Journal of Circuits, Systems, and Computers
 World Scientific Publishing Company
A NEW METHOD FOR CDMA SIGNAL MODELING IN NONLINEAR
SYSTEMS
ABBAS POURZAKI
Electrical Engineering Department, Faculty of Engineering, Ferdowsi University,
Mashhad, Iran
a_pourzaki@yahoo.com
KHALIL MAFINEJAD
Electrical Engineering Department, Faculty of Engineering, Ferdowsi University,
Mashhad, Iran
mafinezhad@um.ac .ir
SAYYED-HOSEIN KESHMIRI
Electrical Engineering Department, Faculty of Engineering, Ferdowsi University,
Mashhad, Iran
keshmiri@um.ac.ir
Received (1 February 2006)
Accepted (12 April 2006)
When a CDMA signal is passed through an RF transmitter, nonlinear elements cause spectral
regrowth which result in reduction of spectral efficiency. CDMA signal has a pseudo noise nature;
hence its mathematical treatment is too complex to analysis. In this paper, first it will be shown how
to simplify the complex mathematics of CDMA signal. Then a deterministic signal replaces the
CDMA signal and then the system response to both of them is calculated. In this paper, for the first
time, ACPR is explicitly calculated for both the CDMA and deterministic signals. ACPR is
calculated in terms of the nonlinear system coefficients and input power, and therefore, can be used
in design objectives. In addition, It will be shown that if input power of the deterministic signal is
multiplied by 20 (i.e. correction factor), ACPR error of these kinds of signals in -55dBc is less than
2.2% for the system nonlinearity orders up to 13. This correction factor is obtained by both
theoretical and simulation methods.
Keywords: Nonlinear systems; CDMA signal; deterministic signal; ACPR; correction factor.
1. Introduction
To support the communication systems design and evaluation, simulation techniques
have been utilized for a long period. In the three past decades, analysis and simulation
methods have been computerized and have had a fast development. The growth of new
complex systems has been the motivation for fast growth in using of these techniques.
These systems become complicate with advances in communication techniques such as
spread spectrum modulation.
1
2
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
Behavioral models are proper tools to predict system performance without
complexity of full simulation of the circuit [1]-[4]. Favorite measures for cellular
communication designers include spectral regrowth (see [5],[6]) and ACPR a [7]-[9].
Spectral regrowth cause interference with adjacent channel. Although performance, but
increasing Back-off tends to decrease power efficiency and finally decrease Handset
batteries.
When a CDMA signal is passed through an RF transmitter, nonlinear devices
generate spectral regrowth, and the spectral efficiency gained by using a CDMA scheme
is reduced. Since the spectral regrowth is mostly generated by a nonlinear RF power
amplifier, it is very important for RF system designers to predict the distortion effects of
power amplifiers on CDMA signals. To this end, many techniques have been developed,
each providing good results. Most of them build the baseband equivalent mathematical
model of a nonlinear power amplifier based on the quadrature decomposition technique
[10]. The spreading signal is then put into the power amplifier’s mathematical model and,
by using a Fast Fourier transform (FFT) algorithm, the amplified CDMA output
spectrum is calculated [11],[12]. Though this method can predict the output spectrum
accurately, it only includes mathematic of parameters of preliminary model and has not
the famous parameters of the RF system designers. After surveying the effects of
nonlinear systems on CDMA signal, simple relations are shown. Then it is proved that if
the goal is output power in the main band, responses of two kinds of signals are similar.
In addition, using correction factor ACPR of two kinds of signals are similar. In this
paper, two signals are considered:
Pseudo noise: This signal is an estimation of a real signal that describes CDMA
signals [12],[13]. This signal naturally is noise but its power spectrum function is known
and is uniform over its band.
Deterministic signal: This signal has deterministic nature; i.e.: its amplitude, phase
and frequency are determined. In this paper it is supposed that its power spectrum would
be the same as pseudo noise power spectrum function.
In the next section, nonlinear system response to pseudo noise is shown and more
simple relations will be presented. In the third section, nonlinear systems responses to
deterministic signal (that its spectrum shape is similar to pseudo noise signal) are shown.
In the fourth section for the first time, ACPR with respect to pseudo noise and
deterministic signals is analytically represented. In the fifth section by applying a
correction factor to deterministic signal, its ACPR will approximate CDMA signal’s
ACPR. This factor will be obtained theoretically and compared with simulation.
Conclusion is in the final section.
2. Nonlinear System Response to CDMA Signal
a - Adjacent Carrier Power Ratio
A New Method for CDMA Signal Modeling in Nonlinear Systems
3
The general transmit system can be simplified as Fig. 1, where a nonlinear power
amplifier is sandwiched between bandpass filters (BPFs) [14]. Input signal can be written
as:

