Adaptive Digital Predistortion Linearizer for Power Amplifie

Adaptive Digital Predistortion Linearizer for Power Amplifiers in Military UHF Satellite

By

Jayanti Patel

A thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Science in Electrical Engineering

Department of Electrical Engineering

College of Engineering

University of South Florida

Major Professor: Ravi Sankar, Ph.D.

Lawrence Dunleavy, Ph.D.

Paris H Wiley, Ph.D.

Date of Approval:

March 29, 2004

Keywords: Non-Linear, AM-AM, AM-PM, IMD, Simulation ? Copyright 2004, Jayanti Patel

ACKNOWLEDGMENTS

I would like to thank Dr. Sankar for being my supervisor and allowing me to choose the thesis topic related to my work. I would also like to thank my other committee members Dr. Dunleavy and Dr. Wiley for reviewing my thesis.

I would like to thank Mr. Crowley, Mr. Coleman, Mr. Strickland, Mr. Yates, Dr. Nazemi, Mr. Muir, and Dr. Sills for their advice and assistance during the development of adaptive digital predistorter hardware and simulation model.

Finally, I would like to thank my wife, Christine for her support, and encouragement during my graduate studies.

TABLE OF CONTENTS

TABLE OF CONTENTS (i)

LIST OF TABLES (iv)

LIST OF FIGURES (v)

ABSTRACT (viii)

1.0 INTRODUCTION (1)

Background (1)

1.1

1.2 Motivation and Research Objectives (2)

1.3 Thesis Outline (3)

2.0 POWER AMPLIFIER LINEARIZATION TECHNIQUES STUDY (4)

Transmitter (4)

Satellite

2.1

2.2 Power Amplifier Requirements (4)

2.3 Power Amplifier Characteristics (4)

2.3.1 AM-AM and AM-PM Conversion Effects in Power Amplifier (5)

2.4 Two Tone Test (6)

2.5 Power Amplifier Technology (7)

2.6 Power Amplifier Linearization Techniques (8)

2.6.1 Feedback Linearization Technique (8)

2.6.2 Simple Envelope Feedback (9)

2.6.3 Polar Feedback (10)

2.6.4 Cartesian Feedback (11)

2.6.5 LINC (12)

2.6.6 Combined Analog-Locked Loop Universal Modulator (CALLUM)13

2.6.7 Single Loop Feedforward (14)

2.6.8 Muti-Stage Feedforward (15)

2.6.9 Envelope Elimination and Restoration (15)

2.6.10 RF/IF Predistortion (16)

2.6.11 Digital Predistortion (16)

2.7 Selection of Linearizer Topology for Power Amplifier (17)

2.7.1 Mapping Predistorter (18)

2.7.2 Complex Gain Based Predistorter (19)

2.7.2.1 Predistorter Table (20)

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2.7.2.2 Table Addressing (22)

2.7.2.3 Table Adaptation (23)

2.7.2.4 Delay Adjustment Estimation (27)

2.8 Up-Conversion Topology (29)

2.8.1 AQM Up-Conversion Topology (29)

2.8.2 DDM Up-Conversion Topology (32)

2.8.3 Digital Up Converter (33)

2.8.4 Analog Mixer (35)

2.8.5 Down-Conversion Topologies (37)

2.8.6 Analog Quadrature Demodulator (38)

2.8.7 Direct Digital Down-Conversion (38)

2.8.8 Discussion on AQM Approach versus DDM Approach (40)

3.0 DEMONSTRATION MODEL (42)

3.1 Predistortion Demonstration Model (42)

4.0 SIMULINK SIMULATION MODEL AND SIMULATION RESULTS (49)

4.1 Digital Adaptive Predistortion MATLAB SIMULINK Model (49)

4.2 SIMULINK Model Description (49)

4.3 SIMULINK Model Simulation Results-100 KHz Signal Bandwidth (51)

4.4 SIMULINK Model Simulation Results-30 MHz Signal Bandwidth (52)

4.5 Sensitivity Analysis (57)

4.5.1 Sensitivity to Predistortion Signal Bandwidth (57)

4.5.2 Sensitivity to Feedback Signal Bandwidth (58)

4.5.3 Adaptation Time versus Table Size (59)

4.5.4 Sensitivity to Time Alignment (61)

4.5.5 Sensitivity to Addressing Scheme (63)

5.0 PREDISTORTER HARDWARE DEMONSTRATION SETUP (64)

5.1 Adaptive Digital Predistortion Hardware Demonstration Setup (64)

5.2 Adaptive Predistorter Correction Results for 30 MHz Signal Bandwidth (68)

5.3 Reasons for the Poor Performance of Adaptive Predistorter (72)

5.4 Memory Effects Classification (73)

5.4.1 Reducing Memory Effects (73)

5.5 Comparison of Hardware Model Results with SIMULINK Model

Simulation Results (75)

