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1716

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-29, NO. 1 1 , NOVEMBER 1981

x t )

= r cos[wot

+JI]

P(r)

-

Fig. 1. Quadrature nonlinear model of a power amplifier.

111. PROPOSED FORMULAS FOR THE QUADRATURE

MODEL

In the quadrature model [8] [ 131, if the inpu t is given by

(l) ,

the outp ut is given by the sum of the in-phase and quadra-

ture components (see Fig. 1)

d t )= p[r(t)l

COS [mot

W] (5a)

d t )

=

-Q[r(t)l

sin

[ W O ~ (t)l

(93)

where P r ) and Q r ) areoddfunctions of r with linear and

cubic leading terms, respectively. Actually,

(5)

can be deduced

form

2 )

with

P r )

= A r) cos [@ r ) ] (6a)

Q (r ) = A r ) sin [ @ r ) ] (6b)

Eric proposed to represent P r ) and Q ( r ) as odd polynomi-

als of r , which requires large numbers of terms to fit realistic

TWT data. Kaye,George,ndEric [9 ], and uenzalida,

Shimbo, and Cook [ 101 represented each func tion by a sum

of Bessel functions of th e first kind of ord er 1, which results

in simplifying the calculation of th e outp ut spec trum . How-

ever, a large number of terms is

still

required

to f i t

realistic

data.HetrakulandTaylor [ 111-[ 131 used two-parameter

formulas involving modified Bessel functions of the first kind ,

which will be used later for comparison.

Here, we propose to represent P r ) and Q r ) by he wo-

parameter formulas

P r )

= apr/(l + r2) (7 )

~ r ) a4r3/(1 + pqr212. (8)

Note that for very large r , both

P r )

and

Q ( r )

given above are

proportional

to

l/r, while those given by Hetrakul and Taylor

approach constantvalues.

A useful property of.(7)and (8)

is

that

Q(r)

= -

P(r)

9 )

apP

p-* q,Pp+P4.

Thus, if the spectrum

of P r )

is calculated fo r a given

r t ) ,

the

corresponding pectrum of

Q ( r )

is readily obtainedby dif-

ferentiation. This property

will

be used in Section

V.

IV. FITTING THE FORMULAS TO EXPERIMENTAL

TWT DATA

To establish the accuracy of the formula s proposed in (3),

(4), (7 ) , and8), experimenta l TWT amplitude-phase nd

b

.

E

a

U

0

>

U

m

1.0

0.8

'

0 . 6 .

0 . 4

0.2

0.0 0.0 0.5 1.0 1.5

Input Voltoge normalized )

Fig. 2. TWT amplitude *)

and

phase

0)

ata of Berman and Mahle

[5]

The solid ines are plotted from

(3)

and

4),

and the dashed

lines

from the Berman-Mahle

[5]

phase ormula,and rom the

Thomas-Weidner-Durrani

[ 7 ]

amplitude formula.

0

1 2 3 5

Input Voltage millivolts 1

Fig. 3.

TWT

in-phase

*)

and quadrature

0 )

data

of

Hetrakul

and

Taylor

[

111

1

131. The solid lines are plotted from

(7)

and

8) ,

and

the dashed lines from their own formulas [111-[ 131.

quadratu re data were obtained from three different sources-

Berman and Mahle [SI , Hetrakul and Taylor

[

111-[ 131, and

Kaye, George, and Eric

[

91. These data are plotted in Figs. 2-

6 by asterisks for

A r )

and

P r ) ,

and by circles for @ r) and

Q r).

The u nits used on th e coo rdinate axes are the same units

used in the cited references. In Figs.

2 ,

5 , and 6, the input nd

ou tpu t voltages are normalized to their corresponding alues at

saturation.

A minimum mean-square-error procedure is described in the

Appendix for fitting the formulas of (3), (4), 7 ) , and (8) to

theexperimentaldata.The resultsare plottedby he solid

lines in Figs. 2-6, which show an exce llent fit in each ase. The

values of the

(Y

and

p

parameters, as well as the resulting root-

mean-square (rms) errors, are given in Table I. The

(Y

param-

eters given in the table for @ r ) give a dimension of radians

when substituted n (4); however, the corresponding rms er-

rors are given in degrees. The remaining quantities in the tabl e

have dimensions consis tent with those used in the associated

figures.

The Thomas-Weidner-Durrani [71 ampli tude ormula is

plottedby adashed line n Fig. 2 forcomparison.The re-

sulting rms error was 0.014 normalized volts, which, in spite



IEEERANSACTIONS ONOMMUNICATIONS, VOL. COM-29, NO. 11, NOVEMBER 1981 1717

Input

Voltoge millivolts )

Fig. 4. TWT Vplitude

*)

and phase

0 )

data obtained from Hetrakul

and Taylor

[111-[

133 through the use'of 6 ) . The solid lines

are

plotted from (3) and

4).

