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AO-A042 440
UNCLASSIFIED
OHIO STATE UNIV COLUMBUS ELECTROSCIENCE LAB F/S «
ADAPTIVE ARRAY PEPFORMANCE WITH COOED FM SIGNALS. (U)
MAY 77 I K LAO* R T COMPTON N0001Q-77-C-0156
APNo. — -
WC fit COPY AOAO'124H)
/
ADAPTIVE ARRAY PERfORNANCE MITN COOCO fit SIGNALS
I.K. L«o ind R.T. Coapton, Jr.
Tti* OMo Skrt* Ualvfiity
EbctfoSciMC9 Libofiloiy
B»cirtcwl ftnwurtu,
CalifiiWOtri* 4»I9
_ Technical Report 4618-2
Nay 1977
Contract N00019-77-C-01S6
Department of the Navy
Naval Air Systams Command
Washington, O.C. 20361
0
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The material contained in this report is also used as a Thesis submitted
to the Department of Electrical Enqineerinq, The Ohio State University as
partial fulfillment for the deqree Master of Science.
■ f e #' 40 A rt fwftr>%f0 0* *#*•#•• •«>f* 0t mm m* t
Adaptive arrays
Frequency modulation
Interference rejection
ipi’"”* '» I .
HThis r
47'* A **»»»■
HThis report discusses the use of FM signals with adaptive arrays. The
FM signals contain an extra pseudonoise bi-phase modulation to enable the
adaptive array to distinguish the desired signal from interference. The
performance of the array is studied as a function of reference signal modu-
lation delay, code timing delay, and frequency offsets, iw
A
OD , j*,!',! 1473 eoiTioxo* • NDv *5 If onsoLr tr
UNCLASSlFlEp__
krr i/niT T ti. *sMMc «*iON O' tMi» ■■■
7S1
TABU or commirs
Chapltfi
I li^TROOUCTION
II A HOOIFIID FM SICMAl
III BCFERCNCE SlCflAl
IV THE IMS LOOP
V RESULTS
A. Reference Stgruil wttH FM Mtxiujetion
B . ReTerence Slgnel lfft»out op
VI CONCLUSIONS
REFIRENCCS
I
1
«
16
22
?S
)l
V
S4
Appendix
11
f
I
I
!
f
I
I
iHAPTIIf I
I M ROOK. T I ON
The objective of thu revr«irtii to <J«*v<*lop 4o 4010004 wstom
C4p4blr of rrcoivinq 4 fM < (Mouoical ion siqo4l in t*«o proMHVo of 4
strong toterferoocr Sigo4l <*ltbout tN> oo«^ for 4rt4tl«><) infonMffon
4bout sf'inal nodulatfon or tbo 4oqle of 4rrf«4l. Tbo b4\ic proper*
ties of 40 4it4ptive 4rr4y *'4ve N*oo tnvestiqeteO prevloos1|r[l .?.3]
4od are well established. In this researth we develop 40 approach
for inteqratinq an adaptive array Into an fM coMoiinicat ion systee<.
An adaptive array is an array of antenna e nients followed by
a real time adaptive processor. The antenna array can automatically
place pattern nulls in directions from which undesired signals
(interference, Jaarinq or clutter) arrive and can also provide oain
on a desired signal. Widrew. et al[l), proposed the basic adaptive
array feedback algorithm, the so-called LNS algorithm. Applebaum
[3] also developed an adaptive array control loop which ma»imi<es
the array output siqnal-to*noise ratio.
In this report we consider an adaptive array based on the IMS
algorithm[ I ]. The general structure of such an array (for t»#o ele-
ments) is shown in figure I. The incoming signal from each element,
y{(t), is split into in-phase and quadrature components xf(t). Each
component is multiplied by a weight wj and then summed to pr^uce
the array output,
4
S(t) » I w.x.(t) . (I)
i«l
The error signal »(t) is obtained by subtracting the output of the
array S(t) from a reference Signal R(t).
.(t) « R{t) - S(t)
(2)
I:
II
n
n
The array feedback adjusts the weights w^ to minimize the mean-square
value of t ( t) .
In the LMS array, the signal used for the reference R(t) deter-
mines which received signals will be accepted by the array and which
1
Figure 1. The LHS adaptive array.
will rejected. When a signal received by the array is highly cor-
related with the reference signal, the array feedback retains that
signal in the output. A signal uncorrelated with the reference sig*
nal is nulled by the array. Hence the array can be used to protect
a conriunication system fron interference if the reference signal can
be made strongly correlated with the desired signal but uncorrelated
with the interference. Such a reference signal is usually obtained
by processing the array output in some manner that preserves the
desired signal but destroys the correlation in the interference com-
ponents. For this to be possible, it is necessary that the desired
signal differ in some known way from the interference.
In Chapter 1 1 , we describe a technique for modifying a con-
ventional FM signal so that the array can distinguish between it and
the interference. Chapter III describes a method of deriving a
reference signal for the adaptive array, based on the signal structure
described in Chapter II. Chapter IV discusses the feedback loop band-
width properties of the LMS algorithm. This information is needed
in the following chapter. Chapter V describes the results of some
simulations of an adaptive array with the FM signals discussed in
Chapter II.
2
(HAPTIH II
A NniUriEO FM SIGFlAl
A conventional FH corwiunicatton siqnal inay be Mrftten
D(t) • A cos[-j.t ♦ *»(t)]
(3)
where A is an amplitude corstant, w, is the carrier freouoncy and
•*(t) is a time-varying phase. The instantaneous Frequency, is
i t dt
(A)
In ordinary FM, is linearly related to a modulating signal F(t),
f(t)
(S)
where K is a constant. The signal D(t) then has the form
D(t) * A cos
u^t ♦ K . f(t*)dt* ♦
(6)
where o is the initial phase at t*0. If, for example, f(t) is
sinusoidal at frequency
f(t) = a cos .fl,t
(7)
the instantaneous frequency is
cos u^t
(8)
A-i) is called the deviation (aw * Ka). The phase variation «(t) is
(O ■ sin
m
n
'9
t
and th(' quant
- - no)
m
is CdllfHl the iiKMluiat ion initu. In qcn^ral, tht* bandMidtH ociupif^
hy ti>«* siqnal increases xi’h . .
We are interested in receiwinq desired signals ti>e above
type with the adaptive array. To obtain the reference signal -inter-
ference Signal decorrelation required to all<>« the adaptive array to
null interference, we add an e«tra onase midulatinn i(t) to this
signal. I.e.. we suppose the desired signal has the for»
DIt)
A COS
.t
(Ilf
whP'-e t(t) is a digital wa»efonn with the values 0 and on the
intervals of length T., as shown in Figure 2a. Iguation (II) can
also be written
Fiugre 2. (a) The waveform i(t); (b) the waveform P(t).
- -jwa_
4
(
.t
D(t) A P(t) los (.! ♦ K • • . 12)
•0
where P(t) is ii iJiqitdl waveionn with values •! as shown in Fiqure 2ti.
This modulating waveform P(t) would be added to the desired signal at
the transmitter. We obtain the waveform P( t ) from a maximal length
pseudonoise code generator(4,5j. A typical 0(t) is shown sketched
in Figure 3c. The modulating signal is assumed to be a sinusoidal
waveform (Figure 3a). P(t) in Figure 3b has the effect of changing
the sign of the conventional FM signal on a bit interval depending
upon the PN sequence.
