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Spectrum Modeling of an RF Power Amplifier for TDMA Signals
Nonlinear effects of an RF power amplifier in a TDMA system
One of the critical and costly components in digital cellular communication systems is the RF power amplifier. Theoretically, one of the main concerns in an RF power amplifier design is the nonlinear effect of the amplifier. Quantitatively, no explicit relationship or expression currently exists between the outofband emission level and the nonlinearity description related to the thirdorder intercept point (IP_{3} ). Further, in experiments and analysis, it was discovered that, in some situations, using only IP_{3} is not accurate enough to describe the spectrum regrowth, especially when the fifthorder intercept point (IP_{5} ) is relatively significant compared to the thirdorder intermodulation. In this article, the nonlinear effect of an RF power amplifier in a time division multiple access (TDMA) (IS54 Standard) system is analyzed and an expression is presented of the estimation of the outofband emission levels for a TDMA power spectrum in terms of the IP_{3} and the IP_{5} , as well as the power level of the signal. This result will be useful in the design of RF power amplifiers for TDMA wireless systems.
In recent years, TDMA has been recognized as one of the most efficient and reliable schemes for cellular radio communications.^{1,2} As in other communication systems, one of the critical and costly components in TDMA systems is the RF power amplifier. One of the main concerns in RF power amplifier design is its nonlinearity, which can degrade the quality of the TDMA signal, increasing bit error rate and interference to adjacent channels. The level of nonlinearity is specified in the IS54 standard^{3} by the outofband power emission levels. It is also called spectrum regrowth. Traditionally, the nonlinearity of an RF amplifier is described by using IP_{3} .^{4,5} In experiments and analyses, it was discovered that, in some cases, using only IP_{3} is not accurate enough to describe the spectrum regrowth, especially when the fifthorder intermodulation is relatively significant compared the thirdorder intermodulation. Quantitatively, to the best of our knowledge, there is no explicit relationship or expression between the outofband emission level and the traditional amplifier nonlinearity description for the TDMA signal amplification. The lack of such a relationship makes it difficult for RF power amplifier designers to choose components. In an early effort, the nonlinear effect of an RF power amplifier on code division multiple access (CDMA) systems was analyzed.^{6,7} Continuing this effort into developing the spectrum analysis approach for TDMA signals, the expressions of the estimated outofband emission levels for TDMA signal are derived, and the relationship between an amplifier's outofband power emission levels and its nonlinearity parameters, IP_{3} and IP_{5} , is presented. The results presented in this article allow RF amplifier designers to specify and measure the TDMA signal amplifiers using simple IP_{3} and IP_{5} descriptions. The expression turns out to be simpler and easier to use in the case where IP_{5} may be ignored. In addition, a spectrum comparison between the simulated and predicted results is presented.
MODEL DESCRIPTION
The TDMA Signal Mathematical Model
Generally, the mathematical model of the IS54 TDMA signal can be presented as^{3}
where
h(t)  =  baseband filter that has a linear phase and square root raised cosine frequency response 
A  =  constant, depending only on the minimum TDMA symbol energy 
Φ_{n}  =  absolute phase corresponding to the nth symbol interval 
T_{s}  =  symbol period, equal to 41.15 µs for IS54 standard 
f_{c}  =  carrier center frequency 
_{0 }  =  initial phase 
Re{x}  =  real part of {x} 
g(t)  =  Ah(tnT_{s} )e^{j( 0+ n)} , which is a pulseshaped nonreturntozero (NRZ) function. Its power spectrum density (PSD) can be obtained through a lengthy derivation: 
P_{g } = A^{2} R_{s} H(f)^{2} (2)
where
R_{s } = 1/T_{s}  =  symbol rate 
H(f)  =  frequency response of the baseband filter 
Since the spectrum of a bandpass signal is directly related to the spectrum of its baseband envelope, the PSD of a TDMA signal, s(t), can be expressed as^{4}
where s(t) = Re{g(t) * e^{jωct} } was given in Equation 1.
