CHAPTER 4 DEVELOPMENT AND PERFORMANCE ANALYSIS OF LINEARIZATION TECHNIQUES

Transcription

1 71 CHAPTER 4 DEVELOPMENT AND PERFORMANCE ANALYSIS OF LINEARIZATION TECHNIQUES 4.1 Introduction The comparison of existing linearization techniques show that DPD technique can be of main concern due to its moderate complexity, high flexibility, good IMD reduction and automatic adaptation. As concluded in chapter 3, although DPD shows moderate complexity and high flexibility, Feedforward linearization technique shows more IMD reduction and greater bandwidths over DPD technique. So, a possibility can be explored to propose a new linearization technique in which combination of these two techniques can be used to obtain better performance. In this chapter for the development of a new linearization technique named Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter has been proposed and presented. For the development of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter, firstly a Look up Table based DPD technique has been proposed. But this technique has resulted in more implementation cost and is complex to implement. Thus to develop Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter, Complex Memory Polynomial based Adaptive Digital Predistortion technique has been proposed. For optimization of parameters, a new adaptation technique with much fast convergence, based on the conjugate-gradient method developed by Fletcher and Reeves has been proposed. Then an optimized Adaptive Feedforward Linearization technique has been proposed. These techniques are than combined to develop a novel

2 72 linearization technique named Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter. A new design methodology has been proposed which is carried out in different design steps, so that optimized value of parameters can be obtained. Also performances of all proposed linearization techniques have been evaluated for single, two and three carrier W-CDMA signal. The performance has been evaluated based on IMD distortion reduction, EVM and ACLR improvement. 4.2 Design and Analysis of Look up Table Digital Predistortion Technique Look up Table DPD is illustrated by Fig. 4.1 and uses two, one dimensional LUTs and is based on the concept of maintaining constant loop gain at all power levels. This is achieved by addressing the LUT with the magnitude of the input complex envelope to obtain complex gain scale factor stored in the LUT. The input signal is then multiplied with the complex gain to obtain a predistorted output which is the inverse of the PA. The complex envelope of the input ( V a ) and the output ( V pa ) of the PA are related by V V V 2 pa a G( a ) (4.1) 2 G( a ) V is the complex gain of the amplifier, and represents its AM/AM and AM/PM characteristics and V a is the predistorted signal (Cavers, 1990). The IQ table contains complex gain factors represented as,,im V F Re V V e e e (4.2)

3 73 Fig. 4.1 Illustration of Complex Gain based Predistorter The gain function from the LUT is multiplied with modulated input signal. The resulting complex quantity is based on the envelope of the input signal and can be represented as c t t F t m od m od 2 V V V (4.3) Predistorter Table The gain based Predistorter only requires two one-dimensional LUT. The LUT table can be based on I/Q representation or polar coordinates. Both approaches require additional signal processing to perform complex multiplications. In addition, the polar co-ordinate table also requires polar/rectangular conversions. The gain function from the LUT is multiplied with modulated input signal. The resulting complex quantity is based on the envelope of the input signal and is represented

4 74 by equation 4.3 where the PA. 2 F{ V mod( t ) } represents the inverse transfer characteristics of Also, c t t t V V V (4.4) m o d pd The polar table approach shown in Fig. 4.2 consists of two one-dimensional tables, one containing amplitude gain error and the other table containing phase rotation error. The polar table can be represented as follows: V F R V, Ø V (4.5) e e e The amplitude part corrects for AM/AM distortion, represented below V t gain error (4.6) e The phase table corrects for AM/PM distortion and represented below: e t phase error V (4.7) The output from the polar table is converted back to IQ representation, which adds sight variation from the standard approach in literature where polar table output is used to predistort the modulating input signal. Therefore, the gain function obtained after polar to rectangular conversion from polar tables is identical to the gain function in IQ representation LUT. This gain function is multiplied with modulated input signal. The resulting complex quantity is based on the envelope of the input signal is represented by equation 4.3.

5 75 Assuming a perfect modulator i.e. we can write, a t t t V V V (4.8) mod pd Fig. 4.2 Illustration of Complex Gain based Predistorter Polar Tables Table Addressing The LUT has been addressed by the magnitude of the source signal. The magnitude calculation of the input signal can be given by 2 2 V m o d r e a l V m o d i m a g V m o d (4.9) The calculation of equation 4.9 is the most time consuming operation of the algorithm compared to the other operations in the Predistorter. It has been reported that the accuracy of square function is not critical since it lead to about 2 db of adjacent channel

6 76 degradation when a table based square root function is employed this reduces the burden on DSP and also reduces the adaptation time. A table-based square root is shown in Fig The table methods employ two small LUT, one containing y i and the other containing y i 1 y addressed by the integer part of i I2 + Q 2 address. The y i table gives an absolute square root value for an integer point and the yi 1 yi table gives difference of a square root value between a integer point and its neighbor. The difference square root table value is multiplied by the fractional part of the integer point, thus giving a weighted version of the difference table output. The absolute square root value and weighted difference table output are added to give an approximate square root value of I/Q input. The result is used to address the complex gain look up table (Pandreani et al., 1999). Fig. 4.3 Look up Table address calculation