x ( t )  x o ( t ) cosc t  ( t )   Re x o ( t )e j( t ) e jt

(1)
where c is an angular frequency and (t) is a phase of the carrier, and xo(t) denotes
the equivalent of x(t) in baseband. Due to the nonlinearities of the power amplifier, the
input signal experiences AM–AM and AM–PM distortions, and the output signal can be
represented as:

 

y( t )  Re y o ( t )e j( t ) e jt  Re Fx o ( t ) e j( t ) e jt  Fx o ( t )  cosc t  ( t )  Fx o ( t )  (2)
where yo(t) denotes the equivalent of y(t) in baseband and F(xo(t)) represents the
complex envelope transfer function of the power amplifier. In (2), | F(xo(t))| is equivalent
to AM–AM distortion and  F(xo(t)) to AM–PM distortion.
Our cellular system band is about 2MHz in 1.9GHz which can be considered as a
narrow band. In such narrow band memory effect can be negligible[14]. For a narrowband memoryless system (as the power amplifier that we work on it), F(xo(t)) function
can be presented by the odd order Taylor series:
n
y o ( t )  F( x o ( t )) 
 f 2k-1x o2k-1 (t)
(3)
k 1
Fig. 1. Block diagram of a general transmit system
The even-order terms are filtered out by the second BPF. The complex coefficients
are the results of AM-AM and AM-PM distortion. The complex coefficients can be
obtained from the odd-order complex polynomial fitting of the complex envelope transfer
function.
If x(t) is a CDMA signal, When a rectangular power spectral function (PSF) is used,
the power spectral density (PSD) of the baseband equivalent input signal is
N / 2
PSD xo (f )   o
0
| f | B
| f | B
(4)
where B is the bandwidth of the PSF and input power is Po=NoB. Using Analysis of
Yi[14], output PSD can be written:
4
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
n
PSD yo (f ) 
 a 2k-1D 2k-1 (f)
(5)
k 1
where:
1
a 2k 1 
(2k  1)!
n k

(2 j  2k  1)!
j 0
2 j. j!
2
f 2 j 2k 1P0j
(6)
and
D 2k 1  PSD XO (f )  PSD XO (f )    PSD XO (f )


(7)
2k 1 times
where  denotes convolution. Since PSDxo(f) has a rectangular shape, the calculation
of its convolution is straightforward, and is:
 1
1  P0


D 2k 1 (f )   (2k  2)! B  2

0



2k 1 k q

f 
 (1) r C 2rk 1  (2k  2r  1)  


B
 r 0




(2q  3)B  f  (2q  1)B
1 q  k
(8)
f  (2k  1)B
where q is number of considered band. As can be seen, PSDo is obtained in terms of
input power (Po), frequency and nonlinear coefficients of the system. If we want to
calculate output power (not PSD), it is enough that Eq. (5) is multiplied by 2B. So:
2 k 1 k q
 n 

f  
2
 P0 
 a
 (1) r C 2rk 1  (2k  2r  1)   
 
2 k 1


 
(2k  2)!  2 
B  
 r 0

k 1

 
Pout (CDMA) (f )  
(2q  3)B  f  (2q  1)B



f  (2k  1)B in calculatio n each term
0


(9)
Therefore, output power is independent of bandwidth (B in f normalize f). Note that
B
in each subband, the relation is different.
By expanding Eq. (6), we can get simpler relation for ak (independent of P0) and so
output power will be more convenient:
a 2k 1 