1.0

-

0.8

-u

N

0.6

r

H

;t: 0.4

>

2 0. i

0. e

0.0 0.5

1.0

1.5

I nput Voltoge ( normalized 1

Fig. 5 . TWT in-phase (*) and quadrature 0 ) dataofKaye,George,

and Eric 191.

The

solid lines areplotted from (7) and 8) .

b 0.6

0.4

w

4

2 0.2

0.0

v

2.

2

0.0

0.5 1.0 1 5 2.0

Input Voltoge normoltzed )

Fig.

6 . TWT

amplitude

*)

and phase

0)

data

of

Kaye, George,

and

Eric [9].

he solid lines'are plotted from

(3)

and

4).

TABLE

I

OPTIMUM PARAMETERS ANP'RESULTING RMS ERRORS

OBTAINED

BY

FITTING

3), 4 ) ,

(7), AND (8) THROUGH

VARIOUS EXPERIMENTAL TWT DATA

(FIGS.' )

Data from

Reference

IS1

of the large number of parameters involved, is slightly larger

thanour 0.012vgu e given inTable

I.

The Berman-Mahle

[5] phase function is also plottedby a dashed line through

their own phase data in Fig.

2.

While tha t formula gives a good

fitup o atura tion , large errors re ncounteredbeyond

saturation.

The Hetrakul-Taylor

[

1

1

]

-[

131 in-phase and quadratu re

formulas are plotted hrough h eir own data

in

Fig. 3. The

resulting rms errors for

P r) and

Q r )

are 0.058 V and

0.040

V,

respectively,which re somewhat higher than

our

cor-

responding 0.057 and 0.023 values given in Table I.

V. INTERMPDULATION ANALYSIS

One

of the applica tions intended for nonlinear model of a

power' amplifier is t o be able ,tp find the outpu t signals' as-

sociated with various inp ut signals of practical interest. Here

we consider the case of multiple phase-modulated carriers. The

quadrature m.odel

o f ( 7 )

and ( 8) will be employed in he

anaiysis.

A

two-carrier signal will be considered first because it

yie1d.s a closed-form expression for the output signal. Actually,

th e use of such a signal has been proposed for

a

satel lite corn:

munication system

[

141. Moreover, the results, with the phase

modulatipn suppressed, can be used in conjunction with he

twoitone nonl6earity test 31, [4] 151.

A .

Two Phase-Modulated Carriers

. .

.

Let the input signal be

X( )

=

v,

cos

[ W l f

+

J/l t)] + v cos [o,t+ J / z ( t ) ] .

(10)



1718 IEEE TRANSACTIONS ONOMMUNICATIONS, VOL. C O M - 2 9 , NO. 1 1 , NOVEMBER 1981

with

M k1

1 k23 * * * >

k p ) =

im[

i

k i v i s ) l ds

4 [ P r )+

C? r)l

J I

r s ) d r

(20)

where J k is the Bessel functio n of the first kind of order k .

Substituting

( 7 )

and

(8)

in (20), one can evaluate th e inte-

gration over r throug h he use of (9) and the Hankel-type

integral given in [ 16, (6.565.4), p. 6861 to o bta in

where K O and

K 1

are the modified Bessel functions of the

second kind

of

orders

0

and 1, respectively.

For n = 2, (21) reduces to the closed-form expressions ob-

tained in th e previous section. However, numerical integration

of (21) seems necessary i f n ' > 2. A direc t numerical integra-

tion of (20 ) would, of course, be more difficult since a dou ble

integral is involved.

It can be shown that when n is very large, then the integra l

in 21) can be evaluated in terms

of

a finite sum involving the

exponential integral. This import ant result will be given in a

future paper

[

171.

VI. A FREQUENCY-DEPENDENT QUADRATURE MODEL

Thus far, it has been implied that the characteristics of the

TWT amplifier are independent of frequency over the band of

interest, which is often the situation encountered in practice,

especially whenhelix-type TWT's are employed. However,

when broad-band input signals are involved,' or in cases where

amplifiers and compone nts are used that are not as inherently

broad-band as helix-type TWT's, arequency-dependent

model is needed. Here we make a conjecture, to be explained

later, that enables us to infer such a model from single-tone

measurements.