Since D(t) is the product of a conventional FM signal with
P(t), the spectrum of D(t) may be found by convolving the spectra of
P(t) and the FM signal. It is apparent that the spectrum of the
modHied FM signal is spread out depending upon the bandwidth of
P(t). To illustrate this point we consider a sinusoidally modulated
FM signal of the form
d( t ) = cos f .(.t ♦ r V in .„,f ] '13)
and we assume P(t) is a sqcare wave of frequency .. For small
modulation index r •< Equation (13) can be written
d(t) ^ cos '..j-t - sm .^t sin .^t
= cos Wft • 2 cus(..(. - .^^)t ♦ 2 tos(-t ♦
d(t) has the frequency spectrum shown in Figure 4a. The (complex)
Fourier coefficients of the square wave are given by
which is plotted ds a function of f r»‘quptu y ni 4b. To simiilify
the convolution, three significant oairs of narr. ,»iics of P( ) have
been taken to convolve witf D(.). vir obtain tt«e tinal '•pectrjin of
the modified TM signal in t igure 4c. The presen« « n* i) on the
desired signal broadens thi spei.trun of tfie > onvetit Knal FM sirjnal.
Ihe selection of the bandwidth of b(f) will be disiMssiwi in Chapter
W.
With the phase modulation ;(t) present on l»ie desired signal,
the same modulation can be introduced on the refereii,#* signal. ]f
this is done, the reference signal will be cnrrelatcvl with the
dt'sired signal but uncorrelated with an interference, as long as
;(t) switches more rapidly than the array feedbac*' loops can tract.
In the following Chapter we will discuss how such a reference signal
can be derived.
8
I
I
¥
Cl Am If in
RlFlMEfiCl Sir,Vi
f
. r*
i
I »
T-
• ; 1 •
I »
I
Ideally, when the desired signal ha' the (orm in F(|uation (II),
the reference signal should also be given by:
R(t) = A cos
r»
f(f)df
Jr
4 .•
0
(16)
i.e., it must be identical to D(t). The problian, of course, is that
f(t) and ;(t) are not knowi. at the rereiving site ahead of time. (If
they were, there would be fo need for the antenna!) Instead, it is
necessary to obtain estimates of f(t) and :(t) (which we denote by
f(t) and 't(t)) by demodulating the receiving signal. From these, a
reference signal
R( t) = A cos
.t+K
; T(f)df
^0
(17)
niay be constructed. This reference signal will be suitable only if
f(t) and t(t) are sufficiently good estimates of f(t) and ;(t).
To make use of this technique, it must be possible to demodu-
late both f(t) and ;(t) separately, i.e., to extract each waveform
without interference from the other. It appears that a Costas loop
[6] followed by a baseband delay lock loop[7,8,9] can be used for
this purpose. In the following paragraphs we discuss the Costas
loop (CL) and the delay lock loop (DLL) respectively and show how
the estimates f(t) and t(t) may be obtained.
A block diagram of a CL is shown in Figure 5. To understand
the operation of this loop, assume that the input is a modified FM
signal given by Equation (12)
D(t) = A P(t) cos
■-U
f(f)df
(12)
where the initial phase angle is defined to be zero. The VCO out-
put is split into two quadrature components, ei(t) and ep(t), given
by
9
%
• • U)
r
! LCWPASS <
1 PIL'^EP 1 ,
1 — — ■ i
1
1 LOWPASS A
J
j PILTEP 3 ^
Olt)
90^
; phase
jSH;FT;
tjlf)
W^rV
^fif) '
i
1
1 1 1
wOWPASS ; 'f/’l ;
FILTEP 2 j
Figure b. lostds loop.
e,(t) = ? cos [ j-t ♦ (♦)]
(18)
and
eplt) = Z sin [.^.t ♦ (l)]
(19)
where <t(t) is a time varying phase angle. The input FM signal is
mixed with ei(t) and eo(t) sei>arately. The products are passed
through the lowpass filters to eliminate the sum output terms from
the mixers. The filter outputs become
ef,(t) = P(t) cos
r
1
f(f)dt' - .<(t)i
(?0)
and
10
= P(t) sin
f(f)df -
e^j(t) and multiplied to give
(?1)
= j P^(t) sir 2
The lowpass filter output is
f(f)'lf - (t)
(22)
ein(t) * p P^(t) sir 2
^y! f(f)df - '.(t)
L^o
because thp filter bandwidth is chosed narrow enough to average
P^(t) but not
sin 2
f(f)df - .(t)
(23)
If the modulation P(t) is ideal, then P (t) - (’l)^ = 1 regardless
of whether P(t) is +1 or -1, so no filtering is required. However,
when the signal in Equation (12) is transmitted through a finite
bandwidth, the modulation F(t) produces envelope modulation on D(t)
[10] and the lowpass filter is needed to eliminate spikes that will
occur in efjp(t).
When the CL is operating in lock, the VCO phase -(t) is a good
estimate of the input phase deviation. The phase error.
k! f(f)df - ft(t)
Jo
is small and the VCO input can be approximated by
ejn(t) K f(t’)df - . (24)
It is easily seen that this is the desired mode of operation for
demodulation of FM. If
11
#
.t
(t) ^ f(f)df
^0
the VfO frequency is a qood estimation of the signal frequency. The
signal frequency deviation is proportional to the* modulating signal
and the VCO frequency is proportional to the signal e^j,(t). Thus
ej^(t) is proportional to the signal frequency and can be used as
the deinodulated output for FM inputs.
Moreover, since
• t
. f(f)df = n(t).
^0
then
cos
f(f)df
so Equation (18) becomes
e^j(t) - P(t)
(2S)
i
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i
i.e., t-'f|(t) is an estimate of the PN code, P(t), on the input signal,
ef](t) = P(t). This estimate of P(t), which in general is noisy, can
be cleaned up with a delay lock loop, as shown in Figure 6.
Since P(t) is a maximal length PN sequance, the waveform for
P(t) is known ahead of time at the receiver. In Figure 6 a generator
is used to generate two PN sequences, P](t) and P2(t), which both
have the same waveform as P(t) but which differ in timing by one bit
interval, i.e., P](t) = P2(t - T^). T>(t) is mixed with P](t) and
Pp(t) separately. The products are passed through narrowband filters
Whose outputs approximate the cross correlation functions of the input
signals, that is, Rpp^{t) and Rpp2(r). When the DLL is correctly
tracking P(t), P2(t) is P,/2 ahead of P(t) and Pi(t) is P,j,/2 behind
P(t). Therefore v] and <12 can be expressed in terms of t, the time
strictly speaking, we are referring to the d-c term in V] and Vt
here. We do not consider the noise terms (self noise or thermal
eoise) in the filter output.
]2
1
I
r~
NARROWBAND
PILTE9
P(f)
NARROWBAND
FILTER
TO
SWITCH
CONTROL
Pjlt)
P,lt)
PN CODE
-J '^co
GENERATOR
j CLOCK
v,( ♦» > V,
^ OA-- DC VOLTAGE
LOGIC SWITCH
Figure 6. Delay lock loop.
difference between P(t) and a code timed half way between Pj(t) and
P2(t). Figure 7a shows V](i) and v^(r).’ The sum and difference of
V](t) and V2() are also shown in Figures 7b and 7c respectively.
If ' VT, a fixed DC voltage is connected to the VCO input and
drives the VCO at a pre-determined speed until the timing of the
generated PN sequences and P(t) is close (v5(t) vj). The VCO input
is then switched to the difference voltage v^j which tends to drive
the DLL into lock.
