Concurrent Linearization - CiteSeerX

D E SE U R CU AT FO E FE SU IS

Concurrent Linearization Patrick Roblin, Christophe Quindroit, Naveen Naraharisetti, Shahin Gheitanchi, and Mike Fitton

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ith the explosive growth of the smartphone and tablet markets, wide bandwidth voice and data communication have become ubiquitous. Users expect to use their wireless portable phone/computing devices at any place and at any time. Furthermore, with the today’s economy of scale, yesterday’s high-performance devices are today’s entry-model devices. To make possible this global wide-bandwidth wireless communication and networking while handling the increasing number of users, new wireless communication standards based on high bandwidth efficiency protocols such as orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) have been developed to meet the user capacity requirements. These new communication standards have, however, placed very challenging demands on the RF front-end specifications in terms of bandwidth and

power efficiency for both hand-held devices and base stations. Indeed, a characteristic of OFMD and CDMA signals is that they feature very high peak to average power ratios (PAPR), typically on the order of 10 dB even after crest factor reduction techniques or softclipping. Thus, for example, an 80 W average power base-station power amplifier (PA) should be able to also amplify linearly outburst, with 800 W instantaneous peak power. Special RF amplifiers are then needed to amplify such signals with large PAPR while providing high average power efficiency. However, power efficiency in PA operation comes in practice at the cost of increased nonlinearities. As is well known, nonlinearities lead not only to inband signal distortion but also to outband spectral regrowth, which are strictly regulated. A provider cannot pollute the band of a competitor. Thus, linearization techniques are used to linearize the amplifiers and reduce the inband signal distortion and outband spectral regrowth to acceptable levels.

Patrick Roblin ([email protected]), Christophe Quindroit ([email protected]), and Naveen Naraharisetti ([email protected]) are with the Department of Electrical and Computer Engineering, Ohio State University, Columbus, Ohio 43210, USA. Shahin Gheitanchi ([email protected]) is with Altera Europe, Holmers Farm Way, High Wycombe, Buckinghamsire, HP12 4XF, United Kingdom. Mike Fitton ([email protected]) is with Altera Corp. 101 Innovation Dr., San Jose, California 95134, USA. Digital Object Identifier 10.1109/MMM.2013.2281297 Date of publication: 15 November 2013

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xL

yL

xU

yU

PA RF Input

RF Output

Figure 1. A dual-band PA excited by two noncontiguous bands. In modern base-stations, digital predistortion (DPD) linearization is now the method of choice to linearize the amplifier while delivering the targeted peak power at the cost of a small reduction (typically 1 dB) in amplification gain. Until a few years ago, this was the state of affairs, but things are getting further complicated. First, to handle the increasing number of users, the aggregated bandwidth of multicarrier base-station signals keeps increasing. For example, systems with 100 MHz useful bandwidth requiring 500 MHz of linearization bandwidth are being developed. The increased bandwidth places technical challenges on the DPD performances as it must handle the PA memory effects for wider bandwidths. Furthermore, due to market dynamics, cellular phone service providers may end up operating base stations in different noncontiguous frequency bands. Tremendous saving can therefore be achieved when new generations of base stations are deployed if a single multiband RF power amplifier is deployed instead. This has motivated significant research in the field, as the recent two special sessions covering multiband linearization at the 2013 IEEE MTT-S International Microwave Symposium (IMS2013) in Seattle testify.

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Figure 2. The composite RF signal Re " x L e i~L t + x U e i~U t , (blue) for two complex modulated signals x L ^ t h and x U ^ t h are plotted together with the RF envelope x L e i~L t + x U e i~U t (red line), the peak envelope |x L| +|x U| (green line), the 1/2 average envelope 6|x L|2 +|x U|2@ (purple line), and the minimum envelope |x L| -|x U| (yellow line).

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A review of the techniques and the challenges involved in the design of multiband DPD systems is the object of this article. While a considerable amount of research investigations have been conducted with regard to single-band DPD hardware implementation, due to its novelty, fewer reports are comparatively available on multibands DPD systems, but the interest has been rapidly growing since the first DPD papers reported in 2008 [1]–[22]. As we shall see, new frequency-selective techniques have been developed to complement traditional time-selective DPD techniques to handle the linearization of multiband PAs.

Quasimemoryless Theoretical Foundation for Multiband Linearization An excellent review of PA modeling and linearization has recently been reported in this magazine [23]. The readers are referred to this review for a very thorough introduction to linearization techniques which have been developed for single-band PAs. Note that a single band can consist of multiple adjacent carriers which are amplified by the same power amplifier. In this article, we will focus instead on the modeling and linearization of multiband power amplifiers used with noncontiguous spectra. In such cases, a single amplifier is used to amplify two or more RF bands whose carrier separation typically greatly exceeds the bandwidth of the individual bands. In such applications, the PA is therefore simultaneously excited by signals of different bands. Consider the case of a PA excited simultaneously by two different bands as show in Figure 1. As the carrier separation increases, the effective modulation bandwidth fU - fL becomes very large. Indeed, when two modulated carriers x L and x U are combined, the composite RF signal will be beating at the frequency fU - fL . That is, as is illustrated in Figure 2, the envelope (red line) of the composite RF signal (blue line) will rapidly oscillate at the frequency fU - fL . Note that the envelope of the composite RF signal oscillates between its peak value (green line) and its minimum value (yellow line) when the two RF signals from the two bands adds constructively (in phase) and destructively (out of phase) respectively. The desired carrier separation fU - fL can be very arbitrary large value. In the physical systems reported in the literature it ranges from 50 MHz to 610 MHz. The conventional time-selective PA modeling and DPD techniques illustrated in Figure 3, which have been developed for single-band PAs could then become unduly complex to handle the resulting large effective modulation bandwidth of the composite signal. It becomes therefore beneficial to come back to the Volterra series modeling of the PA to analyze its response to continuous wave (CW) multitone excitations. Traditionally, the nonlinear PA is modeled as a single input, single output (SISO) system, an approximation which is reasonable if the source and load

