Adaptation of a Digitally Predistorted RFAmplifier Using Selective Sampling

R.Neil Braithwaite

(Powerwave Technologies,Santa Ana,CA 92705,the U.S.)

Abstract:In this paper,a reduced-cost method of measuring residual nonlinearities in an adaptive digitally predistorted amplifier is proposed.Measurements obtained by selective sampling of the amplifier output are integrated over the input envelope range to adapt a fourth-order polynomial predistorter with memory correction.Results for a WCDMA input with a 101 carrier configuration show that a transmitter using the proposed method can meet the adjacent channel leakage ratio(ACLR)specification.Inverse modeling of the nonlinearity is proposed as a future extension that will reduce the cost of the system further.

Keyw ords:amplifier distortion;communication system nonlinearities;power amplifier linearization

1 Introduction

Adigital transmitter used in wireless communication applications comprises several stages,including a digital baseband,digital-to-analog converter(DAC),modulator,and power amplifier(PA).An efficient PA often has a nonlinear gain that varies as a function of the input signal envelope.As a result,a linearization method is required to compensate for undesired nonlinearities.

Digital predistortion(DPD)involves introducing a nonlinear gain function into the digitaltransmission path that opposes nonlinearities in the modulator and PA stages.Adaptive DPD measures the residual nonlinearity of the predistorted transmitter and adjusts the DPDcoefficients to reduce distortion in the output signal.The measurement circuitry,referred to as the observation path,includes down-conversion and digitization of the PA output signal and typically comprises demodulator and analog-to-digitalconverter(ADC)stages.The block diagram of a typical transmitter,including adaptive DPD,is shown in Fig.1.

An inherent problem with adaptive linearization is that the measurement system cannot distinguish distortion generated by nonlinearities in the transmitter from distortion induced by data acquisition components in the observation path.Thus,the observation path must be significantly more linear than the desired linearity of the transmitter.It must also have sufficient dynamic range to avoid degrading the output spectral mask.As a result,the measurement circuitry used is often expensive.

As an example of the contribution of the observation path to transmitter cost,ADC can be considered.The price of an ADC increases with the sampling rate and resolution.Atypical zero-IF observation path for a multicarrier WCDMA[1]signal uses two 14 bit ADCs sampled at 122.88 MHz[2].If thenumber of ADCs was reduced to one,the resolution reduced to 8 bits,and the sample rate reduced to 32 MHz,ADC-related costs would drop by a factor of 25[3].

Figure 1.?Block diagram of a typical transmitter with adaptive DPD,including an observation path for measuring the PA output signal.

An inexpensive approach for measuring nonlinearities in a digital transmitter is shown in[4].In this approach,a known calibration signal is transmitted with an amplitude-modulated(AM)component.The AM component excites the nonlinear modes of the PA to generate distortion.In contrast,the measurement circuit contains a cancellation loop that reduces AM variations so that the signal amplitude is nearly constant.Thus,the distortion generated within the measurement circuit is minimal.The dynamic range of the measured signal at the ADC is also reduced.As a result,the requirements of the data acquisition components are relaxed,and the cost of the adaptive system is reduced.The drawback of[4]is that the measurements must be made offline because of the use of a calibration signal.

It is preferable to optimize the system adaptively based on measurements made while transmitting the actual signal.In section 2,a WCDMA signal is sampled selectively to create a probing signal that highlights nonlinear modes of the transmitter.The probing signal allows for online adaptation of the digital predistorter.Section 3 describes the coefficient estimation module,which includes a recursive integration method for a 4th order predistorter.Memory compensation is discussed in section 4,and linearization results for a WCDMAsignal are given in section 5.Section 6 describes inverse modeling of the nonlinearity as a future extension.

This paper is an extension of[5],a conference paper titled“Measurement and correction of residual nonlinearities in a digitally predistorted power amplifier,”by the author,which appeared in the proceedings of the 2010 75th ARFTG Microwave Measurement Conference?IEEE.The remainder of the introduction includes a brief review of past DPD work done by other researchers.

1.1 Digital Predistortion Background

The predistorted baseband signal for the digitaltransmitter in Fig.1 is

where GDPDis the predistortion gain and a nonlinear function ofχ.The predistorted baseband signalis up-converted to produce an RFsignal

where hDAC{}is a reconstruction filter used in the digital-to-analog conversion(not shown in Fig.1),and ωLO(t)is the LO frequency.The output of the PA is

where GPAis the gain of the PA and a nonlinear function of(χRF).

