Behavioral Modeling of Microwave Power Amplifiers Using A Look Up ...
Proceedings of Asia-Pacific Microwave Conference 2007
Behavioral Modeling of Microwave Power Amplifiers Using A Look Up Table Method Yilong Shen*,
John Gajadharsing*,
Joe Tauritz**
*Business line cellular system ** Microwave communication lab NXP semiconductors University of Twente 6534 AE Nijmegen 7500AE Enschede The Netherlands The Netherlands Email: [email protected] Abstract— The possibility of building a microwave power amplifier (PA) behavioral model based on the look-up table principle is investigated. The model so constructed avoids the difficulties in model structure selection and/or its parameter estimation.
I.
INTRODUCTION
There has been considerable interest as of late in behavioral modeling of microwave PAs. PA behavioral models can be either static or dynamic. A static behavioral model assumes that a PA functions as a quasi memory-less input to output mapping, and is directly built from the gain compression (AM/AM) and amplitude-phase distortion (AM/PM) data of the PA. For this reason the static model is equivalently called the AM/AM AM/PM model. In many cases, however, the quasi memory-less assumption is not valid and a dynamic model is preferred. Dynamic models are of special interest in the performance simulation of mobile communication systems, where the long-term memory effects in PA are usually evident. The paradigm of building a dynamic behavioral model is basically a two-step procedure. Step one assumes a predefined model structure; step two estimates the parameters of this structure, typically using optimization. Such model structures are (equivalent to) specific combinations of linear timeinvariant filters and static nonlinearities. Being more general, a dynamic model is also more costly and less robust than its static counterpart. On the one hand, if ad hoc assumptions are made, then the generality of the model is not guaranteed; on the other hand, a model with universal approximation capability tends to suffer problems in the parameter extraction procedure [1]. The building of a dynamic model is closely related to a specific method of PA behavior characterization. The most widely applied one is the swept two-tone measurement [2]-[6]. Time domain characterization methods, though (possibly) less popular and less systematic, are also reported [7]-[10]. In this paper we investigate the possibility of building a dynamic PA behavioral model based on a look-up table principle, seeking to avoid the time-and-knowledge intensive tasks involved in identifying an appropriate model structure
1-4244-0749-4/07/$20.00 @2007 IEEE.
and/or estimating its parameters. We assume for the moment that there is no PM to AM distortion for the device-under-test (DUT), in order to simplify the validation of the modeling principle itself. In practice, if the PM to AM conversion of the DUT needs to be included, the model can be implemented by substituting the 2-D table with a 3-D table (though this generalization may require some efforts). II.
CHARACTERIZATION OF LONG TERM MEMORY EFFECTS
A. Formal Description The input-output map of a PA can be formulated as:
y (t ) = f nl ( x (t ),
∂x (t ) ∂ 2 x (t ) ∂ n x (t ) , ,..., ), ∂t dt 2 dt n
(1)
where x (t ) and y (t ) are the low-pass equivalent complex input and output signals, respectively. T
T
Let x (t ) = x (t ) = A (t )e
jconst
be a triangle envelope
T
signal of period T , while A (t ) is shown in Fig.1, and denote T
the corresponding output signal as y (t ) , we have from (1)
Abs ( y T (ti ) / x T (ti )) = h1nl ( AT (ti ), 4 A max / T, 0,..., 0) Phase( y T (ti ) / x T (ti )) = h 2nl ( AT (ti ), 4 A max / T, 0,..., 0) (2) under the condition that the measurements are made along the T
ascending edge of A (t ) at time points ti ( 1 ≤ i ≤ p ) and
τ m < t i − t 0 ( τ m is the memory period length
of the DUT),
because
dAT (ti ) = 4 A max / T , (3) dt d k AT (ti ) =0 , (4) dt k dϕ (ti ) d kϕ (ti ) = = 0, (5) dt dt k T where i = 1, 2,..., p, k = 2,..., n . The amplitude of A (t )
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∞ GAM ( A) and
Φ ∞PM ( A) , and they offer a better
characterization of the DUT when long-term memory effects are present. B. Examples T
T
The simulated GAM ( A) and Φ PM ( A) of a single stage BJT amplifier at 890.2 MHz and a single stage LDMOS amplifier at 850MHz will be shown in section IV, together with the model validation data. Fig. 1. A triangle envelope signal with period T
III.
