Novel Baseband Equivalent Models of Quadrature Modulated All-Digital Transmitters
NNovel Baseband Equivalent Models of Quadrature ModulatedAll-Digital Transmitters
Omer Tanovic , , Rui Ma , and Koon Hoo Teo Mitsubishi Electric Research Laboratories, Cambridge, MA, 02139, USA, [email protected] Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute ofTechnology (MIT), Cambridge, MA, 02139, USA
Abstract — In this paper an exact baseband equivalentmodel of a quadrature modulated all-digital transmitter isderived. No restrictions on the number of levels of a digitalswitched-mode power amplifier (SMPA) driving input, northe pulse encoding scheme employed, are made. This impliesa high level of generality of the proposed model. We show thatall-digital transmitter (ADT) can be represented as a seriesconnection of the pulse encoder, discrete-time Volterra seriesmodel of fixed degree and memory depth, and a linear time-varying system with special properties. This result suggests anew analytically motivated structure of a digital predistortion(DPD) of SMPA nonlinearities in ADT. Numerical simulationsin MATLAB are used to verify proposed baseband equivalentmodel.
Index Terms — Digital predistortion, all-digital transmitter,baseband equivalent model.
I. I
NTRODUCTION
Behavioral modeling of nonlinearities in power ampli-fiers (PA), and their compensation, has been an activearea of research for more than two decades (see [1]- [4]and references therein). Most of the work has focused ontraditional transmitters employing linear PAs. All-digitaltransmitters (ADT), which use switched-mode power am-plifiers (SMPA) instead of linear PAs, have recently gainedsignificant importance due to their promising potential ofproviding high power efficiency (see [5]- [7] and referencestherein). Surprisingly, not much research has been doneon behavioral modelling of all-digital transmitters. Neuralnetworks based behavioral ADT models were presented in[8]. Main deficiency of such black box modeling is thelack of insight into structure and internal dynamics of thenonlinear system being approximated. An exact basebandequivalent model for a quadrature modulated ADT wasgiven in [9], but only for 2 level pulse encoding scheme(taking precisely ± values), and without an obvious ex-tension to multi-level case. An analog baseband equivalentmodel was recently given in [10]. Effects of pulse encoderare completely ignored in the derivation of equivalentmodel, thus neglecting large amount of quantization noisein the PA input, which spectrally spreads into signal bandonce processed by the passband nonlinearity. This work was done while Omer Tanovic was an intern at MitsubishiElectric Research Laboratories (MERL).
Recently, in [11]-[12], authors showed that passbandnonlinearities can be represented by an equivalent base-band model consisting of a series connection of a shortmemory nonlinear system followed by a special lineartime-invariant system (LTI). They consider quadraturemodulated transmitters in traditional setup, employinganalog upconversion to carrier frequency and linear PAs.Assumption which enables such a compact system descrip-tion is the zero-order hold digital-to-analog conversionof a PA driving signal. By recognizing similar formof the SMPA driving input, we extend that descriptionto all-digital transmitters. Proposed ADT structure doesnot depend on the choice of pulse encoder (e.g. pulse-width modulation or delta-sigma modulation), nor doesit depend on the number of its output levels. It exactlyrepresents ADT when the passband nonlinearity can berepresented by continuous-time Volterra series of fixedmemory. Furthermore, it suggests a novel, non-obvious,analytically motivated DPD structure, which reduces com-putational and hardware complexity of a standard memorypolynomial or pruned Volterra-series based DPD.II. P
ROBLEM F ORMULATION
Detailed block diagram of a quadrature modulated ADTis shown in the figure below. High resolution input se- (cid:45) P (cid:45) ↑ (cid:45) (cid:104) × (cid:54)(cid:45) Re (cid:45) ZOH (cid:45) G (cid:45) F (cid:45) x [ n ] x d [ n ] e j π n X d x c ( t ) x RF ( t ) Fig. 1. Detailed block diagram of a quadrature modulated ADT. quence x = x [ n ] , whose in-phase and quadrature com-ponents are denoted i = i [ n ] and q = q [ n ] respectively,is first processed with a pulse encoder, denoted P , soto assume amplitude levels suitable for driving a lowresolution SMPA. Here we assume that P acts on i and q separately, and produces digital output signal x d = x d [ n ] = i d [ n ] + jq d [ n ] , where i d = P i , and q d = P q .Signal x d is first upconverted to a digital carrier frequency,before being converted to continuous-time domain by a a r X i v : . [ c s . S Y ] F e b ero-order hold (ZOH) digital-to-analog converter (DAC).Upconversion to digital frequency is usually done at thesampling rate of 4 times the