Memoryless nonlinearities are often described using AM-AM and AM-phase modulated(PM)curves where the amplitude and phase components of the gain are plotted as a function of the input envelope.The gain curves produced by the DPD module are represented using a polynomial function of order N:

where bnare complex DPD coefficients.The gain of the predistorted transmitter,which is the combination of the DPD and PA nonlinearities,is also represented by a polynomial:

where G0is the desired(linear)gain of the transmitter,and anare complex values referred to as residual memoryless coefficients.Estimates of the residual nonlinearity anare used to update the DPD coefficients bnin an iterative manner,that is,

where()Tindicates transpose(i)=]Tat iteration i,and＜α＜1.The iterative sequence(6)has converged when

The standard approach to measuring the residual nonlinearities(shown in Fig.1)is to capture the signal from the PA output,down-convert and digitize it to produce an observation signal yo(k),then compute the coefficients anthat minimize[2],[6],[7]

where BFnis a gain basis function and a nonlinear function ofχ(k).For the memoryless case,the gain basis functions are defined by

In general,the observation signal yo(k)is corrupted by nonlinearities within two paths:the transmitter path fromχ(k)to yRF(t),and the observation path from yRF(t)to yo(k).The DPD is intended to compensate for the former.Any nonlinearity in the observation path offsets the steady-state DPD coefficients b and degrades the adjacent channelleakage ratio(ACLR)measured at yRF(t).Thus,the observation path in Fig.1 must be significantly more linear than the desired linearity of the transmitter.The observation path should also minimize other impairments,such as demodulator imbalance,LO phase noise,and quantization noise.

Up to this point,only memoryless PA nonlinearities have been considered.Nonlinear memory is often modeled using delayed digital samples of the input signal.For example,the gain basis functions for a discrete Volterra series would be

whereτlare integer sample offsets,v is an index,and()＊is complex conjugate.In general,the number of basis functions in a Volterra series is too large to be practical.As a result,pruned versions are typically used[8]-[10].A popular pruned basis function set is

which is referred to as a memory polynomial[6],[7],[11],[12].Memory polynomials are also implemented bydelaying the distortion modes as opposed to delaying the gain component only.This produces predistorted waveforms that are a weighted sum ofnχ(k-τ)instead ofnχ(k).Other memory models use delayed samples from both the inputχ(k)and outputχDPD(k)of the predistorter,where the latter forms a feedback loop around a nonlinear kernel.An example using feedback within an artificial neural network can be found in[13].

?Figure 2.Selective sampling extracts the desired probing signal(blue circle)from the WCDMAsignal(green cloud).

While both the pruned Volterra series and artificial neural network models discernably improve distortion cancellation,memory models based on delayed or fed-back digital samples are not compatible with the measurement approach proposed in section 2.The compatible memory model described in section 4 is based on derivatives of the input envelopet,instead of delayed samples.This model is shown in section 5 to improve ACLRperformance over the memoryless DPD model presented in[5].

2 Measurement Approach

Anew approach to measuring transmitter nonlinearities,suitable for WCDMA signals,is proposed here.The motivation is to replace the standard observation path shown in Fig.1 with something much cheaper.To achieve this goal,the required linearity,dynamic range,and sampling rate of the observation path must be reduced.In the standard approach,the WCDMA output signal is captured directly;however,the sampling rate must be several multiples of the Nyquist rate to measure the out-of-band distortion without aliasing[14].In addition,a large dynamic range is needed in the standard approach to measure distortion below an ACLR2 levelof-50 d Bc[1]from a signal with a large peak-to-average power ratio(PAPR),on the order of 7.2 d B.

The proposed approach involves creating a probing signal,similar to the one used in[4],that is extracted from a subset of the sampled WCDMAinput signal.This selectively sampled probing signal has a lower PAPRand sampling rate than the WCDMAsignal.The proposed measurement circuitry reduces the PAPRof the probing signal further,to almost 0 d B,using a cancellation bridge.As a result,the observation path requires far lower linearity,dynamic range,and sampling rate than the standard approach,allowing cost to be reduced.