7 Table Adaptation The two methods of the LUT operation in both access and update are the continuous update and block update. In continuous LUT update method, the input ( mod ( )) V t is delayed to align with feedback V fb t from the PA and the resulting difference ( Verr ( t )) which should only contain the distortion is computed on sample by sample basis. In block update, a block of data of input ( mod ( )) V t and feedback ( V fb t ) are captured and the DSP is used to align the two signals using cross-correlation, followed by taking the difference ( Verr ( t )) of the two signal which should only contain the distortion. The block processing is performed at a fixed time interval. t t t Verr Vmod V fb (4.10) Fig. 4.4 Linear Convergence I/Q table There are various techniques described in the literature for adaptation of LUT entries,

8 78 such as linear convergence, secant method, rotate and scale, and steepest decent method (annexure II). The method of adaptation selected determines the speed of convergence, stability of the system and computation load on the DSP. The linear convergence is based on classical feedback theory, and it is computationally simplest and the least stable for adaptation LUT entries. The error ( Verr ( t )) in linear convergence is modified by the adaptation constant and resulting V ( ) e t is summed with the previous entry in the table Fi ( k ) {Re( Ve ),Im( V e)}. The new entry in the table is Fi ( k 1) {Re( Ve ), Im( Ve )} and is stored at the magnitude envelope address of ( V ( t )).This iteration update occurs every time the modulating signal envelope mod passes through a given table entry. The subscript i represents a specific entry in the table and k represents the kth iteration. The adaptation constant is generally selected to be less than unity and controls the rate of convergence. If the adaptation constant is large then there exists a possibility that the table entries will not converge, but oscillate and result in an unstable system. Secant adaptation method is based on a straight line approximation. For a given function f(x), the secant convergence algorithm is depicted by a geometrical representation in Fig The function f(x) is being approximated by a straight line be which is an extrapolation based on the two points x i and xi 1. The line passing through the x-axis at xi 1 gives the new value. Fig. 4.5 also shows that the secant line be deviates from the ideal line jk resulting in a small error. It can be seen that the triangles abe and dce are similar. Therefore, a b a e d c (4.11) d e

9 79 f ( xi ) f ( xi 1) x x x x i i 1 i 1 i 1 (4.12) Rearranging equation 4.12, the new value is given by, x i 1 x i f ( xi )( xi xi 1) f ( x ) f ( x ) i i 1 (4.13) Applying the secant method of convergence for adaptation of look up table entries is given by the following equation (Cavers, 1990), F ( k 1) i F ( k 1) e ( F ( k )) F ( k ) e ( F ( k 1)) i g i g e ( F ( k )) e ( F ( k 1)) g g i (4.14) where Fi ( k ) is the kth iteration of look up table entry i and eg is the quantization error at the PA output. The rotate and scale method of adaptation is used for polar tables and is similar to the linear convergence method described above. The equation 4.10 is rearranged to give gain and phase error, err t t t V V V (4.15) m od fb err V t V t V t (4.16) mod fb The gain and phase look up table entry update at kth iteration is given by F ( k 1){ Gain( V )} F ( k){ Gain( V )} a V (4.17) i e i e err

10 80 Fig. 4.5 Illustration of Secant method F ( k 1){ Phase( V )} F ( k){ Phase( V )} a V (4.18) i e i e err Since, the table size is small; the envelope address is unlikely to directly fall on a table entry. Therefore either a linear interpolation between the table entries of adjacent address may be required or a larger size look up table to improve the IMD performance Delay Adjustment Estimation The propagation delays in transmit and receive path results, shown in Fig. 4.6, in the sampled feedback signal Vf n being of later time interval then the input complex signal V n.this delay has to be accurately computed so the time aligned mod Vmod V n and n can be compared to generate the error vector. If the delay is not computed accurately, then the adaptation tables will have noise distortion component in the tables resulting in a less accurate inverse table. Therefore the distortion products generated by the PA will not be cancelled resulting in a non optimal IMD correction. A simple f

11 81 method for compensation of delay in feedback sample f V n is to delay the input sample Vmod n by the required number of samples before a comparison is made between the input and the feedback samples. There are several techniques described in the literature to compute the delay which exits in the forward and the feedback paths of the PA chain. A common technique to determine the delay requires the use of DSP which computes cross correlation between the input V n and the feedback mod V n samples f to determine the delay. This method requires a block of input and feedback samples to be stored in capture memories at a periodic interval. The DSP computes the magnitude of baseband input and feedback samples and then interpolates samples by a predefined factor to increase accuracy of time delay estimation, followed by computing cross correlation of the two series. Fig. 4.6 Delay Processing block diagram

12 82 The cross correlation of V mod and V f in discrete time domain is defined as, n N 1 N R [ n] V [ k n] V [ k] V [ m] V [ m n] (4.19) VmodV f mod f mod f N k n N m 0 where m 0. The sum will be maximum when the two samples streams line up. Therefore delay between the two signals is from the origin to time where the peak occurs in their cross correlation as shown in Fig This delay is not constant and dependent on the modulation rates and amplifier characteristics. The amplifier characteristics change due to temperature, age and voltage. Therefore, the tables are required to be updated continually. Fig. 4.7 Cross Correlation block diagram