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impedances are held constant. The Volterra series modeling of a PA of order r involves a series of multidimensional convolutions of the impulse responses of the PA with the stimuli at its input. When the bandwidth of each band is small compared to the bandwidth separation it makes sense to focus initially on the two carriers before considering their modulation (quasimemoryless approximation). A simplified Volterra series representation results, then, if we switch from the time domain to the frequency domain. Given a nonlinear system of finite order excited by a finite number of tones, one can verify that a finite number of bases or cross-product terms will contribute to each of the intermodulation frequencies generated. Each of these basis cross products, is then multiplied by a nonlinear gain function (labelled in this paper c ) which is dependent on a fixed number of variables. These variables can be called dc terms since for CW excitations they must be zero-frequency terms like the gain functions c. For example in the case where both x (~) and x (2~) are injected at the input, the dc terms are: x (~) x * (~), x 2 (~) x * (2~), and x * 2 (~) x (2~), which all respectively exhibit zero frequencies: ~ - ~ = 0, 2~ - 2~ = 0, and -2~ + 2~ = 0 The gain functions c are referred to as Volterra gain functions [5]. Both the Volterra gain functions and the dc terms they are function of, are generally complex numbers. The gains are represented by analytic functions of the dc terms. This requires that each complex dc term be paired with its conjugate pair. So for the above example, the Volterra gain functions are all of the form: c [|x (~)| 2, x 2 (~) x * (2~), x * 2 (~) x (2~)] . Note that a function is analytic if it is equal to its infinite power series expansion. Singular functions are therefore excluded. An example of an analytic function is the exponential function that can introduce hard nonlinearities. In summary, a nonsingular system of finite order excited by a finite number of CW tones can be fully described by a finite set of bases and associated Volterra gain functions in the frequency domain. Let us assume now that the PA is excited by two CW signals x L and x U, for the lower and upper band, respectively, such that we have: x RF (t) = Re {x L e i~L t + x U e i~U t} with ~ L the radial frequency of the lower sideband (LSB) and ~ U the radial frequency of the upper sideband (USB). No harmonic is assumed to be injected at the input. The LSB and USB frequency bands (black lines) and the resulting third and fifth inband and out-of-band intermodulations (colored lines) at the output of the PA are depicted in Figure 4. The output of the PA around the fundamental (not considering the output harmonics) will then be, for a system of order K = 5:

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xL

zL

xU

zU

DPD

xL

xU

PA DPD Learning

Figure 3. Linearization of a PA using PD.

c+1 c-1 c-5 IMD-5

c-3

IMD-3

c+3 LSB

USB

c+5

IMD+3

IMD+5

Figure 4. Two bands with associated third- and fifth-order intermodulation products and linearization coefficients. y RF = Re {y -5 e i (3~L - 2~U) t + y -3 e i (2~L - ~U) t + y -1 e i~L t + y 1 e i~U t + y 3 e i (2~U - ~L) t + y 5 e i (3~U - 2~L) t}, where the inband terms y -1 and y 1 and the intermodulation terms y -5, y -3, y 3, and y 5 are given by: 3~ L - 2~ U : y -5 = c -5 (|x L| 2, |x U| 2) . x 3L x *U2 2~ L - ~ U : y -3 = c -3 (|x L| 2, |x U| 2) . 2L x *U 2 2 ~ L : y -1 = y L = c -1 (|x L| , |x U| ) . x L 2 2 ~ U : y 1 = y U = c +1 (|x L| , |x U| ) . x U 2~ U - ~ L : y 3 = c +3 (|x L| 2, |x U| 2) . x U2 x L* 3~ U - 2~ L : y 5 = c +5 (|x L| 2, |x U| 2) . x U3 x L* 2 . The complex gains c k are the Volterra gain functions. These gains are analytical functions of the dc terms |x L|2 and |x U|2 . For the two tones case, a single basis is found per frequency; for example x 3U x *L2 for y 5 . For three equally spaced tones the number of bases will be proportional to the number of odd order terms considered, as we shall see later on. The above expansion assumed that the band separation and the order considered are small enough so that the intermodulation terms do not overlap with the harmonics. Additional harmonics terms would need to be included otherwise. Also, it is to be noted that if ~ U is more than twice ~ L , then the third-order LSB radial frequency is actually ~ U - 2~ L rather than 2~ L - ~ U which is negative. In the above presentation, use was made of complex number x U and x L to represent the amplitude and phase of the RF carriers. An alternate representation is to use x L = I L + jQ L , x U = I U + jQ U , and y k = I k + jQ k An expansion of I k and Q k in terms of I L, Q L, I U and Q U can be readily obtained [5]. For



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102 CCDF Probability (%)

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Figure 5. Three bands with associated linearization coefficients.

Figure 6. CCDF of the three uncorrelated LTE bands used as a test.

examples for the third-order terms y -3 = I -3 + jQ -3 and y 3 = I 3 + jQ 3 we obtain:

Three Bands Case

I -3 = (I 2L - Q 2L) I U + 2I L Q U Q L, Q -3 = - (I 2L - Q 2L) Q U + 2I U I L Q L, I 3 = (I 2U - Q 2U) I L + 2I U Q L Q U, Q 3 = - (I 2U - Q 2U) Q L + 2I U I L Q U . It is be noted that a multiplication by a complex gain c k = a k + jb k is equivalent to an IQ modulation of I k and Q k by the coefficients a k and b k since it can represented by the following matrix multiplication:

Ik a k -b k I 'k E # ; E . (1) = 'G =; Qk bk ak Qk

The above equation involves a 2 by 2 matrix with only two independent coefficients a k and b k . Given there are four degrees of freedoms, two additional degrees of freedom are available for the modeling or linearization of the DPD system [5]. In complex envelope notation the general expressions for the both lower and upper bands are thus: y L = c L, 1 x L + c L, 2 x L* , y U = c U, 1 x U + c U, 2 x U* . These two additional degrees of freedom are not necessary for modeling or linearizing the PAs itself but might be needed in DPD in order to correct for the imbalance and nonlinearities of their in-phase quadrature (IQ) modulator. Indeed, DPD systems normally feature digital an IQ modulator for the RF upconversion. The Chaillot functions [5], a set of single-tone multiharmonic Volterra gain functions, can then be further used to assist with the suppression of the IF harmonic suppression. In the rest of this review we shall assume the DACs and modulator is properly balanced and operating in its linear regime.

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The two bands linearization theory can easily be extended to a larger number of bands. In such a case, interband nonlinearities will play an important role beside inband nonlinearities. We consider first the case of three bands with equal frequency separation as shown in Figure 5. Figure 5 shows the three-band inband nonlinearity and third-order interactions. The important thing in the three-band case is the interaction of the LSB and middle sideband (MSB), which generates additional interband modulation terms. One of these terms is located exactly in the position of the USB if all three bands are equally spaced. Similarly, the interaction of the MSB and the USB generates an interband term that falls into the location of the LSB. Even if they are not equally spaced, those interband modulation terms are in the band of interest and need to be removed. For the interband linearization, the same formulas as in the two bands case can be used. For example, the interaction of MSB and USB via x 2M x U* generates the third-order intermodulation distortion (IMD) term in the location of the LSB, and the interaction of LSB and MSB via x 2M x L* generates the +third-order IMD term in the position of the USB. In addition the three band product x L x *M x U generates the interband term in the location of the MSB. Associated with the above third-order IMD terms are the complex linearization coefficients c 3, L, c 3, M and c 3,M . Similarly the fifth-order IMD terms associated with x 2L x *M2 x U, x *L x 3M x *U and x L x *M2 x 2U can be easily be evaluated for the lower, middle and upper bands respectively. From the above discussion it results that for the three-band system, the following third and fifth nonlinear terms are involved if we limit our analysis to the inband spectra: y L = c L x L + c 3, L x 2M x *U + c 5, L x L2 x *M2 x U y M = c M x M + c 3, M x L x *M x U + c 5, M x *L x 3M x *U y U = c U x U + c 3, U x 2M x *L + c 5, U x L x *M2 x 2U .