By utilizing the Costas loop and delay lock loop together we
can obtain estimates of the modulating signal f(t) and the PN coded
signal P(t). A reference signal for the array can then be generated
as shown in Figure 8. When the DLL is locked, its output code timing
r
r
If
’ The autocorrelation function of a maximal length PN sequency is
derived in [5].
13
r "" - “ViiiTTiiinr
is P./2 off from the desired signal code. To compensate for this
time difference, a time shift of P./2 is included in Figure 8. It
is also noted that there will be a time delay between f(t) and f(t)
<lue to the filtering in the Costas loop. To maintain correlation
between the reference signal and the desired signal, this time delay
must be minimized. The effect of this delay on array performance is
studied in Chapter V.
In addition to using the reference signal as derived in Figure
H, another approach is to simply not include the FM modulation in the
reference signal. Instead, a signal of the form
14
I
n
n
I a
A
Figure 8. Reference signal generation loop.
R(t) = A cos [..!(- 1 + :(t)]
(26)
could be used. This eliminates the problem of time delay between
f(t) and f(t), but we suspect that this approach is suitable only
if the frequency deviation of the modulation term
.t
: f(f)df
JO
in the desired signal is snail, i.e., only for low values of modula-
tion index. Chapter V also discusses this type of reference signal
and shows what performance may be expected.
15
1
(HAPTfR IV
thl lms loop
A', w<' rifntionpd previously, the adaptive array discriminates
between the desired signal and the interference bv correlating the
inLoniing signal with the reference signal. Theoretical ly the cor-
relation between these signals is the average of the product of the
signals over an infinite pfriod of time. In the adaptive array,
iiowever , this averaging process is approximated by a lowpass filter
whose bandwidth is set narrow compared with the lowest frequency of
the baseband desired signal.
The LMS alijorithm is
steepest descent. Changes
direction of the estimated
pguat ion is ( I ,?) :
a feedback rule based on the method of
in the weight vector are made along the
gradient vector. The weight control
d_
dt
W(t)
(27)
where W(t)
is a column vector whose components
are the array weights.
W2(t)
W3( t)
( t ) y
(28)
is a positive constant which we refer to as
and v^[.^(t)] is the estimated gradient vector
error with respect to W. tquation (27) can be
the loop gain constant
of the mean-square
written
W(t) = 2 . (t) X(t)
(29)
where X(t) is the signal
nput vector.
16
X(t)
/xi(t)\
X2(t) I
Xj(t) I
V4(‘V
(30)
(Note that the input siynal components, such as x|(t), each consist
of three components, the desired signal, the interference and thermal
noise.) Equation (29) can also be written in its integral form
t
W(t) = W;(0) ^ 2 l<A , X(f) c(f)df (31)
■’ t'=0
where W^(0) is the initial weight vector. Thus we see that each
weight, w, , is obtained by integrating the product of x^(t) and (t).
A circuit implementation oi Equation (31) is described in [2].
If we substitute Equation (2) into Equation (29) and define
matrices S and ♦ as follow"^:
S(t) = R(t) X(t)
(32)
and
i(t)
/x] ( t)x] ( t)
X2(t)x] (t)
X3(t)x](t)
\x4(t)x] (t)
xi(t)x2(t)
x?( t)x3( t)
X3( t)x^( t)
X4(t)x2(t)
XI ( t)x3(t)
X2(t)x3(t)
X3( t)x3(t)
X4( t)x3(t)
x](t)x4(t)\
x?(t)x4(t)
X3(t)x4(t)
X4( t)x4( t) J
(33)
then the system of differential equations of the weights can be
expressed as
~ W(t) + 2 kft ^(t) W(t) = 2 S(t) . (34)
This equation represents the operation of a (multidimensional) lowpass
filter on S(t). It is a coupled system of differential equations with
time varying coefficients. An exact solution in the general case is
17
difficult to obtciin. How*‘vi‘» , vdriou'j t*-( on i'iu**.( 1 1 , 1? . I 3] hdvo t<'nn
u'.od to obtain im>anitii)tul appi ox u'liit •• solutions. Hern wc* assunio tfio
frequency components other than tiie dc term can be neglected and thus
:(t) is approximately a constant matrix. With constant it is easy
to uncouple the system (34 and to obtain its solutions.
An orthogonal matrix, P, is chosen to diagonalize i.e..
P^ t P -- • =
\
(35)
where j to ^ are eigenvalues, i^e define a transformed weight
vector 'i(t):
'i^( t)
•3(1)
V’4“V
(36)
Applying Lquations (35) and (36) to Equation (34), we obtain the
uncoupled system
n(t) + 2 k^A n(t) = 2 pT S(t)
(37)
The solutions are given by
'li
i(t) =
" I-m(o) -
L 1
-2k^ ' i t
(38)
where ii^(0) denotes the initial value of iii(t) and is the ith
component of the column vector
18
i
q^(0
Val'V
When Liquation (36) 1* used to calcuKitf' the weiqhts from
these r^(t), we find that each wj will consist of a sum of expon-
entials with time constants
(40)
Note that the speed of response of the array is limited by the small-
est eigenvalue of ; in Equation (35). We therefore define the time
constant the array to be
I
2k
j
A *mi n
(41)
where
> .
mm
min { • • 1
j ^
(42)
We also note ttiat the bandwidth of the lowpass filter described in
Equation (34) is determined by the largest eigenvalue of t and thus
the feedback loop bandwidtii is defined by
B.W. - 2 k^ (43)
where
y max =
J
»
Each weight has a dc term and a fluctuating component which result,
respectively, from the dc term in R(t)X(t) and the difference frequency
li
HI— —111 r
19
terms in R(t)X(t).* It is ■ l<',ir that spoctril tomi)onr*nts of ^
X(t) atiovn thr (iitott < riajiiom y ?t/\ h.ivc little effect on t h(*
vvei<)lits. On fht' other tiand. dc i r)mponent'. of ".1^( f O detfetnine
the dc values of the weicjnt ,, .ind rion-yero tri'queru.y components of
t)X( t ) helnw the cutoff fre(|uency 2^-A'ma< P'^oduce weight fluctu-
a t i on or .litter.
In a radio ( ommunii.at inn system the signals x^(t) and l^itj a-e
tiandlimited with stieitral components confined to some region around
a currier frc(|uen(.y. The carrier freguency is assumed to be high
compared to the tiandwidtli. Thus the product f^(t)<^(t) contains power
tn a band around yero freguc>iicy arid also in a tiand around twi<e the
c.irrier freguency. When the reference signal is tiroperly designed,
the desired signal -reference' signal product in Ft't)xi(t) contributes
a large- dc ter'm that deterrines the steady state weights and causes
the array to retain the desired signal in the array output. The
interference s igrial -reference signal should have no dc term and should
have as little power as possible within the feedback bandwidth,
‘’*^A max- When this is the case, the array nulls the i nter ference .
Thus we say that, in the adaptive array sense, two signals are strongly
"correlated" wfmn most of the power in the spectral products they con-
tribute to H(t)X|(t) lies within the feedback loop bandwidth, ?k/\
On the other hancl, two signals are "uncorrel a ted" as long as the power
in these spectral components is mostly outside the loop bandwidth.
As we discussed in Chapters II and III, the proper desired signal
correlation and interference decorrelation can be achieved by the in-
troduction of a phase switching in the desired and reference signals.
