Automatic Linearization and Feedforward Cancellation of Modulated. Harmonics for Broadband Power Amplifiers THESIS

Transcription

1 Automatic Linearization and Feedforward Cancellation of Modulated Harmonics for Broadband Power Amplifiers THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Varun Ratnasamy Graduate Program in Electrical and Computer Engineering The Ohio State University 2015 Master's Examination Committee: Prof. Patrick Roblin, Advisor Prof. Steven Bibyk

2 Copyright by Varun Ratnasamy 2015

3 Abstract The wireless devices industry has experienced exponential growth in the past decade, especially in the smartphone market. The emerging market opportunities in the areas of automotive applications, connected homes, and wearable technologies have contributed substantially to the ever-increasing number of wireless users. This dramatic growth in the number of spectrum users, along with insatiable demand for higher data rates for streaming services (such as Netflix, YouTube, Spotify, and gaming), has fueled the development of ultra-wideband and broadband communication schemes. Schemes such as Long Term Evolution (LTE), Orthogonal Frequency Division Multiplexing (OFDM), and Wideband Code Division Multiple Access (WCDMA) have been developed to utilize the frequency spectrum efficiently and meet the user capacity requirements. Since these high data rate schemes exhibit non-constant envelops with high PAPR, strict linearity requirements are imposed on PA to avoid spectral regrowth and intermodulation distortion. To ensure seamless compatibility, there is a strong trend towards implementing multi-band and ultra-wideband systems that accommodate different frequency bands (or technology such as 2G, 3G, and 4G) simultaneously while ensuring backward compatibility. However, this has raised further concerns about interference from harmonics (of lower bands) when the separation between two bands is sufficiently ii

4 large. In practice, digital predistortion and switchable RF band-pass filters banks are used to tackle the problems of linearity and harmonic interference, respectively. However, due to their bulky and lossy nature the filter banks appear to be an unattractive and inflexible solution. Digitally controlled feed-forward harmonic cancellation is emerging as an alternative solution because of the frequency programming flexibility this filter-less solution brings to multi-band systems. This thesis focuses on joint distortion modelling of Broadband Power Amplifiers (PA) using a 2-D Least Square Cubic Spline, and presents a compensation algorithm for simultaneous automatic linearization and cancellation of modulated harmonics. The algorithm relies on a joint system identification of the nonlinearity, memory effects, and group delay of both the main and harmonic cancellation channels. The synchronized PA linearization depends solely on digital predistortion (using a frequency selective technique), whereas the filter-less cancellation of the modulated harmonics uses both predistortion and feed-forward cancellation. iii

5 Dedicated to Amma, Appa, Poorani and Anoop. iv

6 Acknowledgments I am hugely indebted to my advisor, Prof. Patrick Roblin, for providing me the opportunity to work in the Non-Linear RF Lab. His constant support, precious advice and generous guidance motivated me immensely in my research at The Ohio State University. I am also grateful to Prof Steven Bibyk for serving on my thesis committee and for his questions and suggestions. I would also like to thank Dr. Cengang Xie, Prof Meenakshi Rawat and Prof. Seok Joo Doo for their valuable feedback and suggestions. I would like to thank my lab mates: Alejandro Galaviz Aguilar, Chenyu Liang, Francisco Javier Martinez, Hai Yu and Hsiu-Chen Chang for all their help and feedback during the course period. I would also like to thank Dr. Christophe Quindroit and Dr. Naveen Naraharisetti for programming the FPGA and their notes on the testbed. I would like to thank Rockwell Collins, NSF for their financial and technical support. I would also like to thank Altera Inc., Texas Instruments Inc. and Mini-Circuits for their donations, respectively of the FPGA testbed, transceivers and RF Power Amplifiers used during the research. Finally, I would like to thank my sister and brother in-law for their unconditional support encouragement and motivation during the Master s program. I am immensely grateful to my parents for their love, affection and belief in me. v

7 Vita May, B.Tech. Electrical and Communication Engineering, Karunya University August, present...m.s. Student, Electrical and Computer Engineering, The Ohio State University August, present...graduate Research Associate, Electrical and Computer Engineering, The Ohio State University Fields of Study Major Field: Electrical and Computer Engineering Studies in Non-Linear RF Lab: Prof. Patrick Roblin vi

8 Table of Contents Abstract... ii Dedication... iv Acknowledgments... v Vita... vi List of Tables... ix List of Figures... x Chapter 1: Introduction Need for Linearity and Harmonic Cancellation Digital predistortion Feed-forward Cancellation of Modulated Harmonics Behavior Modelling Figure of Merits Outline Chapter 2: Modelling of Fundamental and Harmonics using Cubic Spline Basis Modelling Theory for Wideband Signals in Multiband Sytems vii

9 2.2 Robust Modelling Modelling Harmonics using Memory Polynomial Cubic Splines Modelling Harmonics using Cubic Splines Chapter 3: Measurement Setup for Capturing Fundamental and Harmonics Modules Measurement Setup Post Processing Chapter 4: Automatic Feedforward Cancellation of Modulated Harmonics Introduction Feed-Forward Harmonic Cancellation System Identification Digital Predistortion Testbed Setup Measurement Results Chapter 5: Conclusion and Future Work Conclusion Future Work References viii

10 List of Tables Table 1. Summary of Measurement Results for Third Harmonic Table 2. Summary of Measurement Results for Second Harmonic ix

11 List of Figures Figure 1.1 IMD due to 2 tones input... 2 Figure 1.2 IMD due to dual-band input: (a) Small Δω, (b) Large Δω... 3 Figure 1.3 Spectral Regrowth due to nonlinearity... 4 Figure 1.4 Tradeoff between linearity and PAE... 5 Figure 1.5 Cascaded Gain for linearizing PA... 7 Figure 1.6 General Architecture of the DPD system... 8 Figure 1.7 General Architecture of Feed-forward Technique... 9 Figure 1.8 Characterization of Interfering Harmonics Figure 1.9 Negating Harmonics at Output Figure 1.10 Memory Model for PA Figure 2.1 Concurrent Multiband DPD Figure 2.2 Multiple DPD Modules Figure 2.3 Frequency Selective DPD Architecture Figure 2.4 Dual-Band DPD Architecture Figure 2.5 Weiner Model Figure 2.6 Hammerstein Model Figure 2.7 Memory Polynmial Model for Fundamental Figure 2.8 Memory Polynmial Model for pth Harmonic x