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3.5-dB Compression " Band 1+2+3

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All the gain coefficients c k are now dependent on three dc terms |x L|2,|x M|2 and |x U|2 : 2

2

c k = c k (|x L| , |x M| , |x U| ) .

Simulation Verification Let us test the efficacy of a multiband fifth-order linearization scheme using simulation. Three LTE signals of 5 MHz bandwidth, centered in noncontiguous bands 20 MHz apart are used for the input excitations. The linearization simulations are conducted for simplicity using a memoryless power amplifier model with third-order nonlinearities. The input signal power is selected such that the amplifier is driven into a deep compression of 3.5 dB at the peak input power. Figure 6 shows the complementary cumulative distribution functions (CCDFs) for the three long-term evolution (LTE) bands used and their composite (sum) signal. The peak power is reached when the envelopes of the three bands constructively add. The AM to AM characteristic of the power amplifier is shown in Figure 7. As indicated by the AM-AM curve (Band 1+2+3) the amplifier is memoryless but nonlinear. However the AM-AM characteristics of Band 1, Band 2 and Band 3 seem to individually exhibit strong apparent memory effects as indicated by their hysteresis. Note that, for the sake of clarity these AM-AM characteristics were respectively shifted horizontally by 0.2, 0.4 and 0.6 on the normalized input power axis. However, this is not a true memory effect since the PA is memoryless but originates instead from cross-modulation between the three bands. Indeed, the PA nonlinearities introduce unwanted intermodulation signals between the bands which also superpose with the desired signals in the same frequency range. The spectral regrowth associated with the PA is shown

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1: PA Output 2: 3rd PD 3: 3rd PD+Interband 4: 5th PD+Interband 5: Input Data

30 20 10 0 -10 -20 -30

Figure 7. Simulation results showing the AM-AM characteristics of the test memoryless PA for three LTE signals before (blue and red lines) and after (black lines) PD linearization.

2

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Figure 8. Simulation results showing the spectra obtained at the PA output without PD (1: red line), and with thirdorder memoryless PD without interband (2: blue line), third-order memoryless PD with interband (3: magenta line) and fifth-order memoryless PD with interband (4: black line). The input spectrum (5: black dashed line) is included for reference. on Figure 8 using the top red curve labeled # 1. For comparison the spectrum of the input LTE data is shown using a dashed line labeled # 5. To demonstrate the linearization of the test thirdorder PA, the third and fifth-order predistortion (PD) that partially inverts the PA characteristic in a least square (LS) sense were extracted. First, the results obtained for a third-order PD neglecting the interband terms x 2M x U* and x 2M x L* are shown in Figure 8 using a blue line labeled # 2. At best, a 5dB adjacent channel power ration (ACPR) reduction is observed. The result obtained for a third-order memoryless PD including the interband terms x 2M x U* and x 2M x L* is shown next in Figure 8 using a magenta line labeled # 3. An ACPR reduction of about 10 dB is observed. Finally, the result obtained for a fifth-order memoryless PD including the fifth-order interband terms x 2L x *M2 x U, x *L x 3M x *U and x L x *M2 x 2U is shown in Figure 8 using a black line labeled # 4. An ACPR reduction of at least 20 dB is observed. Referring back to Figure 7, one also observes that the three black lines labeled “Linearized” demonstrate that the proposed memoryless multiband linearization has indeed linearize the signal measured in band 1, 2 and 3. Figure 9 shows the experimental result of the three-band memoryless linearization. Without the interband linearization it would not be possible to fully linearize the PA output for the three band signal (black dashed line in the figure). With the interband linearization, more than 12 dB IMD cancellation was achieved. To investigate the robustness of the DPD system, the mid-channel band can be turned off. The associated CCDF for the resulting two-band signal is shown



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Spectrum Power (dBm)

Slow memory effects can usually be dealt with using adaptation since they involve long time constants, provided a fast enough adaptation is developed.

-10 -20 Before DPD DPD w/o Interband After DPD

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Figure 9. Experimental linearization results for a 96-tone per three-band signal (a) with and without interband linearization. The robustness of the linearization is tested in (b) by turning off the middle band. in Figure 6 using a black line. Figure 9(b) shows the spectrum obtaine when the mid-channel is off. No performance degradation is observed between the two and three band cases, even though the same linearization coefficients are used. Three band linearization including memory effects was recently reported for an arbitrary frequency distribution [18]. In such a case the intermodulations do not necessarily overlap with the bands. However filtering may still not be a solution and multiband DPD will thus be required. Note that a general algorithm which automatically generates the leading terms in a given bandwidth for an arbitrary number of tones and the targeted order has been developed at the Ohio State University demonstrating that the frequency planning can be automatically realized.

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Memory Effects Up to now we have assumed that the inputs tones x L = I L + jQ L and x U = I U + jQ U were of arbitrary phase and amplitude but were not time varying. This approach enabled us to develop a frequency selective scheme to address the piece-wises quasimemoryless modeling and linearization of each tone/band. Obviously we are concerned with the linearization of PA used to amplify modern communication signals which are amplitude and phase modulated. The frequency selective Volterra series formalism described above which is exact for CW multiband signals must now be extended to modulated multiband signals to account for the inertia of the response of the circuit (PA). These effects are loosely referred to as memory effects. Different types of memory effects must be distinguished at that point: 1) fast memory effects, which are typically associated with the rapid response of the PA to the signal given its wide modulation bandwidth (MHz) and 2) slow memory effects such as traps and self-heating, which are associated with the slow response of the PA system to the long term variation of the envelope of the applied modulated signal. Slow memory effects can usually be dealt with using adaptation since they involve long time constants, provided a fast enough adaptation is developed. However, fast memory effects excited by the modulation bandwidth (typically 5–100 MHz) around each carrier are much too fast to be remediated by using adaptation. As reviewed in [23] many band-limited techniques have been proposed for addressing the faster memory effects associated with each band. One of the most convenient and efficient techniques to implement is that of memory polynomials [24], [25]. It is rooted on the Volterra series formalism. As was earlier mentioned, the Volterra series formalism relies on multidimensional convolution using different delays for the excitations. For example, using a discrete time index n, a thirdorder inband term at time n for the upper sideband would involve three delays m, p and q yielding the output product x U (n - m) x L (n - p) x L* (m - q) instead of x U (n) | x L (n) | 2) for a memoryless case. In the memory approximation only the diagonal terms are retained and all the delays are set equal m = p = q. As mentioned in [5] the time selective technique of memory polynomial can be readily combined with the frequency selective technique. For a dual band PA where the intermodulation components can be removed with a filter, the PA output under the memory polynomial approximation is then of the form: y RF (t) = Re " y L e i~L nTs + y U e i~U nTs ,, where the inband terms y L (n) and y U (n) sampled at time t = nTs (with Ts the baseband sampling frequency), are given by:

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y L (n) =

Tremendous saving can therefore be achieved when new generations of base stations are deployed if a single multiband RF power amplifier is deployed instead.

