Solid State Power Amplifier Linearization Using the Concept of Best Approximation in Hilbert Spaces A. Aghasi, A. Ghorbani, H. Amindavar and H. Karkhaneh Department of Electrical Engineering Amirkabir University of Technology, Tehran, Iran Email:
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Abstract— Using the mathematical concept of best approximation in Hilbert spaces a predistorter structure is suggested to compensate both AM/AM and AM/PM conversion characteristics of a solid state power amplifier. To verify the effects of this linearization we generally consider it in a QAM signal transmission in an additive white Gaussian noise channel. The results are compared with the ideal linear case.
I. I NTRODUCTION Broadband wireless systems require bandwidth efficient modulation schemes such as high constellation QAM. The main drawback of such modulations is the large envelope fluctuations, making the system sensitive to nonlinearity of the high power amplifier. So the development of these modulations, needs to obtain techniques of better linearization of power amplifiers. Power amplifiers, in general, exhibit nonlinear distortion in both amplitude (AM/AM conversion) and phase (AM/PM conversion) [1]. There are different useful approaches that try to compensate the effects of nonlinear distortions. One is based on approximating inverses to apply post distortion or predistortion [2]. Another one is composing and inverting orthogonal polynomial representations which is capable of removing selected distortion terms [3]. In this paper at first a modification is made on previous model presented for the nonlinearity of solid state power amplifiers (SSPAs) [4]. We mathematically show that the modified model is more appropriate to express the nonlinear behavior of solid state power amplifier. A predistorter structure and an efficient method of compensation is then proposed and finally to see the performance of presented method we considered the compensated system in a QAM signal transmission in an additive white Gaussian noise channel, comparing the results with linear case. II. S OLID S TATE P OWER A MPLIFIER N ON -L INEARITY M ODEL Ghorbani [4] proposed the following formulas for SSPA model where is a function of input amplitude , representing AM/AM conversion and , also a function of input amplitude , representing AM/PM conversion as:
¾ ¾ ¾ ¾
(1)
(2)
0-7803-9584-0/06/$20.00©2006 IEEE.
The parameters and for are obtained from the measurable input-power and output-power and output phase shift characteristics by curve fitting techniques, this model is widely used [5], [6]. For a GaAs FET power amplifier presented in [4] the coefficients and for a normalized AM/AM and AM/PM are derived and we use it for this paper. We can see that for most SSPAs and are greater than unity and , are usually negative and they express the negative slope behavior of the SSPA after a peak in saturation for AM/AM and AM/PM responses. It can be mathematically shown that in (1) and (2) (3) and (4) This means that for some values close to zero and greater than zero the functions and are both decreasing causing some negative values for and i.e.
(5)
For larger absolute values of and the negative zero crossing of and is more noticeable. To avoid this phenomenon we propose a new but similar form for and by adding a and a term to the numerator of the first term of (1) and (2), which leads to ¾ (6) ¾ ¾ (7) ¾ Within this change the new and satisfy (3) and (8) which avoids and becoming decreasing and taking negative values near zero. According to (6) and (7) the modified model proposed for SSPA AM/AM and AM/PM found different values for the parameters and , in hence the new forms as follow: ¾ ¾ (9) ¾ ¾ ¾ (10) ¾
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TABLE I N EW AND OLD MODEL COEFFICIENTS FOR G A A S FET. old 8.1081 1.5413 6.5202 -0.0718
Ü Ü Ü Ü
new 7.851 1.5388 -0.4511 6.3531
old 4.6645 2.0965 10.88 -0.003
Ý Ý Ý Ý
new 4.6388 2.0949 -0.0325 10.8217
QAM
PD
XMTR
SSPA
Coherent RCVR
AWGN
The previous and new coefficients for GaAs FET model are available in Table I. Fig. 1.
III. THE LINEARIZATION PRINCIPLE
Schematic diagram of the transmission system
A. System and Predistorter Structures Fig.1 shows a block diagram of a QAM transmission system. Using the notations of Fig.1 we assume that
¡
¡
Fig. 2.
(13)
Where denotes the inverse transmission of and and respectively represented modulus and argument of the complex envelope at the PD input. In fact, combination of (11) and (13) leads to ideal PD output as follows:
½
where we defined
½
(14)
(16)
(17)
Referring to the Weierstrass’ theorem [7] a suitably continuous memoryless nonlinearity with input x on a finite interval can be represented in terms of an N-th order polynomial such that for any given error as shown:
(18)
Appropriate sets of polynomials have been found when the signal belongs to a certain class of stationary random processes described in [8], [9]. We wish to obtain an approximate inverse
Functional block diagram of the A&P-PD
(15)
B. Power Series Representation And Compensation of Nonlinearities
´µ
of by choosing the polynomials appropriately, hence is represented in a polynomial form, this analogy also carries through for . To perform the best approximation of and we use the concept of the best approximation in Hilbert spaces [10]. Consider , the Hilbert space of real functions (as vectors of space) square integrable on the interval , and an inner product defined as
(19)
For a , and being a linearly independent sequence of vectors in , the best approximation of with through the choice of the coefficients , is obtained by solving (20) for .