12 Figure 2.9 Gain as a function of the envelop of inputs Figure D Spline Basis Function Figure 3.1 Architecture of Charecterization Testbed Figure 3.2 Characterization Testbed Setup Figure 3.3 PSD of Fundamentals and Harmonics Figure 4.1 Digital Feed-forward Harmonic Cancellation Scheme Figure 4.2 System Idetntification Scheme Figure 4.3 System Idetntification for 2 nd Harmonic Case Figure 4.4 Predistortion and Harmonic Cancellation Scheme Figure 4.5 Results for Third Harmonic Figure 4.6 AM/AM Characteristics for Third Harmonic Figure 4.7 AM/PM Characteristics for Third Harmonic Figure 4.8 Results for Second Harmonic Figure 4.9 AM/AM Characteristics for Second Harmonic Figure 4.10 AM/PM Characteristics for Second Harmonic Figure 5.1 2D DPD for Harmonic Cancellation xi

13 CHAPTER 1 INTRODUCTION The past decade has witnessed dramatic growth in the wireless devices industry, especially in the smartphone market. The emerging market opportunities in the areas of automotive applications, connected homes, and wearable technologies have contributed substantially to the ever-increasing number of wireless users. Connecting everything, anytime, and anywhere has become the order of the day. Adoption of streaming services which require high data-rates, such as Netflix, YouTube, Spotify, and gaming, is driving a boom in data consumption. In particular, mobile data consumption is predicted to expand at a compounded growth rate of 57% every year. Exponential growth in the number of spectrum users, along with insatiable demand for higher data rates for streaming services, drives a need for more spectrums and efficient utilization of existing spectrums. 1

14 These technical challenges have fueled the development of ultra-wideband and broadband communication schemes such as Long Term Evolution (LTE), Orthogonal Frequency Division Multiplexing (OFDM), and Wideband Code Division Multiple Access (WCDMA) that efficiently utilize the frequency spectrum and meet the user capacity requirements [2]-[17]. There is also a strong interest in the implementation of broadband and ultra-wideband systems that accommodate different frequency technologies (such as 2G, 3G, and 4G) simultaneously, to ensure seamless compatibility of products in the competitive wireless market. Thus, the wireless industry is moving towards multiband/wideband applications [2]-[19]. 1.1 The Need for Linearity and Harmonic Cancellation Intermodulation Distortion (IMDs) Intermodulation distortions are the additional frequency components at the output of a non-linear power amplifier (PA) when two or more tones are transmitted through it. The IMDs are generated at a distance of frequency between the tones. Fig. 1.1 and Fig. 1.2 show the IMDs for a two-tone input and dual-band input, respectively. Figure 1.1. IMD due to two-tone input. 2

15 For the dual-band transmitters shown in Fig. 1.2, the IMDs can be classified into three categories: In-Band Intermodulation, Out-of-Band Intermodulation, and Cross Modulation. (a) (b) Figure 1.2. IMD due to dual-band input: (a) Small Δω, (b) Large Δω. 3

16 Spectral Regrowth Broadband signals consist of an infinite number of tones in the bandwidth. Thus, when broadband signals are transmitted through a nonlinear PA, the IMDs at the output fall both in the bandwidth and in the adjacent channel. This phenomenon of spectral spreading is referred to as spectral regrowth. Fig. 1.3 depicts the spectral regrowth for a 10MHz LTE signal. Spectral Regrowth Spectral Regrowth Figure 1.3. Spectral regrowth due to nonlinearity. Linearity Power amplifiers form an indispensable part of the wireless communication system. PAs are usually characterized by their linearity (P1dB), gain, and efficiency. It is well-known that a trade-off exists between efficiency and linearity. Various architectures like Class F, Doherthy PA, etc. have been used to implement highly efficient PA s. However, these architectures exhibit comparatively high nonlinearity. Nonlinearity causes spectral regrowth both within the bandwidth, which decreases the BER, as well as outside the 4

17 bandwidth, which causes distortions and interference in adjacent channels, violating the out-of-band emission requirements mandated by regulatory bodies. Figure 1.4 below illustrates a simple explanation of these processes. Figure 1.4. Tradeoffs between linearity and PAE. Although carrier aggregation schemes like LTE and WCDMA address the problem of spectral efficiency, these high data rate schemes create very challenging demands on the RF front-end specifications [2]-[7], [10], [16]-[18]. Moreover, these signals also exhibit non-constant envelops with very high PAPR (approx. 10dB). Thus, strict linearity requirements are imposed on PAs to avoid spectral regrowth and intermodulation distortion products. Harmonic Cancellation Implementing broadband and ultra-wideband systems that accommodate different frequency bands simultaneously allows seamless compatibility between existing 5

18 standards. However, this has raised further concerns about interference from the harmonics of lower bands when the separation between two bands is sufficiently large [2], [3], [16], as shown in Fig. 1.2 (b). Thus, it is essential to eliminate the harmonics generated by the lower band signals in order to avoid interferences in the higher bands. Possible Solutions Linearity: A highly linear power amplifier can be used. However, this would result in a substantial decrease in the efficiency of the PA, which is associated with rapid battery drain and heating problems, due to higher dissipation of power. Different modulation schemes with increased frequency spacing can be used. However, this results in less spectral efficiency. PAs can operate at back-off instead of at saturation. However, this results in a substantial decrease in the efficiency of the PA, as seen in Fig Feed-forward Linearization and Feed-back Linearization are viable solutions, but are lossy in nature. Digitally predistortion (DPD) is the most widely used linearization technique because of its ease of implementation, robustness, and relative simplicity. Harmonics Cancellation: Switchable RF band-pass filter banks are used to filter the harmonic interference when separations between the bands are small. However, for large separations filtering is not an option. 6

19 Digitally controlled feed-forward harmonic cancellation is emerging as an alternative solution, because of the frequency programming flexibility this filterless solution brings to multi-band systems. 1.2 Digital Predistortion In DPD the base-band signal is digitally processed to create an expanded nonlinearity that complements the non-linear characteristics of the PA, as shown in Fig The cascaded gain experienced by the signal due to the predistorter and the PA should ideally be a scalar value. Thus, high linearity and efficiency can be achieved as the PA can be operated up to its saturation region. It can be observed in Fig. 1.5 that in order to compensate for the IMDs due to the nonlinearity of PA, the characteristic curve of DPD must be an inverse of that of the PA [1], [7], [10]. f DPD (x(t), t) = f 1 PA (x(t), t) Figure 1.5. Cascaded gain for linearizing PA. 7