Accordingly, the choice of the feedback loop bandwidth and the code
f rec)uen(,y** (f.) must be compatible. In the following paragraphs
the design tradeoff betweer these parameters is discussed.
Since the desired signal is transmitted through a finite band-
width, the modulation P(t) produces envelope distortion on D(t),
especially in the region of a bit transition where the envelope rolls
over[10]. The slope at which the envelope rolls over is determined
by the system bandwidth but not by the code freguency. To minimize
envelope distortion we could broaden the system bandwidth. However,
our goal here is to transmit the signal in Eguation (11) through a
conventional TM bandwidth. For that reason, we choose the code fre-
guency near the low end of the audio band.
* The difference frequency terms in f also contribute to the fluctu
ating component of each weight.
**The code freguency is f,j, T^', where T^ is the bit interval.
where is the lowest audio frequency component in the modulation,
for example, 100 Hz.
On the other hand the feedhact loop tiandwidth has to be narrow
ccHupared with f.. Otherwise the array weights will track the switch-
ing of the biphase modulation and could modulate a CW interference
signal to make it match the biphase modulated reference ‘■iqnal. Then
the decorrelation between ](t) and R(t) will not occur. Hence
B-W. = - 2T,f. . (46)
In addition, to prevent the array weights from responding to the
audio modulation, we also want
B.W.
(47)
Combining Equations (46) and (47), we have
B.W. = <' min * 2nf , ,2nfp^i'p f . (48)
It is clear that the eigenvalues 'j depend upon the power of
the incoming signals. (See Equation (33j.) The bandwidth of the
feedback loop is proportional to which is determined by the
strongest interference signal. The array has to be designed to
operate over a range of signal powers. Thus the loop gain constant
2k^ must be chosen to enforce the inequality (48) for the strongest
I(t) to be received.
21
(HAPTfP V
RESULTS
In this (.Luipter wp studv thp performance a two element aUao-
tive array witLi two different t/pes of reference signals, as discgssr>d
in CLiapte-' III. First, the reference signal is assumed to be obtained
iiy [irocessinc; f tie array output as shown in F igure R. Then, a simpler
leterenre sigrial as in Euuation (?6) is assumed, i.e., the EM modula-
tion I , not included. We consider only ttie case where the interfer-
ence is a C.W signal and the desired signal is a coded FM signal with
CW modulation.
The desired signal is assumed to arrive at the array from an
angle (see Figure 9) producing element signals
yij(t) - A sin [,^jt c . .-,in ^,t t :(t)] (49)
and
X^lt) X3(t) Xglt) x,(t)
Q.H. » QUADRATURE HYBRID
Figure 9. A two-element array.
72
1
y3^j(t) = A sin [ * -‘(0 * <4]
(SO)
where
’d " 'd • (^U
d
I is the spacing between elements, which is assumed a half wavelength
at frequency d- ‘d ^"^PP'^pace wavleenqth at frequency d» ^
is amplitude constant, t- i^ the modulation index, and is the
modulatinq frei)uency. The subscript "d" denotes the desired signal
portion of y](t) and y2(t).
In order to study th( effects of Doppler shift or transmitter
frequency inaccuracies, we assume the desired carrier varies around
a fixed frequency c. We refer to this variation as the frequency
offset and define the perccntaqe frequency offset by
frequency offset = x 100 . (52)
'"C
As discussed in Chapter III, the reference signal code will be
generated by a delay lock loop. In general, because of tracking in-
accuracies, a time difference • may exist between the codes in the
desired signal and the reference signal This time difference is
referred to as code timing offset. We define the percentage code
timing offset by
. i time difference i ,r>n
code timing offset = .--v- -,. t — x 100
bit interval
(53)
The effect of code timing offset on the array performance will also
be studied.
• r
. I ■
f ''
The interference signal is assumed to propagate into the array
from an angle f'.j, producing element signals (see Figure 9)
y^ ^( t) = B sin lo^t
(54)
1 , 1
and
I
I
r
yp.(t) = B sin (w^t - y^)
(56)
23
where
21 sin j
r. IS in fimplitu'le constant and ^ is the free-spare wavelength* at
freguern / The subscrijt " i " denotr-s the interference signal
portion of vi(t) and I” results described below, will
be assumed egual to q, i.r the interference signal is exactly on
on the desired signal design frequency.
The total element signals arc
,V|(t) = yi^^(t) 1 y,^(t) (57)
and
The inphase and quadrature components (see Fiqure 9) are thus
l(t) = A
sin
[ ■dt ^
1'. sin
■m^ *
:(t)j 1
B sin
(59)
2(t) -- A
cos
[■■dt *
sin
■mt "
t(t)] r
B cos ft
(60)
3(t) = A
sin
[..gt +
sin
■mt ^
:(t) -
r B sin(.;ft-
' i )
(61)
4(C) * ^
cos
[‘d^ ^
!■! sin
"m^
4'(t) -
t- B cos(...ft-
1 i ) •
(62)
* This 'j has no relation to the eigenvalues discussed in Chapter IV.
?4
iMMMiidMb
••iWr*--*-
I
In the results desuribed below, A is set equol to 1 and B=10. This
dioice makes the input siqnal -to- i nterference ratio -?0 dl3. Further-
more we assume = 0" and = 60 .
With the input signals defined above, we will discuss the array
performance with the two kinds of reference signals in the following
sub-sections, A and B. The results in sub-section A will be based on
theoretical derivations while those in sub-section B will also utilize
simulation results.
A. Signal with FM Modulation
In this sub-section the reference signal has the form
R(t) = A sin [..,j,t + I sin ..^(t-to) + <f(t-i)]
(63)
f.
[
[
where it is assumed that the Costas loop preserves the exact waveform
of the modulating signal but its output is delayed by a time t^, and
also the code timing is offset by a time i.
To evaluate the array performance with this type of reference
signal we first calculate the steady state weights that will result.
The steady-state weights are given by[2]
ss
where
and
/“ssA
'ss2
= }'
•k =
’^ss3
/'xjTtJxJfiJ
xTTtTTfTtT
x^Ttir^Tty
X4”(tlfr^TtT
xJTtTx^Ttl
xJTtJxJTtJ
XgTryr^TtT
x^TtlY^TtT
x^TOTgTtT
x^tlY3TtT
xYrnY3TtT
x7rtTx3TtT
x{(t)x^(t)\
r^TtW^TH'
x~jrtjx^
xjrnx^ j
(64)
(65)
25
I
s
/ R(Ox'^ro^
i RTtTxy O
tf,f. }
where the overhar denotes an infinite time averafje. Thus all fre-
quency components of x^rr)''jrO m : are dropped except the dc terms.
From the output power of the desired siqnal and the interfer-
ence can tie found and thus the output s i gna 1 - to- i nterference ratio
IS obtained.
Consider first the case where the coded PN sequence is assumed
synchronized at the receiver, i.e., '=0. The reference siqnal then
becomes
P(t) = A sin [,j.t + sin „^,(t-tp) < (t)]
(67)
It is shown in Appendix A that the steady state weiqht vector in this
case is given by
ss
- . 1 068
.5
> .1068/
(68)
where J,(X) is the Bessel function of order zero. We see that the
weiqhts'^all contain the fac tor
so the absolute magnitude cf the array response will vary with and
tp according to
J
0
sin
‘Tn_yo \
)
26
but there will be no change in pattern (relative response versus
angle) with t^ or p. This result is understandable since changing
1- and t does not change t. Moreover, the reference signal R(t) can
be written in the form
R(t) = .,Ri(t) + .2R2(t)
(69)
where R](t) is perfectly correlated with the desired signal and
R^(t) is uncorrelated with the desired signal. Increasing B or t^
increases and decreases
The resulting attenuation of the desired signal (which is
denoted by ADS) is plotted in Figure 10 versus the modulation phase
delay (..in,tQ/2) with B as a parameter. When B is small (b '< ^/2),
the narrowband FM signal has most of its power concentrated in the
carrier. The desired signal is highly correlated with the reference
signal regardless of t^ and thus ADS is negligible for all t„. When
(p ■' n/2), ADS becomes significant. At some values of to, there is
no correlation between the desired and reference signals at all.