P

/ c L,p (|x L (n - p)|2,

p =0

|x U (n - p)|2 ) . x L (n - p) y U (n) =

P

/ c U,p (|x L (n - p)|2,

p =0

|x U (n - p)|2 ) . x U (n - p),

Output Envelope (Normalized)

where P is the depth of the memory effects accounted for. Note that the LSB and USB Volterra gain functions c L, p and c U, p are now indexed by the delay index p. The combination of the time-selective memory polynomials with the frequency selective technique has been actively explored and has yielded excellent results [6], [7], [18], [19], [20]. Examples of results obtained with this combined technique are given in the section “Testbed for Dual-Band Linearization.”

Volterra Gain Function Representation and Inversion Representation

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Original PA Output Predistorter Output Linearized PA Output Original PA Output

zed

ari

e Lin

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1

Figure 10. AM/PM simulations for an OFDM signal showing the one-band PA response (blue dots, top), the DPD linearizer response (green dots, bottom) and the combined DPD and PA response (red dots, center) for the quasi-exact inverse technique [5]. The PA model was implemented using B-splines and was extracted from measured data.

In the previous sections, we have presented the general frequency and time selective theories, which are used for modeling the PA as well as perform its linearization using DPD. When modeling the PA or extracting the DPD algorithm, the Volterra gain functions c L, p (|x L|2, |x U|2) and c U, p (|x L|2, |x U|2), for the case of two bands, must be extracted for each delay p. Various techniques are possible for representing these Volterra gain functions. The most common approach relies on polynomials of the envelopes squared |x L|2 and |x U|2 or even the envelope |x L| and |x U|) for improved nonanalytical fitting [6]. The representation of the Volterra gain function is not, however, limited to Maclaurin series. Look up table (LUT) spline [19] artificial neural network (ANN) representation can also be used. Nonetheless, polynomial expansions are

profitably used for the Volterra gain functions due to their simplicity and high performances. Polynomials work particularly well for the DPD linearization. This is illustrated in Figure 10 where the PA response (blue dots, top) and the DPD linearizer response (green dots, bottom) are compared. Clearly, whereas the gain function for the PA saturates, that of the DPD stage diverges at large inputs. Polynomials, which are prone to divergence, are therefore very comfortable with the DPD gain behavior.

1

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Figure 11. Distribution of the envelope |x L| and |x U| at the PA input (a) and the envelope |y L| and |y U| at the PA output (b) for the LSB and USB respectively for two independent LTE signals.

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Figure 12. Distribution of the envelopes |x L| and |x U| at the PA input and the envelopes |y L| and |y U| at the PA output for the LSB and USB, respectively, for a specially synthesized pair of multisines. Normalized input envelopes with an average power |x L|2 +|x U|2 in the same power range are plotted using the same color group on (a). An increment of 0.2 in normalized envelope is used. The corresponding normalized PA output envelopes are shown using the same color on (b). The input and output envelopes are normalized relative to the peak envelope in each band. Still, there is room for improvement with polynomials with a careful choice of the polynomial basis selected. Orthogonal polynomials have been shown to be useful in single band DPD linearization [26], [27]. Similarly, it is possible to extend these results to two bands or more [20], [21]. Indeed, owing to the statistical independence of the various bands making up the composite signal, the orthogonal polynomials developed for a single band can be directly applied without any modification. It is sufficient to generate a tensor product of them to apply them in the multiband case. When using the orthogonal polynomials, the numerical condition number of the matrix used in the linear LS matrix solution is found to greatly decrease. Still a question arises on the way orthogonal polynomial benefits the linearization, given that orthogonal polynomials are obtained using a linear superposition of the various power terms. For ideal theoretical digital signal processing (DSP) systems with unlimited floating or fixed point accuracy, no effective improvement is to be expected. However, when using a reduced number of bits as is the case in practical DSP systems, due to its increased efficiency, the orthogonal representation provides substantial improvements when measured in terms of normalized mean square error (NMSE) performance and reduced number of iterations [20], [21]. Further improvements can be obtained using an iterative method developed to prune the twodimensional (2-D) DPD model to reduce the needed number of coefficients [16], LUT, cubic-spline, and B-spline can also be used as an alternative technique for representing the Volterra gain functions c L, p (|x L|2, |x U|2) and c U, p (|x L|2, |x U|2) [19], [5]. Spline provides a slight improvement in

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NMSE for the power amplifier and to some extent for the DPD linearization. As for the polynomial representation, it is possible to directly extract the spline coefficients using a linear LS inversion of a matrix (to be reported elsewhere). However, in 2-D, due to the scarcity of the data at high power when the output powers of both bands are high at the same time, the conditioning of the spline matrix is very poor and sometimes infinite. Excellent results are still obtained in the region where data are available while unstable results are obtained outside the range of extraction. The scarcity of the data is illustrated in Figure 11 for two independent LTE upper and lower bands. To address the scarcity of data at high power it is possible to develop a 2-D multisine with the correct CCDF, which will map the complete (|x L|, |x U|) space [22]. This is illustrated in Figure 12, where the 2-D envelopes in (a) are seen to fully map the 2-D rectangular space for a more robust extraction. On the other hand, it is observed that the 2-D envelopes in (b) are not fully mapping the 2-D rectangular coordinate. This arises obviously from the saturation of the PA, as is also indicated by the different color mapping for corresponding input power ranges. Note that the saturation at the output of the PA is seen to take place between the straight line |y L| +|y U| = 1 and the circle line |y L|2 +|y U|2 = 1. Both the input and output PA data have been normalized. The result is that the DPD extraction from output to input will be not fully mapped and still prone to divergence in the regions where there is no data. This problem can be resolved by the use of polar coordinates over rectangular coordinates when using splines for extracting the DPD gains.

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|y U| =

|x L | and |x L|2 +|x U|2 (2) | xU | . 2 2 |x L| +|x U|

One can easily verify that |y L| and |y U| satisfy the saturation property |y L| 2 +|y U| 2 = 1 as required. The resulting extrapolation of the Volterra gain functions is illustrated in Figure 13 for the case of a memoryless PA model. The phases of y L and y U in saturation can themselves be interpolated from the data near saturation. The validity of the methodology is visually verified in Figure 13 by the smooth extrapolation provided by the the extrapolated data (black circles) and the extracted cubic-spline (lines) relative to the measured data (red dots). Since it is not practical to train a 2-D predistorter for all possible 2-D distribution of peak envelope events, ensuring for a physical extrapolation of the PA envelope response outside the range of extraction provides for a more robust 2-D spline representation. This extrapolation method may help give spline representation a competitive edge compared to the classical more robust polynomial extraction. Spline have the ability to more efficiently handle harder nonlinearities when the PA operates in deep compression [19]. Driving the PA in stronger compression usually yields benefit in terms of power-added efficiency. For completeness, let us note that ANNs have not yet been reported for multiband DPD but could also be used to achieve higher performance like in single band DPD [28]. ANNs do provide continuous derivatives of infinite orders and naturally exhibits graceful degradation; however, the ANN learning is usually time consuming. To conclude, let us note that both direct [30] and indirect [25], [29] learning approaches have been

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implemented to extract the DPD algorithm. Both of these methods usually lead to the same results by reducing the spectral regrowth from 15 to 20 dB. The results presented in the next section are based on the direct learning forcing some continuity in the coefficient extraction and enabling us to control the convergence. With the direct method, the DPD outputs are furthermore not required, simplifying the field programmable gate array (FPGA) implementation. The gain function representation and model inversion discussed in this section have provided us with some insights in the PA response. For example, in Figure 13 we have observed that the saturation of the PA took place above |y L| +|y U| = 1 and close to the

1 0.8 ]yL]2

| y L| =

The power of such testbeds, compared to the VSG/VSA solution lies on their modular nature and flexibility in linearization system design methodology.

