Abstract--In wideband communication systems, memory effect manifests itself as inter-symbol interference (ISI), which combined with the nonlinearity, complicates the nonlinear compensation. We propose a digital feedforward linearization method to compensate for the nonlinearity of a power amplifier (PA) with memory. The PA is modeled with a Wiener model, which is a linear time-invariant (LTI) system followed by a memoryless nonlinear system. The advantage of the proposed scheme is that the compensation is implemented in baseband by using a digital signal processing (DSP) technique. Simulation results are given to verify the validity of the proposed scheme. Index Terms--digital signal processing, feedforward compensation, high power amplifier, nonlinear system. I. INTRDUCTION Orthogonal frequency division multiplexing (OFDM) is currently used in various applications such as digital broadcasting, wireless local area network (WLAN), broadband internet access, and mobile communications. OFDM-based systems have high spectral efficiency, and robustness against ISI and multipath fading. However, they are sensitive to nonlinearities due to the high peak-to-average power ratio (PAPR). The nonlinear effects include inter-modulation distortion (IMD), out-of-band spectral re-growth and reduction in signal-to-noise ratio (SNR), etc. To solve such problems, many linearization techniques have been continuously reported in literature [1]-[3]. In addition, memory effect cannot be ignored any longer when the signal bandwidth becomes wide. The memory effect means the previous input signal of the PA as well as the current input affects the current output. We propose a digital feedforward compensation scheme for nonlinearity combined with memory effect. Although the feedforward method has been employed for wideband system and shown the best performance among feedback, feedforward, This study was financially supported by research fund of Chungnam National University in 2008. Junsu Kwon is studying in the Department of Information & Communications Engineering, Chungnam National University, Daejeon City, Korea (e-mail: [email protected]). Changsoo Eun, Dr., is a professor in the Department of Information & Communications Engineering, Chungnam National University, Daejeon City, Korea (e-mail: [email protected]). ICITA 2009 ISBN: 978-981-08-3029-8 LTI Memoryless Nonlinearity () x n () un () zn Fig. 1. The Wiener model. and predistortion, it has such problems as large physical size and great power consumption. The proposed scheme can be implemented by a simple circuit which reconfigures the signal routes for the nonlinearity identification and the error cancellation. This paper is organized as follow. In section II, we introduce the Wiener model to represent a nonlinear amplifier with memory. In section III, we describe the structure of the proposed feedforward compensator and the numerical formulas for adaptive algorithm. In section VI, we show the performance of the proposed feedforward scheme through computer simulations. Finally, we conclude our work in section V. II. PA MODEL WITH MEMORY Taking the memory effect into account, we adopted a Wiener model among finite-memory nonlinear models. It consists of the cascade connection of an LTI system followed by a memoryless nonlinear system (see Fig. 1). In this paper, the memoryless nonlinear system is implemented by using the Saleh’s traveling wave tube amplifier (TWTA) model [4]. The output of the L th -order LTI finite impulse response (FIR) block of the Wiener model is given by 1 0 () ( ) L l l un axn l − = = − ∑ (1) where l a ’s are the impulse response values, and L is the duration of the impulse response. Let () un be a complex envelope which can be expressed as ( ) () () j n u un ne φ ρ = (2) where () u n ρ is the amplitude and () n φ is the phase. The output signal () zn of the Wiener model is expressed as {() [ ( )]} () [ ( )] u j n n u zn A ne φ ρ ρ +Φ = (3) Digital Feedforward Compensation Scheme for the Nonlinear Power Amplifier with Memory Junsu Kwon, Changsoo Eun, Member, IEEE 169 The 6th International Conference on Information Technology and Applications (ICITA 2009)
Transcript
Abstract--In wideband communication systems, memory effect manifests itself as inter-symbol interference (ISI), which combined with the nonlinearity, complicates the nonlinear compensation. We propose a digital feedforward linearization method to compensate for the nonlinearity of a power amplifier (PA) with memory. The PA is modeled with a Wiener model, which is a linear time-invariant (LTI) system followed by a memoryless nonlinear system. The advantage of the proposed scheme is that the compensation is implemented in baseband by using a digital signal processing (DSP) technique. Simulation results are given to verify the validity of the proposed scheme.
