Analysis on in-band distortion caused by switching amplifiers

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Published in IET Microwaves, Antennas & PropagationReceived on 15th February 2013Revised on 21st August 2013Accepted on 2nd October 2013doi: 10.1049/iet-map.2013.0096

T Microw. Antennas Propag., 2014, Vol. 8, Iss. 5, pp. 351–357oi: 10.1049/iet-map.2013.0096

ISSN 1751-8725

Analysis on in-band distortion caused byswitching amplifiersWonhoon Jang, Nelson Silva, Arnaldo Oliveira, Nuno Borges Carvalho

Instituto de Telecomunicações, Campus Universitáio de Santiago, Aveiro 3810-193, Portugal

E-mail: [email protected]

Abstract: A hypothetical model is built to explain an in-band distortion mechanism of the class F amplifier. Using the model it isanalytically proved that static non-linearities of switching amplifiers do not cause meaningful in-band distortion when theamplifiers are driven by ideal 1-bit digital RF signals. It is also found that short-term memory effects significantly contributeto in-band distortion. The analysis and the mechanism are tested and supported by MATLAB simulations. A systemcomposed of a field-programmable gate array and a class F amplifier has been built and the in-band distortion mechanism isverified by measurement.

1 Introduction

Power efficiency enhancement in wireless communicationsbecomes an important issue as more power-hungry devicessuch as smart phones and tablets appear, and more efficientspectrum usage loads heavier burden on efficient poweramplification. Accordingly, a lot of research efforts havebeen taken in enhancement techniques; 1-bit ΔΣ and/orpulse-width modulation (PWM) plus switching, linearamplification with non-linear components, Doherty, envelopetracking etc.Among the techniques, the 1-bit modulation plus switching

scheme can be realised with a simple amplifier architecture.However, it imposes limitation in power efficiencyenhancement bounded by low coding efficiency andextremely tight requirements for tunable bandpass filters, soit is not very promising if it is realised in a conventionalway. Nevertheless, the scheme has not been exploited fully,and can appropriately be extended in other advancedarchitectures [1] for the future transmitters such as all-digitaltransmitters that provide high reconfigurability [2]. Thus, itneeds more attention.All-digital transmitters employing the 1-bit modulation plus

switching scheme have been attempted to be built recently in[3–5]. However, all the reported transmitters seem tounexpectedly exhibit large increase of in-band distortion atthe outputs. Especially, the increase is more severe for thetransmitters employing ΔΣ modulation. On the other hand,various classes of switching amplifiers [6–8] have beendesigned mainly focusing on maximising power efficiencyaround intended carrier frequencies by using continuouswave (CW) excitations. Not much attention has been paid tomaximising operation bandwidth, and thus the resultingdesigns normally end up with narrow operation bandwidth.Relatively narrow operation bandwidth compared with theinput signal bandwidth is one of the main causes for the

unexpected increase of in-band distortion and accordinglydegrades error-vector magnitude (EVM).Another cause of in-band distortion could be baseband

effects [9, 10], which are well known to generate asymmetricspectral regrowth. However, in this paper, the authors try toanalytically explain an in-band distortion mechanism ofswitching amplifiers, without considering baseband effectsbecause all asymmetric spectral regrowths observed in [3–5]are not significant around the desired channels. Theexplanation is supported by simulation and measurement.

2 Analysis on in-band distortion

In this section, the Wiener–Hammerstein model is used as ahypothetical model in order to explain a phenomenon ofin-band distortion in a switching amplifier when it is drivenwith a digital RF signal. The model is chosen based on anassumption that baseband effects caused by a switchingamplifier are not significant. The assumption is supportedby observations on the measured frequency spectra reportedin [3–5]. Excluding baseband effects, the Wiener–Hammerstein structure seems to be a model that is properand sufficient to explain the phenomenon even though othermodels also can be used.The model is shown in Fig. 1 where h1(t) and h2(t) are

bandpass filters and account for frequency-dependentcharacteristics. The non-linear function in the middlecaptures static non-linearities of the amplifier. The non-linearfunction can be described with an odd-order polynomial as

y(t) =∑nk=1

ck xk t + Dtk( )

, k:odd (1)

where ck represents the kth order real number coefficient andthe coefficients capture amplitude-to-amplitude modulation

