3-D Fourier Series Based Digital Predistortion Technique for Concurrent Dual-Band Envelope Tracking With Reduced Envelope Bandwidth

Transcription

1 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 3-D Fourier Series Based Digital Predistortion Technique for Concurrent Dual-Band Envelope Tracking With Reduced Envelope Bandwidth Yiqiao Lin, Christophe Quindroit, Haedong Jang, Member, IEEE, and Patrick Roblin, Senior Member, IEEE Abstract In this paper, the design, implementation, and measurement results of a new digital predistortion (DPD) method for a concurrent dual-band envelope tracking (ET) power amplifier (PA) system is presented. The PA gain is represented using a set of 3-D orthogonal Fourier series basis functions, which accounts for the distortions introduced by the two signal bands, as well as the supply modulation voltage. Here, the new Fourier series basis is shown to substantially outperform the traditional memory polynomial approach in terms of its linearization capability for subsequent data for which it was not trained for (prediction) due to its well-defined numerical rank and stability. The new DPD system was implemented using a 10-W peak gallium nitride (GaN) ET PA operating with a dual-band input based on long-term evolution and WCDMA signals center frequency (1.89 and 2.2 GHz) spaced by 310 MHz. The linearization results are compared to the 3-D memory polynomial in two steps: extraction and prediction. In the DPD coefficient extraction phase, tested under the same signal, the traditional memory polynomial and the Fouries series reach close results in both normalized mean square error (NMSE) and adjacent channel power ratio (ACPR). In the prediction phase, however, the proposed method provides a significant performance improvement. A performance improvement as high as 17 db in terms of NMSE and 4.6 db in terms of ACPR in comparison to the conventional dual-band 3-D memory polynomial method implementing ET. Furthermore, the well-defined numerical rank of the Fourier series approach allows for a substantial reduction in coefficients using principal component analysis without detrimenting performance. Index Terms Concurrent dual-band, digital predistortion (DPD), envelope tracking (ET), power amplifiers (PAs). Manuscript received September 02, 2014; revised November 27, 2014, January 27, 2015, and May 13, 2015; accepted June 27, This work was supported by the National Science Foundation (NSF) Collaborative under Grant ECS Y. Lin was with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH USA. She is now with NXP Semiconductors, Smithfield, RI USA ( lin.837@osu.edu). C. Quindroit was with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH USA. He is now with LPA Concepts, Martillac 33650, France ( quindroit.1@osu.edu). H. Jang was with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH USA. He is now with the Infineon Technologies North America Corporation, Morgan Hill, CA USA ( jang.131@osu.edu). P. Roblin is with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH USA ( (roblin.1@osu. edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMTT I. INTRODUCTION T HE increasing demand for cost-effective multi-standard and multi-band base-stations has fostered the development of multi-band power amplifiers (PAs). A central concern of base-station PA design comes from the linearity and efficiency requirements of the transmitter, which translate into signal integrity and operational costs. Today, modern wireless communication standards employ high data-rate and non-constant envelope schemes such as wide-band code-division multiple access (WCDMA), worldwide interoperability for microwave access (WiMax), and orthogonal frequency division multiplexing (OFDM) in long-term evolution (LTE) transmission, introducing high envelope peak-to-average ratio (PAR) to the signal resulting in highly inefficient PAs when using traditional fixed power supplies. Supply modulation schemes such as envelope tracking (ET) [1], [2] have been developed to address these efficiency concerns, and applied successfully to dual-band PAs for modern digital modulation schemes, demonstrating improved efficiencies [3], [4]. There is much interest in extending this approach to multi-band transmitters. The simplified block diagram of a concurrent dual-band ET system is depicted in Fig. 1. It comprises two up-conversion units where each band is up-converted via different modulators before being combined and amplified while the drain bias point of the PA is modulated. Although the ET method can provide improvements in efficiency, they also introduce several drawbacks. One of the main issues, for dual-band ET, is the bandwidth (BW) required for the supply voltage to track the dual-band