2-D LOOK-UP TABLE BEHAVIORAL MODEL
Fig.2 shows the schematic of a 2-D look-up table model for approximating a behavioral model as (9). Please note (9) assumes no PM to AM distortion. When this assumption is not satisfied, ϕ (t ) is equally important as A(t ) and the model expands from 2-D to 3-D tables.
∂A(t ) ∂ 2 A(t ) Abs ( ~ y (t ) / ~ x (t )) = h1nl ( A(t ), , ) dt dt 2 ∂A(t ) ∂ 2 A(t ) Phase( ~ y (t ) / ~ x (t )) = h2 nl ( A(t ), , ) dt dt 2
Fig. 2. 2-D Look-up table model schematic for IMD3 simulation
is allowed to be negative (see fig.1), so its phase keeps
(9)
T
constant and (5) is satisfied, even when A (t ) changes polarity. If t i − t 0 < τ m , then (4) will no longer be satisfied for higher order derivatives and this in turn will lead to invalidation of (2). Define T GAM ( A) = h1nl ( A, 4 A max / T, 0,..., 0)
Φ TPM ( A) = h 2nl ( A, 4 A max / T, 0,..., 0) Comparing (6) with (2), it is easy to see that G
Φ
T PM
( A)
are
actually
T
.
Denote the (i, j)th entry of the R AM i, j
Abs ( y (ti ) / x (ti ))
τ m < ti − t0 . Φ
T PM
As
T approaches ∞ , G
∞ miAM ,0 = G AM ( Ai ) * Ai , ∞ miPM ,0 = Φ PM ( Ai ) ,
and
T
j+ miAM , j + = GAM ( Ai ) * Ai ,
T
j+ miAM , j − = GAM ( − Ai ) * Ai ,
( A) and
T
j+ miPM , j + = Φ PM ( Ai ) ,
( A) with A as a swept parameter will approach the
PM i, j −
m
static AM/AM and AM/PM characteristics, and are, hereafter, ∞
Φ ∞PM ( A) = h2nl ( A, 0,0,..., 0)
.
for T ≠ ∞ . For this reason G
T AM
(8)
(13) (14) (15)
The determination of the row and column indexes is straightforward. IV.
T PM
(12)
− A max ≤ A1 ≤ A2 ≤ ... ≤ Ap ≤ A max .
If the DUT has non-negligible memory effects, we expect
Φ TPM ( A) ≠ Φ ∞PM ( A)
=m
(11)
j + = −1× j − = 1, 2,..., q , 0 < Tq < Tq−1 < .... < T1 ,
(7)
T ∞ GAM ( A) ≠ GAM ( A)
PM i, j +
(10)
where
∞
denoted as GAM ( A) and Φ PM ( A) . Consequently, ∞ GAM ( A) = h1nl ( A, 0, 0,..., 0)
respectively. For
PM miAM , j and mi , j are then given by (10)-(15)
Phase( y T (ti ) / x T (ti )) , respectively, for A = AT (ti ) and T AM
and m
AM/AM and
illustrative convenience, we define 1 ≤ i ≤ p and −q ≤ j ≤ q ,
( A) and
T
PM i, j
AM/PM matrix as m
(6)
T AM
p×(2 q +1)
( A) and Φ ( A) with double variable parameters A and T are a generalization of
MODEL VALIDATION
A. The single stage BJT amplifier
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T
T
The GAM ( A) and Φ PM ( A) of this amplifier are shown
in Fig.3 and Fig.4. IMD3 simulation results using different methods are shown in Fig.5. In Fig.6 the ACPR results are compared, for an EDGE signal stimulus, with a symbol rate of 270.833 KHz and 16 samples per symbol. In total 3072 symbols are simulated. B. The single stage LDMOS amplifier T
T
The GAM ( A) and Φ PM ( A) of this amplifier are shown in Fig.7 and Fig.8. Fig.9 shows the simulated IMD3 results. Fig.10 shows simulated phase responses for a multi-tone input in the time domain. The difference between the amplitude responses is not so large and is therefore omitted here. In both cases, the following phenomena are observed which are attributed to long-term memory effects, T
Fig. 4. Simulated
Φ TPM ( A) of a BJT amplifier
T
1) The shape of GAM ( A) and of Φ PM ( A) diverge from ∞
∞
those of GAM ( A) and Φ PM ( A) gradually as T decreases; T
2) The twist (either up or down) of the left tails of GAM ( A) T
and Φ PM ( A) ( T ≠ ∞ ) grows as T decreases (the twist is on the left side since the observations are made on the ascending instead of the descending slope of the triangular envelope); 3)
T T G AM ( A) ≠ GAM (− A) T T Φ PM ( A) ≠ Φ PM (− A)
( A → 0, T ≠ ∞) .
