Minimizing Spectral Leakage of Nonideal LINC Transmitters by

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 445

Minimizing Spectral Leakage of Nonideal LINCTransmitters by Analysis of Component Impairments

Antoine Choffrut, Barry D. Van Veen, Fellow, IEEE, and John H. Booske, Senior Member, IEEE

Abstract—LInear amplification with Nonlinear Components(LINC) is a distortion-free transmitter architecture under idealconditions. However, in a realistic transmitter, imbalances suchas dc offset, differential gain, and differential phase shift presentin the quadrature modulators result in generation of out-of-bandspectral components, even assuming that both amplifiers haveidentical characteristics. The level of distortion not only increaseswith the magnitude of the imbalances but also depends, to alarge extent, on their relative phase arguments. The contributionof this paper is to provide a simplified characterization of thespectral leakage in the LINC transmitter in terms of quadraturemodulator imbalances, with identical or mismatched amplifiers.This characterization allows identification of the transmitter withminimum spectral leakage under the constraint that the quadra-ture modulators perform equally well when used individually,i.e., in a transmitter composed of just the quadrature modulatorfollowed by an amplifier. With reasonably small imbalances inthe quadrature modulators and realistic amplifier mismatch, thespectral leakage at the output of a LINC transmitter driven bytypical modulation schemes can be reduced by up to 10 dB, simplyby a wise pairing of modulators.

Index Terms—Interface suppression, LINC, nonlinearities,power amplifiers, spectral analysis, traveling wave tubes.

I. INTRODUCTION

S PECTRAL efficiency is a major issue in telecommuni-cation transmission due to limited bandwidth availability.

Spectrally efficient modulation schemes typically employ sig-nals with a varying envelope, which in turn requires amplifiersto operate over a large portion of their transfer curves. However,amplifier nonlinearities generate out-of-band spectral content,which degrades spectral efficiency. The effects of nonlinearitiesmay be reduced by backing off the output power level andoperating in the linear regime. Output power backoff usuallyhas the undesirable effect of reducing power efficiency, whichis typically maximized near saturation where the amplifiernonlinearities are greatest.

Manuscript received November 2, 2004; revised September 9, 2005,November 20, 2005, and December 5, 2005. This work was supported by theU.S. Air Force Office of Scientific Research (AFOSR) under Grant 49620-00-1-0088 and by DUSD (S&T) under the Innovative Microwave Vacuum Elec-tronics Multidisciplinary University Research Initiative (MURI99) program,managed by the AFOSR under Grant F49620-99-1-0297. The review of thispaper was coordinated by Prof. E. Bonek.

A. Choffrut is with the School of Mathematics, University of Minnesota,Minneapolis, MN 55455 USA (e-mail: [email protected]).

B. D. Van Veen and J. H. Booske are with the Department of Electrical andComputer Engineering, University of Wisconsin-Madison, Madison, WI 53706USA (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TVT.2006.889564

Fig. 1. LINC transmitter (signals are denoted by their complex envelopes):An SCS generates two phase-modulated baseband signals, which are upcon-verted in two modulators and then amplified using two power amplifiers. Theamplified signals are summed in a power combiner. The output is a linearlyamplified modulated version of the input signal.

Numerous techniques have been proposed to obtain highoutput power levels and high power efficiency with limitedsignal distortion [1]–[4]: Feedback, feedforward, and predistor-tion linearizers are three widely used methods to compensatefor nonidealities of power amplifiers. The feedback methodsamples the distortion present at the output of the transmitterto make corrections to the input signal [5], [6]. The majordrawback of this technique is the limitation on the input band-width to approximately 10% of the carrier frequency, whichis imposed by the feedback loop [4]. Feedback linearizers areusually restricted to high and lower very high frequencies [1].In a feedforward linearizer, an auxiliary amplifier is used togenerate a signal that is 180◦ out of phase with the distortion atthe output of the main amplifier. The outputs of both amplifiersare then summed to produce a distortion-free signal [6]–[8].This technique works well, especially with wideband signals;however, the implementation is complex, and the presence ofthe auxiliary amplifier, although it usually consumes much lessthan the main amplifier, degrades the power efficiency of thetransmitter. In a predistortion linearizer, a nonlinear deviceis placed before the amplifier to compensate for amplifiernonlinearities [9], [10]. Predistortion linearizers are simple toimplement but require an accurate model for the amplifiercharacteristics. Analog predistorters are limited to relativelymild nonlinearities, while digital predistorters are typicallybandwidth limited.

LInear amplification with Nonlinear Components (LINC) isa type of linearizer that achieves both spectral and power effi-ciency with any amplifier characteristics and at any drive level[11], [12]. A block diagram of the LINC transmitter is shownin Fig. 1. The process starts with a band-limited source signalwith complex amplitude s(t). A signal component separator(SCS) adds to and subtracts from s(t) a complex envelope

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446 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007

wideband signal e(t) (not shown in the diagram). This results intwo phase-modulated signals si,k(t)(k = 1, 2), whose complexenvelopes have constant magnitudes. These component signalsare upconverted and amplified in two separate branches. Eachbranch is composed of a quadrature modulator, followed by anamplifier. The upconverted signals, whose complex envelopessm,k(t)(k = 1, 2) have constant magnitude as well, drive theamplifiers at a fixed point on their transfer curves. Hence, ineach branch, the gain is constant, and amplification is linear.The outputs of the two amplifiers, with complex envelopesso,k(t)(k = 1, 2), are summed in a power combiner, and thewideband signal e(t) introduced by the SCS cancels out. Asa result, the output of the transmitter, with complex envelopesout(t), is an upconverted linearly amplified replica of sourcesignal s(t).

LINC is ideally distortion free but, in practice, is sensitive tocomponent impairments: Examples of impairments include theerror introduced to the component signals si,k(t) by the SCS[13]–[18], imbalances in the quadrature modulators corruptingthe envelope of the inputs of the amplifiers sm,k(t)(k = 1, 2)[19]–[22], gain imbalance between the two RF paths causingthe amplifiers to be operated at different levels on their trans-fer curves [23]–[26], phase delay between the two RF paths[23]–[26], and power amplifier mismatch [19], [20], [23], pre-venting total cancellation of the wideband component e(t).All of these result in nonlinear distortion at the output.Misalignments in the quadrature modulators and amplifier mis-match are arguably the causes of error in the LINC archi-tecture that are the most significant and the most difficult tocompensate [19].

In the semi-ideal case of the LINC transmitter with identicalamplifiers, the total distortion present at the output of thetransmitter is the sum of terms contributed by each branch andcross terms generated as a result of interactions between thebranches. The direct contributions from the branches increaseas modulator imbalances increase in magnitude. However, thebehavior of the cross terms is a complicated function of therelative phase angle between these imbalances. Different pairsof modulators may yield performances differing by up to25 dB in out-of-band spectral energy, even though all deviceshave equivalent tolerance. Hence, analysis of the effects ofcomponent impairments is an important aspect of understand-ing LINC transmitter performance. When the amplifiers havemismatched characteristics, a fraction of the wideband signale(t) remains at the output and contributes to the spectral leak-age. This particular contribution can be practically eliminatedby backing off the amplifier with highest output power.