1
(2k  1)!
n k

j0
(2 j  2k  1)!
2 j. j!
2
f 2 j 2k 1P0j
1
(2k  1)!
(2  2k  1)!
(2 * 2  2k  1)!
f 2k 1 
f 2 2k 1P0 
f 2*2 2k 1P02  
(2k  1)!
1
2
8
2
A New Method for CDMA Signal Modeling in Nonlinear Systems
 (2k  1)! f 2k 1  k (2k  1)f 2 2k 1P0 
k (2k  1)( 2k  2)( 2k  3)
f 2*2 2k 1P02  
4
2
5
(10)
If we want the first term in | | be dominant, f2k-1 should be grater than the others:
f 2k 1  k (2k  1)f 2k 1P0
k (2k  1)( 2k  2)( 2k  3)
f 2k  3 P02
4
f 2k 1 
(11)

Therefore, with considering first term in Eq. (10) we will have:
a2k-1=(2k-1)!f22k-1
(12)
By using Eq. (12), we can determine upper limit of Po (system coefficients are known):
CAP
ID=C2
C=10000 pF
RES
ID=R1
R=4000 Ohm
TLIN
ID=TL2
Z0=50 Ohm
EL=49.5 Deg
F0=1.9 GHz
CAP
ID=C1
C=10000 pF
RES
ID=R2
R=5300 Ohm
2 C
4
S
1
CAP
ID=C4
C=0.91 pF
B
3
RES
ID=R4
R=250 Ohm
RES
ID=R5
R=630 Ohm
RES
ID=R3
R=225 Ohm
CAP
ID=C5
C=1.27 pF
IND
ID=L1
L=9.34 nH
GBJT
ID=GP1
DCVS
ID=V1
V=2.5 V
TLIN
ID=TL1
Z0=50 Ohm
EL=64 Deg
F0=1.9 GHz
E
IND
ID=L2
L=0.8 nH
TLIN
ID=TL3
Z0=50 Ohm
EL=90 Deg
F0=2 GHz
CAP
ID=C3
C=10000 pF
Fig. 2. Power amplifier circuit designed for 1.9GHz
Po 
Po 
f 2k 1
k (2k  1)f 2k 1
4f 2k 1
k (2k  1)( 2k  2)( 2k  3)f 2k  3
(13)

If Po satisfies the above conditions, we can use Eq. (6) instead of Eq. (12).
In this paper, the nonlinear system is a power amplifier that designed for 1.9GHz with
2MHz bandwidth. The circuit of this power amplifier is depicted in Fig. 2. Transfer
function curves (Pout-Pin) are illustrated in Fig. 3, which shows that the power amplifier is
strongly nonlinear. Using Matlab, polynomial coefficients up to 13 (v out-vin) is calculated
(Table 1)[15].
6
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
Table 1. The best polynomials that describe power amplifier
f
x1
x2
x3
x4
x5
x6
x7
x8
x9
1
3.462
2
3.462 .3639
3
4.896 .3639 -3.891
4
4.896 .5588 -3.891 -.3676
5
5.492 .5588 -8.482 -.3676 6.318
6
5.492 .3252 -8.482 .8231 6.318 -1.368
7
5.700 .3252 -11.49 .8231 16.61 -1.368 -9.668
8
5.700 .1298 -11.49 2.744 16.61 -6.811 -9.668 4.582
9
5.699 .1298 -11.48 2.744 16.54 -6.811 -9.511 4.582 -.1142
x10
x11
x12
x13
10
5.699 .1335 -11.48 2.686 16.54 -6.511 -9.511 3.986 -.1143 .4019
11
5.621 .1335 -8.861 2.686 -9.057 -6.511 91.46 3.986 -171.7 .4019 104.4
12
5.621 .0200 -8.861 5.386 -9.057 -28.05 91.46 78.61 -171.7 -115.7 104.4 66.34
13
5.607 .0200 -8.144 5.386 -19.51 -28.05 156.3 78.61 -368.4 -115.7 375.0 66.34 143.9
50
40
Output power (dBm)
30
20
10
0
-10
-20
-30
-40
-50
-100
-80
-60
-40
-20
Input power (dBm)
Fig. 3. Transfer functions of the power amplifier.
0
20
40
A New Method for CDMA Signal Modeling in Nonlinear Systems
7
Table 2. Back_off in 3dB error for various nonlinearities.
n
Back_off(dB)
3
7
5
6.5
7
6.9
9
6.8
11
7
13
6.6
Fig. 4 shows the output power in term of frequency. This figure shows that if we use
Eq. (6) instead of Eq. (12), in low power, output powers are approximately similar.
However, if power increases the error can not be negligible.
Input power = 12dBm
40
Output power (dBm)
20
CDMA
Approximate
0
-20
-40
-60
-80
-100
-6
-4
-2
0
Frequency(B)
2
4
6
Fig. 4. Effect of input power increment on output power spectrum.
3. Nonlinear System Response to Deterministic Signal
If input of the nonlinear system is a deterministic signal, input spectrum shape can be
considered such as Eq. (5). So amplitude of x(t) is P0 (in frequency domain). The
relation between x and yo is yet such as Eq. (3). By applying Fourier transform, we have:
n
Yo (f ) 
 f 2k-1X 2k-1(f)
k 1
(14)
8
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
where
X 2k 1  X(f )  X(f )    X(f )