carrier frequency. Under thisassumption, mixing in digital domain becomes equivalentto interleaving the in-phase and quadrature components,together with their negative counterparts. We denote thedigital upconversion system as X d (Fig. 1). Continuous-time (CT) pulsed signal x c = x c ( t ) is fed into SMPA,which is denoted by G in Fig. 1. We are particularlyinterested in the case when G can be described by thegeneral CT Volterra series model [13], with memory notlarger than the sampling period of signal x d .Output of the nonliner system G is passed through ananalog bandpass filter F , in order to produce the RF signal x RF = x RF ( t ) to be radiated by antenna. Here we assume F to be an ideal bandpass filter with bandwidth equal tothe sampling rate of the input signal x , and centered at thecarrier frequency f c . This is to avoid modeling the wholerange of the output signal’s spectrum which would makethe linearization bandwidth very large, and would put asignificant burden on system design.ADT, as a system, maps baseband DT signal x intopassband CT signal x RF . If baseband equivalent model isused for DPD design, it is more beneficial to find the mapbetween x and the downconverted and sampled versionof x RF , which we denote as ˆ x . Therefore, instead ofconsidering the ADT system given in Fig. 1, it is naturalto consider a system depicted in Fig. 2, and denoted S .Here D denotes an ideal demodulator, which runs at thesampling rate of baseband input signal x [ n ] , and M is usedto denote the ZOH converter. (cid:45) P (cid:45) X d (cid:45) M (cid:45) G (cid:45) F (cid:45) D (cid:45) x [ n ] x d [ n ] ˜ x d [ n ] x c ( t ) y ( t ) x RF ( t ) ˆ x [ n ] Fig. 2. Series connection of ADT and a demodulator system.
The SMPA driving signal x c , i.e. input into G , is apiecewise constant signal, which assumes only a finitenumber of amplitude levels corresponding to the choiceof the pulse encoder P . This is a consequence of the zero-order hold operation of digital-to-analog converter M . Itis now easy to see that series interconnection DFGM ,in Fig. 2, has very similar structure to the analogous onein [11]. The only difference is that sampling rates of theinput and output signals are not equal. Signal ˆ x runs ata sampling rate of the baseband signal x , while x d runsat a higher sampling rate. Let K be the ratio of thesampling rates of x d and x . It was shown in [11], thatthe equivalent of the composition system DFGM can berepresented as a series interconnection LV of a Volterratype DT nonlinear system V , and an LTI system L . Forthe system shown in Fig. 2, the equivalent L should be atime-varying system, due to the above mentioned mismatch between input and output signal rates. In the followingsection we decompose this equivalent linear subsystem intoa series interconnection of a multi-input single-output LTIsubsystem L and a downsampler.III. M AIN R ESULT
Under assumptions introduced in the previous section,we give a general description of an ADT system.
Complete characterization of system S in Fig.2 :Let V be a single-input multi-output system mapping acomplex signal x [ n ] = i [ n ]+ jq [ n ] into a vector valued realsignal v [ n ] = (cid:2) v [ n ] . . . v N [ n ] (cid:3) T , components of whichare all Volterra monomials composed of i [ n ] and q [ n ] ,up to degree M and depths m i and m q . Let K be adownsampling by factor of K subsystem, where K wasdefined in the previous section. System S , as given inFig. 2, can be decomposed into a series connection ofsystems P , V , L and K , as given in Fig. 3, where L isa multi-input single-output LTI system with long memory,and good approximation by finite impulse response (FIR)filters (Proof is omitted due to space limitations.). (cid:45) V V (cid:45) FIR FIR (cid:45)(cid:45) P (cid:45) (cid:103) (cid:65)(cid:65)(cid:65)(cid:85)(cid:1)(cid:1)(cid:1)(cid:21) FIR N (cid:80)(cid:80)(cid:113)(cid:45)(cid:45) V N (cid:45) ˆ x [ n ] x [ n ] K (cid:45) V L
Fig. 3. Baseband equivalent model of S . Here we emphasize some consequences of this result.Block diagram representation of S in Fig. 3, stressesparallel structure in the LV part of the system. Each V k inFig. 