We analyze the effects of component impairments by in-troducing a simplified semi-analytic expression for the outputspectrum based on the one derived in [19].1 The expressionobtained in [19] is primarily used for quick computation but isintractable for analysis. Our expression involves fewer terms,and these terms are expressed using new parameters with aclear physical interpretation—they represent the gain, its rate of

1In [19] and in this paper, the term “spectrum” refers to the averageperiodogram.

change, and the “curvature” of the amplifier characteristic at theoperating point. In the case of identical amplifiers, our analysisshows that the effects of modulator dc offset and differentialgain/phase are decoupled. Thus, it is legitimate to study theeffects of dc offset and differential gain and phase in the quadra-ture modulators independently. Our simplified expression alsoreveals how to pair nonideal devices in order to optimize,i.e., minimize, spectral leakage. In [19], the accuracy of themethod is demonstrated for various, but arbitrary, combinationsof impairments in the quadrature modulators. Our simplifiedexpression allows computation of the full extent of possiblespectral leakage for a given level of imbalance in the quadraturemodulators. Furthermore, our analysis can be adapted to a verylarge class of power amplifiers, as discussed in Section II-C.We illustrate our results using class-C amplifiers (as usedin [19]) and measured traveling-wave tube (TWT) amplifiercharacteristics.

In Section II, we introduce notation and describe the princi-ple of the LINC architecture. Section III restates [19, eq. (43)]with minor changes in the notation from [19]. Section IVderives our simplified accurate expression for the LINC out-put spectrum, assuming identical amplifiers. In Section V,we derive a further simplified approximate expression for theoutput spectrum. This expression is used in Section VI, wherewe describe a method for pairing quadrature modulators insuch a way that spectral leakage is minimized, assuming thatthe quadrature modulator errors have a given tolerance. Theeffectiveness and reliability of the method are verified by meansof numerical simulations, assuming TWT amplifier charac-teristics and using two typical modulation schemes: 1) π/4-shifted differential quaternary phase-shift keying (DQPSK) asused in North American digital cellular systems and 2) avariation thereof, i.e., offset 4-quadrature-amplitude modula-tion (4QAM). Section VII discusses the applicability of ourmethod to different types of amplifiers. We also illustrate theeffectiveness of the method in the case of class-C amplifiers(with respect to the analysis, class-C amplifiers “behave” likeTWT amplifiers, as explained in Section II-C). Finally, wedemonstrate the impact of our analysis on the design of LINCtransmitters.

II. LINC PRINCIPLE

With a few indicated exceptions, we employ the complexbaseband equivalent representation of real passband signals. Wenow describe each operation performed in the LINC diagram,as depicted in Fig. 1.

First, we introduce the notational conventions used through-out this paper.

• a denotes the complex conjugate of a.• X(n) denotes the n-fold convolution of X : X(n) = X ∗X(n−1), where X(1) = X .

• Mean [a] denotes the average of a taken over all simula-tions.

• a ≈ c signifies that a− c is “negligible” relative to aand c.

• a ∼ c signifies that a and c are of the “same order ofmagnitude” (e.g., |a|/|c| ∈ [0.2, 5]).

CHOFFRUT et al.: MINIMIZING SPECTRAL LEAKAGE OF NONIDEAL LINC TRANSMITTERS 447

Fig. 2. Modulator is impaired by differential gain gd, differential phase shiftφd, and dc offsets pI , pQ. In the I-branch, the gain differs from unity by gd,and the oscillator is off by φd. In the Q-branch, the gain differs from unity by−gd, and the oscillator is off by −φd.

A. SCS

Denote by r = maxt |s(t)| the maximum magnitude of thecomplex envelope of the input signal s(t) to be transmitted.This can be estimated, prior to implementation of LINC, basedon a sufficiently long simulation run of the signal generatedfrom the same modulation scheme as the one used in the trans-mission: r ≈ maxt |ssim(t)|. The SCS generates two signalswith phase-modulated constant amplitude envelopes, i.e.,

si,1(t) = s(t) + e(t) and si,2(t) = s(t)− e(t) (1)

where the broadband signal e(t) is defined as

e(t) = js(t)√r2/ |s(t)|2 − 1. (2)

This definition allows the possibility of division by zero whenthe envelope of the signal vanishes. In practice, this problemcan be avoided by introducing a small bias term α > 0, i.e., byrequiring |s(t)| ≥ α. The use of the bias is arbitrarily rare, pro-vided that α is sufficiently small. In addition, as α decreases, thedistortion due to the bias decreases. Therefore, by choosing thebias term to be sufficiently small, its impact on the performanceof the amplifier is expected to be negligible.

B. Upconversion in the Quadrature ModulatorsWith Imbalances

Impairments of a quadrature modulator are typically de-scribed in terms of the “differential gain” gd and the “differen-tial phase shift” φd between the I- and Q-branches, and the “dcoffsets” pI and pQ in the I- and Q-branches [19], [20], [27].These parameters reflect the difference of operation on theI- and Q-components of the input signal, as depicted in Fig. 2.First, the I- and Q-components, which are denoted as x(t) andy(t), respectively, are offset by an amount pI and pQ. Then,they are multiplied by constants gI = 1 + gd and gQ = 1− gd.Finally, these signals modulate carrier waves at frequency fc

that are, in the ideal case, 90◦ out of phase. In reality, the localoscillators are offset by a small error of 2φd.

The input–output relationship of the quadrature modulator inbranch k = 1, 2 is given as follows: Using Fig. 2

zm(t) = ((1 + gd)x(t) + pI) cos(2πfct+ φd)

− ((1− gd)y(t) + pQ) sin(2πfct− φd) (3)

where the input has envelope x(t) + jy(t). It can be shown [20]that the upconverted signal has a complex envelope given by

sm,k(t) = uksi,k(t) + vksi,k(t) + pk (4)

where the baseband input signal in branch k = 1, 2 has complexenvelope si,k(t), and

uk ≈ 1− φ2d,k + jgd,kφd,k (5)

vk ≈ gd,k + jφd,k (6)

pk = pI,k cos(φd,k) + pQ,k sin(φd,k)

+ j (pI,k sin(φd,k) + pQ,k cos(φd,k)) (7)

where the angle φd,k is given in radians. The approximationsassume small yet realistic errors gd,k, φd,k, pI,k, and pQ,k.For example, in the simulations, we use the following numer-ical bounds [19]: 0.0058 < |gd,k| < 0.015; the angle |φd,k| isbetween 0.3◦ and 5◦. Thus, we have uk ≈ 1, |vk| 1, and|pk| 1.

C. Amplification

The upconverted signals of (4) are then amplified in twopower amplifiers, which are assumed to be memoryless andwith frequency-independent transfer curves. The input–outputrelationship for amplifier k = 1, 2 is assumed of the form

so,k(t) = Gk (|sm,k(t)|) · sm,k(t) (8)

where sm,k(t) and so,k(t) are the envelopes of the (modulated)input and output of amplifier k, respectively, and Gk(r) isthe complex gain when the amplifier is driven by an inputwith instantaneous envelope magnitude r. We discuss next themodel used in this paper for the power amplifiers. We employa different model than [19]. Consequently, our expressions forthe output spectrum are much more transparent.

The LINC architecture has the great advantage of operat-ing the amplifiers over a small part of their transfer curves.Therefore, it is only necessary to have an accurate model of thegain function about the operating point. For most amplifiers, asecond-order polynomial in r2 achieves this. The model usedin [19] is

Gk(r) = g1,k + g3,kr2 + g5,kr

4 (9)

where the coefficients g1,k, g3,k, and g5,k are complex-valued.It is this particular model that allows Sundström [19] to derive


TABLE INUMERICAL VALUES OF |c1|, |c3|, |c5|, AND |c1 + c3|.

THESE PARAMETERS CONTROL THE BEHAVIOR

OF THE TRANSMITTER

an expression of the output spectrum. However, this expressioninvolves weights that are functions of the coefficients g1,k, g3,k,g5,k, and r. The interpretation of these weights in this model iscomplicated. To simplify interpretation, we write the amplifiermodel as

Gk(r) = c1,k + c3,k(r2 − 1) + c5,k(r2 − 1)2 (10)

where r = 1 is the operating point, and

c1,k = g1,k + g3,k + g5,k

c3,k = g3,k + 2g5,k

c5,k = g5,k. (11)

Note that at the operating point (r2 = 1), our model parametershave the following physical interpretation: c1,k is the powergain; c3,k is the rate of change of the gain; and 2c5,k is the“curvature” of the gain. The quantities c1,k, c3,k, and c5,k willalso lead to a simplified interpretation of the output spectrum,as shown in (19)–(25). In fact, only the quantities |c1|, |c3|,|c5|, and |c1 + c3| have significance in describing the qualitativebehavior of the transmitter.

