1942
IEEE TRANSACTIONS ON COMMUNICATIONS,
DECEMBER
1974
Correspondence
Lkear Amplkcation with Nonlinear Components
,
D. C.COX
Sa(t)
Abstract-A technique for producing bandpass linear amplification with nonlinear components (LINC) is described. The bandpass
signal first is separated into two constant envelope component signals. All of theamplitudeandphase
information of the original
bandpass signal is contained in phase modulation on the component
signals. These constant envelope signals can be amplified or translated in frequency by amplifiers or mixers'which have nonlinear
input-output amplitude transfer characteristics. Passive linear
combining of the amplified and/ortranslated componentsignals
produces an amplified and/or translated replica of the original signal.
I. INTRODUCTION
{COMPONENT1
E ( t ) cos w o t
TFig.
LINC amplifier.
.I.
and
Conventionalsolid-statelinear
power amplifiers are difficult to
build a t low microwave frequencies and impossible to build a t high
microwave and millimeter wave frequencies. In many transmitter
applications,such as the amplification of single-sideband (SSB)
signals or the amplification of a multiplexed set of several separately
modulated low-level carriers, a linear overall input-output relationship is required in the transmitter power amplifiers.
Nonlinear solid-state power amplifiers are readilyavailable a t
low microwave frequencies and constant-amplitude phase-lockable
signalsources
(GunnandImpatt
diodes andmagnetrons)are
available in the high microwave and millimeter wave region. Also,
for high-power applicationsin
the microwave region, nonlinear
electron tube amplifiers and power oscillators aremore readily
available than are linear amplifiers. A technique that makes use of
these available nonlinear amplifiers or phase-lockable oscillators to
produce bandpass linear amplification with nonlinearcomponents
(LINC) is described in this correspondence. The overall input-tooutput transfer function of these LINC amplifiers is linear over a
wide range of input signal levels but the internal R F amplifying
devices can be highly nonlinear or, in fact, even constant-amplitude
phase-locked oscillators.
+ 4(t) I
~ 1 , ( t )= i ~ , / 2 ) sin
~wot
SzG(t)= (E,/2) sin
Coot
- + ( t )1.
I
(4)
Note that SI,( t ) and Sza( t ) can be represented by constant-amplitude vectors rotating in opposite directions -with increasing values
of + ( t ) as in Fig. 2. With E (1) 2 0 the vectors occupy onlyquadrants
1 and 2. Stability requirements discussed later furtherrequire b ( t ) 5
~ / 2 Since
.
SI, and SZa can be separated, they can be amplified separat,ely in nonlinear amplifiers. In fact, the two amplifiers illustrated
in Fig. 1 can be two power oscillators which are phase locked or
injection locked to &,(t) and S,,(t). Subtracting GSz,(t) from
GSla(t), where G is the identical amplifier gain, yields the amplified
output
GXl, ( 2 ) - GSZ,( t )
=
GE, sin 4 ( t ) cos wot
=
GS, ( t ).
(5)
A similar argument can be carried through for the most general
bandpass signal [I]
11. T H E BASIC PRINCIPLE
The basicprinciple for thisLINC is separatingthebandpass
input signal which may have either or both amplitude and phase
(frequency)variations (i.e., AM and/or angle modulation) into
two component signals S1and 8 2 which are constant amplitudewith
variations inphase
only. Thesetwo
constant-amplitude anglemodulated signals can be amplified separately by anyamplifier with
sufficient bandwidth regardless of itsamplitude
linearity. The
amplified component signals are passively combined to produce an
amplified replica of the input signal.
