s + !2 )in out = sK 2 + Ks + K!
2
(42)
where K is the PLL open loop gain and !2 is the 3 dB frequency of the PLL loop lter. When there is a step change in frequency due to the doppler shifted signal reaching the receiver then !(t) = !step U (t) , where U (t) is
V. PLL Time Response
The weighting function w(t) is a function of the type Fig. 11: Typical weighting function for step response of of PLL electronics used: analog or digital and the PLL's PLL. 105
the Heavyside step function. Then L[!(t)] = !step
0
s
(m/s)
(44)
s2
1 Km
- 40 Veff
The input phase to the PLL is then given by = !step : in
- 20
(43)
- 60
3 Km
- 80 - 100
5 Km 7 Km
- 120 and the response of the output frequency change is 10 2 10 3 10 4 10 5 10 6 10 7 !step (s + !2 ) : (45) Bounce Number, n L[!out] = out = sout = sK(s2 + Ks + K!2 ) Fig. 12: E ective velocity function as a function of number Next we see that the PLL step response can be related of signal bounces for a nite bandwidth PLL with a closing to the received doppler step since an incremental increase velocity of v = ,100 m/s. in the RF frequency at the input antenna causes an incremental decrease in the IF frequency at the input to the 120 PLL, so that !step ! ,!d. Hence, 9 Km 7 Km 100 5 Km K!d(s + !2 ) : out = sout = s(, (46) 80 s2 + Ks + K!2 ) 3 Km Veff 60 (m/s) Although we may choose K and !2 arbitrarily, let us set it 40 to a value that will give a good compromise between speed 1 Km and noise bandwidth. In particular let us choose !2 = K=4. 20 Taking the inverse Laplace transform, we nd
9 Km
, Kt2 U (t) e fout = ,fd 1 , e, Kt2 + Kt Bounce Number, n 2 = ,fdw(t): (47) Fig. 13: E ective velocity function as a function of numHence, we get the functional form of the weighting function ber of signal bounces for a nite bandwidth PLL with a used in equation 11 for the nite bandwidth PLL, plotted separating velocity of v = +100 m/s. in g. 11
10 2
, Kt2 U (t) e w(t) = 1 , e, Kt2 + Kt (48) 2 The maximum occurs at tmax = 4=K . Now if the time it
takes for the signal to traverse the distance between the platforms in one direction is much less than the time for the step response to peak, then w(t) can be approximated by a linear function and will be close to zero in value, and
w(t) KtU (t): (49) So that the value of (t , n ) may also be approximated as v K (t , n ) n t n+1 (50) c (t , n ) = 11 + + vc K ( n+1 , n ) t n+1 Where the time to traverse the one way path is Tpath = n+1 , n . Hence, we may consider the e ective relative
velocity of the platforms at the beginning of each transmission to be veff = vK ( n+1 , n ) (51) In Fig. 13 the test case of two platforms closing on each other is again considered. In this gure the unapproximated e ective velocity, veff = vw(t), of equation 48 is plotted as a function of bounce number. The time
10 3
10 4
10 5
10 6
10 7
between bounces, n , is accounted for by equation 28. The platforms are assumed to be closing at a speed of +100 m/s from various distances ranging from 1 Km to 9 Km and the open loop bandwidth of the phase lock loop is K = 105 rad/s. The times at bounce number 107 for initial ranges of 1 Km and 9 Km are 9.9 s and 89 s, respectively and the platforms are at near collision. As can be seen from this graph it is possible to have e ective velocities that are both greater than or less than the actual velocity, due to the response of the PLL (Fig. 11). The most important property of the e ective velocity function is that it decreases to zero for large bounce numbers when the platforms are approaching each other (v > 0). This monotonic approach to zero occurs at the same time that our in nite bandwidth expression for the system frequency response begins to diverge towards in nity, see Fig. 8. Therefore, it becomes reasonable to assume that it should be possible to stabilize the frequency response as a function of time. This would allow phase conjugating arrays to be arranged as coupled electromagnetic systems that operate in a well behaved manner. In the case of platforms that are moving away from each other the effective velocity function is given in Fig. 13, and stability is always guaranteed.