It is possible to transform the complex input signalχ(k)into

where bothλ(k)andθ(k)vary with time,andρis a constant.This transformationχ(k)=f{ρ,λ(k),θ(k)}is similar to a conversion from rectangular to polar coordinates,except that the origin is offset by the constantρ.Equation(11)is an exact transformation that converts the signal into a form that makes the selective sampling,described below,easier to implement.

Consider a subset of the input samplesχ(k)=,,,whereλois a constant.This subset can be viewed as a selective sampling process where a circle within the I-Q space is chosen.An example for a WCDMA signal is shown in Fig.2.The circular trajectory is specified as a function of the angleθ(Fig.3).The trajectory is used as a probing signal to highlight nonlinearities within the transmitter,which appear as elliptical deformations in the RFoutput yRF(t)(Fig.3).In a typicalimplementation,several probing signals with differentρvalues are tested.These probing signals create circles at various power levels within the I-Q space,although only one circular trajectory is shown in Fig.2.

Elliptical trajectories at the output are caused by slopes in the gain of the transmitter.Deformations of the circular trajectory due to slopes in the AM-AM(center)and AM-PM(right)curves occur along the horizontal and 45 degree axes,respectively.

▲Figure 3.Baseband input and RFoutput signals,selectively sampled.

The deformation of the circular trajectory provides information about the nonlinear gain of the transmitter around the output operating point GoρχLO(t).Any slope in the gain curvescreates an elliptical trajectory at the PA output.For example,a downward slope in the AM-AM curve,,compresses the circle along the horizontalaxis,as shown in Fig.3(center).Aslope in the AM-PM curve,,shears the circle,thereby compressing or expanding the output trajectory along the 45 degree axis,as shown in Fig.3(right).Thus,a complex measurement of the nonlinear gain at a specific power level is extracted from elliptical deformations of the circular trajectory.

?Figure 4.Atransmitter using an RFcancellation loop,detector,and selective sampling to measure residual nonlinearities.

The deformation in the circular trajectory is measured using a bridge circuit comprising a cancellation loop and a square law detector(Fig.4).The cancellation loop output is

where

The AM component within the measurement system is minimized by the cancellation loop.With the AM component removed,the nonlinear modes of the measurement system are not stimulated,and the dynamic range of the detector output signal,is reduced(for the selected samples whereλ=λo).

The selective sampling module,shown in Fig.4 following the detector,contains a sample/hold circuit and an ADC.The sample/hold captures detector valuesγdetcorresponding to time instants tswhenλ=λo.The cancellation loop and selective sampling reduce the resolution and

is specified as a function ofθ.If the cancellation loop is balanced and the transmitter is linear,the selectively sampled signalγdet(θ)is constant as a function ofθ.Misalignment of the cancellation loop creates a first harmonic variation as a function ofθ.Nonlinear gain(elliptical deformation)in the transmitter creates second harmonic variations.These three cases are shown in Fig.5.

Although the use of harmonics of γdet(θ)for measuring nonlinearities is believed to be new(outside of the author's previous work[4],[5]),Cavers used similar first and second harmonics in[15]to measure the offsets and imbalances in modulator circuits for the specialcase ofχ(k)=λoexp(jθ(k))(no cancellation loop).The author sampling rate required of the ADC,allowing for the use of lower cost components.Cost is discussed further at the end of this section.

Figure 5.?The selectively sampled outputs of the detector in three cases.

The selectively sampled output of the detector recommends reading[15]to obtain a better understanding of the proposed technique.

A compact method of representing selective samples of the detected signalγdet(θ(k))is to accumulate the measurements within look-up-tables(LUTs).LUTs of the accumulated zero,first,and second-order moments of γdet(θ(k)),denoted by L0,L1,and L2,respectively,are

where i is the bin index and the quantization ofθis

The mean and variance ofγdetfor bin i are

and

respectively.The first and second harmonics ofγdet(θ)are measured by demodulating the mean LUTas a function ofθ.The demodulated signal becomes

where the first and second harmonics correspond to m=1 and m=2,respectively.Memoryless measurements(Гθ,Г2θ)are obtained for each value ofρtested.