Thus, our analysis and the subsequent method to minimizespectral leakage are valid for any type of amplifier that exhibitsbehavior accurately described by (10) near the operating point.This covers practically all types of amplifiers. For example,a typical model for TWT amplifiers is given as follows [28]:amplitude A(r) = αAr/(1 + βAr

2) and phase shift Φ(r) =αP r

2/(1 + βP r2), where αA = 1.8, βA = 0.81, αP = 0.27π,

and βP = 0.81 are typical values [29] (after normalization).This model is well approximated by a second-order polynomialin r2 over the range 0.9 ≤ r ≤ 1.1. Corresponding numeri-cal values for the parameters of interest (|c1|, |c3|, |c5|, and|c1 + c3|) are as shown in the first column of Table I. Thesimulations in this paper are for a class-C amplifier and twoTWT amplifiers. The model parameters in Table I for theclass-C amplifier are equivalent to those of [19], while theparameters for the TWT amplifiers are obtained from measure-ments of AM–AM and AM–phase modulation (PM) transfercurves taken on actual TWT devices. A detailed investigationof these data is published in [30]. The corresponding numericalvalues for the parameters of interest are shown in the last threecolumns of Table I. All cases are normalized for an operating

point r = 1. Note that all the parameters shown in Table Iare of the same order of magnitude. This property is usedin the process of simplifying the expression for the outputspectrum.

D. Power Combiner

The output of the power combiner is

sout(t) = so,1(t) + so,2(t). (12)

In the ideal case, the quadrature modulators operate perfectlyon their input signals so that sm,k(t) = si,k(t) for k = 1, 2, andthe amplifiers, which are identical, are operated at an equal andconstant complex gain G(r). In this case

sout(t)=G(r) · ((s(t) + e(t)) + (s(t)− e(t)))=2G(r) · s(t).(13)

Thus, the output is a linear replica of the input.

III. FIRST EXPRESSION OF THE OUTPUT SPECTRUM:CASE OF IDENTICAL AMPLIFIERS

In this paper, we define the spectrum as the average peri-odogram, i.e., the average of the squared magnitude of Fouriertransforms of finite-time simulations.

A. Nomenclature

Recall that the parameters in (10) are normalized so thatthe operating point corresponds to r = 1. Thus, the maximummagnitude of the input signal is 1. We denote the Fouriertransforms involved in the calculation of the output spectrumas follows:

sout(t)↔Sout(f) (14)

s(t)↔S(f) (15)

e(t)↔E(f) (16)

(si,k(t))n ↔S

(n)i,k (f), n = 1, 2, . . . . (17)

B. Output Spectrum

The expression for the output spectrum Pout(f) derived in[19] assumes that the amplifiers are identical with the sameoperating point r, i.e., G1(r) = G2(r) = G(r) = g1 + g3r2 +g5r

4. The computation for the output spectrum is carriedout as follows: 1) The output signal sout(t) is expanded inthe time domain in terms of source signal s(t), signal e(t),constant-amplitude envelope signals si,k(t), and the powersand complex conjugates thereof, keeping only low-order terms.2) The Fourier transform Sout(f) is expanded in terms ofthe Fourier transforms of s(t), e(t), and so forth. 3) Next,|Sout(f)|2 is expressed in terms of the square and cross termscomposing Sout(f), some of which are known to average tozero [31] and thus may be ignored when estimating the outputspectrum [19].


We rewrite this expression after the normalization r = 1using the parameters c1, c3, and c5; thus

Pout(f)

= |2c1+Wv1,1+Wv1,2|2 ·|S(f)|2

+|Wv1c,1+Wv1c,2|2 ·∣∣∣S(−f)

∣∣∣2

+|Wp0,1+Wp0,2|2 ·δ(f) +|Wv1,1−Wv1,2|2 ·|E(f)|2

+|Wv1c,1−Wv1c,2|2 ·∣∣∣E(−f)

∣∣∣2

+|Wp2,1|2 ·∣∣∣S(2)

i,1 (f)∣∣∣2+|Wp2,2|2 ·

∣∣∣S(2)i,2 (f)

∣∣∣2

+2Re[Wp2,1Wp2,2S

(2)i,1 (f)S

(2)i,2 (f)

]

+|Wp3,1+Wv3,1|2 ·∣∣∣S(3)

i,1 (f)∣∣∣2+|Wp3,2+Wv3,2|2 ·

∣∣∣S(3)i,2 (f)

∣∣∣2

+2Re[(Wp3,1+Wv3,1)(Wp3,2+Wv3,2)S

(3)i,1 (f)S

(3)i,2 (f)

]

+|Wv5,1|2 ·∣∣∣S(5)

i,1 (f)∣∣∣2+|Wv5,2|2 ·

∣∣∣S(5)i,2 (f)

∣∣∣2

+2Re[Wv5,1Wv5,2)S

(5)i,1 (f)S

(5)i,2 (f)

](18)

where δ(f) is the delta function and the weights are given by

Wp0,k = pk(c1,k + c3,k) (19)

Wp2,k(r) = pkc3,k (20)

Wp3,k(r) = pk2c5,k (21)

Wv1,k(r) = δuk(c1,k + c3,k) (22)

Wv1c,k(r) = vk(c1,k + c3,k) (23)

Wv3,k(r) = vkc3,k (24)

Wv5,k(r) = vk2c5,k (25)

and δuk = |uk| − 1. The weights are now given in terms ofsimple functions of the impairments pk, vk of the quadraturemodulators and the parameters c1,k, c3,k, c5,k modeling thepower amplifiers. Further, due to the normalization r = 1, theseimpairments and nonlinearities are entirely captured in theweights and do not affect the spectra involved in Pout(f). In thenext section, we further simplify (18) to a form that is amenableto analysis.

IV. SIMPLIFIED EXPRESSION OF THE OUTPUT SPECTRUM

The purpose of this section is to derive an expression forPout(f) that is even simpler than (18). Recall that we assumeidentical amplifiers. The case of mismatched amplifiers is ad-dressed in Sections V and VI-C. In Section IV-A, we analyzethe spectra involved in Pout(f) in order to rearrange terms in(18). Next, in Section IV-B, we use rough estimates of theimpairments of the quadrature modulators and the nonlineari-

Fig. 3. Examples of spectra contributing to the spectral splatter, as indicatedin (18), assuming π/4-shifted DQPSK (shown on the left half for frequenciesbelow fc) and offset 4QAM (shown on the right for frequencies above fc).k = 1, 2 refers to the branches of the LINC transmitter.

ties of the amplifiers in order to compare the weights given in(19)–(25) and identify low-power terms that can therefore beneglected in (18). The process of simplification is illustratedhere using typical estimates for the impairments [19] and theamplifier parameters in Table I. This process can be adapted toany LINC transmitter. Note that we assume identical amplifiersin this section. Only orders of magnitude for the quadraturemodulator impairments and amplifier parameters are required.In the case at hand, we are able to separate the contributionsto the spectral leakage of two independent parameters, namely1) p, which is the dc offset, and 2) v, characterizing the impactof differential gain gd and phase shift φd. This is carried out inSection IV-C.