Consider first a constant-phase bandpass signal
S,(t)
=
E ( t ) cos oot
(1)
with a real envelope E ( t ) 2 0 as the input to a LINC amplifier
component separator (see Fig. 1 ) . The component separator produces the two constant-amplitude signals, &,(t) and Sza( t ) , related
, t o S,(t) as follows. Substitute
E ( t ) = E, sin 4 (1) ,
(2)
where the constant E, equals the maximum value of E ( t ) and the
equation defines 4 ( t ) , into (1) to yield
111. AN IMPLEMENTATION OF A SIGNAL
COMPONENT SEPARATOR
The first step required in this LINC amplifier signal component
separator is to obtain the envelope E ( t ) and a constant-amplitude
angle-modulated term p ( t ) = K cos Coot e ( t ) 1. As illustrated
in Fig. 3, p ( t ) is obtained by limiting S ( t ) and E ( t ) is obtained
either from a linear envelope detector or froma synchronous detector.
The next step, illustratedin Fig. 4, is to separate SI ( t ) and SZ( t ) .
Consider first the feedback path around the amplifier with a very
large voltage gain, Gl = - Vo/V,. A phase-shifted p ( t ) is modulat.ed
by Vo(t) to produce
+
Ol(t)
=
K sin Coot
+ e ( t ) + klVo(t)].
The mixer output
Paperapprovedby
the Associate Editor for Communication, Electronics of t h e I E E E Communications Society for publicationwlthout
oral presentation. Manuscript received December 1 1 , 1973: revised July
26, 1974.
The author is with Bell Laboratories. Holmdel, N.J. 07733.
p ( t ) o 1 ( t ) = K2sin Coot
is filtered to yield
+ e t t ) + klVo(t)]cos Coot+ e ( t ) ]
(9)
1943
CORRESPONDENCE
The largest value, which occurs when VOis small and sin klVo(t) =
klVo(t),iskl.Thus,ifG~>>[(R1 R z ) / ( K z R 2 ) ] ( ~ / k(12)
1 ) , becomes
+
-KzRz .
E ( t ) = -Rl sin klV01t)
which is the same form as ( 2 ) . The approximation can be made as
good as required by making GZsufficiently large. The size of GZwill
be dictated by the distortion limits (see Section V) placed on the
overall LINC amplifier. If kl and K are sufficiently large, then GI
could be less than unity and the
baseband amplifier could be replaced by a passive summingnetwork. This should increase the
realizable bandwidth of such a component separator. If K , RI, and
Rz are chosen such that (KZ/2)(Rz/R1)= E,, then
% P
Fig. 2. Signal component vector diagram.
LIMITER
(a)
klVO(t) =
-+w.
(14)
The outputs from the component separator of Fig. 4 [see (9) ] then
become the desired signal components
4-b
ENVELOPE
+ e(t) - +(t)]
K sin [wet + 0 ( t ) + + (1) ]
& ( t ) = Ksin
OZ( t )
=
Coot
=
(2K/Em)Sz(t)
=
( 2 K / E m SI
) ( t ).
Of course, the feedback loop must be designed to satisfy phase shift
and gain conditions required for stability [a].
Note that if the phase modulator in the feedback loop does not
produce a linear phase change as a function of modulating voltage
Vo (i.e., if kl is a function of Vo), then the high-gain feedback loop
will compensatefor this imperfection by distorting VO(t) so that
sin [ k l ( V o ) V o ( =
t ) ]V l ( t ) and k l ( V o ) V o (=t )+ ( t ) . The only requirement is that the two phase modulators shown in Fig. 4 have
the same modulation characteristic kl(Vo). (If a phase modulator
with a sin-’ characteristic could be realized then the feedback loop
could be eliminated.) The matched modulator requirement can be
removed by providing a second similar feedback loop with its own
high gain amplifier, phasemodulator,etc.,forproducing
Oz(t)
directly from E ( t ) . The second loop would be identical to the one
shown but driven by -E(t) to produce a phase modulator output
of 0 2 ( t ).
(C)
Fig. 3. Detectors and limiters.