To show the cost savings of the proposed measurement approach,the approach is compared to the standard observation path.A zero-IF observation path used in[2]is chosen as the bench mark.It uses two 14 bit ADCs sampled at 122.88 MHz to measure a two-carrier WCDMAsignal with a 101 carrier configuration and captures the output signal in a 16 K sample buffer.In contrast,only one ADC is needed for the proposed approach,and the measurements are stored in LUTs L0,L1,and L2.Each LUT has 64 bins for a total storage of 192 bins.The dynamic range required by the proposed approach is negligible because of the circular probing signal and the cancellation loop.However,it will be assumed that an 8-bit ADC is used.

The ADC sampling rate for the proposed approach depends on how the selective sampling module in Fig.4 is implemented.If sampling asynchronously(whenλ=λo),then the sample/hold must be placed before the ADC.The required sampling rate for the ADCis determined by the inverse of the minimum time between selective samples.Because samples can be ignored,the sampling period can be made arbitrarily long,limited primarily by the maximum hold time of the sample/hold device.Thus,it is possible to reduce the sampling rate below 1 MHz to allow the use of an ADC that is integrated within a micro-controller(part of the estimation module).

An alternative implementation is to apply the ADC before the sample/hold(which becomes a digital interpolation).The detected output is sampled at about twice the Nyquist rate of the WCDMA signal so that the time instants ts whenλ=λocan be interpolated accurately.Because the Nyquist rate for the 101 WCDMA signal is around 15 MHz,the ADC with 32 MHz sample rate(mentioned in the introduction)would be sufficient.As mentioned in the introduction,reduction in sampling rate,resolution,and number of ADCs reduces the cost by a factor of 25 over the bench mark system.

The drawback of the proposed approach is that the acquisition time needed to measure the nonlinearity and adapt the DPD is increased compared to the standard approach.This is due to the slower accumulation of samples within the selective sampling process and,as discussed in section 3,the fact that measurements are serially obtained from severalρvalues in order to estimate the DPD coefficients.

However,in the standard approach,16 Ksample blocks are captured infrequently,and large blocks of incoming data between captures are ignored to reduce the DSP requirements in the estimation module.In addition,it is possible to speed up the proposed approach using parallel measurement circuits with differentρ values,at an increased cost.There is a trade-off between cost and acquisition speed in both the proposed and standard approaches.

It is worth noting that Cavers used both a detector-based observation path(similar to the proposed approach)and the standard over-sampled observation path in[15]so that the errors in the modulator and demodulator could be measured independently.This is an example where it is more important to have severaldifferent measurement methods available rather than debate the relative merits of the individual methods used in isolation.

3 Coefficient Estimation

This section describes the estimation of the DPD coefficients b from the memoryless measurements(Гθ,Г2θ).

The relationship between(Гθ,Г2θ)and the residual memoryless coefficients a in(5),for the case of N=4,is approximated by

where

The residual memoryless coefficientsare used to update the DPD coefficients,as shown in(6).However,only two measurements(Гθ,Г2θ)are available for a given value of ρ.It is necessary to integrate measurements from several values ofρ to estimate all four DPD coefficients.

Aset of recursive equations[16]is used to update the DPD coefficients b.Measurements(Гθ,Г2θ)from several values ofρare combined using

where()His conjugate transpose,(0)=[1 0 0 0]T,S0=,Ri=0.0001 I2x2,and Qi=0.0002 I4x4.The matrix Siis the error covariance of.Experiments show that it is beneficial to reset Sito S0after a few cycles of theρset while retaining the current estimate of b.This is likely due to the approximation used for the matrix M in(23),which assumes that[a0a1a2a3]≈[1 0 0 0].

The estimation module would be implemented,typically,in a micro-controller because the amount of computation needed to convert(Гθ,Г2θ)into coefficient updates using(23)-(27)is modest.In contrast,the standard approach for estimating coefficients involves auto-and cross-correlations of the gain basis functions(truncated to 16 K samples)and the cross-correlations of the output capture with the gain basis functions.These correlations are often computed using a high performance DSPchip.In general,micro-controllers are less expensive than DSPchips.

4 Memory Compensation

This section discusses memory compensation within the DPDmodule,including how to measure the PA memory using a cancellation bridge and selective sampling(as was done in section 2 for the memoryless case).In general,it is desirable to use the lowest order DPDmodel that makes it possible to meet the WCDMA specifications.

That is,if the memoryless DPD is adequate,it can be used.However,if additional correction is required,the gain model can be extended to include memory correction.