20 Figure of Merits Instantaneous nonlinearity of memoryless PAs is characterized by the amplitude modulation of the input to the gain, and the phase modulation of the gain to the amplitude modulation of the input, i.e. the AM/AM and the AM/PM, respectively [1], [4], [7], [10]. DPD System A typical block diagram of a DPD system is shown in Fig The return path acts as an observation path for the DPD, where the system compares the input signal to that of the down-converted output signal. The DPD system then extracts coefficients for the inverse model based on this observation, using either direct or indirect learning techniques, and applies the extracted coefficients to the predistortion block [1], [4], [7]. Figure 1.6. General architecture of the DPD system. 8

21 1.3 Feed-Forward Cancellation of Modulated Harmonics The Traditional Feed-Forward Technique The traditional feed-forward technique [1] involves two steps: 1. Signal Cancellation: This involves subtraction of the input signal from the attenuated PA output. 2. Distortion Cancellation: This involves the subtraction of the amplified extracted signal from the PA output to achieve the amplified signal. A conventional schematic for the feed-forward technique is shown in Fig The signal cancellation loop corresponds to step 1, and the distortion cancellation loop corresponds to step 2. Figure 1.7. General architecture of the feed-forward technique. Digitally Controlled Feed-forward technique for Harmonic Cancellation Similar to the traditional feed-forward technique, the digitally controlled feed-forward technique for harmonic cancellation involves two steps [3]: 9

22 1. Extraction of interfering harmonics. This involves identification of the harmonic model in the digital domain. Figure 1.8. Step 1: Characterization of interfering harmonics. 2. Negating harmonics at the PA output with an amplitude and phase controlled (usually 180⁰) modelled harmonic signal. Figure 1.9. Step 2: Negating harmonics at Output. 10

23 1.4 Behavioral Modelling Memory vs Memoryless Behavioral modelling can follow two approaches, as discussed in [1], [4], [7], and [10]: 1. Memoryless modelling: This involves modelling using static characteristics i.e. the AM/AM and AM/PM curves of the PA. Here, the output depends on instantaneous input. 2. Memory modelling: Nonlinear memory effects are due to gain modulations, which are a function of present and past envelops of the input (i.e. G ( x1 2 )). Memory effects can be modelled by applying filtering before and after the memoryless block, as shown in Fig Types of Memory Effects: Memory effects can be classified into two categories. Fast memory effects originate from the intrinsic transistor, as well as the bias and matching networks. They require advanced instantaneous linearization. They are usually observed above 1MHz [1], [10], [15], [16]. Conversely, slow memory effects encompass temperature effects, traps, and power supply responses. They require adaptive linearization techniques. They are usually observed below 1MHz [1], [10], [15], [16]. 11

24 Figure Memory model for PA. Why is memory modelling essential? As the signal bandwidth gets wider, the RF PA exhibits a history-dependent behavior (or memory effect) [8]-[12]. The cause of these memory effects can be associated with the thermal time constants of devices and the frequency-dependent nature of the biasing networks. Memory effects cause dispersion on a memoryless curve, and also result in unequal spectral regrowth. Thus, the DPD performance can be improved when the memory effects are taken into consideration while modelling multiband scheme power amplifiers [8]-[10]. 1.5 Figures of Merit NMSE and ACPR are two different figures of merit used to quantify and evaluate the performance of the algorithm used in this work for modelling fundamental and harmonic outputs, and for implementing the predistorter to successfully linearize the fundamental signal and cancel the harmonic. 12

25 NMSE: A normalized mean square error (NMSE) defines how well data is calculated when compared to the actual measured data as shown in equation (1.1). Typically a small NMSE value indicates a good modelling technique and DPD algorithm. where: ACPR: NMSE = N 1 n=0 y meas(n) y model (n) 2 N 1 y meas (n) 2 n=0 ymeas: is the actual measured data at the output ymodel: is the modeled data using the modelling technique (1.1) The adjacent channel power ratio (ACPR) is defined as the ratio of the signal power in the signal bandwidth to the signal power in the adjacent channel. ACPR is used to confirm whether a signal complies with the mask requirements of the FCC. Each standard has a different ACPR specification, and a typical ACPR value is -50dBc. where: ACPR(dBc) = 10log 10 ( P c P adj ) Pc: is the channel power of the signal. Padj: is the signal power in the adjacent channel. 13 (1.2)

26 1.6 Outline The thesis is organized as follows. Chapter 2 presents insight into robust modelling and cubic spline predistorters for power amplifiers. A detailed explanation on modelling fundamentals and harmonics using a Cubic Spline basis is included. In Chapter 3, a note on the characterization procedure for fundamental and harmonics is presented, along with a brief description of the developed low-cost FPGA based measurement testbed. Chapter 4 presents the algorithm for the simultaneous linearization and cancellation of modulated harmonics of broadband power amplifiers; the automatic algorithm for harmonic cancellation is also investigated experimentally for the first time. A proof of concept is given for the second and third harmonic cancellation. The thesis is concluded in Chapter 5 with some experimental observations and performance evaluations of the algorithm. Chapter 5 also gives a brief account of the scope for future developments. 14

27 CHAPTER 2 MODELLING OF FUNDAMENTAL AND HARMONICS USING CUBIC SPLINE BASIS FUNCTIONS 2.1 Modelling Theory for Wideband Signals in Multiband Systems Frequency Selective Modelling for Multiband Systems Linearizing multiband systems using conventional DPD techniques involves a lot of challenges [4]-[10]. Efficient modelling using the conventional DPD technique (instantaneous adaptive feedback) requires capturing the whole spectrum at the output, implying a need to increase the bandwidth of the DPD (i.e. the DPD module has to sample at a very high rate). However, this is limited by the sampling speed of the DSP controller, since expensive high-sampling ADCs/DACs are required. 15