6.0 PREDISTORTERS FOR POWER AMPLIFIERS WITH MEMORY (76)

6.1 Adaptive Digital Predistorter for Power Amplifiers with Memory (76)

6.2 Adaptive Volterra Predistorter (76)

6.3 Hammerstein Memory Predistorter (77)

6.4 Nonlinear Tapped Delay Line Predistorter (78)

6.5 Memoryless Predistorter with Feedforward for Linearizing Power

Amplifiers with Memory (80)

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7.0 CONCLUSION AND FUTURE WORK (82)

Conclusion (82)

7.1

Work (83)

7.2

Future REFERENCES (84)

iii

LIST OF TABLES

Table 5.1 SIMULINK Model Simulation and Memory-less Adaptive

Predistorter Results (75)

iv

LIST OF FIGURES

Figure 1.1 Conventional UHF Satellite Transponder Architecture (1)

Figure 1.2 US Military SATCOM Downlink Bands and Russian VOLNA Bands2 Figure 2.1 Power Amplifier Distortion Characteristics (4)

Figure 2.2 Illustration of Compression and Intercept Points (5)

Figure 2.3 Illustrates IMD Products due to Conversion Effects [2] (6)

Figure 2.4 Illustration of Harmonic Distortion (7)

Figure 2.5 Illustration of Performance Improvement of a Power

Amplifiers with a Linearizer (8)

Figure 2.6 Illustration of Simple Feedback to Linearize Power Amplifiers (9)

Figure 2.7 Illustration of Envelope Feedback to Linearize Power Amplifier (10)

Figure 2.8 Illustration of Polar Feedback to Linearize Power Amplifier (10)

Figure 2.9 Illustration of Cartesian Feedback to Linearize Power Amplifier (11)

Figure 2.10 Illustration of LINC Method to Linearize Power Amplifiers (12)

Figure 2.11 Illustration of Constant Envelope Signals (13)

Figure 2.12 Illustration of CALLUM Feedback to Linearize Power Amplifier (14)

Figure 2.13 Illustration of Feedforward Technique to Linearize Power

Amplifier (14)

Figure 2.14 Illustration of EER Technique to Linearize the Power Amplifier (15)

Figure 2.15 Illustration of Simple Predistortion Technique to Linearize

Power Amplifier (16)

Figure 2.16 Digital Predistortion (17)

Figure 2.17 Mapping Predistorter (19)

Figure 2.18 Illustration of Complex Gain Based Predistorter (20)

Figure 2.19 Illustration of Complex Gain Based Predistorter-Polar Tables (22)

Figure 2.20 Look-Up Table Address Calculation (23)

Figure 2.21 Linear Convergence –I/Q Table (24)

Figure 2.22 Secant Method (26)

Figure 2.23 Delay Processing Block Diagram (27)

Figure 2.24 Cross Correlation Block Diagram (28)

Figure 2.25 AQM Up Conversion Topology (29)

Figure 2.26 Filtered DAC Output (30)

Figure 2.27 AQM Upconversion Output (30)

Figure 2.28 Quadrature Modulator Compensation Circuit (31)

Figure 2.29 Direct Digital Modulator (32)

Figure 2.30 Digital Quadrature Modulator (33)

Figure 2.31 Digital Quadrature Modulator (34)

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Figure 2.32 Digital Quadrature Modulator (34)

Figure 2.33 Analog Mixer (35)

Figure 2.34 Mixer Frequency Conversion (36)

Figure 2.35 Mixer Distortion Terms (37)

Figure 2.36 AQD Down Conversion Topology (38)

Figure 2.37 Digital Down Conversion (39)

Figure 2.38 Spectral Images of RF Signal from Under Sampling (39)

Figure 2.39 Quadrature Digital Down Conversion (40)

Figure 2.40 Complex Baseband Output of the Predistorter (40)