0.6 0.4 0.2 0 1

1 0.5 ]xU]2

0.5 0 0 (a)

]xL]2

Spline

1 0.8 ]yU]2

Alternatively, the quasi-exact inverse of the memory polynomial used for the PA could be used. Implicit in all our discussions so far is the assumption that the same model could be used for the linearization and the PA modeling. However, it is possible for the case of the one-band memory polynomial model to obtain a quasi-exact inverse of the PA model as shown in Figure 10 [5]. These results can also be extended in the dual band case, as shall be reported elsewhere. As for the one band case, the requirement is that the inversion be performed in the region where the two Volterra gain functions 2 2 2 2 c L, p (|x L| , |x U| ) and c U, p (|x L| , |x U| ) are simultaneously invertable. As an alternative to the measurement of the PA in a full rectangular coordinate, the extrapolation of the Volterra gain functions can also be used. Using the saturation property |y L|2 +|y U|2 = 1 observed by the PA, one can deduce that the envelopes in saturation are given by

0.6 0.4 0.2 0 1

1 0.5 ]xU]2

0.5 0 0

]xL]2

(b)

Figure 13. A 3-D plot of envelopes |y L| (a) and |y U| (b) at the PA output versus the envelopes |x L| and |x U| at the PA input for memoryless PA model. Compared are the measured data (red dots), the extrapolated data (black circles) and the cubic-spline fit (lines).



83

radios can also be combined together [13]. This Dual-Band z1 is the method adopted DPD1 D/A PA Coupler Up– in the rest of this article Conveter to demonstrate that the z2 D/A DPD2 dual or multiband band x2 linearization scheme is Upper Side Band independent of the carLower Side Band y1 rier spacing. In all the Analyzing Stage A/D cases, the multicarrier RF DDC Down– Time Alignment signal is amplified by the Conveter y2 Coefficient same PA. A/D Extraction A schematic of a DPD Upper Side Band concurrent dual-band system is illustrated in Figure 14. Schematic of a concurrent dual-band system dedicated to PA linearization. Figure 14. The baseband signals are predistorted 2 2 by both DPD units and then passed through its respec| y | + | y | = 1 . circle line The importance of the L U tive digital-to-analog converters, up-converted to RF average envelope of |y L|2 +|y U|2 associated with the frequency, power combined, and finally passed to the average power (purple line in Figure 3) suggests that PA. In order to extract the coefficients of both DPD the saturation in the PA is in part a thermal process. units, a small fraction of the transmit signal is fed back On the other hand, deep level traps in a GaN HEMT and transferred to the baseband via a down-frequency based PA will be more affected by the peak envelope converter and analog-to-digital converter(s). After (green line in Figure 3) as they charge during the time alignment, the model parameter extraction unit infrequent peaks and slowly discharge when the PA compares the input to the captured output data and returns its operation to the average power. extracts the coefficients that are updated in both DPD units. Two different solutions are usually proposed in Testbed for Dual-Band Linearization the literature to design the DPD testbed. The first one Various schemes can be used in DPD to generate and relies on an vector signal generator (VSG) and a veclinearize a dual-band signal. For example, a digital IF tor signal analyzer (VSA). The second one relies on a can be generated for each carrier at the baseband level high-speed digital hardware circuit such as FPGAs or in the FPGA/DSP and the composite signal upconas application specific integrated circuits (ASICs). Both verted to RF. However this has the disadvantage of test-bench solutions are discussing in the following being limited in terms of carrier spacing by the FPGA/ sections. DSP sampling clock [1]. Alternatively, the NCO (numerically controlled oscillator) usually implemented in the digital up converter Testbed Based on Vectorial Sources (DAC) can be used to upconvert separately the digital Most of the research conducted in current communibaseband signals just before generating the I and Q cation labs has relied on a computer-driven testbed using VSGs to generate the modulated RF signals and signals. This enables one to synthesize much larger a VSA to analyze the signals as shown on Figure 15. digital IF while using the same local oscillator (LO) [15]. Such a high-performance testbed enables one to To obtain arbitrarily large carrier spacing, the output obtain rapid results. All of the processing is done on of different synchronized modulated RF sources or the computer, typically using MATLAB. For multiband DPD, several VSGs will be required. The waveforms will need to be time synchronized and also DPD Predistorted phased locked if the intermodulation products are to Learning Signals be linearized or if more than two bands are investiVector Signal gated. A disadvantage of such testbeds is their inherVector Signal ent high cost as well as the difficulty in implementing and testing real-time adaptive algorithms. In addiGenerator Analyzer tion, the viability and practicability of the algorithms xL xU yL yU used are best tested when implemented in a DSP and PA RF Input RF Output FPGA testbed like in real systems deployed by cellular phone service providers. This option, which has been less explored, is discussed in more details in the Figure 15. PA linearization testbed realized with a VSG and next section. a VSA. x1