Index Terms--digital signal processing, feedforward
compensation, high power amplifier, nonlinear system.
I. INTRDUCTION Orthogonal frequency division multiplexing (OFDM) is
currently used in various applications such as digital broadcasting, wireless local area network (WLAN), broadband internet access, and mobile communications. OFDM-based systems have high spectral efficiency, and robustness against ISI and multipath fading. However, they are sensitive to nonlinearities due to the high peak-to-average power ratio (PAPR). The nonlinear effects include inter-modulation distortion (IMD), out-of-band spectral re-growth and reduction in signal-to-noise ratio (SNR), etc. To solve such problems, many linearization techniques have been continuously reported in literature [1]-[3].
In addition, memory effect cannot be ignored any longer when the signal bandwidth becomes wide. The memory effect means the previous input signal of the PA as well as the current input affects the current output.
We propose a digital feedforward compensation scheme for nonlinearity combined with memory effect. Although the feedforward method has been employed for wideband system and shown the best performance among feedback, feedforward,
This study was financially supported by research fund of Chungnam National
University in 2008. Junsu Kwon is studying in the Department of Information & Communications
Engineering, Chungnam National University, Daejeon City, Korea (e-mail: [email protected]).
Changsoo Eun, Dr., is a professor in the Department of Information & Communications Engineering, Chungnam National University, Daejeon City, Korea (e-mail: [email protected]).
ICITA 2009 ISBN: 978-981-08-3029-8
LTI MemorylessNonlinearity
( )x n ( )u n ( )z n
Fig. 1. The Wiener model.
and predistortion, it has such problems as large physical size and great power consumption. The proposed scheme can be implemented by a simple circuit which reconfigures the signal routes for the nonlinearity identification and the error cancellation.
This paper is organized as follow. In section II, we introduce the Wiener model to represent a nonlinear amplifier with memory. In section III, we describe the structure of the proposed feedforward compensator and the numerical formulas for adaptive algorithm. In section VI, we show the performance of the proposed feedforward scheme through computer simulations. Finally, we conclude our work in section V.
II. PA MODEL WITH MEMORY Taking the memory effect into account, we adopted a
Wiener model among finite-memory nonlinear models. It consists of the cascade connection of an LTI system followed by a memoryless nonlinear system (see Fig. 1). In this paper, the memoryless nonlinear system is implemented by using the Saleh’s traveling wave tube amplifier (TWTA) model [4]. The output of the Lth-order LTI finite impulse response (FIR) block of the Wiener model is given by
1
0( ) ( )
L
ll
u n a x n l−
=
= −∑ (1)
where la ’s are the impulse response values, and L is the duration of the impulse response. Let ( )u n be a complex envelope which can be expressed as
( )( ) ( ) j n
uu n n e φρ= (2)
where ( )u nρ is the amplitude and ( )nφ is the phase. The output signal ( )z n of the Wiener model is expressed as
{ ( ) [ ( )]}( ) [ ( )] uj n n
uz n A n e φ ρρ +Φ= (3)
Digital Feedforward Compensation Scheme for the Nonlinear Power Amplifier with Memory
Junsu Kwon, Changsoo Eun, Member, IEEE
169
The 6th International Conference on Information Technology and Applications (ICITA 2009)
where [ ]A ⋅ and [ ]Φ ⋅ are respectively AM/AM and AM/PM conversion functions which, in the Saleh’s model, are defined as
2[ ]1
a uu
a u
Aα ρ
ρβ ρ
=+
(4)
2
2[ ]1
p uu
p u
α ρρ
β ρΦ =
+ (5)
where aα , aβ , pα and pβ are adjustable constants whose
values are respectively chosen as aα = 1.96, aβ = 0.99,
pα = 2.53, pβ = 2.82 for simulations in this work.
III. FEEDFORWARD COMPENSATOR DESIGN Fig. 2 shows the block diagram of the proposed feedforward
compensator. We assume that the up/down conversion, A/D, and D/A conversion operations do not affect the signal quality. Their nonlinear behavior can be melt into the finite-memory nonlinear model, if any, which validates our assumption. Also, the error amplifier linearly amplifies signals since its input is small enough for the amplifier to operate in the linear region.