28 Figure 2.1. Concurrent Multiband DPD. Using multiple DPD modules to overcome these challenges is also not a viable solution, as they do not account for cross modulation products. NL z 1 = a k,j x 1 x 1 2(k j) k=0 NU z 2 = b k,j x 2 x 2 2(k j) k=0 (2.1) (2.2) This can be seen in Figure 2.2, which portrays a simple explanation of the same concept. Frequency-selective DPD architectures, as shown in Fig. 2.3, can be implemented to address the challenges associated with compensating multiband systems [5], [6]. The sampling frequency of ADC and DAC can also be greatly reduced by implementing this 16

29 architecture. Figure 2.2. Multiple DPD modules. Figure 2.3. Frequency Selective DPD Architecture. Consider the 2D-DPD frequency-selective model discussed in [10], [16], [19] to compensate for nonlinearities and IMDs of a dual-band transmitter. Each cell is 17

30 responsible for compensating nonlinearities in a particular frequency, band as shown in Figure 2.4. The predistorted input for each band is given by the polynomial model below: NL k z 1 = a k,j x 1 x 1 2(k j) x 2 2j k=0 j=0 NU k z 2 = b k,j x 2 x 2 2(k j) x 1 2j k=0 j=0 (2.3) (2.4) where x1 and x2 are the sample input signals, ak,j and bk,j are the extracted coefficients for the predistorter, z1 and z2 are the sample predistorted signals at PA input, and x1 and x2 are the complex envelops of the input signals for the lower band and upper band, respectively. Figure 2.4. Dual-band DPD architecture. 18

31 2.2 Robust Modelling For wideband communication schemes such as WCDMA and LTE, PA memory effects cannot be overlooked [8]-[12]. Memoryless DPD schemes for wideband PAs (especially those used in base stations) often result in poor compensation. Thus, a robust model for non-linear PAs should always account for the memory effects discussed in Section 1.4. The Volterra series is a general non-linear model that accounts for memory. However, Volterra modelling involves extraction of a large number of coefficients. As a result, the exact inverse (DPD) of the system is difficult to construct. P th order inverse techniques which approximate the inverse can also be used, but they are very complicated. The Wiener model and memory polynomial models are two special cases for the Volterra model that can be used to capture memory effects in a PA [7], model its nonlinearities, and construct an exact inverse of the PA for DPD. Wiener Model The Wiener model consists of a linear filter (h) followed by a memoryless nonlinearity (NL). The output can be represented mathematically as: y 1 (n) = G( x 1 (n) ). x 1 (n) (2.5) where G( x1(n) ) is the memoryless instantaneous gain function (NL). x1(n) designates the output of the linear filter (h): 19

32 M x 1 (n) = h(j). x in (n j) j=0 (2.6) where M is the memory depth and h(j) is the filter coefficient. Figure 2.5. Wiener Model. In the Wiener model the memoryless gain function is identified in the first step. Following that, the input and output signals are de-embedded to extract the filter coefficients h(j). Even though the Wiener model is one of the simplest ways to combine nonlinearities and memory effects, its ability to model PAs is very limited since the output depends non-linearly on the filter coefficients, which makes estimation of linear terms challenging. Hammerstein Model The Hammerstein model is another simple model which can be used to overcome the limitations of the Wiener model. This model consists of a memoryless nonlinearity (NL) followed by a linear filter (g). The output can be mathematically represented as: 20

33 M y 1 (n) = g(j). x 1 (n j) j=0 (2.7) where g(j) is the filter coefficient, M is the memory depth, and x1(n) is the output of the memoryless nonlinearity (NL), as shown below: y 1 (n) = G( x 1 (n) ). x 1 (n) (2.8) Where G( x1(n) ) is the memoryless instantaneous gain function (NL). Figure 2.6. Hammerstein Model. Memory Polynomials A generalized version of the Hammerstein model formulated by choosing different filter coefficients g0 (k),., g0 (n) for different values of order k, and combining them with the power series coefficients to form a 2-D array {ak,j}, is given by: K M 1 y GH (n) = a k,j. x k (n j) k=1 j=0 (2.9) 21

34 A memory polynomial is a reduction of the Volterra series in which only the diagonal terms are kept. Thus, if we select only the combinations x(n) x(n-m) k-1 the generalized Hammerstein becomes the memory polynomial [7]-[12]: K M y MP (n) = a k,j. x(n j). x(n j) k k=0 j=0 (2.10) 2.3 Modelling Harmonics using Memory Polynomials The PA modelling for the inband fundamental signal is given by the memory polynomial shown below [11]: K M y ω (n) = a k,m. x(n m). x(n m) k k=0 m=0 (2.11) where ak,m represent the coefficients of the model, M represents the memory depth and represents the depth of nonlinearity. Since the p th harmonic is theoretically proportional to the p th order nonlinearity of the fundamental [2], [3], [18], we can represent the harmonic using the harmonic memory polynomial model given below: Kp Mp y pω (n) = a k,m. x(n m) p. x(n m) k k=0 m=0 (2.12) where ak,m represent the coefficients of the model, Mp represents the memory depth of the harmonic, and Kp represents the harmonic depth of nonlinearity. 22

35 Figure 2.7. Memory Polynomial Model for Fundamental. Figure 2.8. Memory Polynomial Model for pth Harmonic. 23

36 2.4 Cubic Spline A generalized frequency selective memory polynomial model for a single band PA that takes into account time-selective memory effects [2], [12], [16], [17] is given by: Case 1: Fundamental Case 1: p th Harmonic M 1 y ω (n) = G ω,j ( x 1 (n j) )x 1 (n j) j=0 M 1 or y ω (n) = G ω,j ( x 1 (n j) 2 )x 1 (n j) j=0 M 1 y ω (n) = G pω,j ( x 1 (n j) )x 1 p (n j) j=0 M 1 or y pω (n) = G pω,j ( x 1 (n j) 2 )x 1 p (n j) j=0 (2.13) (2.14) A generalized frequency selective memory polynomial model for a dual-band PA that takes into account time-selective memory effects is given by: 24

37 Case 1: Fundamental M 1 y ω (n) = G ω,j ( x 1 (n j), x 2 (n j) )x 1 (n j) j=0 M 1 or y ω (n) = G ω,j ( x 1 (n j) 2, x 2 (n j) 2 )x 1 (n j) j=0 (2.15) Case 1: p th Harmonic M 1 y ω (n) = G pω,j ( x 1 (n j), x 2 (n j) )x 1 p (n j) j=0 M 1 or y pω (n) = G pω,j ( x 1 (n j) 2, x 2 (n j) 2 )x 1 p (n j) j=0 (2.16) where x1(n) & x2(n) represent the two input signals. Gω ( x1 2, x1 2 ) and Gpω ( x1 2, x1 2 ) represent the gain functions for fundamental and harmonic output, which can be represented as a function of envelops (or squares of envelops) of the input signals at each frequency band. These gain functions are usually represented using various basis functions like conventional polynomials (limited order), orthogonal polynomials, artificial neural networks, or splines (which is used in the current work) [12]-[17]. 25