Therefore the output desired signal power is zero.
Theoretical ly , because of the PN coded phase modulation on the
desired and reference signals, the infinite time average of the pro-
ducts of I(t)R(t) and I(t)D(t) is zero.* There is no interference
power at the array output.
The above results indicate that to maintain low ADS, the modu-
lation delay must be minimized. For small modulation index fi, the
requirement on tg can be eased. If, for example, a system can toler-
ate 10 dB attenuation on the desired signal, and 6=1, the modulation
delay cannot exceed one tenth of the minimum audio cycle.** If P=l,
the limit on t^ is one quarter of the minimum audio cycle.
Now we consider the opposite case where there is no modulation
delay, i.e., tQ=0, but there is code delay i. The reference signal
is
* See the footnote on Page 57.
**The modulating signal we envision is an audio signal whose maximum
frequency component gives the minimum audio cycle.
(70)
R(t) = A sin + . sin
In this case the steady state weight vector is found in Appendix B
to be
('' \
I -.1068
1 1
Y .1068/
Rp(')
(71)
where Rp(i) is the autocorrelation function of the PN code. For a
PN sequence of long period, Rp(0 can be approximated by
f
1
Rp(0 =
1 + t:
0 > 1 1
-T. < t ' 0
i — —
(72)
0 elsewhere
L
The steady state weight vector in Equation (71) is a product of a
column vector and a factor Rp(r). This column vector is shown in
Appendix B to be a function of the signal amplitudes and arrival
angles which are assumed constant in this case. Hence the array pat-
tern is fixed for all i. However, the absolute magnitude of the
array response varies with the code timing offset since t affects
the correlation between the desired and reference signals. Note that
1- is not a parameter here because we assume the FM modulation in the
reference signal to be periect.
The attenuation of the desired signal versus the percentage
code timing offset is shown in Figure 11. It is seen that the ADS
increases as the code timing offset increases. When the timing is
offset by one bit interval, there is no correlation between the
desired and reference signals. Thus the desired signal is attenu-
ated infinitely by the array.
The theoretical results based on an infinite time average show
that the interference signal in this case is not correlated with the
29
Figure 11. Effect of ’he code timing offset on the output
desired sinnal attenuation.
I
i
I
I
I
reference signal at all.*
ference signal completely.
Therefore the array nulls out the inter-
From Figure 11, it i' seen that the code tiniiru) offset must
he kept small for satisfactory array performance. In the delay lock
loop, the code timing tracking error must be kept less than 68 of
a bit interval if 10 dB desired signal attenuation can he tolerated
in a system.
In the next sub-section, we examine the performance of the
array when the reference signal does not contain the FM modulation.
^ • Reference Signal Wi thout
FM_ Modu lotion
In this sub-section the performance of the adaptive array is
studied when the reference signal is a simple bi -phase modulated
signal of the form
R(t) = A sin [oof^t + :(t-i)] , (73)
where the frequency wr is equal to ui^., t(t-T) is a PN sequance which
has the same waveform as that in the desired signal except for the
time difference i. For simplicity, in this sub-section we assume
■♦(t) to be a square wave at frequency i)^. The incoming signals are
defined in Equations (59) to (62).
To study the array performance with the reference signal given
above, the IMS algorithm is utilized to find the approximate solutions
to Equation (64). The discrete form of the IMS algorithm is simulated
on the computer for this purpose. Each weight is adjusted according
to the rule[l]
Wi(j+1) = w^(j) - 2kp>(j)x^(j) (74)
where j is the sample index and k^ is the digital feedback loop gain
constant, kp is chosen so the digital feedback loop is stable and
has the lowpass filter characteristics as discussed in Chapter IV.
The relation between kp and the analog loop gain constant k/\ has
been found[10] to be
* See the footnote on page 57.
31
1 £
- - - ^
max^S
where Tc is the sampling period. From kp, the digital feedback loop
wncrc ic 'o tiic luainpi my i uu . i i um f
bandwidth and its time constant are[10]
Unfortunately , if typical design frequencies are used with this
digital algorithm, excessive computer time is needed before the
weights reach the steady state. For example, consider f^'|p = 100 Hz,
f^=f^ = 10 MHz, 15=2.5 10‘8, A=l, B=10, 0-=60". The maximum
and minimum eigenvalues of the covariance matrix in Equation (65)
can be shown to be*
= (A^ + B^) * I A“^ + B^ + 2A^B^ cosn (sine
r:
d '
With numerical values substituted into Equation (78) we have =200
and Satisfying the bandwidth requirements of Equation (48)
gives"’’
Bd ■ I
For Bq = 2tt10, we find kp = 3.927 x 10"^ and i[j = 6.3694 x 10^ T5.
The matrix f that results for the signals defined in Equations (59)
to (62) is derived in Appendix A, Equation (85).
I
I
I
llciH C tof th(‘ .irrc^v the rf'soonsr tii.'ie (^<1/ 5 tinp ( ons frinfA ,■
is 3.1847 X 10'^^ iterdtions. W i Ui the size of tfi(- computer proqrjm
noeded in this simulation each iteration reouires about 3.6 x 10"
seconds), imprac t ica 1 1 y loruj i omputer runnin'j t iums result. In order
to iru reuse the simulation speed, large values of kp and scaled fre-
quencies must be used. However, large values of kp result in a wider
feedback loop bandwidth. A compromise is made on fmin '’^d f. to
satisfy the bandwidth reiiuirement in Lquation (48). Tnus, in' these
simulations, we choose ^ 4 kHz, c" d^ •'D ^ -00005,
it = 2.1978 X lO"' seconds, Bq = x 10^) rad/sec and p =
5(3000T^. With these choices the computer time is greatly reduced
and the inequality (48) is still met. The corresponding analog time
constant in the real array is found by Equations (41) and (75) to be
.01098 seconds.
rhe initial values of the weights are chosen to be
/
W(0) =
0
0
(80)
\o/
At the beginning of the simulation, each weight in the array goes
through a transient. After the weight transient has ended, and the
weights have become stationary, m successive samples* of each weight
are stored on a magnetic tape. From these weights, the average out-
put power of the interference or the desired signal is calculated.
A computer program has been written to perform the tedious cal-
culations. The values of the modulation index , the code timing
offset ' and the desired carrier frequency offset can be varied so
that their effects on the array performance can be studied.
* The steady state array weights have fluctuating components due to
r* the spectral lines of the products, l(t)R(t) and D(t)R(t). The
value of m is so chosen that m samples form a suitable average
over these fluctuations.
. r
i
33
J
First, we consider the transient pattern behavior of the array.