IV. FREQUENCYTRANSLATIONWITHIN
LINC AMPLIFIER LOOP
LOW PASS
FILTER
(15)
A
MODULATOR
In many microwave or millimeter wave transmitter applications
it is desirable to translate a signal in the 10’s or 100’s of MHz to a
higher frequency and linearly amplify it. This is not possible within
the current state of the millimeter wave solid-state device art. It
can be done, however, by separating the low-frequency signal into
components O1( t ) and 02 ( t ) , translating thecomponents in frequency
with the samemixing oscillator and amplifying as indicated in Fig. 5
to produce
01m(t) = (2K/Em) sin [(WO
+
+ e o ) - +(t)l
WI)~
and
PHASE
MODULATOR
TKsin
o,(t)
’
The mixers and amplifiers can be nonlinear. The outputis the ampliw.
fied input translated to W O
+
kot+8(t)I
Fig. 4. Signalcomponent separator.
(4KG/Em)Sm(t)= (4KG/Em)E(t)COS
V,(t) = (K2/2) sin [k,Vo(t)]
(10)
which has the proper slope for closed loop stability as long as
I k,VO(t) I i T/2.
Assume that the amplifier input impedance is large compared to
Rl and RZso that
Combining V o ( t )= -GlVi with (10) and (11) yields
vo(t)
+
Gz
(K2/2) Rz sin k1Vo ( t )
(R, Rz)VO( t )
+
.
(12)
Stability requires that E (1) be restricted so I k’Vo(1) I 5 ~ / 2 .With
( t ) ) is 2kl/x.
this restriction the smallest value of [sin k1V0( t )]/ ( VO
[(WO
+
+ e(t)l.
W I ) ~
(17)
Frequencytranslationwithina
LINC amplifier may find application in point-to-point and satellite microwave and millimeter
wave repeaters and for amplifying the multiplexed sets of modulated
carriers used in future high-capacity mobile radio base stations.
V. SOMECOMMENTS ON DISTORTIONANDNOISE
Noise and distortion ( N and D ) affect the output differently depending on where they areintroduced into the LINC
loop. A detailed
analysis has not been made but several aspects can be seen from the
equations in the previous sections. First, since a LINC amplifier
is linear, the signal-to-noise ratio at the input will be preserved
in the output.
N or D added within the LINC can be considered in two classes.
Symmetrically added N or D affects both & ( t ) and Sz(t) identically
and usually occurs before the component separator. Assymetrically
1944
IEEE TRANSACTIONS ON COMMUNICATIONS,
DECEMBER
1974
BAND PASS O m ( t )
FILTER
cos
W,
t
OSCILLATOR
BAND PASS
Fig. 5 .
added N or D affects only one component
component separator or amplifier.
Frequency translation within a LINC amplifier.
signal and occurs in a
Symmetrically Added Noise or Distortion
Symmetrically added N or D may occur in the limiter or envelope
detector (Fig. 3). Assume that there exist amplitude fluctuations
on the limiter output of K = K , ( l
6,) where 6, << 1. Then from
(13) with
+
‘C
Level of input E ( t ) . Again, the LING amplifier does not ,enhanct
noise which enters a component signal path.
Effects of Finite Gain in the Signal Component Separator Feedbacl
Loop
The effect of finite gain in the feedback loop is obtained by using
(13) in (12), and (9) to yield the LING output.
K,2Rz/2R1
(21:
and higher powers of 6a neglected, the LING output is
Then, consider the distortion produced on a two-tone test Signal 0
the form
S2(t) =
Thus, 6, will appear on the output as an amplitude variation of 26,.
Phase fluctuations on the limiter output will appear on the LING
amplifier output unchanged.
Forthe
envelope detector,with E(1) = E, ( t ) E n d ( t ) , the
N or D , End ( t ) , appears. in the LING amplifier output in the same
proportion as at the detector output.
+
=
+ sin wtt
-
E z ( t ) COS
sin wzt’
[wet
+ e z ( t )1
(22:
+
where wo = (w1 WZ) /2, and we = (a1 - 02) /2.
Substituting (22) into (21), writing theoutput as a Fourie
series, and solving for the signal power S and distortion power L
with ( Rl Rz)/ (klGIR1) < 1 yields
+
Asymmetrically Added Noise or Distortion
Phase fluctuation in a single component path is one type of asymmetrically added N or D . Consider the output as
S a ( t ) = (GEm/2) {sin [mot
+ + ( t )+ & ( t ’ ) ]
- sin Cwot - + ( t )
where
end
+ e,(t) + e n d l l
(19)
If R1 = Rz then klGl of 100 reduces distortion so that S/D =
50 dB. Thus, the loop gain requirements on this implementatio~
of a LING amplifier are not toosevere.