38 Figure 2.9. Gains as function of the envelop of inputs. Why C-Spline? The conventional polynomial possesses the following demerits: A highly nonlinear PA requires a high order polynomial expansion. Extracting high order polynomials can result in an ill-conditioned regression matrix. As a result, small errors require frequent system updates. High order yields a highly oscillatory solution for scattered data. Due to the global scope of coefficients, the possibility of error can also increase. Piecewise polynomials representations remediate these issues. 26

39 Merits of C-Spline: Since C-Spline is a piecewise polynomial of lower order, it provides for a more robust representation. No oscillations from scattered data due to lower order representation. Since coefficients have a local scope, the errors introduced only create local distortions. 1-D Least Square Cubic Spline A generalized frequency-selective memory polynomial model for a single band PA, which takes into account time selective memory effects, is given by [12], [13], [16], [17]: M 1 y ω (n) = G ω,j ( x 1 (n j) 2 )x 1 (n j) j=0 (2.17) Where x1(n) represents the input signal and Gω ( x1 2 ) represents the gain functions. The gain function for each memory delay index j can be represented using the following cubic spline basis function φi( x 2 ): N 1 G ω,j ( x 1 2 ) = c i,j i ( x 1 2 ) i=0 (2.18) where ci,j is the weight coefficient of the 1-D basis which is represented using a vector form. Φi( x1 2 )=[φ0( x1 2 ), φ1( x1 2 ),, φn-1( x1 2 )] where each element is computed from a set of splines defined along the entire region. 27

40 The basis function representation of the gain can now be substituted in equation (2.17): M 1 N y ω (n) = x 1 (n j) c i,j i ( x 2 ) j=0 i=0 (2.19) The coefficients can then be extracted using least square methods. A typical 1D cubic spline basis (4X4) as a function of envelop is shown in Fig Figure D Spline Basis function. 2D Least Square Cubic Spline A generalized frequency-selective memory polynomial model for a dual band PA, which takes into account time selective memory effects, is given by [16], [17]: M 1 y 1 (n) = G 1,j ( x 1 (n j) 2, x 2 (n j) 2 )x 1 (n j) j=0 28

41 M 1 y 2 (n) = G 2,j ( x 1 (n j) 2, x 2 (n j) 2 )x 2 (n j) j=0 (2.20) where x1(n) & x2(n) represent the two input signals at each band. G1,j ( x1 2, x2 2 ) and G2,j ( x1 2, x2 2 ) represent the gain functions for the obtained memory delay index j, which can be represented as a function of squares of the envelops of the input signals at each frequency band. The gain function can be represented using the 2D basis function φi ( x1 2, x2 2 ) as: N N G 1,j ( x 1 2, x 2 2 ) = c 1,i,j i,j ( x 1 2, x 2 2 ) j=0 i=0 N N G 2,j ( x 1 2, x 2 2 ) = c 2,i,j i,j ( x 1 2, x 2 2 ) j=0 i=0 (2.21) where c1,i,j and c2,i,j are the weight coefficients of the 2-D basis for lower and upper band, respectively. The basis function representation of the gain can now be substituted in equation (2.20) to obtain the output: M 1 N N y 1 (n) = x 1 (n j) c 1,i,j i,j ( x 1 2, x 2 2 ) j=0 j=0 i=0 M 1 N N y 2 (n) = x 2 (n j) c 2,i,j i,j ( x 1 2, x 2 2 ) j=0 j=0 i=0 29

42 (2.22) The coefficients can again be extracted using least square methods. The 2D basis function is synthesized using the 1D basis function s vectors (for each band) along its envelops. Φi( x1 2 )=[φ0( x1 2 ), φ1( x1 2 ),, φn-1( x1 2 )] Φj( x2 2 )=[φ0( x2 2 ), φ1( x2 2 ),, φn-1( x2 2 )] (2.23) where each element is computed from a set of splines defined along the entire region. The tensor product of the two vectors generates the 2D basis function vector form: Φij( x1 2, x2 2 )={ [ φ0( x1 2 )φ0( x2 2 ), φ0( x1 2 ) φ1( x2 2 ),, φ0( x1 2 )φn( x2 2 )] [φ1( x1 2 )φ0( x2 2 ), φ1( x1 2 )φ1( x2 2 ),, φ1( x1 2 )φn( x2 2 )] [φn( x1 2 )φ0( x2 2 ), φn( x1 2 )φ1( x2 2 ),, φn( x1 2 ) φn( x2 2 )] } = Φi( x1 2 )* Φj( x2 2 ) (2.24) 2.5 Harmonic Modelling using Cubic Splines 1D Least Square Cubic Spline A generalized frequency-selective harmonic memory polynomial model for a single band PA, which takes into account time selective memory effects, is given by: 30

43 M 1 y pω (n) = G pω,j ( x 1 (n j) 2 )x 1 p (n j) j=0 (2.25) Similar to equation (2.19), equation (2.25) can be represented using spline basis functions and gain coefficients. These gain coefficients can be calculated using least square methods and substituted to form equation (2.26). M 1 N y ω (n) = x 1 (n j) p c i,j i ( x 2 ) j=0 i=0 (2.26) 2D Least Square Cubic Spline A generalized frequency-selective harmonic memory polynomial model for a dual band PA, which takes into account time selective memory effects, is given by: M 1 y pω (n) = G pω,j ( x 1 (n j) 2, x 2 (n j) 2 )x 1 p (n j) j=0 (2.27) Similar to equation (2.22), equation (2.27) can be represented using spline basis functions and gain coefficients. These gain coefficients can be calculated using least square methods and substituted to form equation (2.29). M 1 N N y pω (n) = x 1 (n j) p c 1,i,j i,j ( x 1 2, x 2 2 ) j=0 j=0 i=0 (2.29) 31