We assume that , 1.2, f, 20 KHz and that there is no frequency
offset or code timing offset. The reference signal is given in
fquation (73). The starting weight vector has been set equal to W(0)
(Equation (80)), so the initial pattern is omnidirectional . A desired
signal is assumed incident on the array from broadside ( fj=0*"') and an
interference signal from i=:60'’. The results are shown in Figures 12-
17. Figure 12 shows the initial pattern. Figure 13 shows the pattern
after 2 time constants, and so forth, up to Figure 16 which shows the
pattern after 4 time constants. Figure 17 shows the steady state
pattern at t--. It can be seen that the final array response is about
65 dB weaker in the interference direction than in the desired signal
direction. Note that the input s i gna 1 - to- i nterf erence ratio is -20
dB. figure 18 shows the transient behavior of the array weights.
Figures 19a and 19b show the steady-state attenuation o^ the
desired signal (ADS) and the output signal -to-interference ratio
(SIR), respectively, as a function of the modulation index . with
the code frequency f. as a parameter. The input signals are defined
in Equations (59) to (62). The reference signal is given above. We
assume the code timing is synchronized at the receiver (i=0) and there
is no frequency offset, . ...|=.;^j^=455 KHz. In Figure 19a the ADS is
plotted versus h. The data shown have been obtained three different
ways. First, the solid curve has been obtained from theoretical cal-
culations. These calculations are contained in Appendix D. Next,
the points marked as "x" and "o" have been obtained by computer simu-
lations of the LMS loop, whose bandwidth is fixed at 2rr(1.45 x 10^)
rad/sec. The x's are obtained with f, = 20 KHz and the o's with f. =
200 Hz. It is seen that regardless of the switching frequency f.,
the theoretical ADS agrees closely with the simulated values. Tfiis
is understandable because the correlation between the desired and
the reference signal is independent of f^,. The theoretical steady
state weight vector is found in Appendix D to be
i' ^
y.l068 J
Jo(B)
(81)
The absolute magnitude of the array response is scaled by the factor
J„(i ) but 1- does not affect the relative array pattern, h changes
the correlation between the desired and reference signals and the
result is that all the weights are scaled by the factor J„(h). At
some particular values of . , which decorrelate R(t) and D(t) com-
pletely, the weights become zero and hence ADS becomes infinite.
34
■V
5T3B1
Dlt)
I
i
o“
180®
Figure 12. Initial omnidirectional pattern.
Pattern computer at f=455 KHz.
= .1
kp = .000005
= -20 dB
0. = 60'’
"d - 0
fin = 4 kHz
f, = 20 kHz.
r
35
I
I
(
r
D(t)
0“
100®
figure 14. Pattern after 2 time constant.
Pattern computed at f=455 KHz.
i
I
'1
D(t)
I
0”
180*
Figure 15. Pattern after 3 time constant.
Pattern computed at f=455 KHz.
38
•!— Wi '*l"i
-90
D(t)
I
0°
180“
Figure 17. Final pattern. Pattern computed
at f=455 KHz.
40
theoretical
MODULATION INDEX [3
Figure 19a. Effect of the modulation index on the output
desired signal attenuation.
1
i
In genet'dl, as the modulat on index increases, the desired signal
bandwidth increases and th< reference signal without IM becomes a
poorer estimate of D(t), S" there is less correlation between them,
and the output ADS increas'-s.
Figure 19b shows the corresponding output SIR derived from the
computer simulation of the LMS algorithm. (The output SIR as obtain-
ed from a theoretical calculaton based on Fguations (64) to (66) is
not meaningful , because the infinite time average conipletely decor-
relates the interference and reference signal. As a result there is
no output interference pow' r in that model.) If we compare these 2
output SIR curves with the corresponding ADS curves in Figure 19a,
it appears that the output interference power is constant for all
We would expect this rtsult, since the correlation between I(t)
and R(t) is independent of . . The output SIR decreases when the
output desired power decreases, because . affects the output desired
power .
Moreover, for a fixed . the array suppresses the interference
much more when f^ = 20 KHz than f. = 200 Hz. This result occurs
because the higher f,, the more the spectral components of the pro-
duct R(t)I(t) are spread beyond the bandwidth of the feedback loop.
For f. = 200 Hz, the first harmonic of R(t)i(t) at 200 Hz lies within
the feedback loop bandwidth. As a result, this harmonic contributes
to the weight jitter and causes the array not to null the interfer-
ence as wel 1 .
From the above results, if the attenuation of the desired signal
must be held within 10 dB, the value for B must be chosen less than
1.8. Therefore this type of reference signal is suitable only for
narrowband FM. It is also seen that the array will null the inter-
ference signal effectively provided the correlation product of I(t)
R(t) does not have signifitant power within the feedback loop band-
width, that is, when the code frequency f,,. is high enough. (Recall
however, that f. must not exceed fmin! (See Equation (45)),)
Next, we consider the effect of the square wave timing offset
on the array performance. We assume no frequency offset on the
desired carrier. Figures i Oa and 20b show the attenuation of the
desired signal and the out(ut signal-to-interference ratio as a
function of the percentage square wave timing offset with b as a
parameter. The conditions of the input signal are the same as those
in Figure 19. The switchiig rate is set equal to 20 KHz. In Figure
20a, for a fixed r the ADS increases as the timing offset increases.
If a maximal length PN cod« is used in this simulation instead of
Figure 20b. Effect of the code timing offset on the output
signal-to-interference ratio.
46
STS
I
f the '^qu<^re Wfive, the desirid ''.iqn.il at t^■n^dtlo^l i •, approx iriate i y
7 dB* (for 0) at bO roil< timiruj offset rather than 36 dR (.
as showti for the sq.jare wave. To understand this statement, consider
ihe simple case where . 0. If ttiere is no fM mrrdulation on the de-
irt'd sicjiial, both the des red and ttie tefereruc signals arr> bi-
lihase modulated sine waves Ttie ( rv>ss c orre iat ion between these two
sivjnals is egu.rl to the au' ocorrel a t i on function of the PN code or
ihe square wave, both of w* ich arf> shown in '"iQure 21. We see that
t.ne correlation becomes zei o at a 1/2 bit offset tor ttie square
wave, while it does not liei ome zero until a one hit offset for the PN
.code. Due to the symmetry property of the square wave, the ADS fron:
60 to 100 square wave tiiinq offset is just the mirror image of
that from 0 to 60 . This portion of the curves is not shown in
figure 20a. Of course, if a maximal length PN cole is used, the
sviimetry property no longei exists. For a fixed square wave timing
offset, the ADS increases .is i becomes large. This effect is to be
expected, since the reference signal is a simple bi-phase modulated
sine wave, which becomes a poorer estimate of the FM desired signal
as f. increases.
21. The autocorrelat ion function of a square
wave ( ) and of a maximal length
PN code ( ). (T, is the bit interval.)
n is the number of stages in the shift
register used to generate tfie code.
If i-.-O and a maximal length PN code is used, this case is identical
to that in F igure 1 1 .
In riqurp 20b, we see that the output SIR drops as the square
wave timinq offset increas<s. The decrease of the SIP is primarily
due to the desired signal beinq attenuated. Since the phase modula-
tion is not present on the interference signal, the correlation be-
tween the interference and reference signal is not affected by the
square wave timing offset. Furthermore, with f. set at 20 KHz, the
output interference power 15 negligibly small.
from the above results, we see that for a maximal length PN
code sequence, the code timing difference between the desired and
reference signals should not exceed half a bit.