A. Analysis of the Spectra: Rearranging Terms in (18)

We assume that the modulation scheme is chosen. Thus,spectra (14)–(17) are fixed. Example spectra are shown inFig. 3, assuming π/4-shifted DQPSK (shown on the left, forfrequencies below fc) and offset 4QAM (shown on the right,for frequencies above fc), where W denotes the message signalbandwidth. All simulations are performed in MATLAB, anda square-root-raised cosine pulse shape is used with a rollofffactor of 0.35. Each spectrum is the average of the squared mag-nitude of 128 Fourier transforms corresponding to sequences of128 random symbols each and sampled at 32 samples/symbol.The resulting estimate is smoothed by averaging with the fouradjacent points in the frequency domain, corresponding to 3%of the message bandwidth.

Observe in Fig. 3 that |S(−f)|2 = |S(f)|2 for the mod-ulations used in this paper. This is reasonable: First, notethat |S(−f)|2 = |S(−f)|2. Next, S(−f) is the Fourier trans-form of the time-reversed process s(−t). Since the transmittedsymbols are identically and independently distributed and the


constellations and pulse shape are symmetric, |S(−f)|2 and|S(f)|2 should average to the same values. Thus, in (18), wemay replace |S(−f)|2 by |S(f)|2 and, similarly, |E(−f)|2by |E(f)|2. Correspondingly, certain coefficients can be com-bined. Define

ΓS = |2 c1 +Wv1,1 +Wv1,2|2 + |Wv1c,1 +Wv1c,2|2 (26)

ΓE = |Wv1,1 +Wv1,2|2 + |Wv1c,1 −Wv1c,2|2 (27)

Γ0 = |Wp0,1 +Wp0,2|2. (28)

In addition, Fig. 3 shows that |S(n)i,1 (f)|2 and |S(n)

i,2 (f)|2 arethe same on average for n = 2, 3, 5. We explain in the Appen-dix why this conclusion is reasonable. Therefore, we replace|S(n)

i,2 (f)|2 by |S(n)i,1 (f)|2 in the remainder of this paper.

Substituting |S(f)|2 for |S(−f)|2, |E(f)|2 for |E(−f)|2,and |S(n)

i,1 (f)|2 for |S(n)i,2 (f)|2, factoring, and using the com-

pound coefficients (26)–(28), (18) simplifies to

Pout(f)

= ΓS · |S(f)|2+Γ0 · δ(f)+ ΓE · |E(f)|2

+(|Wp2,1|2+|Wp2,2|2

) · ∣∣∣S(2)i,1 (f)

∣∣∣2

+ 2Re[Wp2,1Wp2,2S

(2)i,1 (f)S

(2)i,2 (f)

]

+(|Wp3,1+Wv3,1|2+|Wp3,2+Wv3,2|2

) · ∣∣∣S(3)i,1 (f)

∣∣∣2

+ 2Re[(Wp3,1+Wv3,1)(Wp3,2+Wv3,2)S

(3)i,1 (f)S

(3)i,2 (f)

]

+(|Wv5,1|2+|Wv5,2|2

) · ∣∣∣S(5)i,1 (f)

∣∣∣2

+ 2Re[Wv5,1Wv5,2S

(5)i,1 (f)S

(5)i,2 (f)

]. (29)

Further simplification of this expression is obtained by studyingthe weighting factors associated with each spectral component.

B. Analysis of the Weights

Our analysis makes use of the following assumptions: Bothamplifiers are assumed to be memoryless, with frequency-independent characteristics, and identical in both branches ofthe LINC transmitter with complex gain G(r) given by (10).Next, the coefficients c1, c3, and c5 are of the same order ofmagnitude, i.e., both types of nonlinearities (rate of changeas well as curvature) have nonnegligible impact on the per-formance of the transmitter. The term c1 + c3 appears in theexpression for Pout(f). We assume that it is also of the sameorder of magnitude as c3 and c5, which amounts to saying thatc1 and c3 do not have magnitudes that are approximately equaland phase difference close to π. These assumptions are satisfiedfor the amplifiers in Table I. For simplicity, we assume theerrors to have the same modulus in both branches of the LINCtransmitter. The simplification carried out in this subsectionrelies on comparison of the orders of magnitudes of the weights.

We assume that |p1| = |p2| = |v1| = |v2| = 0.01. This choiceis consistent with the numerical bounds stated in Section II-B.

We first verify that Wv1,1 and Wv1,2 are negligible relativeto 2c1, thus simplifying (26). Indeed, comparing (22) and c1,the relative sizes of these terms are essentially determinedby δuk. From [19], δuk ≈ −φ2

d,k(1− g2d,k + φ

2d,k/4)/2 ≈

−(1/2)φ2d,k, which is negligible. Similarly, the size of Wv1c,k

(for k = 1, 2) relative to c1 is essentially determined by vk,which is also negligible since c1 and c1 + c3 are of similarmagnitudes. Thus, ΓS can be replaced by 4|c1|2.

Likewise, comparing (21) and (24), Wp3,k is negligible rela-tive toWv3,k because |vk| |pk|2. Therefore, we shall assumethat the contribution of the third-order terms is (|Wv3,1|2 +|Wv3,2|2) · |S(3)

i,1 (f)|2 +2Re[Wv3,1Wv3,2S(3)i,1 (f)S

(3)i,2 (f)]. Fi-

nally, the dc term Γ0δ(f) is negligible relative to ΓS |S(f)|2and can be ignored.

The remainder of this section is dedicated to showing thatwe may replace ΓE by |Wv1c,1 −Wv1c,2|2 in (29). From (22),(23), and (27)

ΓE =(|δu1 + δu2|2 + |v1 − v2|2

) |c1|2. (30)

Numerical computations give |δuk| ≤ 10−5, while |vk| ≈ 10−2

(thus, satisfying the numerical bounds given in Section II-B).Therefore, “unless” v1 ≈ v2, we have |δu1 + δu2|2 |v1 −v2|2, and we may replace ΓE by |Wv1c,1 −Wv1c,2|2. Thisconclusion still holds in the case v1 ≈ v2, but the argu-ment requires a little more care. The key is to show thatif v1 = v2 (for simplicity), then the contribution ΓE |E(f)|2is altogether negligible compared to the contribution ofthe third-order terms (|Wv3,1)|2 + |Wv3,2)|2) · |S(3)

i,1 (f)|2 +2Re[Wv3,1Wv3,2S

(3)i,1 (f)S

(3)i,2 (f)]. Therefore, for all practical

purposes, it makes no difference if we replace ΓE by |Wv1c,1 −Wv1c,2|2 in (18). Later in Section VI-B, we conclude that thecontribution of S(3)

i,k (f) is maximized with v1 = v2 + 0.5π andminimized with v1 = v2 − 0.5π. The case v1 = v2 is halfwayin between, where the cross term becomes zero. In turn, thecontribution due to S(3)

i,k (f) is approximately

(|Wv3,1|2 + |Wv3,2|2) · ∣∣∣S(3)

i,1 (f)∣∣∣2 . (31)

We now compare the weighting factors. With v1 = v2, (30)becomes

ΓE = |δu1 + δu2|2|c1|2 (32)

and by (24)

|Wv3,1|2 + |Wv3,2|2 =(|v1|2 + |v2|2

) · |c3|2. (33)

With |v1| = |v2| |δu1| ≈ |δu2|, we find that |δu1 +δu2|2 |v1|2 + |v2|2 and that the contribution of E(f) isnegligible relative to that of S(3)

i,k (f).