44 CHAPTER 3 MEASUREMENT SETUP FOR CAPTURING FUNDAMENTAL AND HARMONIC SIGNAL The digitally controlled feed-forward technique [3] for harmonic cancellation involves two steps: 1. Extraction of interfering harmonics and identification of the harmonic model in the digital domain. 2. Negation of harmonics at the PA output with an amplitude and phase-controlled modelled harmonic signal. This chapter gives detailed insight into the first step. In order to effectively model the harmonics for system level simulations, it is essential that we capture the fundamental and harmonic signals. The measurement data is then used to extract digital models using the 2D least square Cubic Splines discussed in the previous chapter, which act as 32

45 behavioral models for system level simulations [1]-[3]. The general architecture of a testbed used to capture fundamental and harmonics is shown in Figure 3.1. Figure 3.1. Architecture of characterization testbed. 3.1 Modules The measurement setup used to capture the fundamental and harmonics in this work consists of the following modules: Altera-V GT (FPGA) Texas Instruments TSW 30SH84 (Dual Channel Transmitter) Texas Instruments TSW 1266 (Broadband Receiver) Texas Instruments TSW 3065 (Local Oscillator) Texas Instruments HSMC Board UWB Power Amplifier ZX L USB-4SPDT RF Switch 33

46 Apart from the different modules (and their GUIs) mentioned above, MATLAB and Quartus software are used to help in digital processing and controlling the FPGA from the host computer, using Universal Service Bus (USB) connections. 3.2 Measurement Setup FPGA Using MATLAB and Quartus, the input data is stored in the pre-allocated memory (DDR memory) of the FPGA. The downloaded data can be sent to the dual-band transmitter at a sampling rate of MHz via the HSMC board. Dual Channel Transmitter The dual-channel transmitter consists of two 16-bit DACs which share the reference clock. The sampling frequency of the DACs can be programmed using a GUI on the host computer. For the current work, a sampling frequency of GHz (with an interpolation factor of 4) is used. The IF frequency of each channel can also be programmed using the GUI. The data is then upconverted (based on the LO frequency) using either of the quadrature modulators (I/Q) available in the dual band transmitter and sent to the corresponding channel. The range of upconverted frequencies for each transmission channel is 300MHz to 4GHz. UWB PA The upconverted signal is then sent to the ultra-wideband PA. The output of the PA is then connected to the Broadband Receiver via the RF switch. 34

47 RF Switch The RF switches toggle between the fundamental and harmonic signal. The RF switch also switches between the harmonic LO and the fundamental LO. Figure 3.2. Characterization Testbed Setup. Broadband Receiver The broadband receiver has a bandwidth of about 500MHz. The sampling frequency of the receiver is about MHz. In order to ensure synchronization between the transmitter and receiver, a common reference clock and LO frequency are used. The demodulator in the receiver downconverts the received data, and the ADC then digitizes the downconverted data. The data is then stored in the pre-allocated FPGA memory. 35

48 MATLAB and Quartus are then used to read the data in these pre-allocated memory slots for post-processing. The measurement setup used for capturing fundamental and harmonics is shown in Fig Post Processing Time Alignment Errors in delay calculation lead to significant dispersions in the AM/AM and AM/PM curves [2], [4], [10]. These dispersions can be wrongly interpreted as memory effects [15] [19] and may also hamper the prediction capability of the DPD. Figure 3.3. PSD of fundamental and harmonics. 36

49 The measured spectrum of the fundamental (linearized) and third harmonic of a transmitted input signal is shown in Fig It is evident that the harmonic bandwidth is thrice that of the fundamental signal. From [2], [3], and [18] we know that the fundamental output needs to be synchronized with respect to the fundamental input, and the harmonic output needs to be synchronized with respect to p th power of the input signal. Frequency domain alignment offers better accuracy compared to time domain alignment [16]. Thus, the frequency domain cross-correlation technique discussed in [16] [19] is used to time align the received signals with reference to the input signal. Case 1: Fundamental Signal For the fundamental frequency, the cross-correlation of x (transmitted) and y (received) in the frequency domain (convolution in time domain) is given by: fft(x y) = X(f) Y (f) (3.1) Mathematically we can represent Y(f) as: Y(f) = G X(f)e i(ωτ+ 0) (3.2) where G represents gain, τ represents time delay and φ0 represents phase difference. Thus, (3.1) becomes fft(x y) = G X(f) X (f)e i(ωτ+ 0) (3.3) 37

50 Multiplying the argument of (3.3) by the time domain received fundamental signal gives us the time-aligned fundamental signal. Case 2: Harmonic Signal For the harmonic frequency the cross-correlation of x p (p th power of the transmitted signal ) and y p (received) in the frequency domain (convolution in the time domain) is given by: fft(x p y p ) = X p (f) Y p (f) (3.4) Mathematically we can represent Y(f) as: Y p (f) = H X p (f)e i(ωτ+ 0) (3.5) where H represents harmonic gain, Γ represents time delay and φ0 represents phase difference. Thus, (3.4) becomes fft(x p y p ) = H X p (f) Y p (f)e i(ωτ+ 0) (3.6) Multiplying the argument of (4.6) by the time domain received harmonic signal gives us the time-aligned harmonic signal. 38

51 CHAPTER 4 AUTOMATIC FEED-FORWARD CANCELLATION OF MODULATED HARMONICS 4.1 Introduction Growing demand for higher data rates pushes the development of ultra-wideband and multiband systems. When the separation between two bands becomes large, the harmonics of the lower band s signal start to interfere with the higher band s signal. In order to suppress the interfering harmonic in concurrent multi-band transmission systems, the solution of using bulky and lossy switchable filter banks is a common practice. However, digitally-controlled feed-forward harmonic cancellation is emerging as an alternative solution because it is filter-less and offers significant frequency programing flexibility. 39