106
Under the condition of large n, we therefore have,
VI. Finite Bandwidth PLL Response
8 Since expressions have been derived for the time interval < fa0 + fSA ,fSB n2 if v > 0 2 between bounces, the weighting function and the time pro (60) g = n 2n , 1 : (fSA , fSB ) gression, we may now consider the behavior of platforms if v < 0 2 ,1 A and B when nite open-loop gains are realized in the electronics of each systems PLL's. We combine the results where g0 is a constant initial frequency and = 1 + v=c. of equations 48 and 28 to obtain an expression for ( n ). Which shows that for large n and platforms that are apSimplifying notation by letting ( n ) = un , we have: proaching each other the frequency will diverge in direct proportion to the square of the bounce number so long as the platforms system frequencies are di erent. This implies ,KR n+1 KR ,KR n+1 v un = 1 + c 1 , e 2c + 2c e 2c (52) that for highly dynamic platforms, simple o set-frequency phase conjugation with each platform having a di erent system frequency may not be useful since the frequency where, 1 , 2vc will become quickly unstable. However, in the simpilist (53) con guration = 1+ v : of the local oscillators in a communications 2c link we may force the system frequencies to be identical. Also, equations 18 and 19 may be combined to solve for a In the case where each platform has the same system fre nite di erence equation in fTA (n). Let us further simplify quency of f = f sysA sysB = 2fa0 the solutions reduce to notation by dropping the subscript\TA". Therefore, ( fa0 if v > 0 fn+2 , un+1 un fn = fSA , fSB un+1 : (54) lim g = (61) j v j n n!1 fa0 1 + 2c if v < 0: This equation exhibits an oscillatory behavior with a period of two. The even and odd index solutions therefore do not The detailed response was simulated by considering the soexhibit any oscillations. Since we would like to obtain the lution to the in nite bandwidth PLL derived in the previsolution to this equation for very large values of n, let us ous section. By using the e ective velocity function instead create nite di erence equations for the even and odd index of the physical velocity the response of the PLL's in an values. By making an index change in the function fn of electromagnetically coupled environment may be approxin ! 2n we may write gn = f2n . In the case of the odd index mated. Fig. 14 shows representative curves for 100m/s equation we let n ! 2n +1 and we then de ne hn = f2n+1 . relative motion at 10 Km and PLL open-loop gains of 104 The even and odd index equation become respectively: to 106 rad/sec. First we note that the large n approximation of equation 61 is correctly predicted by our approxign+1 , u2n+1u2n gn = fSA , fSB u2n+1 (55) mate method. Also, we note that early in the frequency hn+1 , u2n+2 u2n+1 hn = fSA , fSB u2n+2 (56) response, the transmit frequency alternates between two values; fa0 is always one of these frequencies without reThese are rst order nite di erence equations with non- gard to direction of motion of the platforms. The value of constant coecients. Note that rst order nite di erence the other frequency is determined by the product of R and equations of the form K . Hence, o sets greater than the one-way doppler frequency are possible as shown in the case K = 105, where yn+1 = an yn + qn (57) the one way doppler frequency would be 2 KHz but 2.2 KHz is actually transmitted. have an exact solution given by There is a tradeo between bandwidth, gain, and PLL nY ,1 nX ,1 nY ,1 center frequency. When the open-loop-gain is small, the yn = y0 am + ar qk (58) system has the advantage that the magnitude of the frem=0 k=0 r=k+1 quency jump from even to odd bounce is small, assuring the frequency jump is within the lock range of the where y0 is the initial value. The expressions for am would that PLL, however, the locking process may be slow. When be u2m+1 u2m and u2m+2 u2m+1 , the expressions for qm the open-loop-gain is large, then the magnitude of the frewould be fSA , fSB u2n+1 and fSA , fSB u2n+2 corre- quency jump between is large, and if care is not sponding to the even and odd index equations 55 and 56. taken there may be a bounces violation the PLL's lock range, This is a dicult expression to work with for large therefore the platforms may looseofphase-conjugation lock. values of n. Therefore, we consider the even and odd A good system design requires that a balance be mainindex equations under the conditions of large n, where 2n + 1 2n + 2 2n for the time dependent coecients. tained between these competing needs. Hence, we nd gn+1 = hn+1 for large n. Hence,
gn+1 =
g,n + fSA , fSB if v > 0 , 1 + vc 2 gn + fSA , fSB 1 + vc if v < 0
(59) 107
3000 2000
K = 10 4
K = 10 4
1000 0 1000 2000 3000 2 10 3000 2000
f-f ao (Hz)
10 3
10 4
10 5
10 6
10 7
10
2
K = 10 5
10
3
10
4
10
5
10
6
10
7
K = 10 5
1000 0 1000 2000 3000 10 2
10 3
10 4
20000 K = 10 6 15000 10000 5000 0 - 5000 - 10000 - 15000 - 20000 2 3 4 10 10 10
10 5
10 6
10 7
10 2
10 3
10 4
10 5
10 6
10 7
10 4
10 5
10 6
10 7
K = 10 6
5
10 5
6
10 6
7
10 7
10 2
10 3
Bounce Number, n
Fig. 14: Even and odd index response curves. Closing platforms (v > 0) are shown on the left and separating platforms (v < 0) are shown on the right. This simulation was run with R = 10 Km and v = 100 m/s.