C. Simpler Expression

Combining the approximations derived in parts A and B ofthis section gives a much simplified expression for Pout(f), i.e.,

Pout(f) = 4|c1|2 · |S(f)|2 + |Wv1c,1 −Wv1c,2|2 · |E(f)|2

+(|Wp2,1|2 + |Wp2,2|2

) · ∣∣∣S(2)i,1 (f)

∣∣∣2

+ 2Re[Wp2,1Wp2,2S

(2)i,1 (f)S

(2)i,2 (f)

]

+(|Wv3,1|2 + |Wv3,2|2

) · ∣∣∣S(3)i,1 (f)

∣∣∣2

+ 2Re[Wv3,1Wv3,2S

(3)i,1 (f)S

(3)i,2 (f)

]

+(|Wv5,1|2 + |Wv5,2|2

) · ∣∣∣S(5)i,1 (f)

∣∣∣2

+ 2Re[Wv5,1W5,2S

(5)i,1 (f)S

(5)i,2 (f)

]. (34)

This expression involves eight summands versus 14 in (18).However, more importantly, the terms involving p and thoseinvolving v are separate. That is, the impact of p and v aredecoupled in (34) and can be legitimately analyzed sepa-rately. The spectral leakage is the sum of the following twocontributions:

1) contribution of dc offset p in the quadrature modulators:

Pout,p(f) =(|Wp2,1|2 + |Wp2,2|2

) · ∣∣∣S(2)i,1 (f)

∣∣∣2

+2Re[Wp2,1Wp2,2S

(2)i,1 (f)S

(2)i,2 (f)

]; (35)

2) contribution of differential gain gd and differential phaseerror φd in the quadrature modulators:

Pout,v(f) = |Wv1c,1 −Wv1c,2|2 · |E(f)|2

+(|Wv3,1|2 + |Wv3,2|2

) · ∣∣∣S(3)i,1 (f)

∣∣∣2

+ 2Re[Wv3,1Wv3,2S

(3)i,1 (f)S

(3)i,2 (f)

]

+(|Wv5,1|2 + |Wv5,2|2

) · ∣∣∣S(5)i,1 (f)

∣∣∣2

+ 2Re[Wv5,1Wv5,2S

(5)i,1 (f)S

(5)i,2 (f)

]. (36)

Thus, the problem of minimizing the spectral leakage withappropriately chosen quadrature modulators becomes one ofminimizing Pout,p(f) and Pout,v(f) separately.

V. IDENTIFYING DOMINANT TERMS

The effects of quadrature modulator errors have been shownto be independent, in view of (35) and (36). We now show thatthe contribution of S(3)

i,k (f) dominates Pout,v(f) and that (36)can be further simplified. Finally, we consider the more generalcase of mildly mismatched power amplifiers. For all practicalpurposes, the contribution of amplifier mismatch is proportional

to |c1,1 − c1,2|2. We thus obtain an approximation for Pout(f)as the sum of three independent dominant contributions.

A. Dominant Terms in Pout,v(f), (36)

Pout,v(f) is the sum of the following three contributions:

1) contribution of E(f):

|Wv1c,1 −Wv1c,2|2 · |E(f)|2 ; (37)

2) contribution of S(3)i,k (f):

(|Wv3,1|2 + |Wv3,2|2) · ∣∣∣S(3)

i,1 (f)∣∣∣2

+2Re[Wv3,1Wv3,2S

(3)i,1 (f)S

(3)i,2 (f)

]; (38)

3) contribution of S(5)i,k (f):

(|Wv5,1|2 + |Wv5,2|2) · ∣∣∣S(5)

i,1 (f)∣∣∣2

+2Re[Wv5,1Wv5,2S

(5)i,1 (f)S

(5)i,2 (f)

]. (39)

We first consider the contributions (38) and (39). An order ofmagnitude for (38) is given by

|Wv3,1|2∣∣∣S(3)

i,1 (f)∣∣∣2 (40)

and the one for (39) is given by

|Wv5,1|2∣∣∣S(5)

i,1 (f)∣∣∣2 . (41)

From (24) and (25), we conclude that |Wv5,k|2 is about 40 dBbelow |Wv3,k|2 (recall that c3 ∼ c5 and that |vk| ≈ 0.01). Onthe other hand, from Fig. 3, |S(5)

i,1 (f)|2 is at most 20 dB above|S(3)

i,1 (f)|2. Thus, we expect (38) to dominate (39).Fig. 4 shows the range of (38) (the collection of dotted lines

at the top) and the range of (39) (the collection of dotted linesat the bottom) for all possible arg(v1)s and arg(v2)s, whilemagnitudes are fixed at |v1| = |v2| = 0.01. The figure is inagreement with the preceding predictions from rough estimatesthat (39) is negligible compared to (38) and, therefore, can beignored in (36).

As for the contribution of E(f), (37) is proportional to|Wv1c,1 −Wv1c,2|2. Fig. 4 shows, as a solid line, the “maxi-mum” contribution of E(f), which occurs when Wv1c,1 andWv1c,2 add their magnitudes. Note that, in contrast to (38) and(39), the term in (37) may completely vanish for particularvalues of v1 and v2.

Fig. 4 shows that, at most, the contribution of E(f) may beequal to the minimum contribution of S(3)

i,k (f). The maximumerror that results from neglecting the contribution of E(f) is3 dB. We may therefore use (38) as an expression for Pout,v(f)since our goal is as simple and practical a description of thespectral leakage as possible.


Fig. 4. Differential gain and phase error in the quadrature modulators produceterms that contribute to the spectral leakage. These terms can be separated intothree groups given in (37)–(39). This figure shows that (38) is dominant.

B. Contribution of Amplifier Mismatch

We now turn to the more general case of mismatched am-plifiers, where the complex gain functions for the amplifiers,which are given by Gk(r) for amplifier k = 1, 2, are notidentical. Now, [19, eq. (43)] is replaced by [19, eq. (44)].With slightly mismatched amplifiers, the weights in Pout(f),Pout,p(f), or Pout,v(f) do not change significantly. The con-tributions due to p and v, namely Pout,p(f) or Pout,v(f), aregiven by (35) and (38). However, now, the contribution ofE(f)may become significant, and this occurs only if |c1,1 − c1,2|2 issignificant. Thus, our estimate of Pout(f) becomes

Pout(f) = |c1,1 + c1,2|2 · |S(f)|2 + |c1,1 − c1,2|2 · |E(f)|2

+(|Wp2,1|2 + |Wp2,2|2

) · ∣∣∣S(2)i,1 (f)

∣∣∣2

+ 2Re[Wp2,1Wp2,2S

(2)i,1 (f)S

(2)i,2 (f)

]

+(|Wv3,1|2 + |Wv3,2|2

) · ∣∣∣S(3)i,1 (f)

∣∣∣2

+ 2Re[Wv3,1Wv3,2S

(3)i,1 (f)S

(3)i,2 (f)

]. (42)

C. Summary

For all practical purposes, we have that the spectral splatteris due to the following three terms.

1) The contribution of dc offset p Pout,p(f) is given by (35).2) The contribution of differential gain gd and differential

phase error φd Pout,v(f) is given by (38).3) The contribution of amplifier mismatch is given by

|c1,1 − c1,2|2 · |E(f)|2 . (43)

VI. EFFECTIVE PROCEDURE FOR SPECTRAL LEAKAGE

MINIMIZATION: CASE WITH FIXED QUADRATURE

MODULATOR IMPAIRMENT MAGNITUDES

|p1|, |p2|, |v1|, AND |v2|In this section, we determine how to minimize the three

primary contributions to the spectral leakage, as summarizedin Section V-C. We assume that the modulation scheme isfixed, the operating point is r = 1, and only the parameterspk, vk are allowed to vary. For clarity of demonstration, wealso assume the impairments in the quadrature modulators tohave equal magnitudes in both branches. Thus, only their phasearguments are allowed to vary. In Section VI-A, we show thatPout,p(f) depends only on arg(p1)− arg(p2) and is minimizedfor arg(p1)− arg(p2) = 0.8π. In Section VI-B, we show thatPout,v(f) depends only on arg(v1)− arg(v2) and is minimizedfor arg(v1)− arg(v2) = −0.5π. In the first two sections, theamplifiers are assumed to be identical, while in Section VI-C,the amplifiers have distinct gain functions. We show that back-ing off one of the amplifiers is enough to essentially eliminatethe contribution of E(f).