351& The Institution of Engineering and Technology 2014

Fig. 1 Non-linear model of a switching amplifier

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(AM–AM). Δtk is the kth order time delay and the delaysaccount for amplitude-to-phase modulation (AM–PM).Even-order terms are ignored without affecting the purpose ofthis paper.For analytical convenience, it is considered that the

polynomial block generates up to third-order non-linearitieswith which it is enough to find out the generationmechanism of in-band distortion. In this paper, the amplifierto be analysed is considered as the class F type, but otherswitching amplifiers such as classes D and E can beanalysed similarly. The model will be embodied using twodifferent types of excitations, a sinusoid and a square wave,so the resulting model will comply with characteristics of aclass F amplifier when it is driven with those kinds ofexcitations. After that, the established model will be drivenwith a ΔΣ-modulated signal and how in-band distortionoccurs will be discussed.First, if the input is a sinusoid such that x(t) = sin ωct, where

ωc is the carrier frequency, the input filter, h1(t) in Fig. 1, isexpected to be trivial, which means that x(t) ≃ x(t). Thepolynomial function generates fundamental and harmonicproducts. The harmonic products are filtered out by h2(t)and the resulting output, y(t), is an amplified sinusoid suchthat y(t) = A sin ωc(t + Δt), where A and Δt are respectivelyan amplitude and a time delay of the resulting sinusoid.With a sinusoidal excitation, the whole model works as alinear amplifier just as an ideal switching amplifier issupposed to operate. y(t) before the output filtering can beviewed as the drain–source voltage of a transistor in a classF amplifier. In Fig. 2, a sinusoid input and thecorresponding y(t) are shown when

y(t) = 8x(t)− 16

3x3 (t) (2)

For now, (2) has only AM–AM characteristics and, later inthis paper, AM–PM characteristics will be also included. Thecoefficients of the polynomial are chosen in such a way thaty(t) mimics a square wave as the order increases and thus themodel operates as a class F amplifier when the excitation is a

Fig. 2 AM–AM effects to a sine wave


sinusoid. The polynomial with the same coefficients as in (2)is used for the next square wave case as well.Next, the model is driven with a square wave. Again h1(t)

in Fig. 1 is ignored because the result is similar to the previoussinusoid case and trivial if it works as a narrow bandpass filterand passes only the fundamental. Accordingly, the input tothe polynomial can be written as

x(t) = 4

p

∑1m=1

sin (2m− 1)v0t[ ]2m− 1

(3)

where ω0 is the fundamental frequency of the waveform. Aninput with m = 20 in (3) to the polynomial in (2) and thecorresponding output waveforms are shown in Fig. 3.The output waveform reaches closer to a square wave as the

order in (3) increases. After filtering by h2(t) in Fig. 1, theresulting y(t) is a sinusoid, which complies with theoreticalbehaviours of a class F amplifier. It is interesting to see that astatic and odd-order non-linear model with only AM–AMability as the polynomial model in (2) becomes a linear-equivalent model when it is driven with a square wave or a1-bit digital signal whose amplitude alternates in positive andnegative territories. The model behaviour is a bit confusing if itis considered in the frequency domain because of so manyharmonic products involved; however, it is obvious tounderstand the behaviour if it is considered in the time domain.So far, the polynomial model in (2) produces AM–AM

only. Now, we are trying to include AM–PM to thepolynomial model. Before including AM–PM, it might beneeded to clarify what is AM–PM in the case of asquare-wave excitation and how to properly modify thepolynomial model. If one strictly followed the meaning ofAM–PM corresponding to a square waveform, thepolynomial model could be

y(t) = 8x t + Dt x| |( )− 16

3x3 t + Dt x| |( )

(4)

where Δt|x| is a time delay depending on input amplitude.However, considering realistic behaviours of a class Famplifier, Δt|x| tends to be constant or trivial over a switch-operation input range. Even with accurate circuit-levelmodels, it is unlikely to observe the literal AM–PM effectscorresponding to square-waveform excitations. In addition, ifa correctly designed class F amplifier is modelled with apolynomial, it is very likely that the time delay of the first

Fig. 3 AM–AM effects to a square-like wave

IET Microw. Antennas Propag., 2014, Vol. 8, Iss. 5, pp. 351–357doi: 10.1049/iet-map.2013.0096