108
- 40 Osc
illa
- 50 PSD - 60 (dBm) - 70
tion
illa
tion
Osc
fo = 2.99 GHz
- 80
-1.0
- 0.5 f
0.5 f
0 f-fo (MHz)
1.0
Fig. 16: Spectrum Analyzer (SA) measurements demonFig. 15: Experimental set up. The signal moves by f=2 strating frequency oscillation. even though a step change of f occurs. A. A Simple Experiment It was desired to perform a simple experiment which demonstrates the phenomena discussed above. Although a fast moving platform traveling at a sustained 100 m/s could not be mechanized in the lab, an experiment that looked at a simplifed version of the problem could be addressed. In the circuit of Fig. 15 four mixers are connected in a loop and no PLL's are used. Two local oscillators are used as the sources for these mixers. These sources are called LO1 and LO2 and have frequencies of fLO1 +f and fLO2 respectively. The values of the frequencies fLO1 and fLO2 are taken to be constrained to fsys = fa0 + fb0 = 2f0 so that it is similar to the case of pure phase conjugation considered in the previous sections. The value of f is added as a step change in the local oscillator frequencies. It can be shown through an analysis similar to the analysis for the original problem that the transmit frequencies form an alternating series: ff0 +f; f0 ; f0 +f; f0 ; : : :g. Hence, the average transmit frequency over one period is given by hfT i = f0 + 2f : (62) This is the value a spectrum analyzer will see due to the instrument's nite integration time. The spectrum analyzers nite integration time therefore plays the part of the nite bandwidth of the PLL in the previous section. Hence, a change in the LO of f should manifest itself as a change in measured frequency of f=2. This is exactly what was observed, as may be seen in Fig. 16. VII. Satellite Based Systems
The self-phasing property of retro-directive arrays may make them attractive for certain space based applications. Consider the geometry of Fig. 17. A planet (earth) is shown being orbited by a satellite with a retro-directive adaptive array, the satellite is at a position r relative to the planet's center. A second array is on the planet at a position RE . The distance between the satellite and the planet-based transceiver is R. If we restrict the orbits to be circular and in the equatorial plane, for simplicity, then we may write = vt=r, where v is the relative speed of the satellite.
The expression for the velocity component along the vector R easily follows from the geometry. This is the line of sight, velocity VLOS , and its magnitude is given by
,
vRE sin vtr VLOS = q , : RE2 + r2 , 2RE r cos vtr
(63)
However, due to the relative motion of the satellite in orbit and the planet's rotation, the di erence in angular velocity results in r E , 2r : v = Gm (64) r t day
Where G is the universal gravitational constant, mE is the mass of the planet, and tday is the time in seconds for one revolution of the planet. Finally, since the planet itself gets in the way of transmission for angles before time ,Tm and after time +Tm we are constrained to look at line of sight velocities only between these times. Again based on the geometry of Fig. 17 we see that
Tm = vr cos,1 RrE
(65)
From these relations we may plot VLOS as a function of time for a number of di erent orbits, ( Fig. 18 (a) ), Finally, the associated accelerations are plotted in Fig. 18 (b). The simplest case to handle is when the satellite is in geosynchronous orbit (GEO). However, for orbits only slightly higher than typical space shuttle nominal altitudes (400 Km) the velocities, VLOS , quickly become comparable to values analyzed earlier in this paper. From Fig. 18 the accelerations, aLOS also are small once one gets past about three to four the normal altitude of a space shuttle. In the case of Low Earth Orbits (LEO), 400 to 1000 Km, the situation requires greater analysis. However, note that in order to maintain phase-lock across a phase conjugating array when large accelerations are present, one must increase the system type of the PLL that is being used in the retrodirective array, i.e., another integration in the feedback loop of the PLL will be required. This will slow the system down and control any possible instabilities.
109
a 6000 4000
t=Tm
SATELLITE ORBIT
VLOS (m/s)
{
V
2000
b c
e f
0
4000
r
6000 40
R
20
0
t=0
RE
d
2000
VLOS
EARTH
a = 400 Km b = 1600 Km c = 2800 Km d = 4000 Km e = 5200 Km f = 6400 Km
20
40
T (min) (a)
E
a
120 100 S
ALOS (m/s2)
t=-Tm
Fig. 17: Geometry for a satellite in a circular orbit about earth.