instantaneous signal envelope. This requirement can be greater than five times the carrier separation frequency. Thus, a larger BW requirement is placed on the supply modulator. To alleviate the constraints on the supply modulator BW, it has been shown that a reduced bandwidth (RB) signal envelope can provide an effective mitigation [5], [6]. The second main issue with ET PAs concerns the linearity. Indeed the supply voltage is modulated based on the input power, and for optimal efficiency improvement, the amplifier needs to be operating near the 1-dB compression point. Consequently, the improvement in efficiency in ET is accomplished at the cost of increased nonlinearity. Therefore, in order to implement a usable dual-band ET PA, the nonlinearity has to be addressed. In IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 2 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 2. Fast beating of the RF envelope (1), compared to the peak envelope (2), average envelope (3), with the RB envelope. Fig. 1. Block diagram of a dual-band ET system. the previous literature, a significant number of approaches have been proposed to linearize concurrent dual-band PAs with conventional constant supply voltage. The original frequency-selective architecture [7], [8] expands the gain of the lower and upper band gains as a weighted sum of products of the upper and lower band envelopes in order to account for in-band and cross-band distortions. Bassam et al. [9] introduced the 2-D digital predistortion (2-D-DPD) memory polynomial method, which added frequency-selective memory effects and assumed that the far away intermodulation bands could easily be removed through filtering. Further approaches by other authors have attempted to improve on the memory polynomial method by improving stability, as well as the reduction of expansion terms for PA characterization. These methods include the 2-D augmented Hammerstein model [10] and the modified memory polynomial [11]. Volterra series-based digital predistortion (DPD) have been used to linearize both dual-band [12], as well as tri-band [13], [14] PAs, again with a reduced number of expansion terms in comparison to the 2-D-DPD memory polynomial method. Moreover, there has been significant interest in cost-effective hardware implementations to the dual-band DPD problem. More practical DPD implementations have been achieved using a look-up table (LUT) and signal-generator by Kwan et al. [15], as well as a field-programmable gate-array (FPGA)-based LUT for DPD by Ding et al. [16]. Quindroit et al. [17] presented an FPGA-based DPD hardware implementation using two sets of orthogonal polynomials, which also improves the DPD stability in comparison to other methods. These approaches, such as the frequency selective [7] and 2-D-DPD [9], which only consider the dual-input dual-output relationship of the PA, become insufficient when directly applied to a concurrent dual-band ET PA. Indeed, applied in this context, the dual-band DPD techniques neglects the impact of the variation of the drain voltage on the ET PA linearization [18]. This paper is organized as follows. Section II introduces the supply modulation scheme used in the concurrent ET system. The conventional 2-D-DPD and 3-D-DPD models for the linearization of concurrent dual-band fixed-supply PA and ET PA, respectively, are then recalled. Section III introduces a new model for DPD of dual-band ET PAs, accounting for the distortions introduced for both the lower and upper band signals and, respectively, as well as the supply modulation voltage. Section IV presents the developed test-bed. In Section V, the proposed method is demonstrated and compared to the conventional 3-D-DPD reported in the literature and a conclusion is presented in Section VI. II. CONCURRENT DUAL-BAND DPD TECHNIQUE FOR THE LINEARIZATION OF DUAL-BAND ET PA Let us define and as the input envelope of the lower sideband (LSB) and upper sideband (USB), respectively, separatedbythefrequency. The dual-band instantaneous envelope of the input signal is written in terms of the input LSB and USB signals as which reaches a peak value of with and. The average amplitude of the instantaneous input envelope is defined as Fig. 2 illustrates the time-domain representation of the instantaneous ( ), peak ( ), and average ( ) dual-band envelopes for an LTE/WCDMA signal combination. Fig. 3 illustrates the frequency-domain representation of the dual-band envelopes for an LTE/WCDMA signal combination. We can observe that and varies slower than.thus,this provides a relaxation of several orders of magnitude on the required supply voltage BW. Further to accommodate the speed limitation on the available envelope amplifier for the supply modulation, we will adopt the RB method proposed in [19] and [20]. This further reduces the BW of the average envelope, yielding the slower envelope showninfig.2. Thus, in this paper, to demonstrate the proposed DPD concept, we select the modulated supply voltage to track. (1) (2) (3)