'Consider a single-tone test in which the input signal to the

TWT has adjustable amplitude r and frequency

f

and,let the

measured amplitude and phase of the output signal be A r , fl

and @ r , f , espectively. Let

be the measured small-signal phase response. Thus,

@ r ,

f

-

@o(f) + 0 as

r

+

0. In fact, this phase difference is what was

labeled @ I ) in Sections 11-IV. Using (6),oneobtains he

in-phase and quad ratu re outp ut signals



IEEE TRANSACTIONS O N C OMMUNI C ATI ONS , VOL. COM-29, NO. 11, NOVEMBER 1981

1714

For

each given

f,

one can fit the formula s of

(7)

and (8) t o

these data as is described in the Appendix. This would result

in he requency-dependentparameters a p ( f ) ,&,cf), aq(f),

and &(f). One now can compute the functions

H p f ) =

W (25)

G p ( f )

= ~ p ( f ) / W ) (26)

Hs 0

=

W

(27)

G g ( f ) = ag(f)/P43'2(f). (28)

Lets define th e normalized frequency-independent

envelope nonlinearities

Po(')

= r/(1 +

r 2 )

(29)

= r3/(1

+

r212,30)

which are obtained from (7) and (8) by setting thea s and p s

to uni ty. It is observed from. (7), (25), (26), (29), and Fig. 1

that he operation performed on a single-tone signal passing

through the in-phase branch can be .divided into three steps:

first, the input amplitu de is scaled by

H p c f ) ;

next, the result-

ing signal passes through the frequency-independent envelope

nonlinearity Po(r); and finally, the output amplitude is scaled

by

G p C f ) .

Three similar stepsapply or heoperation per-

formed in the quadrature ranch. Now, performing each of the

aforementioned frequency-dependent amplitude-scaling opera-

tions may be nterpreted as passing the signal through an

appropriately located inear filte r having the corresponding

real frequency response. This leads to the frequency-dependent

quadrature model shown in Fig. 7. That model is, of course,

valid for single-tone input signals by virtue of the procedur e

used to constr uct it. However, its validity for arbitra ry input

signals is, at this point, a conjecture that remains to be con-

firmed experimentally.

Thebox in the output side of Fig. 7 is a linear, all-pass

network having an amplitude response of unity, and a phase

response of which is definedn (22). or single-tone

input signal, that phase response can be absorbed totallyor

partially into the filters n each of the two branches of the

mode l with out affectin g the outp ut signal. However, for he

mode l o give an cceptable epresentation of the power

amplifier forarbitrary nput signals, it may be necessary to

absorb that phase response into the various filters in a particu-

lar manner. This point needs further investigation.

VII. CONCLUSIONS

Simple two-parameter ormulas have beenproposed for

each of the functions of the amplitude-phase model of a TWT

amplifier, as well as for each of the func tion s of the equivalent

quad rature model. The formulas fit available TWT data very

well. This implies tha t the nonl inear behavior of a given TWT

can be accurately epresented by only four parameters. The

same is also true for the quadrature model formulas proposed

by Hetrakul and Taylor [ 111-[ 131. Those formulas, however,

seem unnecessarily complicated.

The simplicity of the proposed formula s esulted in a closed

form expression f or heoutput signal of a TWT amplifier

Fig. 7. Proposed requency-dependentquadraturemodel

of

a

TWT

amplifier.

Po r ) and Q o r ) are

frequency-independent envelopenon-

linearities given in

(29)

and(30); the H s and the G's,are

real

linear

fiiters whose requency responses are defined tl(25)-(28), and

@o(n

is a linear all-pass network whose phase response is defined in22).

excited by two phase-modulated carriers. Moreover, when the

inpu t signal consists of:,Fore han wo such carriers, it was

shown that he outp ut signal can be representedby a single

integral. When the number of carriers is very large, it can be

shown that t.his integral can be evaluated in terms of a finite

sum involving the exponen tia l integral.Thisresult,however,

was not given here, and will be presented n a futu re pape r

[ 171.

A frequency-dependent quadrature model of a TWT ampli-

fier has also been proposed (see Fig. 7). The various param-

eters of themodel can be obtained byapplyingminimum

mean-square-error curve fitting omeasurements involving a

single tone of variable amplitude and frequency.

APPENDIX

CURVE-FITTING

PROCEDURE

The formulas 3) , 4), 7) , and (8) assume the general form

where

n

= 1, 2, or 3, and u = 1 or 2. (Actually, noninteger

values of

u

were considered in fitting the data in Section IV,

and, emarkably, = 1 or 2 was fo und obe very nearly

optimum.) Given

m

measured pairs, (zi,

r i ) ,

i

=

1,

2,

-.,

m ,

we

need to find

a

and to fit (A l) t o these data. Defining

and employing standard minimummean-square-error curve-

fittingprocedure,oneobtains he required optimum values

of a and

where all the summations are over i = 1-m.

ACKNOWLEDGMENT

The author hanks L. J. Greenstein for useful discussions

and suggestions.



1720

iE ik

TRANSACTIONS

O N

COMMUNICATIONS,

V O L . cOM-29, No.

1 ,

NOVEM BER 1981

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”The effects of tradsponder nonlinearity on binary ‘CPSK

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