Figure 3.1 Breadboard Digital Predistorter (42)

Figure 3.2 Photograph of Breadboard Digital Predistorter (43)

Figure 3.3 Measured PA Chain Transfer Characteristics (44)

Figure 3.4 PA Chain Transfer Characteristics Polynomial Fit (44)

Figure 3.5 PA Chain Inverse Transfer Characteristics Polynomial Fit (45)

Figure 3.6 PA Gain Compression (46)

Figure 3.7 PA Gain Inverse Curve (46)

Figure 3.8 Predistorter Gain Look-Up Table (47)

Figure 3.9 Left-PA Uncorrected, Right-PA Corrected (48)

Figure 4.1 SIMULINK Model of Complex Gain based Adaptive Digital

Predistorter (49)

Figure 4.2 Power Amplifier Gain and Phase Characteristics (50)

Figure 4.3 Power Amplifier Output without Correction (51)

Figure 4.4 Power Amplifier Output with Correction (52)

Figure 4.5 Power Amplifier Output without Correction (53)

Figure 4.6 Power Amplifier Input and Output Magnitude without Correction (54)

Figure 4.7 Power Amplifier Input and Output Phase without Correction (54)

Figure 4.8 Power Amplifier Output with Correction (55)

Figure 4.9 Adaptation Table Gain and Phase Entries when Loop Converges (56)

Figure 4.10 Power Amplifier Input and Output Magnitude when the Loop

Converges (56)

Figure 4.11 Power Amplifier Input and Output Phase when the Loop

Converges (57)

Figure 4.12 Sensitivity to Predistortion Signal Bandwidth (58)

Figure 4.13 Sensitivity to Feedback Signal Bandwidth (59)

Figure 4.14 Sensitivity to Table Size (60)

Figure 4.15 512 Entry Table Size, Adaptation Time 20 Seconds (61)

Figure 4.16 Sensitivity to Input and Feedback Alignment (62)

Figure 4.17 Sensitivity to Linear and Power Addressing (63)

Figure 5.1 Adaptive Digital Predistorter Hardware Setup (64)

Figure 5.2 Adaptive Digital Predistorter using ISL5239 (66)

Figure 5.2 Photograph of Adaptive Digital Predistorter using ISL5239 (67)

Figure 5.3 Class A/B PA Output Uncorrected and Corrected @ 8 Watts (68)

Figure 5.4 Class A/B PA Output Uncorrected and Corrected @ 12 Watts (69)

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Figure 5.5 Class A/B PA Output Uncorrected and Corrected @ 7 MHz

Signal BW (69)

Figure 5.6 Class A PA Output Uncorrected and Corrected @ 20 Watts (70)

Figure 5.7 Class A PA Input/Output- Amplitude and Phase after Convergence..71 Figure 5.8 Class A/B PA Input/Output- Amplitude and Phase after

Convergence (71)

Figure 5.9 Class A PA Response to Sync Pulse (72)

Figure 5.10 Class A/B PA Response to Sync Pulse (73)

Figure 5.11 Class A/B Low Memory PA Output Uncorrected and

Corrected @ 12 W (74)

Figure 6.1 Adaptive Volterra Predistorter Architecture [37] (77)

Figure 6.2 Adaptive Hammerstein Predistorter Architecture [39] (78)

Figure 6.3 NTDL Power Amplifier Model (79)

Figure 6.4 Adaptive NTDL Predistorter Architecture (80)

Figure 6.5 Adaptive Digital Predistorter with Feedforward Architecture (81)

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ADAPTIVE DIGITAL PREDISTORTION LINEARIZER FOR

POWER AMPLIFIER IN MILITARY UHF SATELLITE

Jayanti Patel

ABSTRACT

The existing UHF Satellite Communications (SATCOM) transponders used for military applications use efficient, saturated power amplifiers, which provide one earth-coverage antenna beam. The amplifier is dedicated to small frequency band and only handles a few carriers simultaneously.

The communications capacity needed to support future military forces on the move will require satellite payload power amplifiers to support hundreds of channels simultaneously, with the channels spread over the entire military UHF SATCOM band. To meet the capacity requirements and simultaneously meet the out-of-band emission, power amplifiers will have to be highly linear. The high-efficiency, ultra-linear power amplifier architecture proposed to support the requirements can only be met by use of linearity improvement techniques.