13 83 Other techniques employed for delay estimation are by comparing the slope of the magnitude of the input and feedback to determine direction of delay adjustment as described in detail by Nagata (Nagata, 1989). Another simple technique exploits the properties of the modulation scheme (Wright and Dutler, 1992). Due to its simplicity of implementation, in the present work, delay has been calculated by computing cross correlation between input and feedback signals Results and Discussions MATLAB and C++ are used to implement the Predistorter architecture. Simulations are carried out and Fig. 4.9 and 4.10 show the performance of proposed Predistorter. x 10 4 Magnitude Phase LUT address) 6 8 Fig. 4.8 Gain and Phase entries for Look Up Tables

14 84 Fig.4.9 Magnitude and Phase errors with and without Predistortion Fig Predistorter Performance in suppressing Spectral regrowth: (1) Output without Predistortion and (2) Output with Predistortion

15 85 Fig. 4.9 shows the calculation of magnitude and phase error and the results show that the average value of 3 rd order IMD falls from db to db and the average value of 5 th order IMD falls from db to db. Fig shows the performance of proposed DPD algorithm in suppressing the spectral regrowth. Results also show reduction of EVM from % to % ie. a decrease of about 5.390% is noticed. 4.3 Design and Analysis of Complex Memory Polynomial based Adaptive Digital Predistortion Technique In Complex Memory Polynomial based Adaptive Digital Predistortion (CMPPD) technique each digital data sample is processed in the Predistorter not according to a LUT but according to a polynomial function. This polynomial function can be 3rd, 5th order or more depending on the desired degree of linearization in adjacent and alternate channels. Fig shows a possible implementation based on the adjacent channel power minimization (Bauernschmitt et al., 2001; Stapleton and Costescu, 1992). Predistorter polynomial is fit to the opposite of PA characteristics and is controlled by an optimizer or adapter. The adaptation unit adjusts the polynomial coefficients, which should be in complex form in order to correct both gain and phase non-linearities. Also in Polynomial Predistorters the forward path must be continuous as in LUT based method and therefore is critical. The coefficients can be stored in a small LUT and optimization can be done with a duty cycle or activated just if it is required. The difference compared to the LUT based Predistorter is the requirement of highly reduced size of the LUT. For example it will require only 4, 6 and 8 positions for 3rd, 5th and 7th order polynomial Predistorters respectively.

16 86 Fig Block diagram of Polynomial Adaptive Digital Predistortion based on Adjacent Channel Power Minimization Fig shows the block diagram of CMPPD technique in detail. It is composed of a Predistorter circuit (a squared magnitude calculation unit, a polynomial Predistortion function calculator, a complex multiplier, a delay element), an optimization circuit for polynomial coefficients, DACs, ADCs, a transmitter chain as given in LUT based Predistorter and a feedback path for calculation of adjacent channel power emission (coupler, down-converter, bandpass filter, power detector). The critical forward path is similar to the one in LUT based Predistorter. The difference is in the Predistorter which has a polynomial Predistorter function unit instead of a LUT. The amount of mathematical operations in this unit can be quite high depending on the order of Predistorter. This in turn limits the bandwidth of the modulation signal to be linearized because of limited system clock frequency. As shown in Fig. 4.11, digital I and Q data is taken and each sample pair is modified in the Predistorter unit. The feedback signal is proportional to the unwanted adjacent channel power emission.

17 Derivation of Polynomial work function The proposed CMPPD shown in Fig is based on the Memory Polynomial model discussed in and is a modification of equation In this method the real part (inphase) and imaginary part (quadrature) of the Predistorter function are described by equations 4.20 and K 2k 1 Re, m 2k 1, m ( ) (4.20) k 1 F x n q K 2k 1 Im, m 2k 1, m ( ) (4.21) k 1 F x n q And q represents the memory depth and k represents order of the polynomial. Fig Implementation of CMPPD work function

18 88 The Predistorter output can be written as Q X PD( n) X ( n).. Fq (4.22) q 0 Where Fq FRe, m jfim, m (4.23) And is complex parameter of complex gain multiplier Complex Gain Multiplier The complex gain multiplier can take on two forms: Polar or Rectangular Implementation. The polar representation requires a voltage-controlled attenuator and phase shifter. The rectangular implementation is of the same form as a quadrature modulator. Either of these configurations needs to operate in the linear region where the generated IMD products are significantly lower than those generated by the PA. The complex gain adjusters are required to be insensitive to variations across the operational bandwidth. In this work rectangular complex gain multiplier has been implemented. Once optimized, the complex gain adjuster provides the inverse non-linear characteristics to that of the PA. Ideally the IMD products will be of equal amplitude but in anti-phase to those created as the two tones pass through the PA. The out-of-band filter will sample the adjacent power interference (ACPI). The function of the adaptation algorithm is to adapt the polynomial work function parameters so that the ACPI is minimized.