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term restricts time delays of odd-order terms. That is imposedby designed harmonic impedances looking towards outputnetworks from a transistor. Hence, it is more interesting tomodify the polynomial model so that it generates AM–PMwhen being excited with a CW, which is a real andobservable behaviour of an amplifier, and to check how itbehaves when being excited with a square wave. Thereforethroughout the paper AM–PM refers to the CW case ordifferent time delays among harmonics.Including AM–PM the polynomial model becomes
y(t) = 8x t + Dt1( )− 16

3x3 t + Dt3( )

(5)

Fig. 4 shows the output waveform when a square wave drivesthe model in (5) where ω0(Δt1 − Δt3) = (π/5).It is obvious that Fig. 4 does not demonstrate proper

behaviours of a class F amplifier. From observations onboth output waveforms in Figs. 3 and 4, it is concluded thatthe polynomial with only AM–AM is more appropriate tomimic behaviours of a class F amplifier.Now, as intended for switching amplifiers, an input signal

takes the form of a 1-bit digital RF signal generated by ΔΣ orPWM and digital frequency up-conversion. The resultingsignal can be described as

x(t) = A× 4

p

∑1m=1

sin (2m− 1) v0t + f(t)( )[ ]

2m− 1(6)

where A is an amplitude of the 1-bit digital RF signal and φ(t)represents time-dependent discrete phase changes. If thebandwidth of h1(t) is wide enough to pass the whole inputsignal, that is, x(t) = x(t), the output of the static non-linearblock having only AM–AM characteristics can be derived as

y(t) =∑nk=1

ckAk × 4

p

∑1m=1

sin (2m− 1) v0t + f(t)( )[ ]

2m− 1

{ }k

=∑nk=1

ckAk × 4

p

∑1m=1

sin (2m− 1) v0t + f(t)( )[ ]

2m− 1(7)

= aA× 4

p

∑1m=1

sin (2m− 1) v0t + f(t)( )[ ]

2m− 1

= ax(t)

Fig. 4 AM–PM as well as AM–AM effects to a square-like wave


where α is a constant and equals to∑n

k=1 ckAk−1. x(t) in (7) is

the same as in (6). Note that

4

p

∑1m=1

sin (2m− 1) v0t + f(t)( )[ ]

2m− 1

{ }k

= 4

p

∑1m=1

sin (2m− 1) v0t + f(t)( )[ ]

2m− 1

in the procedure from the first to the second line in (7) since itis a square wave alternating between 1 and − 1 and k is anodd number. Equation (7) appears to be a result of linearamplification. After filtering by h2(t), the originalinformation channel will be reconstructed that explainstheoretical linear amplification by switching amplifierswhen driven by ideal 1-bit digital RF signals.It is interesting to see how AM–PM effects appear when

the model is driven by a 1-bit modulated signal. IncludingAM–PM effects, the resulting y(t) before the output filteringis similarly obtained as

y(t) =∑nk=1

ckAk 4

p

∑1m=1

sin 2m− 1( ) v0t + f(t)+ fk

( )[ ]2m− 1

= aA4

p

∑1m=1

bm

sin 2m− 1( ) v0t + f(t)+ fm

( )[ ]2m− 1

(8)

where φk is the kth order phase modulation. βm and φm are,respectively, an amplitude and a phase change resulted fromphasors summation, corresponding to the (2m − 1)thharmonic. The component around ω0 in (8), αA × (4/π)β1sin (ω0t + φ(t) + φ1), is a result of linear amplification with aconstant time-shift, (φ1/ω0), since the input componentaround ω0 in (6) is A × (4/π) sin (ω0t + φ(t)). Surprisingly,even AM–PM effects do not cause any in-band distortion,which leads to the fact that static non-linearity alone doesnot generate any in-band distortion in response to ideal1-bit digital RF signals. In addition, h2(t) cannot causein-band distortion. Therefore it can be concluded thatbandwidth restrictions (or short-term memory effects) asimposed by h1(t) in the proposed model are the maincauses of in-band distortion in switching amplifiers. Theconclusion will be verified by simulation and measurementin the next sections, because it is not clear to see whetheran analytical form including h1(t) produces in-banddistortion.