3 describes one Volterra monomial, e.g. ( V k x )[ n ] = i [ n ] i [ n − q [ n ] . Therefore, subsystems V k are relativelyeasy to implement since they only involve delays andmultiplication operations. Attention should be payed topossible delays that can appear in subsystems V k . It can beshown that if continuous-time Volterra kernel has maximalmemory τ (over all variables), then its correspondingdiscrete-time analog (group of monomials V k ), has mem-ory (cid:6) τT (cid:7) , where (cid:100)·(cid:101) is a ceil function. Frequency responsesof components of L , depend heavily on the passbandnonlinearity, but they have a common feature: they aresmooth over the spectral band of interest. This makes themwell approximable in spectral domain, by e.g. polynomials.Moreover, finding coefficients of those polynomials can bedone by simple least squares optimization.It is important to emphasize that due to bandpass fil-tering of the PA output, long memory dynamic behaviors now present, which makes band-limited models funda-mentally different from the conventional baseband mod-els. This implies that traditional modeling methods, suchas memory polynomials or dynamic deviation reduction-based Volterra series modeling, might be too general to de-fine this new structure, and also not well suited for practicalimplementations due to long memory requirements (of thenonlinear part) which would require exponentially largenumber of coefficients. Therefore, the structure proposedin this paper does not suffer from issues of long memory ofnonlinear subsystem, but rather has memory depth equiv-alent to that of the passband nonlinearity. Long memoryrequirements are forwarded to the linear subsystems L k ,taking complexity burden off the nonlinear part.Now we discuss potential impact, of the above result, ondigital predistortion design for all-digital transmitters. DPDdesign can be seen as a process of finding the inverse of S (when it exists). In most applications, with an appropriatescaling and time delay, the system S to be inverted can beviewed as a small perturbation of identity, i.e. S = I + ∆ .Using Neumann’s serries, assuming ∆ is small in an ap-propriate sense, the inverse of S can be well approximatedby S − ≈ I − ∆ = − S . Hence the result of thispaper suggests a specific structure of the compensator C ≈ − S . As pointed out in [11], this implies that plainVolterra monomials structure is, in general, not enough tomodel C , as it lacks the capacity to implement the longmemory effects caused by bandpass filtering. Instead, C should be sought in the form C = I − KL X VP , where P is the pulse encoder, V is the system generating allVolterra series monomials of a limited depth and limiteddegree, L is a fixed LTI system with a long time constant, X is a matrix of coefficients to be optimized to fit thedata available, and K is a downsampler. Elements of thecoefficient matrix X can be easily obtained by applyingsimple least squares optimization on the set of input outputpairs of S . IV. S IMULATION R ESULTS
Validation of the above model of system S is done fortwo cases of passband nonlinearity G . In the first case weassume a simple third order analog nonlinearity, consistingof a linear term and one CT Volterra monomial, as givenbelow ( Gx )( t ) = x ( t ) − δ x ( t − τ ) x ( t − τ ) x ( t − τ ) . (1)Time delays in the above expression are taken as τ =1 . T , τ = 2 . T , τ = 0 . T , where T is a samplingtime of a digitaly upconverted DT signal ˜ x d (see Fig.2). Parameter δ is varied in the interval [0.001,0.2].This parameter roughly dictates the amount of passbanddistortion or equivalently the level of linearity of the Fig. 4. Model Validation Error: (a) Example 1; (b) Example 2. transmitter circuit. In the second case we assume ( Gx )( t ) = x ( t ) − δ (cid:90) t h ( τ , τ ) x ( t − τ ) x ( t − τ ) dτ dτ , (2)where kernel h = h ( t , t ) is of memory less than T , andis separable, i.e. h ( t , t ) = h ( t ) · h ( t ) , with h ( t ) =exp( − . t ) and h ( t ) = exp( − . t ) cos( π/ t ) . Inthis case, parameter δ takes values from [0.001,0.015].In both cases, delta-sigma modulator (DSM) of first orderand 5 output levels, was used as pulse encoder. System S is modeled as given in Fig. 3. In both cases, systems V k are Volterra monomials of maximal degree 3, and digitalmemory of up to 4 samples (due to maximal memory ofpassband nonlinearity equal to T ). Coefficients of the FIRfilters are obtained by fitting this model on a training dataof 40960 input-output pairs.Model validation results are depicted in Fig. 4. Verygood modeling error (below 0.1% or -20dB) is achievedin both cases, though an upward trend in error (as δ / isincreased) is noticable from Fig. 4. This is due to the errorin approximating FIR filters, which gets enhanced as levelof distortion increases.. C ONCLUSION
In this paper, we propose a novel structure of thebaseband equivalent model of a quadrature modulatedADT, under assumption that the passband nonlinearity isof Volterra series type of fixed memory. It was shown thatADT can be represented as a series connection of the pulseencoder, discrete-time Volterra series model of fixed degreeand memory depth, and a linear time-varying system withspecial properties. Model does not assume any particularpulse encoding method, nor does it put constraints on thenumber of its output levels. Results suggest a novel, an-alytically motivated DPD structure, which can potentiallyreduce computational and hardware complexity comparedto traditional DPD design techniques.R
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