3 LIN et al.: 3-D FOURIER SERIES BASED DPD TECHNIQUE 3 The memoryless dual-band input output relationship in (4) and (5) can then be revised as Fig. 3. Spectrum of the RF envelope (1), compared to the peak envelope (2), average envelope (3), with the RB envelope. Although, as mentioned earlier, there exists a wide array of methods to implement dual-band DPD with fixed supply voltages, to date there has been comparatively less attention paid to dual-band ET linearization. Recently, Gilabert et al. [18] and Gilabert and Montoro [21] extended the 2-D memory polynomial approach to 3-D by adding an additional degree of freedom for the supply voltage. We can write the 3-D memory polynomial as (7) Since ET is primarily designed for efficiency optimization, the amplifier operates predominantly in its compression region. In this case, ET unavoidably introduces distortions in the PA while varying its supply voltage. Thus, it becomes necessary to combine the ET PA with DPD to linearize the PA while accounting for the dual-band operation, as well as the supply modulation variation. A. DPD for Dual-Band Fixed-Supply PAs First, consider the input output relationship for a dual-band fixed-supply PA as (4) (5) where and are the baseband signal PA inputs, and are the baseband signal PA outputs, and and are the gain for the LSB and the USB, respectively. For accurate PA modeling, and in (4) and (5) must account for the nonlinearities induced by both the in-band intermodulation and also the cross-band modulation. As mentioned previously, there has been significant attention in the literature pertaining to the linearization of dual-band PAs described by the input output relationship of (4) and (5). The conventional 2-D-DPD memory polynomial method, for example, could be expressed as where is the memory delay, is the memory depth, and is the nonlinear order of the polynomial. B. DPD for Dual-Band ET PAs When a nonconstant positive supply voltage is applied through a supply modulator for ET, the dual-band input output relationships of (4) and (5) becomes insufficient. That is, we must take into account both the cross-modulation effect from the adjacent band together with the slowly varying supply voltage. (6) where is the memory delay and is the memory depth. This equation can then be cast into matrix form and solved in a least squares sense, similar to the 2-D memory polynomial case. Another approach was shown by Sarbishaei et al. [22], who augmented the 2-D-DPD fixed supply Volterra model with a polynomial representation of the supply envelope, achieving similar linearization results as the 3-D memory polynomial. The potential problem, however, is the numerical stability issue of the 3-D memory polynomial model. If we cast (8) into the matrix form the set of coefficient solution (8) (9) are computed through the pseudo-inverse (10) The matrix is generally ill conditioned [23] and results in involvement of significant numerical error in matrix inversion. To alleviate the aforementioned numerical instability problem, in this paper we present a new, stable, and generalized framework for the linearization of dual-band ET PAs via DPD. Our choice for this approach is to use a 3-D Fourier series basis to represent the gains and in (7). The 3-D Fourier series basis functions, which are fully orthogonal over a 3-D space [24], is considered in order to better accommodate for the nonlinear cross-modulation originating from the three inputs:,,and. This choice of an orthogonal basis will result in a highly numerically stable matrix representation of the gain in (7), which will, in turn, yield a more stable indirect learning representation of the inverse gain function. Furthermore, the methodology presented can be easily extended to tri-band and beyond ET PAs with minimal effort. III. 3-D FOURIER SERIES BASIS DPD METHOD FOR DUAL-BAND ET PAS As mentioned in Section II, the lower and upper band input output relationship for a dual-band ET PA can be described through (7). This model accounts for in- and cross-band