V. CONCLUSION For the first time to our best knowledge, the possibility of modeling the dynamic behavior of microwave PAs employing a look-up table principle is demonstrated here. The model as built is a natural extension of the quasi-memoryless AM/AM AM/PM model, and promises to avoid the difficulties associated with traditional modeling approaches.
Fig. 5. Simulated IMD3 versus Pin and tone spacing
Fig. 6. Simulated ACPR (EDGE stimuli) versus Pin
Fig. 3. Simulated
G TAM ( A) of a BJT amplifier
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Fig. 7. Simulated
T GAM ( A) of a LDMOS amplifier
Fig. 10. Simulated phase responses under multi-tone stimuli (a) multi-tone input (b) circuit envelope simulation results (c) 2D look-up table model results (d) AM/PM model results
REFERENCES [1]
Fig. 8. Simulated
Φ TPM ( A) of a LDMOS amplifier
Fig. 9. Simulated IMD3 versus Pin and tone spacing
J. C. Pedro and S. A. Maas, “A comparative overview of microwave and wireless power amplifier behavioral modeling approaches,” IEEE Trans. Microwave Theory and Techniques, vol. 53, pp. 1150–1163, Apr. 2005. [2] W. Bösch and G. Gatti, “Measurement and simulation of memory effects in predistortion linearizers,” IEEE Trans. Microwave Theory and Techniques, vol. 37, pp. 1885–1890, Dec. 1989. [3] H. Ku, M. D. McKinley and J. S. Kenney , “Quantifying memory effects in RF power amplifiers,” IEEE Trans. Microwave Theory and Techniques, vol. 50, pp. 2843–2849, Dec. 2002. [4] C. J. Clark, C. P. Silva, A. A. Moulthrop, and M. S. Muha, “Poweramplifier Characterization using a two-tone measurement technique,” IEEE Trans. Microwave Theory and Techniques, vol. 50, pp. 1590– 1602, Jun. 2002. [5] J. K. Vuolevi, T. Rahkonen, and J. A. Manninen, “Measurement technique for characterizing memory effects in RF power amplifiers,” IEEE Trans. Microwave Theory and Techniques, vol. 49, pp. 1383– 1389, Aug. 2001. [6] K. A. Remley, D. M. Schreurs, D. F. Williams and J. Wood, “Extended NVNA bandwidth for long-term memory measurements,” in IEEE International Microwave Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 1739–1742. [7] A. Soury, E. Ngoya, J. M. Nebus and T. Reveyrand, “Measurement based modeling of power amplifiers for reliable design of modern communication systems,” in IEEE International Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 795–798. [8] P. Draxler, I. Langmore, T. P. Hung and P. M. Asbeck, “Time domain characterization of power amplifiers with memory effects,” in IEEE International Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 803–806. [9] A. Zhu, M. Wien, and T. J. Brazil, “A efficient Volterra-based behavioral model for wideband RF power amplifiers,” IEEE Trans. Instrument and Measurement, vol. 50, pp. 882–887, Aug. 2001 [10] J. I. Diaz, C. Pantaleon, I. Santamaria, T. Fernandez, and D. Martinez, “Nonlinearity estimation in power amplifiers based on subsampled temporal data,” IEEE Trans. Instrument and Measurement, vol. 50, pp. 882–887, Aug. 2001.