80 60 40 b
20
c
15
A computer simulation was performed for a satellite at 4000 Km altitude in a circular equatorial orbit.The maximum acceleration is about 10 m=s2 The simulation assumes a 5.99 GHz transmitter and a worst case scenario of a satellite with processing electronics that have in nite open-loop PLL gain and hence maximum e ective velocities, seen by the PLLs. As a result, any portion of the orbit that had VLOS > 0 should have a maximum tendency for instability. The simualtions made no assumptions about the uniformity of VLOS , hence accelerations were allowed. Because of these assumptions about the satellite, no closed form solution could be obtained and only numerical solutions were developed based on equations 18. In Fig. 19 the o set frequency is plotted against satellite transit time. When the satellite enters into view of the retro-directive array on the earth the satellite is moving at over 3000 m/s towards the earth-based system along the direct line of sight. The system's frequency evolution is maximually unstable at this instant. If the PLL's frequency lock-range is designed to be larger than the induced doppler shift then then the two transceivers will become electromagnetically coupled and the tranceivers are locked. After initial contact, the VLOS is decreasing fast enough in magnitude so that even though the system poles are outside the unit circle and hence unstable a frequency instability never becomes apparent. In addition, as the satellite passes directly over ground based transceiver the sense of direction changes from closing to departing. Therefore, the system now moves into a stable operational mode. The satellite increases in speed as it continues to move along in its orbit and this only serves to increase the tendency of the o set frequencies to stablize. The resulting even and odd bounces change frequency on every new signal bounce and the frequency converges towards half of the maximum doppler frequency encountered at the begining of the orbit. Fig. 20
10
5
0
5
10
15
T (min) (b)
Fig. 18: (a) Line of sight velocity vs. time. (b) Line of sight acceleration. 60000 50000
fTA - fao (Hz)
odd bounce
40000
fDop-Max / 2
30000 20000
even bounce
10000 0 30
20
10
0
10
20
30
T (min)
Fig. 19: Satellite transmitter's o set frequency as a function of time. shows how the o set frequency behaves as a function of distance between the two retro-directive arrays. VIII. Conclusions
Phase conjugating arrays o er many of the important advantages of full phased arrays but without the need for phase shifters to produce beam steering. In the case of highly dynamic mobile platforms under constant rectilinear motion, the transmit frequency of each platform is shown to alternate between two frequency values such that platforms A and B are essentially swapping frequencies on every other pass of the electromagnetic signal. At a point in time, perhaps many seconds after the initial lock, the frequencies may stop oscillating and stabilize to constant and nite values so long as the system frequencies are the same, the platforms are moving apart, and nite bandwidth
110
60000
odd bounce
T =-30 min.
50000
T=0
fTA - fao (Hz)
40000
fDop-Max / 2
T = 30 min.
30000
T=0
20000
even bounce
10000 0
T =-30 min.
Arrows indicate time evolution. 4000
5000
6000
7000
8000
R (Km)
Fig. 20: Satellite transmitter's o set frequency as a function of distance from observer on earth. PLL's are used in the link. Although, the closed form results were derived for the case of linear continuous systems numerical simulations show that the general results should also be valid for systems under acceleration such as satellites. Such systems should be readily constructed by using serial-digital-phase- lters (logic based PLL's) for the phase processors of the adaptive arrays. References [1] L. C. Van Atta, \Electromagnetic Re ector", Patent Number 2,908,002, Patent application 1955. [2] E. D. Sharp and M. A. Diab, \Van Atta Re ector Array", IRE Trans. on Antennas and Propagation, vol. AP-8, Number 4, pp.436-438. [3] M. I. Skolnik and D. D. King, \Self-Phasing Array Antennas", IEEE Trans. on Antennas and Propagation, vol. AP-12, Num. 2, pp.142-149. [4] C. Y. Pon, \Retrodirective Array Using the Heterodyne Technique", IEEE Trans. on Antennas and Propagation, vol. AP-12, Num. 2, pp.176-180. [5] S. N. Andre and D. J. Leonard, \An Active Retrodirective Array for Satellite Communications", IEEE Trans. on Antennas and Propagation, vol. AP-12, Num. 2, pp.181-186. [6] E. M. Rutz-Philipp, \Spherical Retrodirective Array", IEEE Trans. on Antennas and Propagation, vol. AP-12, Num. 2, pp.187-194. [7] R. K. Tyson, \Principles of Adaptive Optics", Academic Press,1991, ISBN 0-12-705900-8. [8] C. Pobanz, I. Itho (Chair), \Time Varing Active Antennas, Circuits and Applications", PhD Dissertation at UCLA [9] W.H. Lee, E. V. Harrington, and D. B. Cox \A New IntegratedCircuit Digital Phase-Locked Loop", National Aerospace Electronics Conference, vol. 75CH0956-3 NAECON, 1975, pp.377-383 [10] D. B. Cox, E. V. Harrington, W. H. Lee, and W. M. Stonestreet, \Digital Phase Processing for Low-Cost Omega Receivers", Journal of the institute of Navigation, vol. 22 Num. 3, 1975, pp.221234 [11] D. H. Wolaver, "Phase Lock Loop Circuit Design", Prentice Hall Publishers, 1991, ISBN 0-13-662743-9 [12] R. E. Best, \Phase Locked Loops, 3rd ed.", McGraw Hill Publishers, 1997, ISBN0-07-006051-7.
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