Although it is possible to model memory using a pruned Volterra series based on delayed digital samples of the input signal,such an approach does not allow for selective sampling.A compatible approach defines the nonlinear gain as a function ofand,both of which are referenced to the time sample ts whenλ=λ0.One possible model for the DPD gain is

where spare DPD coefficients associated with the memory correction,P is the polynomial order of the memory,and hω{}is a bandpass filter used to limit the high frequency noise.

The gain modelfor the predistorted transmitter is

where rpare coefficients associated with the memory component of the residual nonlinearity.

Residual PA memory is measured using a cancellation loop,detector,and selective sampling(Fig.4).The selective sampling in this case produces time-aligned triples()at each sample instant ts.This allows the detector output to be expressed as a function of two variables,γdet().

The separable form of(28)allows the DPDcoefficients to be estimated using two sets of LUTs:the memoryless LUTs defined in(15)and(16),and memory LUTs defined in(31)and(32).Anew measure of the memory is needed,which is whereγ0is the expected value ofγdet(average radius of the circle formed by the selectively sampled points),δis a small constant used to prevent a divide by zero,and hω(k)is the filtered derivative hω/?t}sampled at time k.The numerator in(30)measures the correlation of the detector outputγdetwith the filtered derivative hω}.The subsequent estimation and update of the memory DPD coefficients minimizes the correlation.

The accumulated LUTs used for the memory estimation are

The mean LUTis

which provides an estimate of(30)as a function ofθi.Using the mean LUT,the demodulated signal becomes

Memory measurements(ψθ,ψ2θ)are obtained for each value ofρtested.

As in the memoryless case,memory measurements(ψθ,ψ2θ)are integrated over several values ofρ.The

relationship between(ψθ,ψ2θ)and the residual memory coefficients r,for the case of P=4,is approximated by

where M is the same matrix defined in(23).The memory coefficients of the DPD gain in(28)are updated using

where s(i)=[s0s1s2s3]Tfor the iteration i.

The estimations of the memoryless and memory DPD coefficients b and s are decoupled in this implementation.Decoupling does not impact the convergence when E[}]=0 for each angleθiof the selectively sampled input signal.This is a reasonable assumption for a WCDMA signal.It is recommended that the memoryless coefficients be adapted first,in isolation,because the uncorrected memoryless distortion tends to be larger than the memory-related distortion.Both components are adapted concurrently once the residualmemoryless nonlinearity is reduced to a level comparable to the memory component.

The number of basis functions in the memory model can be adjusted using the polynomial order P.In most cases,the order of the memory is less than the order of the memoryless nonlinearity,that is,P＜N.In such cases,a subset of the matrix M is used in(35).If needed,the size of the memory model can be increased without increasing the polynomial order P by using a set of filtered derivatives,hω(n){},that have different frequency responses.Separate memory measurements(2θ)|ω(n)could be obtained for each filter hω(n)and used to estimate additionalmemory coefficients,sω(n).Assuming that the number of filters used is Nh,the total number of memory basis functions available becomes NhP.Incorporating more filters hω(n)into the DPD model in(28)is similar to expanding the memory depth of a memory polynomial.

The DPD model in(28)is designed to be compatible with the proposed measurement system and the selective sampling process.It is based on filtered derivatives ofinstead of delayed values of.However,the difference between a derivative and a delay is minor in practice.For example,hτ{}can be approximated by-,where T is the sampling period.Thus,the basis functions have the form BFv=n-1,which is similar to the memory polynomialshown in(10).The memory depth of this representation can be made large and expressive,like the memory polynomial,but there is no value in exceeding what is needed.

At this point the question might be asked“Which is better?”,the DPD model in(28)or the pruned Volterra models in(9)and(10)?This question is misguided.When selecting the basis function set,it is important to remember that the goal of linearization is to meet the WCDMA specification,and there is no prize for destroying the ACLR specification by 10 d B.In fact,excess margin is money wasted[17].

The DPD module is often implemented in an FPGA.The cost of an FPGA,in general,is related to the number of multipliers used.Thus,to minimize the multiplier usage and reduce cost,the smallest basis function set that meets the WCDMA specification should be used.