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Behavioral Modeling of Microwave Power Amplifiers Using A Look Up Table Method Yilong Shen*,
John Gajadharsing*,
Joe Tauritz**
*Business line cellular system ** Microwave communication lab NXP semiconductors University of Twente 6534 AE Nijmegen 7500AE Enschede The Netherlands The Netherlands Email: [email protected] Abstract— The possibility of building a microwave power amplifier (PA) behavioral model based on the look-up table principle is investigated. The model so constructed avoids the difficulties in model structure selection and/or its parameter estimation.
I.
INTRODUCTION
There has been considerable interest as of late in behavioral modeling of microwave PAs. PA behavioral models can be either static or dynamic. A static behavioral model assumes that a PA functions as a quasi memory-less input to output mapping, and is directly built from the gain compression (AM/AM) and amplitude-phase distortion (AM/PM) data of the PA. For this reason the static model is equivalently called the AM/AM AM/PM model. In many cases, however, the quasi memory-less assumption is not valid and a dynamic model is preferred. Dynamic models are of special interest in the performance simulation of mobile communication systems, where the long-term memory effects in PA are usually evident. The paradigm of building a dynamic behavioral model is basically a two-step procedure. Step one assumes a predefined model structure; step two estimates the parameters of this structure, typically using optimization. Such model structures are (equivalent to) specific combinations of linear timeinvariant filters and static nonlinearities. Being more general, a dynamic model is also more costly and less robust than its static counterpart. On the one hand, if ad hoc assumptions are made, then the generality of the model is not guaranteed; on the other hand, a model with universal approximation capability tends to suffer problems in the parameter extraction procedure [1]. The building of a dynamic model is closely related to a specific method of PA behavior characterization. The most widely applied one is the swept two-tone measurement [2]-[6]. Time domain characterization methods, though (possibly) less popular and less systematic, are also reported [7]-[10]. In this paper we investigate the possibility of building a dynamic PA behavioral model based on a look-up table principle, seeking to avoid the time-and-knowledge intensive tasks involved in identifying an appropriate model structure
1-4244-0749-4/07/$20.00 @2007 IEEE.
and/or estimating its parameters. We assume for the moment that there is no PM to AM distortion for the device-under-test (DUT), in order to simplify the validation of the modeling principle itself. In practice, if the PM to AM conversion of the DUT needs to be included, the model can be implemented by substituting the 2-D table with a 3-D table (though this generalization may require some efforts). II.
CHARACTERIZATION OF LONG TERM MEMORY EFFECTS
A. Formal Description The input-output map of a PA can be formulated as:
y (t ) = f nl ( x (t ),
∂x (t ) ∂ 2 x (t ) ∂ n x (t ) , ,..., ), ∂t dt 2 dt n
(1)
where x (t ) and y (t ) are the low-pass equivalent complex input and output signals, respectively. T
T
Let x (t ) = x (t ) = A (t )e
jconst
be a triangle envelope
T
signal of period T , while A (t ) is shown in Fig.1, and denote T
the corresponding output signal as y (t ) , we have from (1)
Abs ( y T (ti ) / x T (ti )) = h1nl ( AT (ti ), 4 A max / T, 0,..., 0) Phase( y T (ti ) / x T (ti )) = h 2nl ( AT (ti ), 4 A max / T, 0,..., 0) (2) under the condition that the measurements are made along the T
ascending edge of A (t ) at time points ti ( 1 ≤ i ≤ p ) and
τ m < t i − t 0 ( τ m is the memory period length
of the DUT),
because
dAT (ti ) = 4 A max / T , (3) dt d k AT (ti ) =0 , (4) dt k dϕ (ti ) d kϕ (ti ) = = 0, (5) dt dt k T where i = 1, 2,..., p, k = 2,..., n . The amplitude of A (t )
2591
∞ GAM ( A) and
Φ ∞PM ( A) , and they offer a better
characterization of the DUT when long-term memory effects are present. B. Examples T
T
The simulated GAM ( A) and Φ PM ( A) of a single stage BJT amplifier at 890.2 MHz and a single stage LDMOS amplifier at 850MHz will be shown in section IV, together with the model validation data. Fig. 1. A triangle envelope signal with period T
III.
