Adaptation of a Digitally Predistorted RF Amplifier Using Selective Sampling

Release Date:2011-10-29 Author:R. Neil Braithwaite Click:

1 Introduction
    A digital 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 digital transmission path that opposes nonlinearities in the modulator and PA stages. Adaptive DPD measures the residual nonlinearity of the predistorted transmitter and adjusts the DPD coefficients 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-digital converter (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. A typical zero-IF observation path for a multicarrier WCDMA [1] signal uses two 14 bit ADCs sampled at 122.88 MHz [2]. If the number 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].


    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 WCDMA signal 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 digital transmitter in Fig.1 is

 

    where GDPD is the predistortion gain and a nonlinear function of χ . The predistorted baseband signal is up-converted to produce an RF signal

 

 

    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 GPA is 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 bn are 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 G 0 is the desired (linear) gain of the transmitter, and a n are complex values referred to as residual memoryless coefficients. Estimates of the residual nonlinearity a n are used to update the DPD coefficients bn in an iterative manner, that is,

 

    where ( )T indicates transpose b(i) = [b 0  b 1  b 2  b 3]T at iteration i, and 0<α <1. The iterative sequence (6) has converged when a = [a 0   a 1   a 2   a 3]T = [1 0  0  0]T.
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 a n that minimize [2],[6],[7]

 

 

    where BFn  is a gain basis function and a nonlinear function of χ (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 from χ (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 channel leakage 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

 

 

    where τl  are 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 by delaying the distortion modes as opposed to delaying the gain component only. This produces predistorted waveforms that are a weighted sum of  χ (k-τ) nχ (k-τ) instead of  χ (k-τ) nχ (k ). Other memory models use delayed samples from both the input χ (k ) and output χDPD (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].


    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 envelope δχ/δt, instead of delayed samples χ (k-τ) . This model is shown in section 5 to improve ACLR performance over the memoryless DPD model presented in [5].


2 Measurement Approach
    A new 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 level of -50 dBc [1] from a signal with a large peak-to-average power ratio (PAPR), on the order of 7.2 dB.


    The proposed approach involves creating a probing signal, similar to the one used in [4], that is extracted from a subset of the sampled WCDMA input signal. This selectively sampled probing signal has a lower PAPR and sampling rate than the WCDMA signal. The proposed measurement circuitry reduces the PAPR of the probing signal further, to almost 0 dB, 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 signal χ (k ) into

 

 

    where both λ(k ) and θ (k ) vary with time, and ρ  is a constant. This transformation χ (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 constant ρ. 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 samples χ (k )=f {ρ, λ(k ), θ (k )}, where λo is 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 angle θ (Fig. 3). The trajectory is used as a probing signal to highlight nonlinearities within the transmitter, which appear as elliptical deformations in the RF output yRF (t) (Fig. 3). In a typical implementation, several probing signals with different ρ  values 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 thetransmitter. 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.

 

 

 


    The deformation of the circular trajectory provides information about the nonlinear gain of the transmitter around the output operating point Go ρχ LO (t ). Any slope in the gain curves δGtrans /δ  χ   creates an elliptical trajectory at the PA output. For example, a downward slope in the AM-AM curve, δGtrans /δ  χ , compresses the circle along the horizontal axis, as shown in Fig. 3 (center). A slope in the AM-PM curve, δarg {Gtrans}/δ  χ  , 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.
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, γdet =  εRF  2, is reduced (for the selected samples where λ = 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 values γdet  corresponding to time instants ts  when λ = λo. The cancellation loop and selective sampling reduce the resolution and sampling rate required of the ADC, allowing for the use of lower cost components. Cost is discussed further at the end of this section.


    The selectively sampled output of the detector

 

    is specified as a function of θ. If the cancellation loop is balanced and the transmitter is linear, the selectively sampled signal γdet (θ ) is constant as a function of θ. Misalignment of the cancellation loop creates a first harmonic variation as a function of θ. 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 special case of χ (k ) = λo exp( jθ (k )) (no cancellation loop). The author recommends reading [15] to obtain a better understanding of the proposed technique.


    A compact method of representing selective samples of the detected signal γdet (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 of θ is

 

 

    The mean and variance of γdet  for bin i are

 


    and

 

 

    respectively. The first and second harmonics of γdet (θ ) are measured by demodulating the mean LUT as a function of θ. 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 of ρ tested.


    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 WCDMA signal 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 L 0, L 1, and L 2. 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 (when λ =λo), then the sample/hold must be placed before the ADC. The required sampling rate for the ADC is 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 when λ =λo can 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 several ρ values in order to estimate the DPD coefficients. However, in the standard approach, 16 K sample 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 different ρ 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 several different 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 coefficients a  are used to update the DPD coefficients b, 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 of ρ to estimate all four DPD coefficients.


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

 

 

    where ( )H is conjugate transpose, b(0) = [1 0 0 0]T, S 0 = 2I4x4, Ri = 0.0001 I2x2, and Qi = 0.0002 I4x4. The matrix Si  is the error covariance of b. Experiments show that it is beneficial to reset Si  to S 0 after a few cycles of the ρ set while retaining the current estimate of b. This is likely due to the approximation used for the matrix M  in (23), which assumes that [a 0 a 1 a 2 a 3] ≈ [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 DSP chip. In general, micro-controllers are less expensive than DSP chips.


4 Memory Compensation
    This section discusses memory compensation within the DPD module, 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 DPD model 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 of  χ   and ?? χ  /??t, both of which are referenced to the time sample ts when λ = λ0. One possible model for the DPD gain is

 


    where sp are 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 model for the predistorted transmitter is

 

 

    where rp are 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
(θ, hω{??  χ  /??t },γdet ) at each sample instant ts. This allows the detector output to be expressed as a function of two variables, γdet  (θ, ??  χ  /??t ).
The separable form of (28) allows the DPD coefficients to be estimated using two sets of LUTs: the memoryless LUTs defined in (15) and (16), and memory LUTs defined in (31) and (32). A new measure of the memory is needed, which is

 

 

    where γ 0 is the expected value of γdet  (average radius of the circle formed by the selectively sampled points),   δ   is a small constant used to prevent a divide by zero, and hω (k ) is the filtered derivative hω{??  χ  /??t } sampled at time k. The numerator in (30) measures the correlation of the detector output γdet   with the filtered derivative hω{??  χ  /??t }. The subsequent estimation and update of the memory DPD coefficients minimizes the correlation.


    The accumulated LUTs used for the memory estimation are

 

    and

 

    The mean LUT is

 

 

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

 

 

    Memory measurements (ψθ, ψ 2θ) are obtained for each value of ρ tested.


    As in the memoryless case, memory measurements (ψθ, ψ 2θ) are integrated over several values of ρ. The relationship between (ψθ, ψ 2θ) 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 ) = [s0  s1  s2  s3]T for 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 [hω{??  χ  /??t }] = 0 for each angle θi  of 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 residual memoryless nonlinearity is reduced to a level comparable to the memory component.


   

[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.

[Keywords] amplifier distortion; communication system nonlinearities; power amplifier linearization