5 Results

Aproof of concept can verify that a polynomial predistorter tuned using selective samples of a WCDMA signal will converge.It can also estimate the ACLRperformance of the proposed transmitter.

The author has a library of PA characterizations,without linearization,that comprise digitized data captures of input and output signals.A class AB-biased amplifier driven by a two-carrier WCDMA signal is chosen for the proof of concept.The WCDMA signal has a 101 carrier configuration and is crest factor reduced to a peak-to-average power ratio(PAPR)of 7.2 d B.The RFoutput power and center frequency of the PA are 45.3 d Bm and 2.14 GHz,respectively.

The sample rate used in the input and output data captures is 122.88 MHz.The data captures are 614,400 samples long and time-aligned to remove the delay between the input signal and the linear component of the output signal.

The synchronized data captures of the PA input and output signals are denoted byχ(k)and yo(k),respectively.These are used to compute the gain of the PA model,GPA.Note that yo(k)denotes the digitized data capture of the actual uncorrected PA output signal whereas y(k)is used to denote the simulated predistorted PA output signal computed using GPAχDPD(k).

The PAgain GPAand the input data captureχ(k)are used in a Matlab simulation of the digital transmitter to demonstrate the convergence of the DPD algorithm.The structure of the transmitter is shown in Fig.6.The memoryless portion of GPAis represented as AM-AM and AM-PM LUTs.The LUTrepresentation was chosen for the PA gain so that any modeling errors would differ from those associated with the polynomial gain models used in the digital predistortion.GPA is augmented by a memory component estimated from the output capture yousing a least square technique,similar to(7)(see[2],[6],or[7]for details).Once the algorithm has converged,the steady-state value of the DPD gain function GDPD provides an estimate of the inverse nonlinear gain needed to linearize the PA.

The ACLRresults from the Matlab simulation,based on y(k)=(k),are optimistic because noise and some nonlinear behaviors from the actual PA are not modeled.In contrast,the output capture yo(k)contains noise and intermodulation(IMD)products generated by the actual PA.As a result,the actual ACLRperformance can be estimated by linearizing the output data capture yo(k)using the inverse PAgain functionthat is,y(k)=(k).Although this is not the same as predistortion,the difference in ACLR performance resulting from commutingandis minimal for mild nonlinearities[9].

Figure 6.?AMatlab simulation of the digital transmitter and selective sampling process is used to demonstrate the convergence of the DPD algorithm.Memory correction is included in the simulation butis not shown above.

The memoryless and memory DPD coefficient estimations are performed independently as decoupled processes.The memoryless adaptation is performed initially in isolation to reduce the residualnonlinearity.After this,both the memoryless and memory adaptations are performed concurrently.

Let us begin with the estimation of the memoryless portion of the DPD gain in(28),which has four coefficients(N=4).In order to estimate allof the coefficients,several values ofρare selected,ρ=[0.5,0.8,1.1,1.4],and tested sequentially.For a given value of ρ,the signals y(k)andγdet(k)are computed for the current setting of the DPD module GDPD.Avalue forλois chosen(λo=0.35),and selective sampling is applied to determine the sampling instants tswhereλ=λo.Corresponding values ofθ(ts)and γdet(ts)are sampled.The best method for determining tsis to interpolateχ(k)to localize the instants whenχ(k)crosses the circle defined by(ρ,λo).However,for ease of implementation using Matlab,the signalχ(k)is up-sampled by a factor of 4,and samples are selected ifχ(k)is within 0.025λoof the(ρ,λo)circle.θ(ts)and γdet(ts)are accumulated in LUTs in(15)and(16),from which E[γdet(θi)]is computed using(19),then demodulated using(21)for m=[1,2]to produce the memoryless measurementsГθandГ2θ.

?Figure 7.AM-AMand AM-PM curves for the uncorrected and predistorted PAmodels(From[5]?2010 IEEE).

?Figure 8.WCDMAoutput spectra of the originaloutput capture yo(no linearization),the converged predistorted transmitter model y=GPAx DPD(Matlab simulation),and the linearized outputcapture y=GDPDyo.

Fig.7 shows the AM-AM and AM-PM curves for the uncorrected and predistorted PA models.The measurement system and the recursive(25)-(27)produce steady-state DPD coefficients b that flatten the AM-AM and AM-PM curves as desired over the input envelope range where the probability density function(PDF)of|x|has its highest density.