2-D LOOK-UP TABLE BEHAVIORAL MODEL
Fig.2 shows the schematic of a 2-D look-up table model for approximating a behavioral model as (9). Please note (9) assumes no PM to AM distortion. When this assumption is not satisfied, ϕ (t ) is equally important as A(t ) and the model expands from 2-D to 3-D tables.
∂A(t ) ∂ 2 A(t ) Abs ( ~ y (t ) / ~ x (t )) = h1nl ( A(t ), , ) dt dt 2 ∂A(t ) ∂ 2 A(t ) Phase( ~ y (t ) / ~ x (t )) = h2 nl ( A(t ), , ) dt dt 2
Fig. 2. 2-D Look-up table model schematic for IMD3 simulation
is allowed to be negative (see fig.1), so its phase keeps
(9)
T
constant and (5) is satisfied, even when A (t ) changes polarity. If t i − t 0 < τ m , then (4) will no longer be satisfied for higher order derivatives and this in turn will lead to invalidation of (2). Define T GAM ( A) = h1nl ( A, 4 A max / T, 0,..., 0)
Φ TPM ( A) = h 2nl ( A, 4 A max / T, 0,..., 0) Comparing (6) with (2), it is easy to see that G
Φ
T PM
( A)
are
actually
T
.
Denote the (i, j)th entry of the R AM i, j
Abs ( y (ti ) / x (ti ))
τ m < ti − t0 . Φ
T PM
As
T approaches ∞ , G
∞ miAM ,0 = G AM ( Ai ) * Ai , ∞ miPM ,0 = Φ PM ( Ai ) ,
and
T
j+ miAM , j + = GAM ( Ai ) * Ai ,
T
j+ miAM , j − = GAM ( − Ai ) * Ai ,
( A) and
T
j+ miPM , j + = Φ PM ( Ai ) ,
( A) with A as a swept parameter will approach the
PM i, j −
m
static AM/AM and AM/PM characteristics, and are, hereafter, ∞
Φ ∞PM ( A) = h2nl ( A, 0,0,..., 0)
.
for T ≠ ∞ . For this reason G
T AM
(8)
(13) (14) (15)
The determination of the row and column indexes is straightforward. IV.
T PM
(12)
− A max ≤ A1 ≤ A2 ≤ ... ≤ Ap ≤ A max .
If the DUT has non-negligible memory effects, we expect
Φ TPM ( A) ≠ Φ ∞PM ( A)
=m
(11)
j + = −1× j − = 1, 2,..., q , 0 < Tq < Tq−1 < .... < T1 ,
(7)
T ∞ GAM ( A) ≠ GAM ( A)
PM i, j +
(10)
where
∞
denoted as GAM ( A) and Φ PM ( A) . Consequently, ∞ GAM ( A) = h1nl ( A, 0, 0,..., 0)
respectively. For
PM miAM , j and mi , j are then given by (10)-(15)
Phase( y T (ti ) / x T (ti )) , respectively, for A = AT (ti ) and T AM
and m
AM/AM and
illustrative convenience, we define 1 ≤ i ≤ p and −q ≤ j ≤ q ,
( A) and
T
PM i, j
AM/PM matrix as m
(6)
T AM
p×(2 q +1)
( A) and Φ ( A) with double variable parameters A and T are a generalization of
MODEL VALIDATION
A. The single stage BJT amplifier
2592
T
T
The GAM ( A) and Φ PM ( A) of this amplifier are shown
in Fig.3 and Fig.4. IMD3 simulation results using different methods are shown in Fig.5. In Fig.6 the ACPR results are compared, for an EDGE signal stimulus, with a symbol rate of 270.833 KHz and 16 samples per symbol. In total 3072 symbols are simulated. B. The single stage LDMOS amplifier T
T
The GAM ( A) and Φ PM ( A) of this amplifier are shown in Fig.7 and Fig.8. Fig.9 shows the simulated IMD3 results. Fig.10 shows simulated phase responses for a multi-tone input in the time domain. The difference between the amplitude responses is not so large and is therefore omitted here. In both cases, the following phenomena are observed which are attributed to long-term memory effects, T
Fig. 4. Simulated
Φ TPM ( A) of a BJT amplifier
T
1) The shape of GAM ( A) and of Φ PM ( A) diverge from ∞
∞
those of GAM ( A) and Φ PM ( A) gradually as T decreases; T
2) The twist (either up or down) of the left tails of GAM ( A) T
and Φ PM ( A) ( T ≠ ∞ ) grows as T decreases (the twist is on the left side since the observations are made on the ascending instead of the descending slope of the triangular envelope); 3)
T T G AM ( A) ≠ GAM (− A) T T Φ PM ( A) ≠ Φ PM (− A)
( A → 0, T ≠ ∞) .