The literature search revealed many power amplifier linearity improvement techniques. Each technique was reviewed to determine its suitability for the proposed power amplifier architecture.

The adaptive digital predistortion technique was found to be the most suitable in terms of bandwidth, correction achievable, and complication.

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A discussion on common linearization techniques is presented, followed by analysis of the adaptive digital predistortion technique. A SIMULINK simulation model of an adaptive digital predistorter was developed. The simulation results show that adaptive digital predistortion was able to significantly reduce the Inter-Modulation Distortion (IMD) terms generated by a memory-less power amplifier operating in the 240 MHz to 270 MHz range. An actual hardware implementation of adaptive digital predistorter was constructed and the test results show that there was a large reduction in IMD terms generated by a memory-less power amplifier. In the contrary, the results show there is only moderate improvement in IMD performance if the power amplifier has memory. The electrical memory in the power amplifier with memory was minimized, but this resulted only a modest improvement in the IMD performance. Therefore, it was concluded the majority of the memory effect was due to thermal memory.

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1.0 INTRODUCTION

1.1 Background

The existing UHF Satellite Communications (SATCOM) transponders used by the US military use highly efficient, saturated power amplifiers, which provide one earth-coverage antenna beam. The amplifier is dedicated to small frequency band and only handles a few carriers simultaneously. The out-of-band inter-modulation distortion generated by the output of the saturated power amplifier is suppressed by the use of narrow, band-pass filters (see Figure 1.1). This approach can support up to 39 channels through a single earth coverage antenna.

Figure 1.1 Conventional UHF Satellite Transponder Architecture

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The communication capacity needed to support future military forces on the move will require satellite to support hundreds of channels simultaneously with the channels spread over the entire UHF satellite communications band. This capacity and the availability requirements for the next generation satellite can be met by providing multiple downlink beams, which can change direction and channel assignment within the beam. In the multi beam approach each beam can have few channels to hundreds of channels, occupying full downlink spectrum. The multi-beam system requires each power amplifier to operate over the full downlink band of 240 to 270 MHz [1]. 1.2 Motivation and Research Objectives In 1981, at bilateral coordination meeting between US and Russia, US agreed to limit the radiated power within the Russian satellite (VOLNA) bands which are interposed between the US military satellite bands as shown in Figure 1.2. The VOLNA treaty limits the inter-modulation distortion to –52 dB relative to full power in a single channel. The next generation of satellite power amplifiers have to operate over the full downlink band, carry hundreds of channels simultaneously, generate out-of-band emissions level which do not require further filtering and meet the VOLNA emissions limits. Therefore, new power amplifiers have to be highly linear, thereby creating only minimal out-of-band energy when transmitting hundreds of channels simultaneously.
VOLNA MUOS
US Military
Frequency MHz
Figure 1.2
US Military SATCOM Downlink Bands and Russian VOLNA Bands2

The conventional approach of moderately linear power amplifier followed by narrow band filters can be used to implement proposed architecture, but the system would be extremely complex and impractical for a satellite.

The size, weight and power restriction placed on the power amplifier because of satellite application, means that the strict out-of-band emissions limits can only be met with linearity improvements techniques [1].

The literature search revealed many power amplifier linearity improvement techniques. Each technique was evaluated to determine its suitability for the proposed power amplifier architecture. Adaptive digital predistortion technique was found to be the most suitable in terms of bandwidth, correction achievable and complication. SIMULINK model of adaptive digital predistortion was developed to evaluate sensitivity to parameter changes and determine the complexity of the adaptation scheme.

Hardware demonstration models were also built to show to the prospective users the viability of proposed power amplifier architecture.

1.3 Thesis Outline

This section serves as an introduction to the need for ultra linear power amplifier for the next generation of military satellites. Section 2.0 presents power amplifier characteristics followed by a review of different linearization techniques. Each technique was reviewed to determine its suitability for the proposed power amplifier architecture. The digital predistortion techniques are treated in more detail because of its suitability for the proposed power amplifier architecture. Section 3.0 details hardware demonstration model results for a non-adaptive digital predistorter. Section 4.0 details the simulation results for a adaptive digital predistorter and sensitivity analysis of predistorter to various parameter changes. Section 5.0 details results of actual hardware model built for adaptive digital predistorter for memory-less power amplifier. Also, the methods used to detect memory in power amplifiers and techniques used to overcome memory in power amplifiers are presented in this chapter. Section 6.0 presents possible adaptive digital predistorter architecture for a power amplifier with memory. Section 7.0 details conclusions reached and recommendations for future work.