84

Lower Side Band



November/December 2013

Practical FPGA Testbed and Experimental Results

PC Running MATLAB Data Read/Write

MATLAB SIMULINK

1.8 GHz < LO < 2.2 GHz 16-b DAC

As was seen in the prefrom/to vious part, the VSG and 12-b ADC Memory the VSA are commonly USB Interface Up to 122-MHz BW engaged for multiband test setup design. While Synthesis Digital Processing this solution is useful RX1 to quickly test, accuHSMC TX1 rately capture data, and RFin design new algorithms, it is reduced to a laboraTX2 50 X tory usage and cannot RX2 be involved in real-time HSMC application. To address 61.44-MHz the industrial requireADi MSDPD Altera FPGA Ref. Clock ments, DPD systems Stratix IV need to be implemented based on commercial products. Indeed, com- Figure 16. A testbed for concurrent dual-band PA linearization. panies have to evaluate the cost of the solution in diplexer and sent to the two MSDPD RF observation term of hardware foot print, complexity, and efficiency paths. Both signals are down-converted to an interbefore integrating them into their communication mediate frequency (IF) of 184.32 MHz, digitized and systems. Development platforms based on a FPGA stored in the FPGA memory. are therefore well suited for DPD system implementaOn the software side, the received data stored in tion. While a tremendous amount of research has been the FPGA board are downloaded to the controlling conducted with regard to single-band DPD hardware computer, using the USB communication link and implementation, due to its innovation, fewer studthe Altera’s SIL APIs. The signals are then digitally ies have been conducted on multibands DPD system down-converted (DDC) and time aligned with their based on FPGAs [1], [19], [20]. In this section, we prescorresponding input signals. This is all done using ent a dual-band DPD testbed based on commercially available products. The setup for the platform is depicted on Figure  16. On the hardware side, it consists Algorithm Simulations MATLAB of two Analog Devices mixed signal DPD DPD Model MATLAB (MSDPD) demo boards [33], both connected Data Processing and clock synchronized to the Altera FPGA Coefficient Estimation Stratix IV [34]. The Stratix IV FPGA development board includes DDR3 memory banks and standard USB and HSMC connectors. The analog front-end with two LOs enables High-Level Integration Simulink Bit–True it to support two widely separated bands. DPD Model MATLAB For this purpose, the MSDPD enables the Data Processing up/down conversion, filtering, digital-toCoefficient Estimation analog conversion (DAC) and analog-to-digital conversion (ADC) done by 16-bit DACs and 12-bit ADCs respectively. The center System-Level Integration frequency of both bands can be set between Hardware 1.8 GHz and 2.2 GHz, and the signal bandMATLAB DPD Data Processing width allowed is up to 122.88  MHz. Each Coefficient Estimation board has a separate transmission path. The outputs are then combined to drive the PA. On the observation path, the output signal is captured through a coupler, splitted using a Figure 17. Design methodology.

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85

By the end of this process, an image file is generated by the compiler for a target FPGA and then downloaded to the target FPGA device. MATLAB. The processed data are then ready to be analyzed for the DPD coefficient estimation. The power of such testbeds, compared to the VSG/ VSA solution, lies on their modular nature and flexibility in linearization system design methodology. Indeed, in the first development stage, the testbed can used as an VSG/VSA solution to easily test the DPD algorithms in MATLAB. In the second development stage, once the designer is satisfied with the linearization performance, the linearization algorithm can then be implemented in the FPGA. The associated design flow is detailed in the following section.

Design Methodology, from MATLAB to FPGA Implementation The final purpose of linearization system design is the implementation of the algorithms in the FPGA. To achieve this goal, the following design methodology is proposed and summarized on Figure 17. In the first step, the setup is used as a regular VSG/ VSA solution to ensure the DPD and related algorithms meet the expected results and performances while benefiting from the flexibility and simplicity of the MATLAB environment. At this stage, the predistorter system is implemented in MATLAB, and both baseband predistorted signals are written into the FPGA memory to be played. Then the received signal is analyzed, and the DPD coefficients are recursively optimized by writing the new set of data to the FPGA memory. Table 1. Summary of the three scenarios. Lower Sideband Carrier Frequency = 1,890 MHz

Upper Sideband Carrier Frequency = 2,200 MHz

Scenario I

One-carrier WCDMA

Five-carrier WCDMA

Scenario II

LTE

Three-carrier WCDMA

Table 2. Summary of the signal characteristics.

86

Linearization Performances of the DPD Testbed The DPD testbed described has been evaluated to linearize an amplification stage composed of a linear driver followed by a broadband 10 W peak output power PA, based on the NXP Semiconductor GaN HEMT CLF1G0060-10 transistor [39]. The 2-D-DPD model from [6] is implemented. In Figure 14, considering x L, x U and z L, z U, the two input and output baseband signals of the DPD model, the complex baseband input/output relationship of the 2-D-DPD memory model for concurrent dual-band is expressed as

z L ^nh =

M -1 K -1

z U ^nh =

M-1 K-1

k

/ / / c (mL,)k,j $ x L $

xL

k -j

$ xU j 

m =0 k =0 j =0 k

/ / / c (mU,k) ,j $ x U $

xU

k-j

$ x L j , (3)

m=0 k=0 j=0

Five-carrier WCDMA

Scenario III LTE

Signal Type

In the second step, once the DPD system meets the standard requirements, the FPGA implementation starts by using a hardware accurate model of the signal processing component provided by FPGA companies [35], [38]. In the reported results, the Altera developed DSP Builder Advanced (DSPBA) block sets running in Simulink were used. DSP Builder includes basic FPGA block models such as adders, multipliers, or delays and enables one to build and verify the user’s digital system with real hardware parameters without requiring FPGA implementation in the system. Once the DPD model is built, the generated outputs are tested by transferring them to the FPGA memory and playing them to ensure that they provide similar performance to the MATLAB implementation. Moreover, these high-level synthesis tools are able to translate to RTL code directly via an automated process. Finally, the generated RTL code is integrated in the overall communication system using standard interfaces and commercial integration tools [36], [37]. By the end of this process, an image file is generated by the compiler for a target FPGA and then downloaded to the target FPGA device. At this stage, only the signal processing and the coefficient estimation is done in MATLAB, which can furthermore be implemented on a different dedicated signal processor.

1c-WCDMA 3c-WCDMA 5c-WCDMA LTE

Signal 3.84 MHz bandwidth

13.84 MHz

23.84 MHz

10 MHz

Signal PAPR

9.4 dB

10.5 dB

10.2 dB

5.7 dB



(L)

(U)

where c m, k, j and c m, k, j are the coefficients of the model for the LSB band and USB band, respectively, K and M, the nonlinearity order and the memory depth, respectively. The LSB and USB are separated by 310 MHz and centered at 1890 and 2200 MHz, respectively. Table 1 summarizes the three different signal scenarios that have been considered in this article for the lower and USBs. In the first two scenarios, the USB contains a fivecarrier WCDMA signal, each carrier is separated by 5 MHz, and the total signal bandwidth is equal to

November/December 2013



NMSE [dB] = 10 $ log f (i)

/ nN=1 x i (n) - y i (n) 2 , (4) / nN=1 x i (n) 2 p

where i = " L, U , and N is the sample length of the signal in each band. Again, both implementations are really close; lowering the LSB below -40 dB and decreasing by more than 20 dB on USB. As a comparison, the performance metrics for scenario I are given in Table 3. Figures 20 and 21 are comparing the linearization performances for both implementations by showing the output power spectrum of upper and LSBs for the scenario II, respectively.

November/December 2013

Lower Side Band

-40

No DPD MATLAB Impl. FPGA Impl.

PSDLSB (dB/Hz)

-50 -60 -70 -80 -90

-100

1, 86

1, 85

0

0 1, 87 0 1, 88 0 1, 89 0 1, 90 0 1, 91 0 1, 92 0 1, 93 0

-110 Frequency (MHz)

Figure 18. LSB output spectrum for scenario I, 1c-WCDMA. Comparison of the spectrum without DPD (blue), with DPD implemented in MATLAB (red circle) and DPD implemented in FPGA (green triangle).