4 4 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES distortions introduced by the input envelopes and,as well as the supply modulation voltage.now,tocharacterize the dual-band ET model of (7), we expand the gain functions as a weighted sum of the new set of basis functions, which are orthogonal over the space accounting for the three inputs,,and. The basis functions have the representation where we define the 1-D basis function for real as (11) (12) (13) with the three harmonic indices each taking values over the range and the three integers each taking the value 1 or 2. Replacing the notation by a single index,weobtain2 basis functions for the dual-band ET case as follows: (14) application in 3-D-DPD, their actual ability to improve the system stability will be studied in full detail in the remainder of this paper. Now we can rewrite the lower and upper band gain functions in (7) as (23) (24) where is a measure of the nonlinear expansion order. If we further include the time-selective memory effect into the equations, we will have at the th sample (25) (26) (15) (16) where the gain functions with memory delay sample are at a given (17) (18) (19) (27) (20) (21) Note that the basis functions are orthogonal over,and therefore provides a highly independent set to represent the I/O relationships of the dual-band ET PA. One can indeed easily verify the orthogonality of the basis functions over a unit cube in using the normalized envelopes and and the normalized drain voltage ( )as for for and/or and (22) Although the orthogonality property of the Fourier basis used for the gain functions provides a motivation to investigate their (28) with being the memory depth, being the memory delay for the lower and upper bands, and being the memory delay for the RB supply modulation with a skipping integer. The skipping factor is used to account for the slower memory effects associated with the supply modulation of the transistor. Note that different memory depths could be used for the different bands. Finally, assuming total samples and using (27) and (28), the transformations in (25) and (26) can be cast in matrix form as (29) where is a 1 vector, is,and is 1, where.thesetof coefficients for each band and are computed through the pseudo-inverse solution (30)

5 LIN et al.: 3-D FOURIER SERIES BASED DPD TECHNIQUE 5 Fig. 5. Singular values of matrix for Rayleigh ( ) generated data. Fig. 4. condition number versus nonlinear order. It is important to verify that the choice of the Fourier basis in (11) will result in a well-conditioned matrix, particularly in comparison to the conventional polynomial approach, which is prone to numerical stability issues for dual-band ET PAs. The choice of an orthogonal basis over a unit cube in to represent the gain will result in a highly linearly independent set of columns for, optimizing the number of degrees of freedom necessary to characterize the PA input output relationship. To illustrate it, Fig. 4 shows the condition number of the matrix for the 3-D memory polynomial basis, as well as the 3-D Fourier basis for a selection of randomly generated input data as a function of the nonlinear order used. It is clear from Fig. 4 that the 3-D Fourier series approach is highly stable in comparison to the 3-D memory polynomial approach, particularly as the nonlinear order increases. To investigate why the 3-D Fourier series provides a more stable gain matrix, Fig. 5 shows the computed singular values of the matrix for the 3-D Fourier basis, as well as the 3-D memory polynomial, for Rayleigh ( ) generated data. Here, it is evident that the proposed basis results in a that is ill conditioned, but has a well-defined numerical rank, and hence, the pseudo-inverse can be accurately and efficiently computed using just the contributions from the significant singular values - using the truncated SVD representation of [25] [27]. This, in essence, tells us the number of modes necessary to characterize the PA. On the contrary, the slow rolloff of the singular value spectrum of the classic 3-D memory polynomial approach indicates a poorly defined numerical rank, and hence, the linearization becomes prone to numerical errors when performing a pseudo-inverse, as more polynomial basis functions must be used. It was verified that other orthogonal polynomials (Chebyshev and shifted Legendre) also did not exhibit the abrupt drop in singular values like the Fourier series, indicating that they too did not have a well-defined rank. Here, we can see that the 3-D Fourier basis approach provides a stable solution as the number of coefficients increases, while the 3-D memory polynomial does not. Fig. 6. Testbed setup to capture dual-band concurrent signal at harmonic distance. With the modeling of the dual-band ET PA completed, we proceed to use the proposed modeling method to find the appropriate inverse function via the indirect learning method. As a final remark, we note that that the proposed 3-D Fourier basis method could easily be extended to multiband ET applications by modifying the 3-D Fourier basis functions proposed in (11) to an space, where - are the input signal envelopes for each band, and is the normalized supply modulation voltage. That is, we could use basis functions of the form (31) where and, and where the basis functions functions are defined as in (12). IV. IMPLEMENTATION AND TESTBED SETUP The ET setup is based on the testbed introduced in [17] and depicted in Fig. 6. It consists of two Analog Devices mixed-signal digital-predistortion (MSDPD) demo boards, both connected and clock synchronized to the Altera FPGA Stratix IV. The MSDPD enables the up/down conversion, filtering