52 Active harmonic cancellation uses an upper-band channel to send a cancelling signal which is in opposite-phase (a 180 phase shift) to the modulated RF harmonics generated by the broadband nonlinear PA from the lower band signal. The system thus requires the careful characterization of the modulated harmonics [2]-[3], [18]. Further, due to the upper band channel s group delay, nonlinearity, and dispersion, a frequency-selective correction for the gain, phase shift, and time delay must be applied to the cancelling signal so that it exactly cancels the harmonics at the coupler placed after the PA. In a previous study [3], this three-fold adaptation of the cancelling signal was performed manually. In this work, a joint PA distortion modelling and compensation algorithm is developed to ensure that the cancellation is automatic. 4.2 Feed-Forward Harmonic Cancellation Figure 4.1 shows the feedforward harmonic cancellation scheme. In the phase of conceptproving, the upper band channel is only used to send a cancelling signal to sum it up with the signal in the lower band channel at the PA output. The output of the PA contains the amplified fundamental signal and different orders of harmonics caused by the PA s nonlinearity. As indicated in the previous section, the cancelling signal is 180 degrees out of phase from one of the p th order harmonics which is meant to be cancelled. Therefore, the p th harmonic should be cancelled at the combiner after the PA, when it is summed up with its 180 degree out-of-phase version. The negative version of the p th harmonic is generated digitally based on the behavioral model of the PA to predict the real harmonic. 40

53 Figure 4.1 Digital feed-forward Harmonic Cancellation Scheme To achieve effective cancelling results, two key features are required: accurate harmonic modelling, and accurate modelling and compensation for the channel s distortion and delays in the cancelling signal by the upper channel. The first feature can be guaranteed by using effective modelling tools, such as memory polynomials in case of a moderately nonlinear PA [2]. On the other hand, fast and accurate modelling and compensation for the cancelling channel s distortion remains an obstacle to harmonic cancellation s application in practical time-variant transmission systems. Given the distortions and delay of the upper band channel, the cancelling signal needs to be pre-distorted to compensate for those distortions to cancel the modulated harmonic exactly at the PA output. Therefore, an accurate model of the distortions and delay is also required to achieve proper compensation. Previous digitally supported feed-forward harmonic cancellation schemes [3] adopted a manual adjustment of the cancelling signal s amplitude, phase, and time delay based on a trial-and-error method. This work presents a faster and more accurate modelling and predistortion algorithm to enable the automatic feed-forward cancellation of the p th harmonic of the PA. 41

54 4.2 System Identification To conduct the system identification, i.e., to obtain the needed modelling parameters of the two channels, two sets of signals x1 and x2 are transmitted simultaneously through the two channels. As shown in Fig. 4.2, the purple signal RFfund sent through the PA channel is for PA channel identification, and the red signal RFharm sent through the cancelling channel at the frequency band of the p th harmonic is used for the cancelling channel identification. A generalized behavioral model is used to represent the multi-harmonic output of the PA channel main channel, as follows: M 1 y 1,p (n) = G 1,p,m ( x 1 (n m) 2, x 2 (n m) 2 )x 1 p (n m) m=0 (4.1) Figure 4.2. System Identification Scheme. 42

55 Figure 4.3 System Identification for 2 nd Harmonic Case. In (4.1), y1,p is the output of the system at the fundamental or p th harmonic. x 1 (n m) is the input of the corresponding band with the memory delay term m. M is the memory depth. Memory delay terms are included in the model to account for the frequency dependence of the PA nonlinearities in a wideband communication system [2],[3], [7]- [12]. G1,p,m( x1(n-m) 2 ) are the gain functions of each delay channel, which are a function of the envelope squares of the signal in the corresponding channel. These gain functions can be implemented in different forms such as memory polynomials and splines, to represent the higher order nonlinearity caused by the PA. This model thus takes into account both the PA s p th modulated harmonics for cancellation purposes as well as its in-band intermodulation terms for in-band linearization purposes. Although the feedforward cancelling channel is not typically driven in strong saturation, a similar model can also be used, as will be discussed later. There are two requirements for the identification signal x2 in the cancelling channel (RFharm in Fig. 4.2). First, it has to be uncorrelated to the identification signal x1 used for the PA channel. Since the 43

56 identification signal in the cancelling channel occupies the same frequency band as the p th harmonic to be cancelled, the two signals will overlap after the coupler combines them. Being uncorrelated enables them to be separated in the digital domain, to identify the corresponding gain functions of the two channels. Second, because the channel s distortion extends over a p th time in the bandwidth and exhibits amplitude varying within a certain dynamic range, the identification signal should have a bandwidth that covers the entire frequency band that the p th harmonic to be cancelled occupies, and exhibits peaks with a larger amplitude than the p th harmonic of the PA. At the p th harmonic frequency, the received signal is as follows: y 2 (n) = y 2,1 (n) + y 1,p (n) M 1 y 1,p (n) = G 1,p,m ( x 1 (n m) 2, x 2 (n m) 2 )x 1 p (n m) m=0 M 1 y 2,1 (n) = G 1,p,m ( x 1 (n m) 2, x 2 (n m) 2 )x 2 m=0 (n m) (4.2) The component y1,p is the modelled PA p th harmonic introduced in (4.1), and the component y2,1 is the modelled version of the output of the cancelling channel. G1,p,m( x1(n-m) 2, x2(n-m) 2) is the gain function on which the modelling of p th harmonic is based, and G2,1,m( x1(n-m) 2, x2(n-m) 2 ) is the gain function that models the distortions and delay of the cancelling channel. Utilizing the least-square method, the gain functions in (4.2) can be readily extracted. For an accurate representation, we use a cubic-spline 44

57 basis [13]-[17] with 4 bases to represent the gain functions of the two channels. The fidelity of the model can be verified by comparing the modelled signal y2 with the measured one. 4.4 Digital Predistortion The negative version of the modelled y1,p should be sent to the PA output to cancel its p th harmonic, but in consideration of the second key point mentioned in the previous section, this signal has to be pre-distorted in order to compensate for the cancelling-channel s distortion and delay. Digital predistortion (DPD) is used to fulfill this purpose. A block called a predistorter is added in the digital domain before the DAC, to predistort the cancelling signal y1,p. Figure 4.4 Predistortion and Harmonic Cancellation Scheme The parameters of the predistorter are calculated by the method of indirect learning [9]. For this method, the predistorter is an inverse model of the channel, as shown in (4.3): M 1 ( i) 2 2 G2,1, m y2,1( n m) y2,1( n m) m 0 x. (4.3) 45