Upper Side Band

-40 PSDUSB (dB/Hz)

-50 -60

No DPD MATLAB Impl. FPGA Impl.

-70 -80 -90

-100

0

0

24 2,

0

23 2,

0

22

21

2,

2,

0 20 0 2,

0

19 2,

0

18 2,

17 2,

16

0

-110

2,

23.84 MHz for a PAPR for the total signal of 10.5 dB. In the first scenario, the LSB includes a single-carrier WCDMA, 3.84 MHz signal bandwidth with a PAPR of 5.7 dB. In the second scenario, a single-band LTE signal is played with a 10 MHz bandwidth and a PAPR of 10.2 dB. The last scenario (III) evaluated uses the single band 10 MHz LTE signal for the LSB and a threecarrier WCDMA signal for the USB. The four types of signal characteristics are summarized in the Table 2. In order to compare and validate the two proposed design methodologies, the linearization stage has been implemented in 1) MATLAB and in 2) the FPGA. In the FPGA, due to the maximum resource constraints, the DPD is implemented using a 32-bit fixed point accuracy. Regarding the equation (2-D-DPD equation), the maximum nonlinearity is set to K = 6 and the maximum memory depth is M = 4. Coefficients are extracted from 8,000 samples signals, and the linearization performances are evaluated on 262,000 samples. The indirectlearning LS algorithm depicted previously enables us to extract both set of coefficients that are then played for verification until they meet the performance requirements. The PA is used for 1 dB gain compression, and the corresponding average output power is 31 dBm. Figures 18 and 19 compare the linearization performances for both implementations. It shows the output power spectrum of lower and USBs for scenario I, respectively. As expected, both implementations give similar results by reducing the spectral regrowths by approximately 20 dB, ensuring the ACPRs at 5 MHz are lower than -50 dBc in both bands. Note that the power spectral densities (PSDs) compared with and without PDD all exhibit about the same average output power so that no backoff is used for a fair comparison of the ACPR improvements. The second common metric is the NMSE, which gives an insight of the performance inside the band. The NMSE in dB is expressed for each band as follows:

Frequency (MHz)

Figure 19. USB output spectrum for scenario I, 5c-WCDMA. Comparison of the spectrum without DPD (blue), with DPD implemented in MATLAB (red circle) and DPD implemented in FPGA (green triangle). As in scenario I, both implementations for Scenario II give a similar performance. While the spectral regrowth reductions are less than in scenario I, the ACPRs are still lower than -50 dBc in both bands. NMSEs are below -37 dB and -39 dB after linearization

Table 3. Summary of the performances for scenario I. Lower Sideband ACPR -/ + 5 MHz NMSE (dB) (dBc) Without DPD

Upper Sideband ACPR -/ + 5 MHz NMSE (dB) (dBc)

-37.64/-38.86

-26.68 -37.99/-36.27

-16.27

DPD implemented -58.77/-59.27 in MATLAB

-42.85 -53.17/-53.07

-39.36

DPD implemented -58.62/-59.11 in FPGA

-42.52 -52.73/-52.93

-38.12



87

Lower Side Band

-50 No DPD MATLAB Impl. FPGA Impl.

-60 -70

Upper Side Band

-40 PSDUSB (dB/Hz)

PSDLSB (dB/Hz)

-40

-80 -90

-50 -60 -70

No DPD MATLAB Impl. FPGA Impl.

-80 -90

-100

-110

-110

2, 16

1, 85

0 2, 17 0 2, 18 0 2, 19 0 2, 20 0 2, 21 0 2, 22 0 2, 23 0 2, 24 0

0 1, 86 0 1, 87 0 1, 88 0 1, 89 0 1, 90 0 1, 91 0 1, 92 0 1, 93 0

-100

Frequency (MHz)

Frequency (MHz)

Figure 20. LSB output spectrum for scenario II, 1c-LTE. Comparison of the spectrum without DPD (blue), with DPD implemented in MATLAB (red circle), and DPD implemented in FPGA (green triangle).

Figure 21. USB output spectrum for scenario II, 5c-WCDMA. Comparison of the spectrum without DPD (blue), with DPD implemented in MATLAB (red circle) and DPD implemented in FPGA (green triangle).

for the lower and USBs respectively. Table 4 summarizes the linearization performance in both bands. In the two different scenarios that were dealing with different types of signals in each band, the spectral regrowths are reduced, enabling the system to respect the different standard spectral masks. As predicted, the two linearization stage implementations exhibit similar performances. In scenario III, the PA is now more saturated with a 3-dB gain compression, the nonlinearity order is set to K = 8 , and the memory depth order is M = 5. Due to the increased complexity of the model, a high number of multipliers is required, and the model was only implemented in MATLAB so that the test bench is used in its VSG/VSA configuration. Table 5 summarizes the linearization performance in both bands.

Applying the frequency-selective memory-polynomials (2-D-DPD) enables one to improve the ACPRs by more than 16.5 dB in each band and ACPRs are lower than -50 dBc in both bands. NMSEs are improved by more than 26 dB. Compared to memoryless DPD, an improvement of 12.8 dB and 14.4 dB is observed for the LSB and USB, respectively, when accounting for memory effects. However, the drain efficiency of the PA drops by 2% from 15% with no DPD to 13% with DPD. Using cubic-spline, it was also verified that a DPD performance similar to that of memory polynomials can be achieved for dual-band linearization while using fewer coefficients (to be reported elsewhere). An LUT implementation can also be used to save on FPGA DSP resources and rely instead on FPGA lowcost memory resources.

Table 4. Summary of the performances for scenario II. Lower Sideband

Upper Sideband

ACPR -/+ 5 MHz (dBc)

NMSE (dB)

ACPR -/+ 5 MHz (dBc)

NMSE (dB)

-31.78/-34.33

-21.36

-38.28/-36.16

-20.33

DPD implemented in MATLAB -50.46/-52.01

-37.42

-51.33/-50.53

-39.51

DPD implemented in FPGA

-37.20

-50.87/-50.37

-39.25

Without DPD

-50.28/-51.69

Table 5. Summary of the scenarios III performances. Lower Sideband

88

Upper Sideband

ACPR -/+ 5 MHz (dBc)

NMSE (dB)

ACPR -/+ 5 MHz (dBc)

NMSE (dB)

Without DPD

-32.95/-34.98

-14.68

-31.49/-30.39

-15.37

With memoryless DDP

-43.34/-46.73

-29.23

-44.35/-47.52

-28.20

With DPD

-51.15/-51.50

-42.07

-51.90/-52.77

-42.64



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Table 6. Comparison between prior studies. Signal Type (Bandwidth)

fcenter (MHz)