6 6 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES TABLE I SUMMARY OF THE TWO SCENARIOS Fig. 7. FPGA and MSDPD boards as part of the testbed setup. provide a sufficient voltage swing as high as 25 V, the 3-dB cutoff frequency for this amplifier is only of 2 MHz. Hence, due to this hardware availability restriction, we must further reduce the BW of the supply voltage signal that we use in the test-bed setup (5 10 MHz) in order to accommodate the cutoff of the TEGAM 2348 amplifier. Therefore, the additional BW reduction/slew rate limitation adopted from [19] and [20] is performed on the average dual-band envelope, generating an RB envelope (as shown in Fig. 2). The BW of the corresponding shaped supply voltage is reduced from approximately 5 10 to 0.6 MHz. Fig. 8. PA designed for concurrent dual-band ET. and DAC/ADC. The ADC sampling rate is MHz. Both processed baseband signals are sent to their respective MSDPD to be up-converted to 1.89 and 2.20 GHz for the LSB and USB, respectively. Both generated RF signals are then merged together to drive the amplification stage. Both received data are time aligned on a PC through MATLAB, as well as the DPD functions. The drain voltage signal is synthesized using MATLAB and downloaded to the arbitrary waveform generator (AWG) based on the average envelope of the dual-band input signal. The AWG is triggered by the FPGA (Fig. 7) in order to synchronize the drain voltage signal to the RF signal. The drain voltage signal is amplified by the TEGAM 2348 amplifier that is used as the supply modulator and drive the PA. As part of the testbed implementation, a dual-band PA (Fig. 8) was designed and fabricated with a CREE GaN transistor CGH27015F. The input and output matching provides the proper multi-harmonic impedance transformation for simultaneous class-b operation at the two operating frequencies. At MHz, the PA exhibits a peak drain efficiency of 68% at the peak output power of 38.7 dbm. At 2200 MHz, the PA provides a peak drain efficiency of 60% at the peak output power of 38.5 dbm. The large-signal characterization of this PA reported in [6] was used to establish the ET shaping functions. Before continuing, it is important to note that even though the TEGAM 2348 amplifier used as the supply modulator can V. EXPERIMENTAL RESULTS In this section, we demonstrate the application of the proposed 3-D Fourier series DPD method in linearizing a dual-band ET PA. Here, two different input signal scenarios are investigated. In the first scenario, a single-carrier WCDMA signal with 5-dB PAPR and an LTE 10-MHz signal with 9.3-dB PAPR, areusedforthelsbandusb,respectively. For the second scenario, a single-carrier WCDMA signal and a three-carrier WCDMA signal spaced each other by 5 MHz with 5- and 7.5-dB PAPR, are used respectively. Table I summarizes the two different signal scenarios that have been considered in this paper for LSBs and USBs. The 3-D Fourier series DPD approach is compared to the conventional 3-D memory polynomial method proposed in [18]. The 3-D Fourier series utilized a DPD nonlinearity order and memory depth, while the memory polynomial utilized a nonlinearity order and a memory depth. Both approaches utilized a memory skipping factor, which was optimized through empirical testing. First, we examine the model extraction performance for each case, where we seek to extract the correct coefficients in (9) and (29) for the two test signal scenarios. A. Dual-Band ET-PA Model Extraction Table II shows the summary of the model extraction performance in terms of the normalized mean square error (NMSE) and adjacent channel power ratio (ACPR) for the 3-D memory polynomial and the proposed 3-D Fourier basis for the two signal scenarios shown in Table I. Here, it is evident that the memory polynomial and Fourier basis achieve similar model extraction performance, with slightly better NMSE exhibited by the Fourier basis, particularly in the LSB. The AM AM plots for the LSB and USB for each signal case are also shown in Figs. 9 and 10, respectively, where it is again seen that the two DPD methods are highly effective in linearizing the sampled data. With the extracted coefficients obtained, the next step is to implement them within a dynamic environment.