The class AB-biased amplifier requires some memory correction to ensure that the worst case ACLRof the transmitter passes the WCDMA specification with at least 3 d Bof margin(for manufacturing tolerances).The memory model chosen comprises one term,s0.From(28),where N=4 and P=1,the DPD gain becomes

A weighted average of the memory measurementsψθfrom the fourρvalues is used to update the memory coefficient s0,that is,

whereρj=[0.5,0.8,1.1,1.4],ψθ,jis the memory measurementψθobtained forρj,α＜1,and wjis a weight defined by

That is,the weight used for eachρjis determined by the minimum bin value of the L0,d|χ|LUTused in the memory estimation(31).Note thatρ2λoin(38)corresponds with M(1,1)in(23).

Fig.8 shows the output spectra for three cases:the original output capture yo(k),the predistorted memory PA model y(k)=GPAχDPD(k),and the output capture linearized using the converged value of GDPD,that is,y(k)=GDPDyo(k).

▼Table 1.ACLRfor Various Output Spectra

Table 1 contains the ACLR measurements.

?Figure 9.Baseband input and RFoutputsignals for the inverse model.

Comparing the uncorrected capture yo(k)and predistorted memory results y(k)=GPAxDPD(k),the steady-state DPD coefficients reduce the ACLR2 by 18.6 d B to-57.6 d Bc.This is well below the WCDMA specification of-50 d Bc.This result validates the memory measurement and shows that the coefficient estimation converges.It also validates the implementation of the memoryless and memory coefficient estimations as decoupled processes.

The ACLRmeasurements are optimistic because they are based on a simulated model of the PA.

The estimate of the actual ACLR performance is obtained from the linearized output capture GDPDyo(k)and is shown in Table 1.The ACLR2 values pass the WCDMA specification with 3.4 d B of margin,as desired.Thus,the basis function set(N=4,P=1)used to form the memory DPD(37)provides sufficient linearization of the PA for a 101 WCDMAinput signal.The basis function set in(37)is a compact representation(low number of coefficients)allowing the DPD module to be implemented in a smaller FPGA than would be possible if a pruned Volterra series with a large memory depth was chosen for the DPD model.

6 Future Extension

A future extension of the DPD approach includes the use of inverse modeling.In this alternative implementation,the inverse nonlinearity of the transmitter is computed by using the output of the detectorγdetto control the selective sampling.The sampling instances tsare selected when γdet(ts)=γo.The precision required by the measurement system is reduced because the detector output at the selectively sampled instants tsis constant by definition.For example,a single bit comparator can be used to control the sampling,where the sample times tsare indicated by changes in the output state of the comparator.The elliptical deformations associated with residual nonlinearities appear in the selected input samplesχ(ts)and result in a time-varying value ofλ(ts),as shown in Fig.9(a).

The selected samples ofχ(ts)are used to compute(ρ,λ,θ).The measurement LUTs are created based on(θ,λ)instead of(θ,γdet).The LUT capturing the first-order moment ofλ(instead ofγdet)as a function of quantized values ofθis

where the sampling and quantization are defined by

The mean ofλfor bin i is

The demodulated signal for the inverse nonlinearity becomes

The integration of the measurements uses the same equations as the previous implementation in(25)-(27),except that the sign of the coefficient update is reversed.

The nonlinear gain of the transmitter is indicated by an elliptical trajectory at the input,instead of the output as shown in Fig.3.

7 Conclusion

A method has been proposed where selective sampling of a WCDMAsignal is used to obtain a probing signal that highlights nonlinearities within a transmitter.The measurement circuitry comprises a cancellation loop,detector,sample/hold,and an ADC.

The cancellation loop and selective sampling reduce the linearity,resolution,and sampling rate required of the ADC.LUTs defined by(15)and(16)provide a compact representation of the detected signalγdet(θ),reducing the storage requirements compared with an over-sampled capture of the PA output signal.A DPD model with memory correction that is compatible with the proposed measurement system has been presented.Results show that a compact DPD model can linearize a class AB biased PA sufficiently to meet the ACLR specifications for a 101 WCDMA input signal.The approach reduces the cost of the observation path,estimation module,and DPD module compared to a standard transmitter with DPD based on a pruned Volterra series with a large memory depth and adapted using output captures based on over-sampled data.