Reference

LSB

USB

ACPR (dBc) LSB w/o / w DPD

USB w/o / w DPD

D ACPR LSB LSB/ w/o / w USB DPD

USB w/o / w DPD

D NMSE LSB/ USB

-45/ -55.8

-41.1/ -53.1

10.8/ 12

-30.84/ -43.77

-26.73/ -41.88

12.93/ 15.15

45/53

-39.2/ -51.1

8/11.9

-19.05/ -41.61

-26/ -39.4

22.56/ 13.4

48/58

-38.34/ -54.55

10/ 16.21

-30.04/ -41.05

-26.19/ -42.51

11.01/ 16.32

2c-WCDMA 2c-WCDMA -36.0/ (13.84 MHz) (13.84 MHz) -55.5

-37.1/ -58.4

19.5/21.3 -/-

-/-

-/-

2c-WCDMA (8.84 MHz)

2c-WCDMA (8.84 MHz)

-39.0/ -55.6

-37.7/ -56.7

16.6/ 19

-/-

-/-

-/-

LTE (10 MHz)

2c-WCDMA (8.84 MHz)

-32/ -45

-32/ -45

13/13

-/-

-/-

-/-

1c-WCDMA (3.84 MHz)

-39.6/ -55.6

-35.6/ -53.7

16/ 18.1

-/-

-/-

-/-

1c-WCDMA (3.84 MHz)

-42.48/ -52.06

-41.2/ -56.2

9.58/ 15

-21.97/ -40.12

-20.69/ -39.5

18.15/ 18.81

-31.88/ -52.54

14.73/ 20.66

-17.38/ -38.16

-19.1/ -34.72

20.78/ 15.62

LSB

USB

1c-WCDMA (3.84 MHz)

WiMAX 2-D-DPD 1,900 2,000 (5 MHz) Bassam et al. [6]

1c-WCDMA (3.84 MHz)

WiMAX (10 MHz)

DPD-widely spaced carriers Braithwaite [9]

1-D-LUT Ding et al. [11]

1,935

1,861

1,985

1,959

2-D-enhanced 1c-WCDMA Hammerstein 1,800 2,411 (3.84 MHz) Moon et al. [32]

2-D-modified DPD Liu et al. [16]

2c-WCDMA (8.84 MHz) 1,960

3c-WCDMA -36.05/ (13.84 MHz) -50.78

Lower Side Band at 1,890 MHz

1.2 1

1.6 dB

PA Without DPD

0.8 0.6 0.4

PA with DPD

0.2 0

0

0.2

0.4

0.6 Input

0.8

1

1.2

Figure 22. Output versus input for the LSB obtained from the PA (black dots) and the DPD+PA system (red line). For reference, the AM-AM calculated based on the low-power memoryless gain is also included (green dots). A compression of 1.6 dB is observed at peak power.

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Upper Side Band at 2,200 MHz

1.4 Normalized Output

1.4 Normalized Output

1,900 2,000

880

NMSE (dB)

1.2

PA Without DPD

1

3.5 dB

0.8 0.6 0.4 0.2 0

PA with DPD 0

0.2

0.4

0.6

0.8

1

Input

Figure 23. Output versus input for the USB obtained from the PA (black dots) and the DPD+PA system (red line). For reference, the AM-AM calculated based on the low-power memoryless gain is also included (green dots). A compression of 3.5 dB is observed at peak power.



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With the explosive growth of the smartphone and tablet markets, wide bandwidth voice and data communication have become ubiquitous.

Figure 24 shows the AM/AM plots for the PA response ^|y inst |versus|z inst| h and the PA + DPD response ^|y inst|versus|x inst| h . We can observe that the PA is in the saturated mode and that at the peak output power the PA goes into 3-dB compression relatively to the extrapolated low-power AM to AM response (green dots). Note that the LSB and USB PA responses exhibit 1.6 and 3.5 dB of compression, respectively, when amplified concurrently, as can be seen in Figures 22 and 23. This section has presented the practical design of a testbed dedicated to advanced dual-band DPD algorithm development, which can be used for both research and development. Using this platform, a 2-DDPD predistorter compensating for the inband concurrent dual-band PA nonlinearities can be accurately and quickly implemented in a FPGA and was found to yield a performance similar to a MATLAB implementation and other published studies.

Instantaneous Output

2.5 3 dB

2 PA Without DPD

1.5 1

PA with DPD 0.5 0

0

0.5

1 1.5 Instantaneous Input

2

Figure 24. Output versus input (AM-AM) for the combined signals (instantaneous envelope) obtained from the PA (black dots) and the DPD+PA system (red line). For reference the AM-AM calculated based on the lowpower memoryless gain is also included (green dots). A compression of 3 dB is observed at peak power. In Table 6, we summarize and compare the different figure of merits (ACPR, NMSE) for different frequency scenarios that have been published. As can be seen, the low-cost FPGA testbed presented in this section exhibits a performance comparable to that reported in the literature. The associated AM to AM response for the lower and USBs are shown in Figures 22 and 23. The memory effects associated with the two bands are removed as expected. To gain further insights in the dual-band DPD operation, we propose to recast the dual-band linearization into that of a single-band system. This is done by up-converting the signals of both bands and combining them into a single instantaneous baseband signal for the average carrier frequency (fL + fU) /2. For this purpose the instantaneous input x inst, DPD z int and output y inst complex envelopes are calculated using



-jrDft -jrDft x inst ^nh = x ovs + x ovs L $e U $e ovs ovs -jrDft -jrDft z inst ^n h = z L $ e + zU $ e -jrDft -jrDft  y inst ^n h = y ovs + y ovs L $e U $e



In this article we have presented a review of the state of the art regarding the modeling and linearization of multiband power amplifiers used for the amplification of signals with noncontiguous spectra. The interest in multibands DPD system has been rapidly growing since the first dual-band DPD papers reported in 2008, and a comprehensive body of work is already available. Supportive theories, simulation, and experimental results were presented to demonstrate the concept and practical implementation of multiband DPD systems. However, a number of challenges remain to be addressed to further improve the performance of multiband DPD and apply it to practical scenarios. Mostly the dual-band DPD system has been studied so far but a few multiband band DPD systems have also been reported [15], [18]. As the number of noncontiguous bands increases, the algorithm complexity increases. The use of the peak and average power envelopes should then prove useful to maintain the same order of complexity as 2-D-DPD while increasing the number of bands above 2-D. Crest factor reduction techniques for composite multiband signals should also be optimized for multiband application. Finally, cooperative schemes involving multiband DPD and power-added efficiency techniques such as envelope tracking and Chirex remain areas that could be explored. Many of all the techniques that have been developed for multiband DPD might also find application in single-band DPD systems of wide bandwidth.

Acknowledgments (5)

ovs where x ovs and y ovs are the oversampled signals of i , zi i x i, z i and y i, respectively, i = " L, U , and Df = fU - fL is the frequency-spacing between both bands.

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Conclusion

We would like to thank the Altera Corporation for the support provided for this research. Partial support for this work was also provided to the first author by the National Science Foundation under grant ECS 1129013. We would like to thank Analog Device for providing us with the MSDPD boards customized for this demonstration testbed. We are grateful to NXP for providing

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the GaN power amplifier used for the experimental work. Finally, we are grateful for the insighful suggestions made by the reviewers.

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