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2.0 POWER AMPLIFIER LINEARIZATION TECHNIQUES STUDY

2.1 Satellite Transmitter

The satellite transmitter section consists of channel filtering/limiter at Intermediate Frequency (IF) followed by an Up-Converter, which translates the filtered signal to desired carrier frequency. The power amplifier amplifies the signal to the required power level before being fed to the antenna

2.2 Power Amplifier Requirements

In addition to the operating bandwidth of 30 MHz, from 240 to 270 MHz and linearity requirement that generates Inter-modulation distortion (IMD) products of less then –52 dB in the VOLNA bands.

Another requirement is that the average power per amplifier would be 12 Watts with peak power of 120 Watts, with efficiency of approximately 20%. The input drive level of -16 dBm was selected for maximum power and –23 dBm drive level was chosen for minimum channel capacity.

2.3 Power Amplifier Characteristics

The three main classes of linear amplifiers are A, AB and B. Class A is the most linear

Figure 2.1 Power Amplifier Distortion Characteristics

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and least efficient. The amplitude dependent characteristics of a power amplifier can split into three regions. The cut-off region is when the amplifier is not conducting, the linear region is where the amplifier starts conducting and signal amplification occurs, and finally the saturation region where the amplifier output starts to flatten (see Figure 2.1). The main characterizations of power amplifier are the second and third-order intercept point, 1 dB gain compression point and input back-off. Figure 2 illustrates, when the input is increased, the second harmonic will increase in proportion to square of the input signal and the third harmonic will increase in proportion to cube of the input signal. Thus, the second and the third harmonics will increase at a greater rate than that of the fundamental component. There comes a point where the harmonic components equal the fundamental. The signal level at which the second harmonic is equal to the fundamental is called the second order intercept point and the point at which the third harmonic is equal to the fundamental is called the third order intercept point.

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Input Voltage

O u t p u t V o l t a g e

Figure 2.2 Illustration of Compression and Intercept Points

It is possible that this intercept point may be beyond the maximum output power of the amplifier. In this case, points are shown by dotted line to where intersection occurs. The intercept point indicates the linearity performance of the amplifier and is a fixed quantity from which the distortion level at a particular operating point may be predicated. The 1 dB compression point is defined as the point at which the output power level has dropped 1 dB below the ideal output power. Input back-off is defined as the ratio of the signal power measured at the input to the power amplifier to the input signal power that produces the maximum signal power at the amplifier's output.

2.3.1 AM-AM and AM-PM Conversion Effects in Power Amplifier

The nonlinear relationship between the input power and output power present in the power amplifier is referred to as AM-AM conversion. Another effect is conversion from

amplitude modulation on the input signal to phase modulation on the output signal. This is known as AM-PM conversion. Figure 2.3 shows the IMD terms generated by this two conversion effects [2].

IMD products due to

Amplitude Nonlinearity

Phase Nonlinearity

Figure 2.3 Illustrates IMD Products due to Conversion Effects [2]

The amplifier used in this design is nonlinear and assumed to be memory-less[4] i.e. the transfer function is not frequency dependent. Therefore, real-valued, nonlinear and memory-less function can be expanded into a power series as follows:

V o(t) = a0 + a1 . V in(t) + a2 . V in(t)2 + a3 . V in(t)3 + a4 . V in(t)4 + a5 . V in(t)5 (2.1)

2.4 Two Tone Test

A standard two-tone test is used to assess the amplitude and phase distortions present in a power amplifier. In the two-tone test the envelope of the input signal is varied throughout its complete range so the amplifier is tested over its whole transfer characteristics. Input signal is represented by:

V i n(t) = v cos(ω1t) + v cos(ω2t) (2.2)

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So the output voltage is

V o(t) = a1v[cos(ω1t) + cos(ω2t)] + a2v2[cos(ω1t) + cos(ω2t)]2

+ a3v3[cos(ω1t) + cos(ω2t)]3 + a4v4[cos(ω1t) + cos(ω2t)]4

+ a5v5[cos(ω1t) + cos(ω2t)]5 + a6v6[cos(ω1t) + cos(ω2t)]6

+ a7v7[cos(ω1t) + cos(ω2t)]7 + (2.3) Each product term in equation 2.3, other than the fundamental generates number of distortion products. In general, the even order terms IMD terms will be well out-of-band of interest where as the odd order IMD terms may fall in-band (see Figure 2.4). It is understood that the IMD distortion causes major problems to a communication system as opposed to harmonic distortion. The harmonic distortion is far away from the fundamental signal and thus much easier to suppress by use of filters.