19 Adaptation using Conjugate-Gradient Method There are different ways of optimizing polynomial Predistorter coefficients. For example it can be based on minimization of adjacent channel emission or minimization of EVM at the PA output (Baudoin and Jardin, 2000). After selecting the system structure, the adaptation algorithm should be selected. There are mainly two groups of methods applied in polynomial Predistortion systems: direct search methods and gradient methods. Direct search methods are robust and have low computational load compared to gradient methods but they can not converge as fast as gradient methods. In this work conjugate- gradient method has been used. The slowest convergence rate of the steepest descent method can be increased by choosing the search directions in a more sophiscated way. For example, as the solution progress, linear combinations of previous search directions might be used to form new search directions. As effective approach of this form is the conjugate-gradient method. The first step in the conjugate-gradient method is a steepest descent step. That is, starting with an initial guess 0 x, set d k k g where g f ( x ) next, for k=0 compute 0 0 x x d k 1 k k k (4.24) here is the optimal step length obtained by performing a line search along direction d k, k starting at point k x. To generate the kth search direction, k d For 1 k n, the following linear combination of k g and k 1 d can be formed ( g ) ( g ) g g k T k k 1 T k 1 (4.25)

20 90 d g d k k k 1 k (4.26) Note that computing effort than computing k d for the conjugate-gradient method involves only slightly more k d for the steepest descent method. Suppose the function f ( x ) is quadratic with Hessian matrix H. Then the first n search directions of the conjugategradient method can be shown to be H-orthogonal or conjugate in the following sense. k T j ( d ) Hd 0, k j (4.27) When the objective function f is quadratic with a positive-definite Hessian matrix, the conjugate-gradient method converges to the minimum x* in at most n iterations (Luenberger, 1973). Practically f ( x ) may not be quadratic. Thus, for the more general case, the following adaptation of the conjugate-gradient method developed by Fletcher and Reeves quoted in (Schilling and Harris, 2007) can be used. Like the steepest descent method, it terminates when the error criterion is satisfied or a maximum number of iterations has performed Fletcher-Reeves Algorithm 1. Pick 0, m 0, and x 0 n R. set j=0 and k=0 2. Do { k k k (a) Compute g f ( x ) and g (b) If (k<1) set d k g k, else compute

21 91 { ( g ) ( g ) g g k T k k 1 T k 1 d g d k k k 1 k } (c) Find k 0 to minimize ( ) ( k k F f x d ) (d) Compute x x d k 1 k k (e) Set k=k+1 and j=j+1 (f) If (k=n), set k=0 and x x 0 n } 3. While ( j m) and ( ) and 1 ( k k x x k m x ) The one-dimensional minimization in step (2c) of algorithm can be performed using any line search technique. Notice that after n iterations, the iterative procedure is reset in step (2f), which ensures that every nth step is a steepest descent step. This is important for nonquadratic objective functions. Steady progress toward a local minimum of f(x) is assured because the value of the objective function is guaranteed to decrease every nth step and will not increase during the intervening n-1 steps. The steepest descent step is sometimes called a spacer step (Luenberger, 1973). A detailed discussion of conjugate direction methods in general can be found in (Polka, 1971). If the conjugate-gradient method is compared with the steepest descent method by applying both techniques it has been found in (Schilling and Harris, 2007) that the

22 92 conjugate-gradient method finds the optimal value in only 9 iterations, compared to 302 iterations of the steepest descent method. Consequently, an increase in speed of more than 33 times has been observed Results and Discussions The set up shown in Fig. 4.13was used to implement the Predistorter and to optimize the work function. For measurements and optimization, two tone ramps at f GHz and f GHz with power varying from 0dBm to 13dBm centered around 1.950GHz is applied at the input. The RF Predistorter design has been completed in a 7- step design process to develop an optimized design. The design begins with an optimization using the ACLR minimization technique. The output signal from the PA is subtracted from the input reference signal. The resultant error signal consists of only the distortion generated by the PA. The work function coefficients are then optimized so as to minimize the error signal. The input to the work function is the squared envelope of the incoming signal. Fig Set up used to implement the CMPPD

23 93 A group delay is required to compensate for the delay from the envelope detector, and a delay is required in the feedback path to compensate for the delay from the upper branch. These delays are calculated as 0.1nsec and 3.8nsec respectively. In the fifth step of the design a pilot tone f GHz at -50 dbm has also being applied to optimize the design for more IMD products. For simplicity of design all passive components are assumed to be ideal. The Low Pass Filter used is of Butterworth type with F 200MHz, A 1dB, F 500MHz and A 75dB. The design has been pass pass competed using the following step by step procedure: stop stop Step 1: Spectral plots in the initial state Fig Initial Spectrum at output of CMPPD calculated during Step 1

24 94 Fig Optimized Spectrum at output of CMPPD calculated during Step 1 Step 2: Optimization of coefficients based on IMD reduction Fig Output Spectrum at Predistorted Power Amplifier for CMPPD calculated during Step 2

25 95 Fig Variation of Lower Order Carrier to IMD ratio for CMPPD calculated during Step 2 Fig Variation of Upper Order Carrier to IMD ratio for CMPPD calculated during Step 2

26 96 During this step the minimum value of Carrier to IMD ratio was calculated as dbc when A/D converter has no bits = 6. Thus in the next step A/D bits were set at 6. Step 3: 3rd-order coefficient sensitivity about optimum Fig Optimized Spectrum at Error Port CMPPD calculated during Step 3 Fig Variation of real part of Complex Gain Adjuster parameter (α) for CMPPD calculated during Step 3

27 97 Fig Variation of imaginary part of Complex Gain Adjuster parameter (α) for CMPPD calculated during Step 3 During this step the value of complex gain adjuster parameter is calculated as j Step 4: 5th-order coefficient sensitivity about optimum Fig Initial Spectrum at output of CMPPD calculated during Step 4