(15)
2k 1 times
X(f) and Y(f) are frequency domain representations of input and output signals,
respectively. These relations are similar to Eq. (5) and Eq. (7) except: i- fk is substituted
by ak and ii- voltage of input and output are replaced with the power.
Then we can use Dy(f) in Eq. (8) for calculation of Y(f); Because as Po has a
rectangular spectrum with amplitude of No , X(f) has a rectangular spectrum with
amplitude of Po . Therefore, we can replace Po in Eq. (8) by Po to calculate X2k-1:
2 k 1

 P0 
k  q
f 
1
r 2 k 1 



  (1) Cr  (2k  2r  1)  

X 2 k 1 (f )   (2k  2)!  2 
B 

 r  0

0
f

(
2
k

1
)
B


(2q  3)B  f  (2q  1)B
(16)
Finally we can obtain output power as:
 n

Pyo (f )  Y (f )  
f 2k-1 X 2k-1 (f) 


 k 1

2

2
(17)
Now we can compare nonlinear system responses to both the CDMA and
deterministic signals (Fig. 5). In a linear system it is expected that system responses be
similar as seen in Eq. (5) and Eq. (17). So in nonlinear responses of main band, output
power of both kinds of signals is similar where output power is less damaged by
nonlinearities. However, in the sidebands, spectrums are different considerably.
Increasing the input power leads to increases nonlinearities and so the different in main
band output. In Table 2 Back-off (in 3dB difference) for various nonlinearities are shown.
This table shows that 7dB Back-off is enough to reduce error to less than 3dB.
A New Method for CDMA Signal Modeling in Nonlinear Systems
9
Input power = 15dBm
50
Output power(dBm)
CDMA
Deterministic
0
-50
-100
-6
-4
-2
0
Frequency(B)
2
4
6
Fig. 5. Output power response of CDMA and Deterministic (n=5).
4. ACPR for CDMA and Deterministic Signals
One of the measures of nonlinearities is ACPR. It can be presented as:
3B
ACPR 
 Pyodf
B
B
 Pyodf
B
(18)
10
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
n=5
-25
CDMA
Approximate
-30
ACPR (dBc)
-35
-40
-45
-50
-55
-60
-65
-5
0
5
Input power (dBm)
10
15
Fig. 6. Effect of simplification of CDMA formulas on ACPR for five terms nonlinearity.
To calculate the ACPR in CDMA response, we should integrate output power in
main and adjacent bands. Therefore, from Eq. (5) we have:
ACPR 
n

k 1

1
P 
 a 2k 1 (2k  1)!  2o 


2 k 1 k  2


r
r
2 k 1
 (2k  2r  2) 2 k 1
  (1) 2 k 1 C (2k  2r  4)
 r 0
n 
1
 Po 
2 *  a 2 k 1
 