Even though analysis does not indicate any requirements whicl
will prevent LING from being successful, it has shown the inheren
sensitivity to balance (end and &). Thus, realization of an amplifie
with low distortion and wide dynamic range requires careful design
(20)
VI. LABORATORY DEMONSTRATION OF THE
FEASIBILITY OF COMPONENT
SEPARATION AND
RECOMBINATION
is the N or D. The output can be rewritten as
‘COS
+
1
@ ( t ) cos
[Wet
+ & ( t ) f ‘@,dl.
The sign of (1 cos end)1’2is always positive but the sign of (1 cos Ond)i’2 depends on the quadrant of end.
One effect of endis a phase fluctuation in the output of f e n d . Also,
for very small values o f . E ( t ) , the second term in the brackets defines a minimum LING amplifier output level with envelope = &a.
A small modulation of the desired envelope E ( t ) is also produced
by end.
As an example consider end uncorrelated with E ( t ) and with a
peak amplitude of 0.02 radians. This produces a peak fluctuation
in the outputenvelope of about E, x 10-2 which 1s of the sameorder
as the noise in the component signal path. Thus, the LINGamplifier
does not enhance phase noise which enters a component signal path.
A similar consideration of an amplitude fluctuahion, 6, << 1, in
a component signal path shows that for large signal levels, 6, produces a small output phase perturbation of the order of 6,. It also
modulates the envelope by a factor of approximately 1 $6,. When
the peak value of E ( t ) / E , becomes of the same order as the peak
value of 6,, extraneousphase andamplitude fluctuationson the
output become quite large andthe
amplitudefluctuations
and
phase fluctuations due to 6, become overpowering at about the same
+
for a two-tone test signal with maximum envelope swing.
The key problems in realizing LING are producing the constan
envelope signals and maintaining thephase and amplitude balancc
A nonoptimum realization Of two component separators (similar t
Fig. 4) and a combiner were assembled from available laborator,
components to demonstrate the feasibility of the LING techniqut
Envelope detectors and limiters were not included. The operatin
frequencies, bandwidths, etc., were determined by available corn
ponents. No attempt was made to optimize any component sinc
the only objective was to demonstrate that the twoconstant-ampli
tude phase-modulated signals could be generated and combines
to produce a specified envelope varying narrow-band signal.
A two-tone test envelope [fe = 10 kHz = we/27r in (22)] wa
applied tothe two narrow-band component separators (fo = 7
MHz = w 0 / 2 7 r ) . The “two tones” a t f~ = fo
fe and fz = fo out of the combiner [see (22) ] were examined with a spectrur
analyzer. At maximum output for this nonoptimized assembly th
strongest spurious signal was 22 dB below the level of the two equr
amplitude “tones,” fl and fz. At a 9 dB lower output the stronges
spurious signal dropped to 32 dB below the level of the tones. Thi
+
1945
CORRESPONDENCE
simple experiment demonstrated the feasibility of the LINC tech- above
and below threshold operation, demonstrate the accuracy of
nique, but did not give any indication of the minimumspurious
the approximateexpressions.
signal level possible from a well-designed amplifier.
REFERENCES
I. INTRODUCTION
[ I ] M. Schwartz, W. R . Bennett,and S. Stein, CommunicationSystem
New York:McGraw-Hill. 1966, sec. 1-6, 1-7. 1-8,
andTechniques.
Simulationresults are presented in Snyder [l, pp. 98-1041 for
the F M demodulator described by
and ch. 4.
[ 2 ] J. G. Linvill and J: F. Gibbons, Transistors and Active Circuits.
New
York: McGraw-H111. 1961.
J. J.
D’Azzo and C.H. Houpis, Control System Analysis and Synthcsis.
New York: McGraw-Hill, 1960.