The proposed feedforward compensation algorithm is shown in Fig. 3. The key idea behind this method is that the output of the overall system is used as a means of checking the error between the undistorted and distorted signals. It is also possible to compensate for any distortion components caused by other system elements than the PA. In addition, the compensation is performed in baseband by using a DSP technique without the help of analog devices such as attenuator and phase shifter.
In Fig. 3, input signal ( )x n is split into three branches to drive the PA, to generate the estimated distortion signal ( )v n , and to be compared with the final output signal ( )y n . The role of the finite-memory nonlinear model is to estimate the nonlinear distortion component so that the output signal can be compensated by removing the distortion component with the estimated signal. The finite-memory nonlinear model can be described by the following Volterra series
1 1 1
(1) (2),
0 0 01 1 1
(3), ,
0 0 0
( ) ( ) ( ) ( )
( ) ( ) ( )
L L L
k k lk k l
L L L
k l mk l m
v n h x n k h x n k x n l
h x n k x x l x n m
− − −
= = =
− − −
= = =
= − + − − +
− − − +
∑ ∑∑
∑∑∑ (6)
where L is the memory length, kh , , k lh and , , k l mh are the Volterra kernels [5]. Here, we use a third-order Voterra series model. Equation (6) can be rewritten in a matrix form as
( ) ( )Tv n n= hx (7)
DAC PA
DAC
DSP ADC
Input signal
Oscillator
Up-conversion
Down-conversion
Erroramplifier
Output signal
Up-conversion
Fig. 2. Block diagram of the feedforward compensator.
( )x n ( )y n
( )v n
( )z n
( )e n
Fig. 3. Block diagram of the feedforward compensation algorithm.
where h is the Volterra kernel vector, and ( )nx is the input vector, which are defined as
Subtracting the output of the finite-memory nonlinear model
denoted by distortion signal ( )v n from the output signal of the PA, we obtain the final output signal which is compared with input signal through the feedback path. Ideally, the input and the final output signals should be identical if the nonlinearities are perfectly compensated. Our goal is to obtain the Volterra kernels so that the error signal ( )e n , the difference between the input and output, approach zero. Then the input signal and
170
output signal bear a linear relation. A cost function ( )J n to be minimized is defined as
{ }22( ) [ ( )] ( ) [ ( ) ( )]J n E e n E x n z n v n= = − − (10)
Once the Volterra kernel vector minimizing the cost function is obtained, the optimal nonlinearity compensation is attained. The Volterra kernel vector is adaptively updated by the variable step size least mean squares (VSS-LMS) algorithm [6] until it converges, and the update rule is given by
( 1) ( )
( ) ( ) ( )n n
μ n n e n+
= +h h x (11)
where ( )nμ is the variable step size, which is given by
2( 1) ( ) ( )n n e nμ αμ γ+ = + (12)
where α , γ are suitable constants which determine the update amount.
IV. SIMULATION RESULTS The examination for the PA linearization is, typically, to
confirm the suppression of the spectral re-growth and improvement of bit error rate (BER) performance. We demonstrate the validity of the proposed digital feedforward compensator through computer simulations.
For simulation, we use a 2.5 GHz mobile WiMAX signal based on OFDM technique with QPSK modulation. The parameters for VSS-LMS algorithm are set to α = 0.97, γ = 4.8 × 10-4. The impulse response values of the FIR filter of the Wiener PA model are (used in [7]):
a = [0.7692, 0.1538, 0.0769] (13)
The AM/AM conversion output versus the input signal is
shown in Fig. 4. It can be seen that one input corresponds to multiple outputs due to the memory effect. Fig. 5 shows the nonlinear distortion component, error signal, modeled by the VSS-LMS algorithm as discussed in the previous section.
Fig. 6 shows the signal constellation for QPSK symbols with and without compensation. Note that it shows the received signal without noise to examine the degree of distortion by the nonlinear PA with memory. The error vector magnitude (EVM) represents the normalized deviation between the measured symbols and the desired signal point. As a result of the compensation, the EVM has improved from 0.3819 to 0.0636.