28 98 Fig Variation of 3 rd order coefficient of Memory Polynomial work function of CMPPD calculated during Step 4 Fig Optimized Spectrum at output of CMPPD calculated during Step 4

29 99 Fig Variation of 5 th order coefficient of Memory Polynomial work function of CMPPD calculated during Step 4 Fig Comparison of Initial and Optimized Spectrum at Error Port of CMPPD calculated during Step 4

30 100 During this step 3 rd and 5 th order coefficients of Memory Polynomial work function are calculated as 3 j j1.556 and 5 j j1.263 respectively. Step5 : Signal Cancellation Loop optimization In this step a pilot tone f GHz at -50 dbm has also applied and then monitored at the output of the Error Port. Fig Initial Spectrum at Error Port CMPPD calculated during Step 5

31 101 Fig Optimized Spectrum at Error Port CMPPD calculated during Step 5 Fig Variation of real and imaginary parts of Complex Gain Adjuster parameter (α) for CMPPD calculated during Step 5

32 102 During this step value of complex gain adjuster parameter was calculates as j Spectrum of Predistorter at Output Power (dbm) freq, GHz Fig Spectrum at output of CMPPD calculated during Step 5

33 103 Step6: IMD optimization using Signal Cancellation Loop Fig. 4.31Variation of real and imaginary parts of Complex Gain Adjuster parameter (α) for CMPPD calculated during Step 6 During this step value of complex gain adjuster parameter was calculates as j Fig rd and 5 th order IMD improvement for CMPPD calculated during Step 6

34 104 Fig Initial and Optimized Spectrum at the output of CMPPD calculated during Step 5 The final value of complex gain adjuster parameter is calculated as j Also 3 rd, 5 th and 7 th order coefficients of Memory Polynomial work function are calculated

35 105 as 3 j j0.058, 5 j j1.587 and 7 j respectively. 4.4 Design and Analysis of Adaptive Feedforward Linearization Technique As discussed in article 3.8, the concept of Feedforward was invented by H.S. Black of Bell Telephone Laboratories. His idea for Feedforward was simple: reduce the amplifier output to the same level as the input and subtract one from the other to leave only the distortion generated by the amplifier. Amplify the distortion with a separate amplifier and then subtract it from the original amplifier output to leave only a linearly amplified version of the input signal. Black's idea for negative feedback was spawned from his simple Feedforward concept: feed an attenuated version of the amplifier output signal back to the input in anti-phase and combine it with the input signal. Use the same amplifier (rather than a separate amplifier as in Feedforward) to amplify the difference signal thus producing a linearly amplified version of the input signal. The advantage of the feedback solution is that it is automatic and required no manual adjustment as operating conditions changed. Its disadvantage, of course, is its potential for instability. The Feedforward configuration shown in Fig consists of two circuits, the signal cancellation circuit and the error cancellation circuit. The purpose of the signal cancellation circuit is to suppress the reference signal from the main amplifier (or PA) output signal leaving only amplifier distortion, both linear and non-linear, in the error signal. Linear distortion, in contrast to non-linear distortion described already, is due simply to deviations of the amplifier's frequency response from flat gain and linear phase (Cavers, 1994). Feedforward technique can also compensate for memory effects, since distortion due to memory in the main amplifier is also included in the error signal and thus ultimately canceled in the linearizer output.

36 106 Fig Concept of Feedforward Technique to linearize Power Amplifier In order to suppress the reference signal, the values of the sampling coupler and fixed attenuation are chosen to match the gain of the main amplifier so that the PA output signal can be reduced to approximately the same level as the reference signal. The variable phase shifter ahead of the PA is then adjusted to place the PA output in anti-phase with the reference. The variable attenuation serves the fine tuning action of precisely matching the level of the PA output and the reference. The delay line in the reference branch, necessary for wide bandwidth operation, compensates for the group delay of the main amplifier by time aligning the PA output and reference signals before combining. The purpose of the error cancellation circuit is to suppress the distortion component of the PA output signal leaving only the linearly amplified component in the linearizer output signal. In order to suppress the error signal, the gain of the error amplifier is chosen to match the sum of the values of the sampling coupler, bed attenuator, and output coupler so that the error signal is increased to approximately the same level as the distortion component of the PA output signal. The variable phase shifter ahead of the error amplifier is then adjusted to place the error in anti-phase with the PA output. The variable

37 107 attenuation again serves the fine tuning function of precisely matching the level of the error signal and the distortion component of the PA output. The delay line serves the same purpose as in the signal cancellation circuit. The error amplifier must be chosen such that it linearly amplifies the error signal while still providing the required output power, otherwise uncorrectable IMD shows up in the linearizer output. This usually dictates the use of a linear class A amplifier with sufficient back-off. Note that any bandwidth limit, manifested as incomplete distortion suppression, is imposed either by imperfect delay matching or by linear distortion in the error amplifier, the variable attenuators l phase shifters, or the couplers and combiners. The crux of the proper operation of the Feedforward circuit is the proper adjustment of the attenuation and phase in the signal and error cancellation circuits such that good IMD suppression is maintained over time. Variations of component characteristics with temperature and time as well as changes in operating conditions such as input power level and supply voltage all necessitate readjustment. For these reasons Black, himself, essentially abandoned Feedforward in favor of feedback. He found, using vacuum tube amplifiers that "every hour on the hour-24 hours a day-somebody had to adjust the filament current to its correct value. In doing this, they were permitting plus or minus 1 db to 112 db variation in amplifier gain. In addition, every six hours it became necessary to adjust the battery voltage, because the amplifier gain would be out of hand. Even with modern solid state amplifiers, changes with temperature and time are still significant enough, with respect to the accuracy requirements of the Feedforward circuit, to necessitate adaptation. After its invention in 1923, Feedforward was essentially ignored until Seidel, also at Bell Laboratories, investigated the use of Feedforward for microwave frequency TWT amplifiers in the late sixties and early seventies (Kahn, 1952).