Comments on “Survivability Analysis of Command and
Control Communications Networks-Part I”
where
F = [ O0
ROBERT E. LAWRENCE
In the above paper,’ Dr. Frank points out many problems commonly encountered in the analysis of command and control comof the
munication (C3) systems. In particular,hemakesmention
erroneous results which arise from having “ignored the network’s
structure once communication probabilities have been calculated.”
An example of the difficulties which arise due to network structure is given by considering a simple system (see Fig. 1 ) whose link
availability is assumed to be 90 percent. As Dr. Frank points out, it
is incorrect to say that theprobability that neither A nor B receives
the message is given by the calculation
(1.0 - 0.81) (1.0 - 0.73)
+ (0.9) (1.0 - 0.81) (1.0 - 0.9)
- 0.9)
1,
G = [ (2k)1/2
and
n = -kC.ZN
x^(t) represents an estimate of z ( t ) and (1) is called the estimator
equation. r ( t ) is the observable form of
r ( t ) = C sin
Coot
+ Mzl ( t ) ] + n ( t )
(3)
where n ( t ) is a white Gaussian noise process with spectral densityN .
The desired message is the second component of the process generated by
~ ( t=
)
Fz(t)
+ Gw(t)
where w ( t ) is a white Gaussian process, independent of n ( t ) ,with
unit spectral density. The spectrumof the desired message is
0.051.
However, the suggested correct solution is also incorrect.
The proper solution is given by
(1.0
‘1,
-k
=
0.1171.
Missile Complex
A
The demodulator described by (1) and ( 2 ) is based on small error
approximations tothe
conditional densityfunction
derived by
Kushner [a]. The demodulator is a special case of the extended
Kalman filter [ 3 , p. 3381.
Equation (2) is called the gain or variance equation. The steadystate solution of ( 2 ) is
Commnd
Port
Missile Complex
B
Fig. 1. Exampleillustratingthe
effect of overlappingcommunication
paths.
Paper approved by the Associate Editor for.Communication, Systems
Disciplines of the IEEE Communlcatlons Society for pubhcatlon wlthout oral presentation. Manuscript received June 5, 1974; revised July 2 ,
1974.
The author is with Automation Industries, Inc., Silver Spring, Md.
1 H. Frank, I E E E Trans. Commun., vol. COM-22, pp. 589-595, May
1974.
where
y = (1
p
=
+ 2pW2)
112
nL/k.
Foroperationabovethe
threshold (m2Vll< $), (4)and (6)
accurately approximate the steady-state
phase and message estimate
variances,respectively. Unfortunately, the approximations do not
show the nonlinearthresholdingphenomenon
that characterizes
nonlinear demodulators.
Threshold Behavior of a Quasi-Optimum FM
This correspondence presents a technique to obtain approximate
Demodulator
expressions, valid both above and below the operatingthreshold, for
boththe phase and message estimate variances. The technique
extends the work on the phase estimate variance by Snyder [1, pp.
F. S. NAKAMOTO AND N. J. BER.SHAD
97-1021 based
on
the Fokker-Planck
technique
discussed by
Abstract-An analytical technique is presented here forobtaining Viterbi [4].
The analytical techniquepresented here consists of two parts.
approximate expressions for both the phase and message estimate
variances for anFM demodulator. Numerical computations, for both The first part begins with the steady-state differential equations for
the density functions
of the phase and
message errors. The projection
is applied to obtain approxitheorem for conditional expectations
mate solutions to the differential equations. The net result is the
Paper approved by the Associate Editor for Communication Theory
of theIEEE
Communications Society for publication .without oral
functional form for the phase and message errorvariances. The
presentatlon. Manuscript received December 5, 1973; revised June 14,
is present in both
second part dealswith a jointstatisticthat
1974. This work was supported in part by the Offlce of Naval Research
under Contract N00014-69-A-02000-9004.
variance expressions. A Gaussian assumption is made to relate the
The authors are with theSchool of Engineering, University of Califorjoint statistic to the individual
variances.
nia, Irvme, Calif. 92664.