7 LIN et al.: 3-D FOURIER SERIES BASED DPD TECHNIQUE 7 TABLE II SUMMARY OF THE MODEL EXTRACTION PERFORMANCE OF THE SIGNALS IN TABLE IN COMPARISON WITH THE CONVENTIONAL MEMORY POLYNOMIALS I Fig. 11. Singular values of matrix for the 1cWCDMA-3cWCDMA signal using memory polynomial and Fourier series basis. Fig. 9. LSB AM AM performance for (1cWCDMA and 3cWCDMA) model extraction. Fig. 12. LSB for 1cWCDMA-3cWCDMA linearization using full SVD spectrum (all coefficients). Fig. 10. USB AM AM performance for (1cWCDMA and 3cWCDMA) model extraction. That is, we must use the extracted coefficients to linearize (for verification) a new longer input dataset of the same signal statistics, as in Table I. Before proceeding, we will illustrate how the well-defined numerical rank of the Fourier series can be exploited to reduce the number of coefficients from an original value to a small number via singular value decomposition (SVD). B. Coefficient Reduction Using SVD and Principal Component Analysis As alluded to in Figs. 4 and 5, the 3-D Fourier series gain matrix has a well-defined numerical rank. Hence, we should be able to reduce the number of modeling coefficients used for linearization via principal component analysis (PCA). Fig. 11 depicts the SVD of the matrix in (9) and (29) for signal scenario II (1cWCDMA-3cWCDMA). Here, we define the total number of coefficients for the memory polynomial, as

8 8 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 13. USB for 1cWCDMA-3cWCDMA linearization using full SVD spectrum (all coefficients). Fig. 15. USB for 1cWCDMA-3cWCDMA linearization using full SVD spectrum (all coefficients). Fig. 14. LSB for 1cWCDMA-3cWCDMA linearization using full SVD spectrum (all coefficients). Fig. 16. LSB spectrum for 1cWCDMA-3cWCDMA linearization using full SVD spectrum (all coefficients). well as the Fourier series cases as. For memory polynomial,, and for the Fourier series,. However, as expected, the numerical rank of the Fourier series approach is actually, substantially smaller than. On the contrary, the memory polynomial numerical rank is that of the number of coefficients since the spectrum does not exhibit any sharp dropoff to near-zero singular values. In general, it is observed for a memory polynomial of various orders that the singular values have a slow rolloff indicative of an ill-determined numerical rank. SVD can be effectively used as a method to reduce the number of modeling coefficients from to,where is the number of chosen significant singular values (e.g., its numerical rank). This can be done without significant loss in accuracy for well-determined rank matrices. To accomplish this, (29) can be written in terms of its SVD as Fig. 17. USB spectrum for 1cWCDMA-3cWCDMA linearization using full SVD spectrum (all coefficients).

9 LIN et al.: 3-D FOURIER SERIES BASED DPD TECHNIQUE 9 TABLE III SUMMARY OF THE FULL SVD SPECTRUM NEW FOURIER SERIES LINEARIZATION PERFORMANCE IN COMPARISON WITH CONVENTIONAL MEMORY POLYNOMIALS, FOR THE SIGNALS IN TABLE I (32) where and are the left and right singular vectors of,and is the matrix of singular values, and superscript is the conjugate transpose. Note that when the full SVD is computed, is, is,and is. If we consider just the largest singular modes, the truncated SVD version of (32) can be written as (33) where,and,,and are the truncated SVD matrices, which when multiplied out give the rank- representation for in (29). We can subsequently solve (33) for the principal component coefficients via pseudo-inverse, reducing the number of coefficients needed for linearization. This methodology is known as PCA [28]. It leads to a more compact and lower cost DPD implementation for both the coefficient extraction and the real-time linearization. C. 3-D-DPD Linearization Using Extracted Coefficients Finally, we compare the linearization results for the memory polynomial and Fourier series methods when using the extracted,aswellastheir truncated counterparts (as described in Section V-B). Here, for brevity, we will only analyze the linearization of signal scenario II (1cWCDMA and 3cWCDMA). 1) Linearization Using All Extracted Coefficients: First, we examine the linearization of scenario II when utilizing all coefficients for both the memory polynomial and Fourier series cases. The NMSE and ACPR as a function of iteration are shown in Figs Here, the first three iterations are used to take in and extract the coefficients for a given set of signal II data, as was done previously. Afterwards, a new data set of signal II is input to the PA, and the extracted coefficients are used to linearize this new input. From Figs , it is clear that the Fourier series DPD method provides superior linearization in comparison to memory polynomial when new data is loaded in a result of the larger degree of numerical stability of this approach. Figs. 16 and 17 show the LSB and USB spectrum when linearizing with all coefficients, where it is seen that the Fourier