38 108 Seidel constructed a Feedforward amplifier which achieved distortion suppression of 38 db over a 20 MHz band. The setup employed an automatic control scheme for the variable attenuation and phase in the error cancellation circuit. The control scheme was based on driving a mechanical attenuator phase shifter with an error signal derived by comparing the amplitude and phase at two different points in the error cancellation circuit of a pilot tone inserted after the main amplifier. No control scheme was utilized for the variable attenuation and phase in the signal cancellation circuit. In this way he was able to maintain time independent distortion suppression over a period of several months. Several patents concerned with Adaptive Feedforward systems then started to appear in the mid-eighties, and many more appeared in the early nineties. In these patents two general methods of adaptation both with and without the use of pilot tones, namely adaptation based on power minimization and adaptation based on gradient signals have been discussed. The control scheme for the former, adjust the complex vector modulator in the signal cancellation circuit so as to minimize the measured power of the error signal in the frequency band occupied by the reference signal. In the error cancellation circuit, the frequency band is chosen to include only that occupied by the distortion. Once the optimum parameters have been achieved, deliberate perturbations are required to continuously update the coefficients. These perturbations reduce the IMD suppression. In this work adaptation using conjugate gradient method has been used and completed in 8-step design process. The adaptation algorithm has already been discussed in section The input signals for the complex correlator are the error signal and the reference signal. The error signal is derived by subtracting the input signal from the PA output signal. The error signal, if properly aligned, should contain only the resulting distortion generated by the PA.

39 109 Fig Feedforward technique used in the proposed work The reference signal is the input to the Feedforward Linearizer. The objective of the correlator is to optimize the complex gain adjuster so as to ensure that the two signals are uncorrelated Results and Discussions Following set up was used to implement the Feedforward Linearizer and to complex gain multiplier parameters. For measurements and optimization, two tone ramps at f GHz and f GHz with power varying from 0dBm to 13dBm centered around 1.950GHz is applied at the input.

40 110 Fig Set up used to implement the Adaptive Feedforward Linearizer The Adaptive Feedforward Linearization design has been completed in an 8-step design process to develop an optimized design. The design begins with calculation and optimization of cancellation loop swept coefficients, followed by calculation and optimization of error loop swept coefficients. Finally optimization was carried out considering both the loop using minimum ACLR criterion. In this case a group delay of 3.8 nsec has been calculated. The design has been competed using the following step by step procedure:

41 111 Step1 :Calculation of Signal Cancellation Loop coefficients Fig Variation of Lower Side Band cancelled power with Complex Gain Adjuster parameter used in Signal Cancellation Loop in Step 1 Fig Variation of Upper Side Band cancelled power with Complex Gain Adjuster parameter used in Signal Cancellation Loop in Step 1

42 112 The minimum value of Lower and Upper Side Band cancelled power is calculated as 0.83dBm and 0.81dBm respectively. Step 2: Optimization of coefficients in Signal Cancellation Loop Fig Spectrum of cancelled power with in Signal Cancellation Loop in Step 2

43 113 Fig Variation of cancelled power in Signal Cancellation Loop in Step 2 During this step the value of complex gain adjuster parameter (α) used in signal cancellation loop is calculated as j0.003.

44 114 Step 3 : Optimization of coefficients using Complex Correlator Cancelled Signal Fundamentals versus Iteration # 0-10 Cancelled Signal Level (dbm) Iteration # Fig Variation of cancelled power in Signal Cancellation Loop in Step 3 During this step the value of complex gain adjuster parameter (α) used in signal cancellation loop is calculated as j0.003.

45 115 Step4 : Adjustment of Error Cancellation Loop gain Fig Spectrum at Signal Cancellation Loop in Step 4 Fig Spectrum at output of Auxiliary Amplifier in Step 4

46 116 Fig Spectrum at output of Power Amplifier in Step 4 Step5: Optimization of coefficients for Error Cancellation Loop Fig Variation of 3 rd order IMD with Complex Gain Adjuster parameter used in Error Cancellation Loop in Step 5

47 117 During this step, the minimum value of 3 rd order IMD power is calculated as dB. Step 6 : Optimization of Error Cancelation Loop coefficients using IMD minimization Fig Initial Spectrum at Feedforward output in step 6 Fig Spectrum at output of Power Amplifier in step 6

48 118 During this step the value of complex gain adjuster parameter (β) used in error cancellation loop is calculated as j Step7 : Optimization of Error Cancellation coefficients using ACLR minimization Fig Spectrum at output of Power Amplifier in step 7 Fig Optimized Spectrum at Feedforward output in step 7