58 G (i) 2,1,m( y2,1(n-m) 2 ) is the gain function of the inverse model, i.e. the predistorter. This gain function can be calculated from y2,1 obtained from the dual-band system identification. Next, we predistort the cancelling signal y1,p by substituting y1,p for y2,1 in (4.3) while using the gain functions of the predistorter G (i) 2,1,m( y1,p(n-m) 2 ) just extracted: M 1 m 0 ( i) z 2 G2,1, m y1, p ( n m) ( y1, p ( n m)). 2 (4.4) z2, as shown in Fig. 4.3, is the predistorted cancelling signal, which will yield the desired y1,p signal in the combiner after experiencing all the distortions and delays from the cancelling channel. We use the same predistortion procedure for the desired fundamental signal at the same time, to reduce the in-band intermodulation caused by the PA. 4.5 Testbed Setup The cancellation of the second and third harmonics generated by a Mini Circuit ZX L+ PA is done as a proof of concept. Figure 4.2 (a) shows the system diagram of the testbed setup. A set of 10 MHz bandwidth LTE signals is used as the fundamental signals transmitted in the lower band through the PA channel, and a set of 30 MHz bandwidth WCDMA signals are transmitted in the upper band through the cancelling channel in the system identification procedure. Digital predistortion of the upper band cancelling signal is done within MATLAB and the predistorted data is then stored in an Arria V FPGA, which is used to pass the digital data to two 16-bit DACs at a sampling rate of 307.2MHz, incorporated in a dual-band transmitter Texas Instrument board 46

59 TSW30SH84. Then the fundamental signal in the lower band and the cancelling signal (or identifying signal) in the upper band are transmitted simultaneously by the dual band transmitter. The output signal of the combiner is received by the TSW1266 receiver board after passing a set of filters and switches used to tuned the IQ demodulator to the desired band. Within the receiver, the received data are down-converted, sampled, and digitized by the ADC s at a sampling rate of 614.4MHz and stored in the FPGA, before being passed to MATLAB for analysis. 4.6 Measurement Results In Fig. 4.5, the right part shows the comparison of the original third harmonic signal to the cancellation result. The third harmonic is suppressed by 31 db. Since the upper band does not impact the lower band, the lower band signal can still be linearized by the digital predistortion method, as shown in Fig. 4.5, Fig. 4.6 and Fig. 4.7 show the AM/AM and AM/ PM characteristics at fundamental. Figure 4.5. Results for Third Harmonic 47

60 Figure 4.6 AM/AM Characteristics for Third Harmonic After DPD Before DPD Figure 4.7. AM/PM Characteristics for Third Harmonic 48

61 Table I summarizes the measurement results. The harmonic model NMSE is the normalized mean square error of the modelled 3rd harmonic, i.e., y1,3 compared to the actual 3rd harmonic. The harmonic before and after cancellation is the harmonic strength relative to the fundamental signal. TABLE 1 SUMMARY OF MEASUREMENT RESULTS (3 RD HARMONIC) Evaluation Criterion Measured Value Harmonic Model NMSE Harmonic before cancelling Harmonic after cancelling Harmonic cancellation Main Channel NMSE after DPD ACPR after DPD (LSB,USB) db -28 dbc -59 dbc 31 db db -49.5,-50.3 db Figure 4.8 Results for Second Harmonic 49

62 Figure 4.9 AM/AM Characteristics for Second Harmonic After DPD Before DPD Figure AM/PM Characteristics for Second Harmonic 50

63 Similarly, in Fig. 4.8 the right part shows the comparison of the original third harmonic signal to the cancellation result. The second harmonic is suppressed by 30 db. Since the upper band does not impact the lower band, the lower band signal can still be linearized by the digital predistortion method, as shown in Fig Fig. 4.9 and Fig show the AM/AM and AM/ PM characteristics at fundamental. Table II summarizes the measurement results. The harmonic model NMSE is the normalized mean square error of the modelled 2 nd harmonic, i.e., y1,2 compared to the actual 2rd harmonic. The harmonic before and after cancellation is the harmonic strength relative to the fundamental signal. TABLE 2 SUMMARY OF MEASUREMENT RESULTS (2 ND HARMONIC) Evaluation Criterion Measured Value Harmonic Model NMSE Harmonic before cancelling Harmonic after cancelling Harmonic cancellation Main Channel NMSE after DPD ACPR after DPD (LSB,USB) db -29 dbc -59 dbc 30 db db -49.5,-49.3 db 51

64 CHAPTER 5 CONCLUSION & FUTURE WORK 5.1 Conclusion This thesis developed a new algorithm for the simultaneous linearization and cancellation of modulated harmonics of broadband power amplifiers. The algorithm relies on joint system identification of the nonlinearity, memory effects, and group delay of both the fundamental and the harmonic channel, using a 2-D Least Square Cubic Spline (2-D LSCS) basis. The identification of the nonlinearity, memory effects, and group delay and modelling of the predistorter (inverse model of channel) is done using a 2-D Least Square Cubic Spline (2-D LSCS) basis, which is directly measured from the captured data. To leverage the advantages of this quick and robust algorithm, a low cost FPGA-based testbed was developed to perform filter-less cancellation of the modulated harmonics. The testbed is able to capture PA output at the fundamental and the harmonic frequencies. Using the inverse models extracted from the measured data for each channel, 52

65 the setup is able to take advantage of both the methods of predistortion and the feedforward cancellations for the compensating distortion in broadband PAs. The algorithm presents a fast and accurate digital solution for both filter-less cancellation of harmonics and linearization of the fundamental signal at the output of PAs, and thus alleviates the need for bulky filter banks in transmitters. Experimental verification with a broadband PA yields a reduction of 31dB and 30dB for the third and second harmonics, respectively, below the main channel. Simultaneously, the linearization provides 50dBc ACPR and -41.1dB NMSE at the fundamental frequency. 5.2 Future Work FPGA Implementation The algorithm can be implemented completely on an FPGA or an ASIC, which can be adopted for practical, portable ultra-wideband transmitters [19]. A study on the design methodology for FPGA implementation and its performance can be conducted to verify the practical ability of the algorithm. Simultaneous Cancellation of Modulated Harmonics The algorithm can also be used to characterize, model, and cancel multiple harmonics generated by the PA. However, active multiple harmonics cancellation requires multiple channel transmitters to send a cancelling signal in the opposite phase of each harmonics 53

66 generated. A testbed should be developed to address this issue, and its performance should be evaluated. 2D DPD for removal of Harmonic Interference A new algorithm for 2D DPD removal of Harmonic Interference should be developed to avoid the use of combiners in PA output. This would further help in reducing the size of ultra-wideband transmitters and increasing their performance. A testbed implementing the 2D DPD should also be developed, and its performance should be evaluated. Figure D DPD for Harmonic Cancellation. 54