A. Minimizing the Contribution of DC Offset p, Pout,p(f)

Here, we wish to minimize Pout,p(f) given in (35). Since|p1| and |p2| are fixed, from (20), |Wp2,1|2 and |Wp2,2|2 arealso fixed. Thus, we need to find arg(p1) and arg(p2) that, onaverage, minimize the cross term

Re[Wp2,1Wp2,2S

(2)i,1 (f)S

(2)i,2 (f)

]. (44)

Use (20) to write

Wp2,1Wp2,2S(2)i,1 (f)S

(2)i,2 (f)

=(p1e

jθ2(f)p2

)|c3|2

∣∣∣S(2)i,1 (f)S

(2)i,2 (f)

∣∣∣ (45)

where

θ2(f) = arg[S

(2)i,1 (f)S

(2)i,2 (f)

]. (46)

Note that

Re(p1e

jθ2(f)p2

)= |p1||p2| cos (arg(p1)− arg(p2)− θ2(f)) .

(47)

Hence, (44) is bounded in magnitude by |p1||p2||c3|2|S(2)

i,1 (f)S(2)i,2 (f)| and depends only on arg(p1)−arg(p2). The

minimum is achieved whenever

arg(p1)− arg(p2) = θ2(f)± π. (48)

Although θ2(f) changes from one realization of symbols toanother, numerical experiments demonstrate that this variabilityis small over random realizations. We have observed fromexperiments that a reliable metric is

θave2 (f) = arg

(Mean

[S

(2)i,1 (f)S

(2)i,2 (f)

]). (49)


Fig. 5. θave2 (f), as defined in (49).

Fig. 6. Range of outcome of Pout(f) with variable angle between p1 and p2,while the magnitudes are held equal and constant.

This represents the phase argument of the average of

S(2)i,1 (f)S

(2)i,2 (f). The quantity θave

2 (f) is depicted in Fig. 5.θave2 (f) shows relatively little variation (less than 0.1π) about

the value −0.2π over the frequency band 1.5W < |f − fc| <4W for both modulation schemes, where W is the messagebandwidth. Hence, with

arg(p1)− arg(p2) = 0.8π (50)

we anticipate minimum spectral leakage in this frequencyrange.

Fig. 6 shows the output spectra for −π ≤ arg(p1)−arg(p2) ≤ π varying in increments of 0.1π. The lower solidline represents the minimum spectral leakage scenario, whichis obtained when (50) is satisfied. For completeness, we notethat the maximum spectral leakage scenario is represented bythe upper solid line and corresponds to arg(p1)− arg(p2) =

Fig. 7. θave3 (f), as defined in (54).

−0.2π. This indicates that spectral leakage can be reducedby up to 25 dB simply by wisely pairing up two misalignedmodulators.

It should not come as a surprise that θave2 (f) gives an indica-

tion of the value of arg(p1)− arg(p2) that minimizes spectralleakage over a significant portion of the spectrum. However,it is remarkable that in fact, arg(p1)− arg(p2) = θave

2 (f)± πgives “precisely” the value that minimizes spectral leakage atany frequency f .

B. Minimizing the Contribution of Differential Gain gd andDifferential Phase Error φd, Pout,v(f)

Here, we wish to minimizePout,v(f) given in (38). Since |v1|and |v2| are fixed, from (24), |Wv3,1|2 and |Wv3,2|2 are fixed.Hence, as before, we need to minimize

Re[Wv3,1Wv3,2S

(3)i,1 (f)S

(3)i,2 (f)

]. (51)

This occurs when

arg(v1)− arg(v2) = θ3(f)± π (52)

where

θ3(f) = arg[S

(3)i,1 (f)S

(3)i,2 (f)

]. (53)

As with θ2(f), θ3(f) depends on each realization of symbolsbut shows little variance over all realizations. The statistic weuse to characterize θ3(f) is

θave3 (f) = arg

(Mean

[S

(3)i,1 (f)S

(3)i,2 (f)

]). (54)

This represents the phase argument of the average over all

realizations of S(3)i,1 (f)S

(3)i,2 (f). In the range 2W < |f − fc| <

4W , θave3 (f) is within ±0.1π of the value 0.5π, as shown in

Fig. 7. Note that θave3 (f) shows greater variation over f than

θave2 (f), so the frequency range over which the spectral leakage


Fig. 8. Range of outcome of Pout(f) with variable angle between v1 and v2.

is minimized will be slightly smaller. We anticipate minimumspectral leakage on 2W < |f − fc| < 4W when pairing mod-ulators such that

arg(v1)− arg(v2) = −0.5π. (55)

Fig. 8 shows the output spectrum for arg(v1)− arg(v2)varying over −π to π in increments of 0.1π while the mag-nitudes are held equal and constant. The upper and lowerbounds for spectral leakage are plotted using solid linesand correspond to arg(v1)− arg(v2) = +0.5π and arg(v1)−arg(v2) = −0.5π, respectively. Again, these results are accu-rately predicted from θave

3 (f). In this case, the spectral leakagecan be reduced by 10–20 dB over the range 2W < |f − fc| <4W by optimal pairing of modulators. It can also be verifiedthat arg(v1)− arg(v2) = θave

3 (f)± π gives precisely the valuethat minimizes spectral leakage at frequency f .

C. Minimizing the Contribution of AmplifierMismatch [see (43)]

The contribution of amplifier mismatch is proportional to|c1,1 − c1,2|2. A very elementary solution to reducing this termis to back off the amplifier with a larger gain, e.g., amplifier 1.Suppose then that amplifier 1 is operated at r = λ (amplifier 2is still operated at r = 1). Recall that c1,1, c3,1, and 2c5,1 are thegain, rate of change, and curvature of the gain, respectively, ofamplifier 1 (at r = 1) and c′1,1(λ), c

′3,1(λ), and 2c′5,1(λ) denote

the gain, rate of change and curvature of gain of amplifier 1operated at λ. Assuming mild amplifier mismatch, λ remainsclose to 1. It is reasonable to assume that the physical behaviorof the amplifier will not be significantly different at r = λ sothat if c1,1, c3,1, and c5,1 are of the same order of magnitude,the same is true of c′1,1(λ), c

′3,1(λ), and c′5,1(λ). Thus, the

Fig. 9. LINC transmitter is impaired with amplifier mismatch and quadraturemodulator errors (|p1| = |p2| = 0.015, |v1| = |v2| = 0.008). The thick solidline depicts the contribution of amplifier mismatch that would be present ifamplifier 1 were not backed off. The dotted lines represent the spectral leakagewhen amplifier 1 is backed off to eliminate (43) and for all possible arg(p1) −arg(p2) and arg(v1) − arg(v2). The solid lines are close to the minimum andmaximum spectral leakage. They represent the simulations where arg(p1) −arg(p2) and arg(v1) − arg(v2) are chosen, following the method describedin Section VI-A and B. The amplifiers are operated at r = 1, and the input toamplifier 1 is scaled by a factor of λ = 0.7978 + j0.0143.

minimization procedures described in the previous subsectionsremain applicable for Pout,p(f) and Pout,v(f).

In order to eliminate the contribution of amplifier mismatch,we must solve for the operating point for λ that satisfies

λG1(λ) = G2(1). (56)

That is, the gain has changed from G1(1) to G1(λ), andthe factor λ in front of G1(λ) accounts for the fact that themagnitude of the input has been reduced to λ.