(
2
k

1
)!
k 1 
 2



 (19)


r
r
2 k 1 
 (1) 2 k 1 C (2k  2r  2)

r  0

2 k 1 k 1


Now we compare ACPR results of original and simplified formulas, i.e. Eq. (6) and
Eq. (12) respectively (Fig. 6). In Fig. 6 nonlinearity order is five. As Eq. (14) indicates,
in low power two curves have overlap. But at upper powers, curves are distant.
To calculate the ACPR of deterministic signal, we should integrate Eq. (17). So we have:
3B

Pyodf
ACPR  B
B
3B
Y
(f )df
 B
(20)
B
 Pyodf  Y
B
2
B
2
(f )df
A New Method for CDMA Signal Modeling in Nonlinear Systems
11
n=5
0
-20
CDMA
Deterministic
ACPR(dBc)
-40
-60
-80
-100
-120
-20
-10
0
10
Input power (dBm)
20
30
20
30
a
n=11
0
-20
CDMA
Deterministic
ACPR(dBc)
-40
-60
-80
-100
-120
-20
-10
0
10
Input power (dBm)
b
Fig 7 ACPR variation in terms of Pin. (a) is for three term of nonlinearities and (b) is for five terms.
where Y(f) is substituted from Eq. (17). Now we can compare ACPR of CDMA and
deterministic signals in term of various input powers. In Fig. 7(a) these curves calculated
by third order nonlinearities and Fig. 7(b) considered five order.
5. Correction Factor
12
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
As seen in Fig. 7, difference of two curves are constant in low power and changes only
when power become moderately high and nonlinearity increase strongly. Therefore, we
can revise the inputs such that this difference becomes compensated. Now we calculate
correction factor (c) such that:
Pi(CDMA)  c Pi(det er)
(21)
which Pi(CDMA) denotes CDMA input signal and Pi(deter) denotes deterministic input
power. The aim is that ACPR of two signals will become similar using this factor; i.e.:
ACPR ( CDMA )  ACPR (det er)
(22)
Now, we calculate both sides of Eq. (22):
3B
 Py o(CDMA ) df
B
B
3B
 Py o(deter) df

 Py o(CDMA ) df
B
B
(23)
 Py o(deter) df
B
B
As can be seen in Fig. 7, in low power distance are constant. Nearly all nonlinear
systems such as power amplifiers show only third order nonlinearity effect in low power
(as power increases other nonlinearity effects appear and then curves distant will not be
constant). Therefore the correction factor is defined in low power. So using Eq. (17) and
Eq. (20) we have:
f 32 Pi3( CDMA )
f12 Pi (CDMA )  3f1f 3 Pi2(CDMA )  ...

1
20
f 32 Pi3(det er)
f12 Pi (det er)  56 f1f 3 Pi2(det er)  ...
(24)
For Correction factor it is needed that input power be low such that second and later
terms of bottoms can be ignored. i.e.:
Pi ( CDMA ) 
f1
3f 3
5f1
6f 3
Pi (deter) 
(25)
Then in this conditions, for calculating correction factor we have:
f 32 Pi3( CDMA )
2
1 i ( CDMA )
f P