V. CONCLUSION For the first time to our best knowledge, the possibility of modeling the dynamic behavior of microwave PAs employing a look-up table principle is demonstrated here. The model as built is a natural extension of the quasi-memoryless AM/AM AM/PM model, and promises to avoid the difficulties associated with traditional modeling approaches.
Fig. 5. Simulated IMD3 versus Pin and tone spacing
Fig. 6. Simulated ACPR (EDGE stimuli) versus Pin
Fig. 3. Simulated
G TAM ( A) of a BJT amplifier
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Fig. 7. Simulated
T GAM ( A) of a LDMOS amplifier
Fig. 10. Simulated phase responses under multi-tone stimuli (a) multi-tone input (b) circuit envelope simulation results (c) 2D look-up table model results (d) AM/PM model results
REFERENCES [1]
Fig. 8. Simulated
Φ TPM ( A) of a LDMOS amplifier
Fig. 9. Simulated IMD3 versus Pin and tone spacing
J. C. Pedro and S. A. Maas, “A comparative overview of microwave and wireless power amplifier behavioral modeling approaches,” IEEE Trans. Microwave Theory and Techniques, vol. 53, pp. 1150–1163, Apr. 2005. [2] W. Bösch and G. Gatti, “Measurement and simulation of memory effects in predistortion linearizers,” IEEE Trans. Microwave Theory and Techniques, vol. 37, pp. 1885–1890, Dec. 1989. [3] H. Ku, M. D. McKinley and J. S. Kenney , “Quantifying memory effects in RF power amplifiers,” IEEE Trans. Microwave Theory and Techniques, vol. 50, pp. 2843–2849, Dec. 2002. [4] C. J. Clark, C. P. Silva, A. A. Moulthrop, and M. S. Muha, “Poweramplifier Characterization using a two-tone measurement technique,” IEEE Trans. Microwave Theory and Techniques, vol. 50, pp. 1590– 1602, Jun. 2002. [5] J. K. Vuolevi, T. Rahkonen, and J. A. Manninen, “Measurement technique for characterizing memory effects in RF power amplifiers,” IEEE Trans. Microwave Theory and Techniques, vol. 49, pp. 1383– 1389, Aug. 2001. [6] K. A. Remley, D. M. Schreurs, D. F. Williams and J. Wood, “Extended NVNA bandwidth for long-term memory measurements,” in IEEE International Microwave Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 1739–1742. [7] A. Soury, E. Ngoya, J. M. Nebus and T. Reveyrand, “Measurement based modeling of power amplifiers for reliable design of modern communication systems,” in IEEE International Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 795–798. [8] P. Draxler, I. Langmore, T. P. Hung and P. M. Asbeck, “Time domain characterization of power amplifiers with memory effects,” in IEEE International Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 803–806. [9] A. Zhu, M. Wien, and T. J. Brazil, “A efficient Volterra-based behavioral model for wideband RF power amplifiers,” IEEE Trans. Instrument and Measurement, vol. 50, pp. 882–887, Aug. 2001 [10] J. I. Diaz, C. Pantaleon, I. Santamaria, T. Fernandez, and D. Martinez, “Nonlinearity estimation in power amplifiers based on subsampled temporal data,” IEEE Trans. Instrument and Measurement, vol. 50, pp. 882–887, Aug. 2001.
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