49 119 Fig Variation of Complex Gain Adjuster parameter (β) used in Error Cancellation Loop in Step 7 During this step the optimum value of complex gain adjuster parameter is calculated as j Step8: Optimization of coefficients using ACLR minimization Fig Variation of Complex Gain Adjuster parameters α and β used in both loops in Step 8

50 120 Fig Optimized Spectrum at Feedforward output in Step 8 The final optimized values of complex gain adjuster parameters for signal cancellation loop and error cancellation loop are calculated as j0.003 and j0.982 respectively. 4.5 Design of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter Some of the work using Feedforward combined with Adaptive Digital Predistorter has been reported in (Hoyerby and Hansen, 2009; Proctor and Mucenieks, 1999). In (Proctor and Mucenieks, 1999) a LUT-based Predistorter and Feedforward correction signal processing mechanism has been proposed. The input signal to the RF amplifier is stored for comparison with the measured the RF output. In each of Predistorter and Feedforward signal processing paths, the magnitude of the complex waveform of the input signal is extracted to derive a read-out address to a dual-port RAM which stores weights to be multiplied by the input signal. In the Predistortion signal processing path,

51 121 the product is coupled to the RF PA. In the Feedforward correction loop, the product is amplified by an auxiliary Feedforward RF amplifier and coupled into the amplified output signal path of the RF PA. In (Hoyerby and Hansen, 2009) a linearization scheme for an RF PA combines an adaptive Predistorter modulator with a Feedforward error correction loop, which cancels noise imparted by Predistorter modulation to the amplified signal, and minimizes distortion in the RF amplifier's output to a level that allows the use of a low cost auxiliary RF error amplifier in the Feedforward loop. The Predistorter correction mechanism produces a Predistorter signal based upon the input signal and is adaptively adjusted by an error signal extracted from the output of the main RF PA. The input signal is supplied to a work function generator unit and to a subtraction unit, which is also coupled to receive a fractional portion of the amplifier output signal and outputs the RF error component. The RF error component is coupled to a Predistorter function generator, which is driven by the work function generator unit. The Predistorter modulator uses the output of the Predistorter function generator to predistort the input signal by a compensation characteristic equal and opposite to the distortion expected at the output of the main RF amplifier. When subjected to the transfer function of the RF amplifier, the Predistorter signal injected into the input signal path effectively cancel the amplifier's anticipated distortion behavior. The Predistorter is made adaptive by tracking the error signal and coupling this error signal to the Predistorter function generator Proposed Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter In this section a novel adaptive linearization technique named Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter has been proposed. The proposed

52 122 linearizer is a combination of Complex Memory Polynomial based Adaptive Digital Predistortion Technique proposed in section 4.3 and Adaptive Feedforward Linearization Technique proposed in section 4.4. The RF Predistorter is embedded in the signal cancellation loop of the Feedforward Linearizer. The Predistorter consists of a complex gain adjuster, which controls the amplitude and phase of the input signal. The Predistorter is based on a work function that interpolates the inverse AM/AM and AM/PM nonlinearities of the PA. An envelope detector is used to extract the incoming amplitude modulation; this signal is then used as an input into the polynomial work function. The error signal from the signal cancellation loop of the Feedforward Linearizer is used to adapt the Predistorter coefficients. The advantages of embedding a RF Predistorter inside a Feedforward Linearizer are that the IMD reduction requirements of the Feedforward Loop alone are reduced. This reduces the component sensitivities across the band of frequencies. The net result is the overall efficiency improvement of the PA. Fig Set up used to implement Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter

53 Results and Discussions Two tones are applied at f GHz and f GHz around f 1.950GHz at power level of 13dBm. Pilot Tones are also used for optimizing the complex gain adjuster coefficients in both loops. A Pilot Tone at f GHz is injected at the input of the Feedforward Linearizer and then monitored at the output of the signal cancellation loop. The first Pilot Tone ensures that the signal cancellation loop achieves optimum reduction of the fundamental component. The residual signal contain only the distortion created by the PA. A second Pilot at f GHz is injected in the upper branch of the first loop and monitored at the output of the Feedforward Linearizer. The second Pilot Tone is used to ensure that the error cancellation loop achieves optimum reduction of the PA's distortion. The advantage of embedding an Adaptive Digital Predistorter inside the Adaptive Feedforward Linearizer is that the resultant error signal from the first loop is used to optimize the Predistorter work function. Minimization of the adjacent channel power at the Error Port is an effective technique for optimizing the work function coefficients. The complete design has been completed in 10-step design process as follows:

54 124 Step1: Linear Coefficients optimization using Complex Correlator Fig Variation of real part of Complex Gain Adjuster parameter (α) used in Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 1 Fig Variation of imaginary part of Complex Gain Adjuster parameter (α) used in Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 1

55 125 Fig Variation of Group Delay for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 1 The minimum value of group delay is calculated as nsec. Fig Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 1

56 126 Fig Optimized Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 1 During this step the value complex gain adjuster parameter used in Adaptive Digital Predistorter is calculated as j0.001.