1 2 3
20 3 i (det er)
2
1 i (det er)
(26)
Pi2(deter)
(27)
f P
f P
So:
Pi2(CDMA ) 
1
20
Now we can substitute Eq. (21) and extract correction factor:
A New Method for CDMA Signal Modeling in Nonlinear Systems
13
Table 3. Standard ACPR in some systems
IS-95
Base Station
-42 dBc
Handset
-55 to -42 dBc
IS-54/IS-136 Hansset
-48 to -30 dBc
PDC Handset
-62 to -47 dBc
Table 4 Maximum of ACPR with error of 3dB
n
ACPR(dBc)
3
13.5
5
0.1
7
-21.1
9
-20.5
11
-19.2
13
-15.9
Table 5. ACPR error in -55dBc for CDMA and
deterministic with correct coefficient
n
Error(dB)
Error(%)
3
.0004
.0097
5
.095
2.2
7
.084
1.95
9
.085
1.98
11
.083
1.92
13
.086
2.0
c  20  4.4721
(28)
Therefore, correction factor, for input power of deterministic signal, is 4.4721 and for
input voltage is 2.1147. This factor is independent of system coefficients and bandwidth.
Note that in above calculation it is supposed that third order nonlinearity effects will
appear before higher nonlinearities such as fifth order. If power increases and violate
from conditions of (25), error in ACPR will appear, but it will be shown that for common
power amplifiers this error in standard ACPR
sss is negligible.
14
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
n=3
0
CDMA
Deterministic
-10
-20
-30
ACPR(dBc)
-40
-50
-60
-70
-80
-90
-100
-20
-10
0
10
Input power (dBm)
20
30
a
n=7
-10
CDMA
Deterministic
-20
-30
ACPR(dBc)
-40
-50
-60
-70
-80
-90
-100
-20
-15
-10
-5
0
Input power (dBm)
5
10
15
b
n=13
0
-10
CDMA
Deterministic
-20
ACPR(dBc)
-30
-40
-50
-60
-70
-80
-90
-100
-20
c
-15
-10
-5
0
5
Input power (dBm)
10
15
20
A New Method for CDMA Signal Modeling in Nonlinear Systems
15
Fig 8 ACPR various shapes in term of Pin and in various nonlinearities orders with corrected deterministic
signal.
This correction factor is 20 for input power. This coefficient is independent of
system coefficients and bandwidth. ACPR curves in input power are shown in Fig. 8. Fig.
8(a) to Fig. 8(c) show various nonlinearity effects in nonlinearity of three, seven and 13.
This figure shows that the result of ACPR calculation at standard levels (see Table 3) is
satisfactory. If we consider error of 3dB, this error is occurred at power far away from
standard levels. This related ACPR in various nonlinearity orders is shown in Table 4.
In Table 5 the error of ACPR from CDMA and corrected deterministic signals at IS95 standard level (-55dBc) are inserted. From Table 5 it is obvious that maximum of
error is occurred at n=5 that is .12dB (equal to 2.8%) that is insignificant.
6. Conclusion
In this paper we analyzed and surveyed the effects of nonlinear systems on pseudo noise
(CDMA) and deterministic signals. Three new subjects were discussed: i- simplifying the
complex relation of CDMA response, ii- ACPR analysis of CDMA and deterministic
responses, and iii- Describing needed condition for substitution of pseudo noise with
deterministic signal.
Firstly, nonlinear system response to pseudo noise (CDMA) is calculated and a
simpler relation replaces complex relation. It is shown that if input power is low, the
response of the nonlinear system for both signals are moderately equal. If power is not
low, the large error will appear.
The response of nonlinear system to deterministic signal with uniform spectrum
(similar to PSD of CDMA) is calculated. As shown in this paper, system response to
CDMA and deterministic signal are nearly similar in main band. For the considered
power amplifier wich is nonlinear, output power Back-off about 7dB results in good
estimation when CDMA is replaced by deterministic signal.
In this paper for the first time, ACPR is analytically extracted for two kinds of signals
and the results are compared. It seemed that up to high level input power, the difference
between ACPR of the two kinds of signals is constant and so we could define a
correction factor for deterministic signal. Therefore, ACPR of CDMA can be calculated
from ACPR of the deterministic signal with acceptable error. Theoretically and using