In Fig. 7, the BER curves for an additive white Gaussian noise (AWGN) channel are shown. The proposed feedforward compensator achieves an improvement of about 7.5 dB at BER = 10-4 compared with the un-compensated system. The difference of the performances between the compensated system and the ideal linear system is within 2 dB at BER = 10-4.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Normalized input amplitude
Nor
mal
ized
out
put a
mpl
itude
distorted signal
linear
Fig. 4. AM/AM response.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Normalized input amplitude
Nor
mal
ized
out
put a
mpl
itude
Fig. 5. Error signal estimated by Volterra model.
Fig.8 shows the power spectral densities (PSDs) of the input
signal, the distorted output signal, and the compensated output signal. With feedforward compensation, we can confirm that the spectral re-growth is suppressed by more than 20 dB.
Finally, we conduct a two-tone test to check the suppression of IMD. The two-tone test signals are of frequencies f1 = 2.4995 GHz and f2 = 2.5005 GHz. In Fig. 9, we show the results where it can be seen that the third-order IMD at frequencies (2f1-f2) and (2f2-f1) is suppressed by more than 40 dB.
V. CONCLUSION We proposed a digital feedforward method to compensate
for the nonlinear PA with memory, which circumvents the issues originating from the use of analog components. It may readily be implemented by using application-specific integrated circuit (ASIC) or field programmable gate array (FPGA). Experimental results show that the proposed scheme
171
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
Real
Imag
Fig. 6. QPSK constellation for the mobile WiMAX signal.
(gray points: without compensation, black points: with compensation)
0 5 10 15
10-4
10-3
10-2
10-1
100
Eb/N
0 [dB]
BER
inputwithout Feedforwardwith Feedforward
Fig. 7. BER performance for the mobile WiMAX signal in AWGN channel. improves the out-of-band spectral re-growth and the performance of BER for a 2.5 GHz mobile WiMAX signal which uses OFDM. It implies that the proposed feedforward compensator effectively compensates for the PA nonlinearity with memory such that it may improve the PA efficiency.
REFERENCES [1] A. S. Wright, W. G. Durtler, "Experimental Performance of an Adaptive
Digital Linearized Power Amplifier," IEEE Trans. Vehicular Technology, vol. 41, No. 4, pp. 395-400, Nov. 1992.
[2] Bo Shi, L. Sundstrom, "Linearization of RF Power Amplifier Using Power Feedback," IEEE Trans. Vehicular Technology, vol. 2, pp. 1520-1524, July 1999.
[3] N. Pothecary, "Feedforward Linear Power Amplifiers," ArtechHouse, 1999.
[4] Adel A. M. Saleh, "Frequency-Independent and Frequency-Dependent Nonlinear Models of TWT Amplifiers," IEEE Trans. Commun., vol. COM-29, pp. 1715-1720, Nov. 1981.
2.5 2.6 2.7 2.8 2.9 3
x 109
-80
-70
-60
-50
-40
-30
-20
-10
0
Frequency
Pow
er d
istrib
utio
n [d
B]
original input
without compensation
withcompensation
Fig. 8. Spectral distributions of the mobile WiMAX signal.
Fig. 9. Spectral distributions from the two-tone signal test with 2.4995 GHz and 2.5005 GHz. (dotted line: without compensation, solid line: with compensation)
[5] Changsoo Eun, Edward J. Powers, "A New Volterra Predistorter Based on
the Indirect Learning Architecture IEEE Trans. Signal Processing, vol. 45, pp. 223-227, Jan. 1997.
[6] R. H. Kwong, E. W. Johnston, "A Variable Step Size LMS Algorithm," IEEE Trans. Signal Processing, vol. 40, pp. 1633-1642, July 1992.
[7] Changsoo Eun, Edward J. Powers, "A Predistorter Design for a Memory-less Nonlinearity Preceded by a Dynamic Linear System," Proceedings of the IEEE Global telecommunications conference, pp. 152–156, Nov. 1995.
[8] Junseok Oh, Min Kim, Jongman Gim, Changsoo Eun, "Adaptive Feed-forward Compensation for High Power Amplifier Nonlinearity Using a Digital Signal Processing Technique," ICEIC2008, June 2008.