As an example, we assume that amplifier 1 has thecoefficients of TWTA 1, and amplifier 2 has the coefficients ofTWTA 2 (see Table I). Solving for λ to satisfy (56) amountsto finding the zero of a polynomial. Standard numericalmethods give

λ ≈ 0.7978 + j0.0143. (57)

The quadrature modulators assume that |p1| = |p2| = 0.015and |v1| = |v2| = 0.008. The thick solid line in Fig. 9 depictsthe contribution of amplifier mismatch that would be presentif amplifier 1 were not backed off. The dotted lines representthe range of the spectral leakage when amplifier 1 is backedoff to eliminate (43) and for all possible arg(p1)− arg(p2)and arg(v1)− arg(v2). The thin solid lines are close to theminimum and maximum spectral leakage. They represent thesimulations where arg(p1)− arg(p2) and arg(v1)− arg(v2)are chosen, following the method described in the previoussubsections. We see that the presence of amplifier mismatchwould have maintained the spectral leakage at the level of (43).


Backing off amplifier 1 results in effective cancellation ofspectral leakage due to amplifier mismatch.

VII. FURTHER DISCUSSION

A. Amplifier Characteristics

Recall from Section II-C that the amplifiers are assumed tobe memoryless and that their transfer curves are assumed to befrequency independent. The numerical values for the gain givenin Table I correspond to the class-C amplifier in [19], whilethose for the TWT amplifiers are obtained from single-tone testmeasurements of AM–AM and AM–PM transfer curves of ac-tual S-band devices [30]. Assessing the operational bandwidthlimitations on physical devices is a difficult task that requiresextensive data, which is not available to the authors. In addition,the operational bandwidth depends greatly on specific hardwarecomponents; thus, it is unlikely that any general statementthat would be valid for both class-C and TWT amplifiers canbe made. Nevertheless, concerning TWT amplifiers, one mayuse as a guideline the work in [32], which suggests 5%–10%relative bandwidths (300–500 MHz for 3–6-GHz carrier fre-quency amplifiers) for another linearization scheme whosedesign assumes frequency-independent transfer functions. It isspeculated (but left for future research to confirm) that similarbandwidths can be expected for this LINC scheme, given thatit shares the common assumption of frequency-independenttransfer characteristics.

We now discuss another aspect of amplifier characteris-tics. Recall that the LINC architecture, with reasonably smallimbalances, operates the amplifiers in the vicinity of theiroperating point. Thus, a second-order polynomial in r2 is morethan sufficient to model the gain over this relatively narrowrange of operation. In the previous sections, we have simplifiedthe expression for Pout(f) and, in turn, characterized pairs ofquadrature modulators, which minimize out-of-band spectralleakage. We have made assumptions as to the physical behaviorof the amplifiers, namely, that c1, c3, c5, and c1 + c3 were ofthe same order of magnitude. Should one deal with a LINCtransmitter whose amplifiers behave differently, the method canbe easily adapted. For instance, with less severe nonlinearities,say, c5 is negligible, all weights involving c5 vanish, and newconclusions as to optimizing pairs can be arrived at.

The simulations presented thus far assume TWT amplifierswith the parameters given in Table I. The conclusions also applyto the solid-state class-C amplifier simulated in [19] since theparameters c1, c3, c5, and c1 + c3 are also of the same orderof magnitude. The simulations in the next subsection assume aclass-C amplifier.

B. Modulator Errors With VariousCombinations of Impairments

In Section V, we have seen that with carefully paired pksand vks, the contributions of certain orders from both branchesnearly cancel out when |p1| = |p2| and |v1| = |v2|. However,in case |p1| �= |p2| and |v1| �= |v2|, the cancellation of thoseterms may be incomplete. Fig. 10 shows the output spectrumof the LINC transmitter, assuming identical class-C amplifiers

Fig. 10. Scenarios A and B (solid lines) assume that the modulators haveimpairments equal in magnitude but with a phase angle difference to maximize(upper curve) or minimize (lower curve) spectral leakage. The reduction inspectral leakage reaches 10 dB. In scenario C (dotted line), the impairmentsof one of the modulators have been reduced in magnitude, but they are chosento maximize spectral leakage. As a result, this transmitter produces a higherlevel of spectral leakage than that in scenario B. In scenario D (dashed line),both amplifiers have reduced impairments in magnitude, but they are chosento maximize spectral leakage. Spectral leakage is even higher than that inscenario C.

and quadrature modulators impaired with dc offset p as well asdifferential gain gd and differential phase φd in four scenarios.

1) Scenario A (upper solid line): The errors in the modula-tors are of the same magnitudes with |p1| = |p2| = 0.03,|v1| = |v2| = 0.02, and phases chosen to “maximize”spectral leakage.

2) Scenario B (lower solid line): The errors in the modula-tors are of the same magnitudes with |p1| = |p2| = 0.03,|v1| = |v2| = 0.02, and phases chosen to “minimize”spectral leakage.

3) Scenario C (dotted line): The errors are |p1| = 0.03,|p2| = 0.015, |v1| = 0.02, |v2| = 0.01, and phases cho-sen to minimize spectral leakage.

4) Scenario D (dashed line): The errors are |p1| = |p2| =0.015, |v1| = |v2| = 0.01, and phases chosen to maxi-mize spectral leakage.

Scenarios A and B show that spectral leakage is reduced byapproximately 10 dB with intelligently paired components.Next, even though the modulator impairments are less withscenario C than scenario B, the leakage for scenario B is lessthan that for scenario C because the errors produced in the twobranches are better cancelled. In scenario D, both modulatorshave smaller impairments than in scenario B. However, theimpairments are such that the errors add their effects, and thenet leakage is greater than that for scenario B.

These results indicate that the relative values of the argu-ments of error parameters are equally if not more important thanthe magnitude of the errors for obtaining minimum spectralleakage.


VIII. CONCLUSION

We derive an expression for the output spectrum of a LINCtransmitter afflicted with impaired quadrature modulators andmismatched nonlinear power amplifiers. This expression is asimplified version of that in [19]: Not only is the number ofcomponents greatly reduced, but also, the weighting factorsinvolved in this expression are expressed in terms of parameterswith a much clearer and transparent physical interpretation.That is, they describe the behavior of the amplifiers near theiroperating point.

With this simplified expression, we describe an analysisof the output spectrum of the LINC transmitter that involvesonly the most significant terms, assuming slightly mismatchedamplifiers and modest modulator errors. The analysis revealsthat the phase of the complex-valued modulator imperfectionshas a large impact on the level of spectral leakage and suggestsa method for pairing modulators to minimize spectral leakage.Using realistic TWT and solid-state amplifiers, we show thatthe optimal pairing of imperfect modulators reduces the spectralleakage by as much as 25 dB.

APPENDIX

Our goal is to assess the performance of the LINC transmitterby measuring the amount of spectral leakage at the output,which is the sum of the outputs of the two power amplifiers.In case of a wide-sense stationary (WSS) random process,the power spectral density (PSD) can be approximated byaveraging periodograms. However, not only is the input signalbest modeled as a cyclostationary process, but it also undergoesstrong nonlinear transformations, first through the SCS and thenthrough the amplifiers. Therefore, it is very difficult to ascertainthe statistical properties of the output to the transmitter. Never-theless, averaging periodograms is the simplest tool to gaugethe performance of the transmitter in terms of spectral leakage;thus, it is used in this paper. In addition, it allows for a clearcomparison with the result of [19].