3 Simulation

The model in Fig. 1 with the polynomial in (2) is used in thissimulation to demonstrate effects of the input filter. It isdriven by a ΔΣ-modulated input whose spectrum is shownin Fig. 5a. If the operation bandwidth of the model is wideenough to take in all the input signals, in other words, h1(t)does not filter out any of the input signals, thecorresponding time-domain signal maintains a square-likewaveform most of the time as partly shown in Fig. 5b.Through the polynomial, the resulting output spectrum isshown in Fig. 5c which is a zoomed-in version around thefrequency range indicated by the dashed box in Fig. 5a andthe spectrum barely contains distortions. The computeddistortion is so small that it is located lower than the bottom


Fig. 5 Simulation results without input filtering

a Input spectrumb Part of the input in the time domainc Input and output spectraDistortion in (c) is located out of the bottom range and not shown here

Fig. 6 Simulation results with input filtering

a Input spectrumb Part of the input in the time domainc Input, output and distortion spectra

Fig. 7 AM–AM characteristics of three non-linear functions: third(line) and seventh (dash) order polynomials, and a hyperbolictangent (dash-dot) function

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range of Fig. 5c and it is not shown. As derived in (7), theoutput is a result of almost linear amplification of the input.The identical signal drives the same polynomial model, but

this time the signal is filtered by h1(t) as the spectrum isshown in Fig. 6a. The information channel is intact afterthe filtering. A part of the corresponding time-domainsignal is shown in Fig. 6b. Note that the input signalenvelope is not constant anymore because of the filtering.Now, it is expected that the static non-linearity plus thenon-constant envelope of the input excitation generate asignificant amount of in-band distortion. In Fig. 6c, azoomed-in version of the output spectrum is shown alongwith the input and the resulting distortion spectra. It isobserved that significant in-band as well as out-of-banddistortions occur. This simulation mimics the situation thatthe operation bandwidth of a class F amplifier issignificantly narrower than the bandwidth of an inputsignal. As concluded in the previous section, it is clearlyvisualised that the input filter plays a key role in generatingin-band distortion.Now, in-band distortion is quantitatively evaluated

depending on operation bandwidth and static non-linearity.Three kinds of non-linear functions are used; third andseventh order polynomials, and a hyperbolic tangentfunction. The AM–AM characteristics of the functions areshown in Fig. 7. The resulting EVMs depending onoperation bandwidth are depicted in Fig. 8. Percentages of


power on the x-axis indicate how much power is passedthrough the input filter compared with the total input power.The noise power level of the input signal has been set at− 45 dBc. The simulation shows similar behaviours to themeasurement results in the next section.


Fig. 8 Simulated EVM against percentage of power passedthrough the input filter, depending on the three non-linearfunctions as in Fig. 7

Table 1 Measured input signal powers inside the operationbandwidth of the amplifier depending on sampling frequenciesand modulation types

Frequencies, MHz Pin_80 MHz, dBm

PWM ΔΣ

7.03125 −6.6 −6.114.0625 −7.3 −6.428.125 −8.6 −6.656.25 −13.4 −9.3

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4 Verification by measurement

An amplifier has been built to verify the analysis andsimulation results in the previous sections. The amplifierconsists of a driver, a class F amplifier and a simplebandpass filter as shown in Fig. 9.The driver is included to boost the output of a

field-programmable gate array (FPGA), because the signallevel from the FPGA is too low to directly drive the class Famplifier. The overall gain is 25 dB and the operationbandwidth is 80 MHz about 900 MHz of the centrefrequency. The maximum drain efficiency and power addedefficiency of the amplifier including the driver are,respectively, 60 and 55% when it is driven with a CW.The built amplifier cannot exactly be separated as the

structure of the model, so the bandwidths of input andoutput bandpass filters cannot independently be adjusted. Infact, one restrains the other. Accordingly, it is ratherreasonable to consider that both bandwidths are equal to theoperation bandwidth of the amplifier. In addition, tomeasure in-band distortion depending on operationbandwidth, the bandwidth of the amplifier should bemanipulated as the model is treated in the previous sections.