57 127 Step2 : Memory Polynomial coefficients optimization using power minimization Fig Variation of Complex Gain Adjuster parameter (α) used in Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 2 Fig Variation of 3 rd order coefficient of Memory Polynomial work function of Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 2

58 128 Fig Variation of 5 th order coefficient of Memory Polynomial work function of Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 2 Fig Initial Spectrum at output of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 2

59 129 Fig Comparison of Initial and Optimized Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 2 During this step the value of complex gain adjuster parameter used in Adaptive Digital Predistorter is calculated as j0.004.the values of 3 rd and 5 th order coefficients of Memory Polynomial work function are calculated as 3 j j0.432 and 5 j j0.483 respectively. Step3 : Pilot Tone optimization in Signal Cancellation Loop In this step a Pilot Tone at f GHz is injected at the input of the Feedforward Linearizer and then monitored at the output of the signal cancellation loop. This Pilot Tone ensures that the signal cancellation loop achieves optimum reduction of the fundamental component and the residual signal will contain only the distortion created by the PA.

60 130 Fig Variation of Complex Gain Adjuster parameter (α) used in Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 3 During this step the value of complex gain adjuster parameter used in Adaptive Digital Predistorter is calculated as j The optimum value of group delay is also calculated as nsec. Step4 : Pilot Tone and IMD power minimization at Error Port Fig Variation of Complex Gain Adjuster parameter (α) used in Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 4

61 131 Fig Variation of 3 rd order coefficient of Memory Polynomial work function of Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 4 Fig Variation of 5 th order coefficient of Memory Polynomial work function of Adaptive Digital Predistorter for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 4

62 132 Fig Initial Spectrum at output of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 4 Fig Optimized Spectrum at output of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 4

63 133 Fig Optimized Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 4 During this step the value of complex gain adjuster parameter used in Adaptive Digital Predistorter is calculated as j0.003.the values of 3 rd and 5 th order coefficients of Memory Polynomial work function are calculated as 3 j j0.031 and 5 j j0.085 respectively. Step5: Pilot Tone optimization in Error Cancellation Loop During this step a second Pilot at f GHz is injected in the upper branch of the first loop and monitored at the output of the Feedforward Linearizer. The second Pilot

64 134 Tone is used to ensure that the error cancellation loop achieves optimum reduction of the PA's distortion. 0.8 Beta Coefficients vs Iteration Interval OPTIM.Beta_Q OPTIM.Beta_I Iteration # Fig Variation of Complex Gain Adjuster parameter (β) used in Signal Cancellation Loop for Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 5 During this step the value of complex gain adjuster parameter used in Signal Cancellation 5 Loop is calculated as j

65 Initial Spectrum at Output Port Power (dbm) m1 m1 freq= kHz dbm(vout[0,::])= freq, GHz Fig Initial Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 5

66 Optimized Spectrum at Output Port 0-50 Power (dbm) m2 freq= kHz dbm(vout[maxiter,::])= m freq, GHz Fig Optimized Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 5

67 137 Step6 : Two Pilot Tones used for optimization of coefficients Fig Initial Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 6

68 138 Fig Initial Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 6 Fig Optimized Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 6

69 139 During this step the value of complex gain adjuster parameters used in Adaptive Digital Predistorter and Signal Cancellation Loop are calculated as j0.003 and j respectively. Step7 : Two Pilot Tones optimization and IMD power minimization During this step both Pilot Tones are used simultaneously. Following results are obtained during this step: Fig Initial Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 7

70 140 Fig Optimized Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 7

71 141 Fig Initial Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 7 Fig Optimized Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 7

72 142 During this step the value of complex gain adjuster parameters used in Adaptive Digital Predistorter and Signal Cancellation Loop are calculated as j0.004 and j0.007 respectively. Also the values of 3 rd and 5 th order coefficients of Memory Polynomial work function are calculated as 3 j j0.019 and 5 j j0.075 respectively. Step8 : Output Pilot Tone removal using re-injected Pilot at Error Port Fig Initial Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 8

73 Optimized Spectrum at Output Port m3 freq= kHz dbm(vout[numiters,::])= Power (dbm) m freq, GHz Fig Optimized Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 8

74 Spectrum at Error Port m2 freq= kHz dbm(verror[maxiter,::])= Power (dbm) m freq, GHz Fig Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 8 During this step the value of complex gain adjuster parameters used in Error Cancellation Loop is calculated as j0.245.

75 145 Step9 : Coefficients optimization using both Pilot Tones and re-injected Pilot 50 0 Initial Spectrum at Output Port m1 freq= kHz dbm(vout[0,::])= Power (dbm) m freq, GHz Fig Initial Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 9

76 Optimized Spectrum at Output Port m3 freq= kHz dbm(vout[maxiter,::])= Power (dbm) m freq, GHz Fig Optimized Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 9

77 147 Power (dbm) m4 Initial Spectrum at Error Port m4 freq= kHz dbm(verror[0,::])= freq, GHz Fig Initial Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 9 Fig Optimized Spectrum at Error Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 9

78 148 During this step the value of complex gain adjuster parameters used in Adaptive Digital Predistorter, Signal Cancellation Loop and Error Cancellation Loop are calculated as j0.139, j and j0.253 respectively. Step10 : Pilot Tones and re-injected Pilot Tone and IMD power optimization Fig Initial Spectrum at Output Port of Adaptive Feedforward Linearizer combined with Adaptive Digital Predistorter in step 10