series approach again yields superior ACPRs in comparison to the memory polynomial. Table III summarizes the findings of linearization via the memory polynomial and Fourier series. Fig. 18. SVD spectrum for using 1cWCDMA-3cWCDMA with significant modes indicated. Fig. 19. LSB NMSE for 1cWCDMA-3cWCDMA linearization using significant SVDs ( coefficients). From Table III, we can see that the proposed method provides a performance improvement ranging from 14.5 to 17.5 db in terms of NMSE and from 1 to 4.7 db in terms of ACPR in comparison to the conventional dual-band memory polynomial method including ET. 2) Linearization Using Reduced Number of Coefficients via PCa: Next, we use PCA (as described in Section V-C1) to reduce the number of coefficients needed to linearize the ET-PA under the signal scenario II. Here, we will reduce the number of

10 10 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 20. USB NMSE for 1cWCDMA-3cWCDMA linearization using significant SVDs ( coefficients). Fig. 22. USB ACPR for 1cWCDMA-3cWCDMA linearization using significant SVDs ( coefficients). Fig. 23. LSB spectrum for 1cWCDMA-3cWCDMA linearization using significant SVDs ( coefficients). Fig. 21. LSB ACPR for 1cWCDMA-3cWCDMA linearization using significant SVDs ( coefficients). coefficients from to,where is the number of significant singular values (see Fig. 18) from the Fourier series case. For the case of the Fourier series basis, this corresponds to the numerical rank of. Figs show the NMSE andacprwhenusing coefficients for both the memory polynomial and Fourier series. Again it is observed that even when truncating a large number of coefficients, the Fourier series provides a higher degree of linearization in comparison to the memory polynomial when a new signal comes in. The LSB and USB spectrums are shown in Figs. 23 and 24. Here, using a reduced number of coefficients, the Fourier series provides significant ACPR improvements in comparison to memory polynomials. As expected, when truncating to, the linearization Fig. 24. USB spectrum for 1cWCDMA-3cWCDMA linearization using significant SVDs ( coefficients). performance for the Fourier series case is comparable to when using the full spectrum.

11 LIN et al.: 3-D FOURIER SERIES BASED DPD TECHNIQUE 11 TABLE IV SUMMARY OF THE TRUNCATED SVD NEW FOURIER SERIES LINEARIZATION PERFORMANCE IN COMPARISON WITH CONVENTIONAL MEMORY POLYNOMIALS, FOR THE SIGNALS IN TABLE I. A table summarizing linearization performance using coefficients (obtained via PCA) is shown in Table IV. From this table, the Fourier series basis in conjunction with PCA is seen to be an attractive option for reducing the number of coefficients for well-determined numerical rank, and provides higher linearization than the comparable 3-D memory polynomial, with less coefficients. From Table IV, we can see that the proposed method provides a performance improvement ranging from 12.4 to 17 db in terms of NMSE and from 1 to 4.6 db in terms of ACPR in comparison to the conventional dual-band memory polynomial method including ET. D. ET-PA Efficiency The measurement of the drain efficiency is conducted for both cases of the dual-band PA. In scenario I (1c-WCDMA and 10 MHz-LTE), with an average output power of 23.5 dbm, an average drain efficiency of the PA of 28.6% is reached for ET after DPD, while 16.0% drain efficiency is measured for constant supply voltage of 25 V after DPD. In scenario II (1c-WCDMA and 3c-WCDMA) with an average output power of 23.5 dbm, an average drain efficiency of 26.3% is reached for ET after DPD, while 16.1% drain efficiency is measured for constant supply voltage of 25 V after DPD. Indeed, it is obvious that the PA efficiency has been improved by utilizing dual-band ET in comparison to a constant supply voltage, with a PA efficiency improvement of about 12 and 10 percentage points for cases 1 and 2, respectively. However the efficiences themselves remain relatively small due to the RB envelope (aforementioned supply modulator limitations), notably limiting the swing of the supply voltage. Larger efficiencies are obtained when the BWs of the RF signals and ET modulation are comparable [6]. Nonetheless, this RB ET testbed enables us to demonstrate the DPD advantage of the new Fourier series basis over the conventional memory polynomial approach. VI. CONCLUSION We have introduced a new DPD methodology for dual-band ET based on a Fourier basis over for PA characterization. The orthogonality of the basis functions over provides a gain matrix with a well-defined numerical rank, and thus, fully characterizes the nonlinearities introduced by the input envelopes and, as well as the supply modulation voltage with a numerically stable solution. Several test scenarios for a dual-band GaN ET PA operating at 1.89 and 2.20 GHz were implemented for validation and comparison with the literature. The general principles of this new method could be easily be extended to multi-band ET PA. Also in