simulation, it is shown that the correction factor of 20 for input power of the
deterministic signal can result in ACPR that is near to ACPR of CDMA signal.
This correction factor was independent of bandwidth and nonlinear system coefficients. It
is shown that we could accept results of deterministic ACPR 30 to 50 dB higher than
standard. Error of these two kinds of ACPR in -55dBc is .0004dB to .095dB (.0097% to
2.21%) which is a very good result.
The other more important result is that in practice it is not necessary to have a CDMA
excitation to predict CDMA ACPR. It is enough to consider the correction factor and
apply to a rectangular deterministic signal (or even a multitone) is calculated. If signal is
not strongly nonlinear, the resulting ACPR will be moderately equal to ACPR of CDMA
16
A. Pourzaki, K. Mafinejad, and S. H. Keshmiri
signal. Therefore, authors of this paper believe that it could help RF system designers to
predict the distortion effects of an RF power amplifier, and it is thought to be useful for
future software applications.
Acknowledgments
I wish to thanks Dr. Saremi and Dr. Nabovati. This work was supported in part by the
Iran Telecommunication Research Center (ITRC).
References
1. M. B. Steer, J. W. Bandler, and C. M. Snowden, “Computer-aided design of RF and
microwave circuits and systems”, IEEE Trans. Microwave Theory and Techniques, 50 (2002)
996-1005.
2. C. Maziere، T. Reveyrand، S. Barataud، J. M. Nebus، R. Quere، A.Mallet، L. Lapierre and J.
Sombrin، “A Novel behavior model of power amplifier based on a dynamic gain approach for
the system level simulation and design”, IEEE MTT-S digest, (2003) 769-772.
3. J. Wood، D. E. Root and N. B. Tufillaro، "A behavioral modeling approach to nonlinear
model-order reduction for RF/Microwave ICs and Systems" ، IEEE Trans. Microwave Theory
and Techniques, 52 (2004) 2274-2284.
4. J. Zhu, and T. J. Brazil, “Behavioral modeling of RF power amplifiers based on pruned
Volterra series ", IEEE Microwave and Wireless Components Letters,14 (2004) 563-565.
5. K. G. Gard, L. E. Larson, and M. B. Steer,” The Impact of RF front-end characteristics on the
spectral regrowth of communications signals”, IEEE Trans. Microwave Theory and
Techniques 53 (2005) 2179-2186.
6. C. Liang, , J. Jong, W. Stark and Jack R. East, “Nonlinear amplifier effects in communications
systems” IEEE Trans. Microwave Theory and Techniques, 47 (1999) 1461-1466.
7. T.S.L. Goh، R.D. Pollard، “ACPR predicting of CDMA systems through statistical behavioral
modeling of power amplifiers with memory” ، IEEE ICCS, (2002) 1189-1193.
8. C. Liu, H. Xiao, Q.Wu, F. Li, “Spectrum analysis of nonlinear distortion of RF power
amplifiers for wireless signals”, IEEE proceedings of ICCT, (2003) 1489-1492.
9. F. Liti, S. Chen and l. Chuing, "Computer simulation of nonlinear effects of RF power
amplifiers based on EVM and.. ACPR for digital wireless communications", Electronics
Letters, 36 (2000) pp.77-78.
10. C. Huang, H. Pai ad K. Chen, "Analysis of microwave MESFET power amplifiers for digital
wireless communications systems", IEEE Trans. Microwave Theory and Techniques, 52
(2004) 1284-1291.
11. V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing, (Wiley, New York, 2000).
12. Q.J. Zhang and K. C. Gupta, Neural Networks for RF and Microwave Design, (Artech
House,Norwood MA, 2000).
13. K. M. Gharaibeh, and M. B. Steer, “Characterization of cross modulation in multichannel
amplifiers using a statistically based behavioral modeling technique” IEEE Trans. Microwave
Theory and Techniques, 51 (2003) 2434-2444.
14. S. Yi، S. Nam، S. Oh and J. Han، "prediction of a CDMA output spectrum based on
intermodulation products of two-tone test"، IEEE Trans. Microwave Theory and Techniques,
49 (2001) 938-946.
15. Y. Yang, J. Yi, J. Nam, B. Kim and M. Park, “Measurement of two-tone transfer
characteristics of high-power amplifiers”, IEEE Trans. Microwave Theory and Techniques,
vol. 49 (2001) 568-571.