Fig. 9 Photo of an amplifier used in measurement


However, that is very cumbersome because eachmeasurement needs a different hardware design. Theproblem is circumvented by fixing the bandwidth of theamplifier and changing bandwidth of an input signal instead.EVMs caused by the amplifier and drain efficiencies are

measured when it is driven by a PWM or ΔΣ signal withvarious sampling frequencies. Changing sampling frequencyvaries signal bandwidth. Using various signal bandwidthsprovides an equivalent way of changing operationbandwidth of the amplifier and avoids difficulties inmanipulating operation bandwidth by hardware means. InTable 1, measured input signal powers inside 80 MHz ofthe operation bandwidth of the amplifier are listeddepending on sampling frequencies for PWM and ΔΣmodulations.The total powers measured up to 5 GHz for the PWM

and ΔΣ modulation signals are, respectively, −5.7 and−5.5 dBm. With the measurements in Table 1, percentagesof power inside the operation bandwidth are calculated withrespect to the total measured powers.Fig. 10 shows measured EVMs depending on percentages

of power inside the amplifier bandwidth with respect to thetotal input signal power. The EVMs only count thosecaused by the amplifier, that is, EVMs caused by the FPGAare compensated from the measurements. In addition, drainefficiencies are measured and shown in Fig. 11. Theefficiencies are measured by only considering signal powerand ignoring coding efficiency. When more than 80% ofinput signal power is taken in by the amplifier, drainefficiency reaches almost to the maximum and EVM is keptlow at the same time. That is because the envelope of a

Fig. 10 Measured EVM against percentage of power inside theoperation bandwidth for PWM (line) and ΔΣ (dash) signals


Fig. 12 Measured output frequency spectra of the ΔΣ signal

a With 14.0625 MHz of sampling frequencyb Zoomed-in version of Fig. 12ac With 56.25 MHz of sampling frequencyd Zoomed-in version of Fig. 12c

Fig. 11 Measured drain efficiency against percentage of powerinside the operation bandwidth for PWM (line) and ΔΣ (dash)signals

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1-bit input signal is maintained nearly constant as a CW if aclass F amplifier has enough bandwidth.Figs. 12a and c show measured output frequency spectra of

the ΔΣ signal when sampling frequencies are 14.0625 and56.25 MHz, respectively. The signal bandwidth in Fig. 12cis four times wider than that in Fig. 12a. Percentages ofpower taken in the operation bandwidth are about 80% inFig. 12a and 40% in Fig. 12c. The zoomed-in spectra inFigs. 12b and d clearly show different signal-to-noise ratios.The difference is about 10 dB.Constellation diagrams shown in Figs. 13a and b,

respectively, correspond to Figs. 12a and c. MeasuredEVMs with respect to Figs. 13a and b are about 3 and10%, respectively. Since the EVM of the input signalgenerated from the FPGA is 2%, actual EVMs caused bythe amplifier are about 1 and 8%.It has been verified that constraints in operation bandwidth

of switching amplifiers can cause a large amount of in-banddistortion. That aspect suggests switching amplifiers bedesigned by maximising not only power efficiency, but alsooperation bandwidth for linearity enhancement. In that way,


Fig. 13 Measured constellation diagrams corresponding to

a Fig. 12ab Fig. 12cMeasured EVMs caused by the amplifier are: (a) 0.7% and (b) 8.2%EVM of the input signal is about 2%

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digital pre-distortion requirements could be eased thanks tothe enhanced linearity.

5 Conclusion

An in-band distortion mechanism of switching amplifier wasexplained analytically by using a hypothetical model. Theanalyses were supported by MATLAB simulations. Atransmitter composed of a class F amplifier and an FPGAwas built and used in measurement. The measurementsverified the analyses and the simulations. The study in thispaper indicates that, in addition to power efficiency,operation bandwidth is also an important figure of merit indesigning switching amplifiers.

6 References

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2 Silva, N.V., Oliveira, A.S., Gustavsson, U., Carvalho, N.B.: ‘Adynamically reconfigurable architecture enabling all-digital


transmission for cognitive radios’. IEEE Radio Wireless Symp.,January 2012

3 Frappé, A., Flament, A., Stefanelli, B., Kaiser, A., Cathelin, A.: ‘Anall-digital RF signal generator using high-speed ΔΣ modulators’, IEEEJ. Solid-State Circuits, 2009, 44, (10), pp. 2722–2732

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7 Kim, J., Jo, G., Oh, J., Kim, Y., Lee, K., Jung, J.: ‘Modeling and designmethodology of high-efficiency class-F and class-F−1 power amplifiers’,IEEE Trans. Microw. Theory Tech., 2011, 59, (1), pp. 153–165

8 Thian, M., Fusco, V.: ‘Analysis and design of class-E3F andtransmission-line class-E3F2 power amplifiers’, IEEE Trans. CircuitsSyst. I, Reg. Pap., 2011, 58, (5), pp. 902–912

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