A functional diagram like Fig. 2 can shed more light onto this PD.
(12)
Similar to [2], to obtain an exact linearization of SSPA, the input-output response of the PD should be equal to the inverse response of the SSPA itself, i.e.
(11)
is the signal complex envelope at the SSPA input, with and as its time-varying modulus and argument respectively. Recalling (9) and (10), the signal complex envelope at the HPA output can be written as follows:
.. .
.. .
.. .
.. .
.. .
(20)
Now we refer to the PD. To approximate by a polynomial, let the independent sequence of vectors be , where increasing provides a better approximation. The coefficients for are calculated through a matrix inversion in (20) and results in , which is the polynomial representation of . In order to find the polynomial expansion of the matrix at the right side of (20) can not be evaluated because there is no analytic form for . So we propose this new method in order to avoid the numerical difficulty associated with
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can write
½
½
we (M −1)d
After a change of variable as (21) can be written as:
. To calculate
or
(21)
d 2 3d
(22)
d
(23)
½
½
Λ 2,2
Λ 2,M/ 2
Λ 1,1
Λ 1,2
Λ 1,M/ 2
d
3d
d 1
Λ 2,1
d 1
or
Λ M/ 2,M/ 2
d M/ 2−1
,
¼
Λ M/ 2,1 Λ M/ 2,2
(M −1)d d M/ 2−1
d 2
Fig. 3. Illustration of transmitted elements and corresponding regions of correct decision
which finally leads to
(24)
½
(25)
The proposed scheme can be applied to other nonlinear systems as well and in comparison with the method proposed in [3], it is faster and reasonably precise. In the last section we are going to compute the bit error rate of 16, 64 and 256-level QAM transmission using polynomial representation of a given PD for different values of and . IV. E RROR P ROBABILITY C ALCULATION To see the linearization effect of A&P-PD, we consider probability of error ( ) for 16, 64 and 256-level QAM transmission. Generally we assume QAM modulation and in a manner similar to [4], [11], we derive a general form for . The diagram of Fig.3 illustrates the region of correct decision ¼ corresponding to the transmitted element being . For a given degree of back-off in dB ( ), from [12] we know that:
¼
¾ ¾
(27)
!
and
(28)
(30)
!
!
!
!
!
!
(31)
(32)
The average symbol error probability is finally obtained from (33) as:
¾ ¾
where for a given SNR, noise variance can be easily obtained. The error probability assuming is transmitted can be written as: ¼ ¼
(29)
¼ ¼ where and denoting the inphase and quadrature components of received signal are in the following form:
(26)
where is the carrier amplitude resulting in saturation and denotes the average transmitted power for this modulation and obtained by:
Using (24) in (20) one can obtain a polynomial representation of degree of as
is extracted using (26) and (27), and the average received power can be written as:
¾ ¾
(33)
For different values of , and , we optimized using Simplex method [13], by choosing the best decision levels,
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0
0
10
10
Nf =Ng =5
Nf =Ng =3
Ng =Nf =1 −1
10 −2
10
Pe
P
e
B0 =3 dB
−2
B0 =3 dB
10
B0 =6 dB
B0 =6 dB Ng =Nf =3
−4
10
−3
10
Linear
−6
10
10
Linear
−4
12
14
16
18
20
22
10
24
SNR (dB)
25
27
29
31
33
34
SNR (dB)
Fig. 4. Plot of probability of error versus SNR for 16-level QAM and A&PPD with different values of Æ and Æ , at different values of backoff.
23
Fig. 6. Plot of probability of error versus SNR for 256-level QAM and A&P-PD with different values of Æ and Æ , at different values of backoff.
−1
10
ACKNOWLEDGMENT This work is supported by Iran Telecommunication Research Center (ITRC) and we wish to thank them.
−2
10
P
e
R EFERENCES −3
10
Nf =Ng =3 B0 =3 B0 =6
−4
Nf =Ng =5
10
Linear
20
22
24
26
28
30
SNR (dB)
Fig. 5. Plot of probability of error versus SNR for 64-level QAM and A&PPD with different values of Æ and Æ , at different values of backoff.
¼ s. Note that in (29) ¼ and ¼ are considered as and respectively. Fig. 4, Fig. 5 and Fig. 6 show the error probability for different values of , and , in 16, 64 and 256-level QAM signal transmission.
V. C ONCLUSION In this paper a modified and enhanced model has been proposed to take into account the effects of amplitude nonlinearity and amplitude modulation to phase modulation conversion of the microwave solid state power amplifiers. We illustrate a novel approach for SSPA linearization in digital radio systems, based on the use of a predistortion circuit whose is obtained by inverse function polynomials applicable to any modulation format of the input signal. Finally the effect of this linearization is described and simulated.
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