In Section IV-A, we have made the observation from Fig. 3that |S(n)

i,1 (f)|2 and |S(n)i,2 (f)|2 are the same, on average, for

n = 2, 3, 5. In this Appendix, we show that if all signals canbe assumed to be WSS, then |S(n)

i,1 (f)|2 and |S(n)i,2 (f)|2 are

the same.Given an exponent n, the PSD of the WSS random process

(si,k)n(t) is equal to the Fourier transform of the autocorrela-tion function [33]

Rsni,k(τ) = Rsn

i,k(θ + τ, θ) = E

[(si,k)n(θ + τ) · (si,k)n(θ)

](58)

where E is the expectation operator. We expand (58) usingsi,1(t) = s(t) + e(t) and si,2(t) = s(t)− e(t). With n = 1, weobtain

E[(s± e)(θ + τ)(s± e)(θ)

]= E

[s(θ + τ)s(θ)

]

±E[s(θ + τ)e(θ)

]± E

[e(θ + τ)s(θ)

]+ E

[e(θ + τ)e(θ)

]

(59)

where the ±s are simultaneously all “+” (if k = 1) or all “−” (ifk = 2). We see that the cases k = 1 and k = 2 only differ in thesigns of certain terms. We obtain similar results with n > 1. Theheart of the matter is that those terms in which k = 1 and k = 2differ can be paired up and their sums shown to be zero. Forinstance, in (59), E[s(θ + τ) : e(θ)] + E[e(θ + τ) : s(θ)] = 0.In the more general case, what is to be shown is that

E [H (s(θ + τ), s(θ))] +E[H (s(θ + τ), s(θ))

]= 0 (60)

whereH is a function of two complex variables, which dependsonly on the magnitudes of the parameters and their phasedifference. Formally, expressing the complex parameters intheir polar forms, H satisfies

H(s, s′) = H(reiα, r′eiα′

)= h(α− α′; r, r′) (61)

where h is conjugate antisymmetric in the first parameter

h(α− α′; r, r′) = −h(α′ − α; r, r′). (62)

For simplicity, we denote by f(x, y, . . .) the probability den-sity of the occurrence of X = x, Y = y, . . ., where X,Y, . . .are random variables, and x, y, . . . are outcomes. Consider

E [H (s(θ + τ), s(θ))]

=∫H

(reiα, r′eiα′

)f(r(θ + τ)=r, α(θ + τ)=α

r(θ)=r′, α(θ)=α′) dα′dr′dαdr

(63)

=∫h(α− α′; r, r′)f(α(θ + τ)− α(θ)=α− α′, α(θ)=α′

r(θ + τ)=r, r(θ)=r′) dα′dr′dαdr.

(64)

With the change of variables ∆α := α− α′, this becomes

E [H (s(θ + τ), s(θ))]

=∫h(∆α; r, r′)f (α(θ + τ)− α(θ)=∆α, α(θ)=α′

r(θ + τ)=r, r(θ)=r′) d(∆α)dα′dr′dr

(65)

=∫h(∆α; r, r′)

( ∫f (α(θ + τ)−α(θ)=∆α

α(θ)=α′, r(θ + τ)=r

r(θ)=r′) dα′)d(∆α)dr′dr (66)

=∫h(∆α; r, r′)f(α(θ + τ)− α(θ)=∆α, r(θ + τ)=r

r(θ)=r′) d(∆α)dr′dr. (67)


Likewise, using (62), we get

E[H (s(θ), s(θ + τ))

]

=∫h(∆α; r, r′)f (α(θ)− α(θ + τ) = ∆α, r(θ) = r

r(θ + τ) = r′) d(∆α)dr′dr (68)

= −∫h(−∆α; r, r′)f (α(θ)− α(θ + τ) = ∆α, r(θ) = r

r(θ + τ) = r′) d(∆α)dr′dr. (69)

With the change of variable ∆α �→ −∆α, this becomes

E[H (s(θ), s(θ + τ))

]= −

∫h(∆α; r, r′)

× f(α(θ + τ)− α(θ) = ∆α, r(θ + τ) = r′, r(θ) = r

)× d(∆α)dr′dr. (70)

The difference between (67) and (70) is the position of r and r′

in the probability density.We finally claim that

f (α(θ + τ)− α(θ) = δα, r(θ + τ) = r′, r(θ) = r)= f (α(θ + τ)− α(θ) = δα, r(θ + τ) = r, r(θ) = r′) (71)

since the symbols are independently and identically distributed.Hence, any realization of these symbols is as likely as thetime-reversed sequence. On the other hand, the pulse shapeis symmetric in time. Therefore, for any r, r′, α, α′, thetransition from s(θ) = reiα′

to s(θ + τ) = r′eiα is as likely asthe opposite transition, from s(θ) = r′eiα′

to s(θ + τ) = reiα.

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Antoine Choffrut received the Diplome d’Ingenieurdegree from the Ecole Superieure d’Electricite, GifSur Yvette, France, in 2000 and the M.S. degreein electrical engineering from the University ofWisconsin-Madison in 2001, under the supervisionof Professors Van Veen and Booske.

He worked toward graduate research with the Uni-versity of Wisconsin-Madison until the summer of2003. Since September 2003, he has been a GraduateAssistant with the School of Mathematics, Univer-sity of Minnesota, Minneapolis.


Barry D. Van Veen (S’81–M’86–SM’97–F’02) wasborn in Green Bay, WI. He received the B.S. degreefrom Michigan Technological University, Houghton,in 1983 and the Ph.D. degree from the Universityof Colorado at Boulder in 1986, both in electricalengineering. He was an ONR Fellow while workingtoward the Ph.D. degree.

In the spring of 1987, he was with the Departmentof Electrical and Computer Engineering, Universityof Colorado at Boulder. Since August 1987, he hasbeen with the Department of Electrical and Com-

puter Engineering, University of Wisconsin-Madison, where he is currently aProfessor. He coauthored Signals and Systems (New York: Wiley, 1999 and2003) with S. Haykin. His research interests include signal processing forsensor arrays, wireless communications, and biomedical applications of signalprocessing.

Dr. Van Veen served as an Associate Editor for the IEEE TRANSACTIONS ON

SIGNAL PROCESSING and on the IEEE Signal Processing Society’s StatisticalSignal and Array Processing Technical Committee and the Sensor Array andMultichannel Technical Committee. He was a recipient of the 1989 PresidentialYoung Investigator Award from the National Science Foundation, the 1990IEEE Signal Processing Society Paper Award, and the Holdridge Teaching Ex-cellence Award from the Department of Electrical and Computer Engineering,University of Wisconsin-Madison, in 1997.

John H. Booske (S’82–M’85–SM’93) received thePh.D. degree in nuclear engineering from the Uni-versity of Michigan, Ann Arbor, in 1985.

From 1985 to 1989, he was a Research Scientistwith the University of Maryland, College Park. In1990, he joined the faculty of the Department ofElectrical and Computer Engineering, University ofWisconsin-Madison (UW), where he is currently aProfessor of electrical and computer engineering.From 2001 to 2005, he served as the Director ofthe UW Materials Science Program. He coedited

Microwave and Radio Frequency Applications (Westerville, OH: AmericanCeramics Soc. 2003) and Microwave and Millimeter-Wave Power Electronics(Piscataway, NJ: IEEE Press, 2004). His research interests include the experi-mental and theoretical study of coherent electromagnetic radiation sources andtheir applications with emphasis in the RF, microwave, millimeter-wave, andterahertz regimes. His current research interests include vacuum electronics,microwave and RF heating of materials, microfabricated terahertz waveguidecomponents, biological/biomedical application of microwave and RF fields,and nanoscale transport of dopants in semiconductors.

Prof. Booske was a Guest Editor of the IEEE TRANSACTIONS ON PLASMA

SCIENCE. He served as a Codirector of the U.S. Department of DefenseMURI99 Consortium on Innovative Microwave Vacuum Electronics and as theDirector of a MURI04 Consortium on Cathode Field Emission and RF windowbreakdown in high-power microwave sources. He was a recipient of the Univer-sity of Wisconsin Vilas Associate Award for research, the U.S. National ScienceFoundation Presidential Young Investigator Award, and numerous teachingawards, including the UW Chancellor’s Distinguished Teaching Award and theBenjamin Smith Reynolds Award for Excellence in Teaching Engineering.

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Minimizing Spectral Leakage of Nonideal LINC Transmitters by