Finally the effect oi the desired signal carrier frequency
offset on the perfonnance nf the array is studied in Figures 23a
and 23b with as paramete* . We assume the desired carrier deviates
from ^-(2-(455 x 10^) rad/sec) (to simulate transmitter frequency
inaccuracies or Doppler shift). Thus the desired signal is given
by
D(t)= A sin [(.(•• .)ti- sin ,^,t + t(t)J
(82)
where - c reference and interference signals are
assumed to be the same as those in Figures 19a and 19b, i.e..
R(t) = A sin [cij t + ; (t)]
(83)
and
I ( t ) = B sin t
(84)
-{(t) is a square wave at f . =20 KHz in this simulation. For B ^ 2,
the Fourier transform of t^ie correlation product D(t)R(t) is shown
in Figure 22. The frequency offset has eliminated the dc term of
D(t)R(t) as seen in Figure 22. This dc term determines the corre-
lation between the desired and reference signals. As Ai.; increases,
the desired-reference product has less power within the feedback
loop bandwidth (2n(1.45 x 10^) rad/sec in this case). Figure 23a
shows the desired signal attenuation as a function of the frequency
offset. We see that for a fixed p the desired signal attenuation
increases as the frequency offset increases. When Am is greater
than the feedback loop bandwidth, the significant component of R(t)
D(t) at Am is filtered out by the feedback loop. Thus the desired
signal suffers a heavy attenuation. For a given percentage frequency
offset, the ADS increases when n increases. Again, this effect is
-20 -16 -12 -8 -8
0 4 8 12 16 20
( UNITS IN 2tt X 10^ )
SEC
Figure 22. The correlation product of the desired
and reference signal. ( u varying).
expected because increasing j- decreases the correlation between the
desired and reference signals.
Figure 23b shows the output s i gnal - to- i nti'rf erence ratio versus
the frequency offset. It is seen that for a fixfd . , the output SIR
decreases as the frequency offset increase';. It we compare the two
MR curves with the 2 ADS curves in Figure 2 3a, it is seen that the
output interference power is constar)t for all freguencv oftsets.
This behavior is what we would expect, because the frequency offset
on the desired carrier does not aMect the correlation between the
interference and reference signals.
From the above results, we (onclude that to maintain high cor-
relation between the reference and desired si'inals, the desired signal
carrier offset must not be larger than the feedtiack loo(> bandwidth.
The feedback loop bandwidth, in turn, is constrained to be less than
the minimum audio modulation frequency (see Fquation (47)). As a
result, this type of system is rather sensitive to frequency offset.
49
ii
Figure ?3a. Effect of the frequency offset on the
output desired signal attenuation.
50
4.4 -.3 -.2 - I 0 .1 .2 .3 .4
I % FREQUENCY OFFSET
V
{.
I
Figure 23b. Fffect uf the frequency offset on the
I output s ignd 1 - to- i nter feren{ e ratio.
I
51
CHAPTER IV
r
*.(
CONCLUSIONS
An approach for inteorating an adaptive array into a conven-
tional FM coninunication system has been suggested. The method con-
sists of adding an extra digital pseudonoise coded phase modulation
to the desired signal in addition to the FM.
Two types of reference signals were proposed. One of them
includes the FM modulation while the other does not. Both types con-
tain the PN coded phase modulation which appears on the desired sig-
nal. To generate such reference signals, the FM modulating signal
and the PN code must be extracted from the received signal. A
method of doing this was suggested.
Theoretical studies and digital simulations of the array per-
formance with such reference signals have been done. The results
indicate first that the performance of the array will be much better
if the FM modulation is included in the reference signal. However,
when the FM is present on the reference signal, the FM modulation
delay must be kept small. The permissible delay depends upon the
modulation index. For example the modulation delay cannot exceed
about one tenth* of the minimum audio cycle** for b=3. The permiss-
ible delay is about a quarter* of the minimum audio cycle for B=1 .
When the FM modulation is not included in the reference signal
(which makes the reference signal simpler to implement), the array
operates properly only for small values of modulation index (b <<
1.8*). This approach is suitable only for narrowband FM.
For both kinds of reference signals, the code timing offset
must be kept within about half a bit if a maximal length PN code
is used.
* These values are based (in the assumption that a system can
tolerate 10 dB attenuation on the desired signal.
** See the footnote on Page 57.
52
Final ly, the array p(rformance for either type of reference
signal is very sensitive to the desired signal frequency offset.
The frequency offset on the desired signal carrier should he less
than the feedback loop bandwidth, which in turn has to be less than
the minimum modulation fre<|uency. For f;-l, the maximum frequency
offset is approximately 60 * of the feedback loop bandwidth. These
results mean that the frequency must be accurately controlled (or
tracked) in such a system.
* We assume a system can tolerate 10 dB desired signal attenuati n.
53
REFERENCES
1. Widrow, B., Mantey, P.E., Griffiths, L.J., and Godde, B.B.,
"Adaptive Antenna Systems," Proc. IEEE, 55, 12 (December 1967),
pp. 2143-2159. ~
2. Riegler, R.L. and Compton, R.T., Jr., "An Adaptive Array for
Interference Rejection," Proc. IEEE, 61, 6 (June 1973), p. 748.
3. Applebaum, S.P., "Adaptive Arrays," Special Projects Laboratory
Report SPL-TR66-1 , August 1966, Syracuse University Research
Corporation, Syracuse, N.Y.
4. Stiffler, J.J., Theory of Syncrhonous Communications, Prentice-
Hall, inc., Englewood Cliffs, New Jersey, (l9?l), p. 178.
5. Birdsall, T.G. and Ristenbatt, M.P., "Introduction to Linear
Shift-Register Generated Sequences," The University of Michigan
Research Institute, Report 90 (October 1958).
6. Ziemer, R.E. and Tranter, W.H., Principles of Communication,
System, Modulation and Noise, Houghton Miff! in Co. , Boston
(1976), p. 134.
7. Gill, W.J., "A Comparison of Binary Delay-Lock Tracking-Loop
Implementations," IEEE Trans., AES-2, 4 (July 1966), p. 415.
8. Spilker, J.J., Jr. and Magil, D.T., "The Delay-Lock Discrimi-
nation— An Optimum Tracking Device," Proc. lER, 9
(September 1961), p. 1403.
9. Spilker, J.J., Jr., "Delay-Lock Tracking of Binary Signal,"
IEEE Trans., SET-9 (March 1963), p. 1.
10. Chan, L.C. and Compton, R.T., Jr., "An Adaptive Array Technique
For AM Signals," Report 4326-3, January 197/, The Ohio State
University ElectroScience Laboratory, Department of Electrical
Engineering; prepared under Contract N00019-76-C-0195 for
Department of the Nav/.
11. Brennan, L.E., Pugh, E.L. and Reed, I.S., "Control Loop Noise
in Adaptive Array Antennas," IEEE Trans., AES-7, 2 (March 1971),
p. 254.
54
I
1?. ►;)l(>s/cir, G.L., "The Stortiastic Properties of the Weights in
rin Adaptive Antenna Array," Report 4063-1 , August 1975, The
Ohio State University ElectroScience Laboratory, Department
of Electrical Engineering; prepared under Contract N00019-75-C-
0179 for Naval Air Systems Conmand.
13. Miller, T.W., "The Transient Response of Adaptive Arrays in
TDMA Systems," Report 4116-1, June 1976, The Ohio State
University ElectroScience Laboratory, Department of Electrical
Engineering; prepared under Contract F3060P-75-C-0061 for Rome
Air Development Center.