future research, additional multidimensional delays such as in [22] and [29] could be implemented with the new basis to further improve the performance. ACKNOWLEDGMENT The authors would like to thank Analog Devices Inc., Wilmington, MA, USA, for donating the Mixed signal digital pre-distortion system (MSDPD) boards, which are used in this study. The authors would also like to thank the Wireless Systems Solutions Group, Altera Corporation, for the donation of the Stratix IV FPGA. REFERENCES [1] D. Kimball et al., High-efficiency envelope-tracking W-CDMA basestation amplifier using GaN HFETs, IEEE Trans. Microw. Theory Techn., vol. 54, no. 11, pp , Nov [2] J. Hoversten, S. Schafer, M. Roberg, M. Norris, D. Maksimovic, and Z. Popović, Codesign of PA, supply, and signal processing for linear supply-modulated RF transmitters, IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp , Jun [3] C.-T.Chen,C.-J.Li,T.-S.Horng,J.-K.Jau,andJ.-Y.Li, Highefficiency dual-mode RF transmitter using envelope-tracking dual-band class-e power amplifier for W-cdma/WiMax systems, in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp [4] A. Cidronali, F. 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Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. Philadelphia, PA, USA: SIAM, 1998,vol.4. [27] V. Klema and A. J. Laub, The singular value decomposition: Its computation and some applications, IEEE Trans. Automat. Control, vol. AC-25, no. 2, pp , Apr [28] P. Gilabert et al., Order reduction of wideband digital predistorters using principal component Analysis, in IEEEMTT-SInt.Microw. Symp. Dig., Jun. 2013, pp [29] A. Kwan et al., Dual-band predistortion linearization of an envelope modulated power amplifier operated in concurrent multi-standard mode, in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2014, pp Christophe Quindroit was born in Corbeil-Essonnes, France, in October He received the M.Tech and M.S. degrees in electronics from the École Polytechnique de l Université de Nantes, Nantes, France, in 2005, and the Ph.D. eegree in electronics from Xlim, University of Limoges, Liomges, France, in He was a Project Engineer with Alcatel-Lucent. He was a Research Engineer with The Ohio State University, Columbus, OH, USA. He is currently a Radio Engineer with LPA Concepts, Martillac, France. His research interests include analog system level modeling, power amplifier (PA) linearization techniques and field-programmable gate-array (FPGA) implementation. Haedong Jang (S 09 M 14) was born in Wonju, Korea, in He received the B.S. degree in electrical engineering from Kangwon National University, Chuncheon, Korea, in 1994, the M.S. and Ph.D. (ABD) degrees in electrical and computer engineering from Inha University, Incheon, Korea, in 2005 and 2008, respectively, and the Ph.D. degree in electrical and computer engineering from The Ohio State University, Columbus, OH, USA, in In 1996, he joined a non-profit governmental organization, Small Business Corporation, Shiheung, Korea, where, until 2007, he was a Product Development Assistant Consultant involved with more than 100 commercialized consumer product developments. In 2014, he joined the Infineon Technologies North America Corporation, Morgan Hill, CA, USA, as an Advanced Development RF Engineer. His main interests include advanced RF power amplifier architecture, nonlinear devices characterization, modeling, and power amplifier linearization. Dr. Jang was a corecipient of the First Place Award of the 2012 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) Student Development of a Large-Signal-Network-Analyzer-Round-Robin Design Competition. Patrick Roblin (M 85) was born in Paris, France, in September He received the Maitrise de Physics degree from the Louis Pasteur University, Strasbourg, France, in 1980, and the M.S. and D.Sc. degrees in electrical engineering from Washington University, St. Louis, MO, USA, in 1982 and 1984, respectively. In 1984, he joined the Department of Electrical Engineering, The Ohio State University (OSU), Columbus, OH, USA, as an Assistant Professor. He is currently a Professor with OSU. He is the founder of the Non-Linear RF Research Laboratory, OSU. At OSU, he developed two educational RF/microwave laboratories and associated curriculum for training both undergraduate and graduate students. He lead-authored the textbook High-Speed Heterostructure Devices (Cambridge Univ. Press, 2002). His current research interests include the measurement, modeling, design and linearization of nonlinear RF devices and circuits such as oscillators, mixers, and power amplifiers. Yiqiao Lin received the Ph.D. degree in electrical engineering from The Ohio State University, Columbus,OH,USA,in2014. She is currently an RF Application Engineer with NXP Semiconductors, Smithfield, RI, USA. Her research interests include RF power amplifier designs for wireless communication systems, digital predistortion, envelope tracking, and Doherety architectures.