3F1

c

Fundamental Spectrum Second Harmonic

Spectrum Third Harmonic Spectrum

Figure 2.4 Illustration of Harmonic Distortion

2.5 Power Amplifier Technology

New device technologies that have been developed for cellular base stations and microwave communications satellites have been surveyed. The power amplifier built with these latest technology devices when subjected to the two-tone test revealed that IMD performance could be as good as –40 dB. Adding 6 dB (to account for multi-tone) and allowing 2 dB degradation (for multistage and environmental effects) shows that power amplifier linearization technique is required which provides at least 20 dB of correction to meet –52 dB IMD specification [1].

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2.6 Power Amplifier Linearization Techniques

To obtain both linear amplification and high power efficiency, a linearizer is required. The linearizer allows the amplifier to be operated at much higher operating point since the distortion generated by the amplifier because of the peaks in input signals can be corrected up to the saturation level of the amplifier as shown in Figure 2.5. Any input signal which drives the amplifier to hard saturation, the resulting distortions cannot be

Figure 2.5 Illustration of Performance Improvement of a Power Amplifiers with a Linearizer

corrected since any increase in input power beyond this point will not result in an increase in output power. The linearization methods reported in the literature can be classified into Feedback, Feedforward, Predistortion and Digital Predistortion (Signal Processing).

2.6.1 Feedback Linearization Technique

The simplest method of reducing amplifier distortion is by some form of feedback. The Figure 2.6 illustrates the use of negative feedback around an amplifier with the effect of distortion n(t). G is the gain of the amplifier and K is the feedback attenuation.

Output: y(t) = G . e(t) + n(t) (2.4)

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Feedback: f(t) = K

y(t) (2.5) Error: e(t) = x(t) – f(t) (2.6) Therefore,

y(t) = K(G . x(t) + n(t))/ (G + K) (2.7)

Pow er Amplifier

Figure 2.6 Illustration of Simple Feedback to Linearize Power Amplifiers

If the amplifier gain is much greater than the feedback ratio G>>K, then K + G approximates to G. So

y(t) = K . x(t) + ( K . n(t) )/G (2.8)

Therefore, the distortion produced by the main amplifier is reduced by a factor K/G. The disadvantage of this approach is that the improvement in distortion performance is at the expense of the gain of the power amplifier and also feedback needs more bandwidth than signal.

2.6.2 Simple Envelope Feedback

Simple envelope feedback has matched envelope detectors coupled to the power amplifiers input and output ports. A differential amplifier forms amplitude error-correcting amplifier based on the detected envelope signals. The resulting error is used to control the gain of the amplifier. This technique has been widely employed to improve the IMD performance of VHF and UHF solid-state power amplifier in the mobile communication industry. The main draw back is that since this technique performs simple amplitude correction, it starts generation IMD products when the envelope operates in the compression region of the amplifier. The delays in the detection and signal processing can cause phase differences between AM and PM processes. This may

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cause asymmetry IM side bands as discussed earlier and may substantially reduce any correction obtained by amplitude feedback process. The analysis has shown that that envelope correction does not provide correction over the operating bandwidth for this satellite application [1][5].

Figure 2.7 Illustration of Envelope Feedback to Linearize Power Amplifier

2.6.3 Polar Feedback

The polar feedback technique combines the envelope feedback with an additional feedback loop to account for phase shift variation through the power amplifier by dynamically adjusting the phase of the Radio Frequency (RF) input. The phase correction shown in Figure 2.8 uses a phased locked loop to maintain a constant phase shift over the amplifier’s dynamic range. The two feedback loops are interdependent, any variation in the AM/AM loop, will produce phase as well as gain variation and similarly AM/PM will interact with the AM/AM loop if the insertion loss of the phase shifter varies. It has been reported in the literature that phase amplifier requires much higher bandwidth, which is a major limiting factor in the performance of the polar feedback [6].

Phase

Figure 2.8 Illustration of Polar Feedback to Linearize Power Amplifier

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