APPENDIX A
In this appendix, we calculate the steady state weights for
' the case where the reference signal is
R(t) = A sin [(.)(-t + r sin M^Ct-to) + <t(t)] . (67)
The in phase and the quadrature array signals are given by Equations
(59) to (62),*
x-|(t) = A sin + i-i sin w^t + (}>(t)] + B sin ui^t (59)
x^(t) = A cos [u)(.t + p. sin ij^^t + (>{t)] + B cos w^-t (60)
\ ^3(t) = A sin [to^t + P sin oj^t + <ti(t) - y^j]
+ B sin(w(.t - y-j ) (61 )
and
I x^(t) = A cos [u^t + 6 sin uj^t + >^(t) - yj]
+ B cos(iD(-t - y^- ) . (62)
The steady state weight vector is found from
!
i
* Here we assume there is no frequency offset on the desired signal
carrier; is thus equal to
56
with t and S defined in Equations (65) and (66) respectively.
For the signals given above we have
2 2
0
a2
rV
b2 .
■5- cos t cos I ^ 0-sinYH-* — sinY^
i 2
'd’2
-A^ . B" . ,
^ cosyj+j cosy^- -^sinv^-j smv ■>
,2
A‘- . B^ . A^ B'
sin ' ^ ^ C0SYq-*^C0SYi
A^ B^
i 2 2
A^ . B^ A^ B^
2 2
2 2
(85)
and the elements of the S matrix can be calculated as follows
X](t)R(t) = AB sin u)(-t sin[o)(.t + 6 sin ,.)^(t-to) + f(t)]
+ sin[w(-t + 6 sin u^t + (}i(t)]
sin[uj,t + 6 sin w^(t-tQ) + <>)(t)] . (86)
Since the PN code is included in the reference signal but not in the
interference, the infinite time average eliminates the first term
in Equation (86),* Thus
Xi(t)R(t) = lim —
V2
-V2
^lim ^
rV2
J-Tm/2
cos[b(1-cos u)^tQ)sin oj^t
+ 6 sin utp,tQ cos io^t]dt
cos[p6 cos(u)^t - y)] dt
(87)
* We assume that If w =u),„, a spectral line of I(t)R(t) will
fall at dc and the above analysis will not apply. The real modu-
lation signal we envision is an audio signal with power distri-
buted continuously between umin ^nd wn,ax. We then choose u>^ to
be less than to prevent a dc term in I(t)R(t).
57
where
•Jf -
cos
“Itl
to)‘
+ sin‘
‘in^o
= 2 sin
, t
m 0
(88)
> = tan
V ,
(89)
and Ty is a time window over which the integration is performed.
Since the integrand is periodic, the infinite time average may be
replaced by an average over one period of the integrand
2 ^
Xift)R(t) = A m
2 2” i-./.
C0s[pB C0s(Mn,t - 0]dt
If we let x=ui^t. Equation (90) becomes
2 r
;(t)R(t) =3-?^ I COs[pB COS(a,^t - y)]dt
2 2n
(90)
Jjj(dB)
(91)
where Jq( ) is the zero order Bessel function. Finally
xTrtTRTty = ^ Jq
Similarly we find
7^(tWUJ = 0
2b si
■ /‘mV
,(t)R(t) = ^ (cos Y(j) J
2b sinl-p
lii t
m 0
and
X4(t)R(t) = 2~ (sin ^^j)
2b si
<-V)
(92)
(93)
(94)
(95)
The elements of the S matrix all contain the zero order Bessel
functions of argument 2Bsi n(ojn,tQ)/2) ,
(96)
The steady state weights can be obtained by substituting the inverse
of <}> in Equation (85) and Equation (96) into Equation (64).
If we let 0H=O°
easily be computed.
, 0-60°, A
The result
=1 and B=10,
is
the inverse
/. 52804
0.
,47195
.21361 \
. 52804
-.21361
.47195
, .47195
-.21361
. 52804
“■ I
\. 21361
.47195
0.
.52804 J
The steady state weight vector corresponding to this special case is
-.1068
/(i) t
26 sin( (D °)
.5
0
L V 2 /
y.l068 /
(98)
59
APPENDIX B
In this dppnndix we assume the PN codes in the reference and
desired siqnals are not synchronized. Thus the reference signal
has the form
R(t) = A sin[.,(-t + e sin .op,t + <t>(t-i)] . (69)
The in phase and quadrature array signals are defined in Appendix A.
To calculate the steady state weights, we must know the matrices
: and S. Since the input signals are the same as those in Appendix A,
the i matrix does not change (see Equation (85)). The elements of
the S matrix can be calculated as follows:
2
xi(t)RftT = r ^cos[^(t) - f(t-i)]
- cos[2u)(.t + 23 sin (o^t + cj>(t) + 4>(t-T)]l .
(99)
The infinite time average eliminates the second term and hence
Equation (99) becomes
xY(tTRrtT = f- pitypTt^ (100)
where P(t) is a periodic PN code of period Tp as shown in Figure 2b.
The limiting process in Equation (100) is then not necessary and
X] (t)R(t) can be simplified to give
(■^P
2 P(t)P(t-T)dt
. f-
(101)
where Rp( i ) is the autocorrelation function of P(t) [5] and is shown
in Figure 21. If the number of stages in the shift register genera-
ting the PN code is large, Rp(i) may be approximated by Equation (72).
We also find
60
I
I
I
I
I
I
I
I
I
I
and
Rp(r)sin yj
(102)
(103)
(104)
With 6jj=0°, 0|=60°, A=1 and B=10, the steady state weight vector
can be computer as follows:
.52804 0.
.47195 .21361\ /.5^
.52804 -.21361 .47195 0
.47195 -.21361 .52804 0.
Rp(x)
.21361 .47195 0.
.52804/ V 0.
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65
APPENDIX D
In this appendix, we calculate the array steady state weight
vector when the reference signal is given by
R(t) = A sin[(i)^t + 'ii(t)] (106)
and the input signals are defined by Equations (59) through (62).
Note that the code timing in the desired and reference signals is
assumed to be synchronized and there is no frequency offset in the
desired signal (fD^=U(.).
The final weight vector is found from Equation (64), Since the
input signals are the same as those in the Appendix A, is the same
as in Equation (97). From the above signals, the elements of the S
matrix can be computed as follows:
X] (t)R(t) = A^ sin[ui(t + e sin + ({-(t)] sin[uj(.t + <t(t)]
+ AB sin u)^ t sin[ o^.t 4'(t)]
(107)
The infinite time average makes the 2nd term zero due to the absence
of the PN code in the interference signal. Equation (107) may be
wri tten
.2 (V2
x'lTtTRTt) = lim J - I cos(a sin .,„t)dt (108)
where T^ is the time window. Let x = ,,)j„t, and sut’Stitute an average
over one period for the limiting process. Then w(- have
2 71 2
ft)R(t) = ^ I cos(p sinx)dx = ^ Jq(b) . (109)
Similarly
66
xyrtwo = 0
(110)
a2
X3TT)RTtl = y- (cos ,j) j^(f-)
(111)
and
x^TtlRTiy = (sin .jj) Jg(iO
(112)
Substituting Equation (97) and Equations (109) through (112) into
Equation (64) yields
! .52804 + .47195 cos
-.21361
ss
.47195 + .52804 cos
.21361
y
■ r ■ ^o'®>
With A=1 and f(j=0. Equation (113) becomes
\
-.1068
ss
J,(H)
V .1068y
67
(113)
(114)