Equivalently, the mathematical model of s(t) can also be described as
s(t) = r(t)cos^{ } (t)cos(2πf_{c} t + ^{ } _{0} ) r(t)sin^{ } (t)sin(2πf_{c} t + ^{ } _{0} )
= r(t)cos[2πf_{c} t + ^{ } (t) + ^{ } _{0} ] (4)
where
and is the baseband envelope of s(t). Its Fourier transform can be derived from the PSD of g(t), P_{g} through a lengthy derivation
where F{r(t)} is the Fourier transform of {r(t)}.
A Power Amplifier's Mathematical Model
Generally speaking, a practical amplifier is only a linear device in its linear region, meaning that the output of the amplifier will not be exactly a scaled copy of the input signal when the amplifier works beyond the linear region. Considering an amplifier as a functional box, it can be modeled by a Taylor series.^{4,5} The Taylor series model is only valid for a memoryless nonlinearity function. For a memoryless amplifier with only a few stages, a Taylor series model is fairly good for predicting the nonlinearity. Therefore, the Taylor series is adopted for modeling RF power amplifiers. Using the TDMA signal equivalent mathematical model s(t) of Equation 4, the output of an amplifier can generally be written as
y(t) = O{s(t)} = F[r(t)] cos {2πf_{c} t + ^{ } (t) + ^{ } _{0 } + Φ[r(t)]}(6)
O{s(t)}  =  operation of amplifier 
F[r(t)]  =  amplitude to amplitude conversion (AM/AM) 
Φ[r(t)]  =  amplitude to phase conversion (AM/PM) 
The functions F[r(t)] and Φ[r(t)] are dependent on the nonlinearity of the amplifier and modeling type.
For the memoryless power amplifier, AM/PM conversion causes only a deterministic constant phase shift to be added to the argument of the signal at the output but has no other effect on its phase,^{8} that is, Φ[r(t)] = ^{ } _{ap} , which is the constant phase shift. Therefore, Equation 6 becomes
y(t) = 0{s(t)} = F[r(t)] cos (2πf_{c} t + ^{ } ) (7)
where
^{ } = ^{ } (t) + ^{ } _{0 } + ^{ } _{ap}
Let (t) = F[r(t)], the Taylor expansion of O{s(t)} can be used to determine (t). Generally, the Taylor model of an RF amplifier can be written as
Here, only the oddorder terms in the Taylor series are considered, since the spectra generated by the evenorder terms are at least f_{c} away from the center of the passband; the effects from these terms on the passband are negligible. Furthermore, as a linear amplifier, the third and fifthorder terms dominate in Equation 8 for distortion. Therefore, in this analysis, the following model is used for an RF amplifier
Substituting the input passband signal s(t) = r(t)cos
(2πf_{c} t + ^{ } ) into y(t) of Equation 9 (after manipulation) produces
where
with
Here, the coefficient a_{1} is related to the linear gain G of the amplifier, and the coefficients a_{3} and a_{5} are directly related to IP_{3} and IP_{5} , respectively. It can be proven after a lengthy derivation that the expression for these coefficients becomes
From Equations 10 to 13, it can be seen that an amplifier's output y(t) is a function of G, IP_{3} , IP_{5} and the input signal s(t). Consequently, using Equation 10 and the PSD of s(t) in Equation 3, the PSD of y(t) can be calculated and the power emission levels can be determined. Therefore, all of the nonlinear effects of the amplifier with the TDMA signals can be evaluated.
THE POWER SPECTRUM DENSITY (PSD) OF THE AMPLIFIED TDMA SIGNAL
Now, the PSD of y(t) can be calculated. Since y(t) = (t) cos(2πf_{c} t + ^{ } ), the PSD of y(t) can be determined by the PSD of (t) as^{4,5}
and then the PSD of (t) can be derived by WienerKhintchine Theorem as^{4}
By definition, R_{ } (*) is expressed as
where E{x} is the mathematical expectation of {x}.
Since
(t) = ã_{1} r(t) + ã_{3} r^{3} (t) + ã_{5} r^{5} (t)
P_{ } (f) can be expressed in terms of the Fourier transform of r(t) through a lengthy derivation as
P_{ } (f)  =  F{R_{ } (*)} = F{E{ (t) * (t+*)}}  
 =  ã_{1} ^{2} F{r(t)}F{r(t)}+2ã_{1} ã_{3} F{r(t)}F{r^{3} (t)}  
 +  2ã_{1} ã_{5} F{r(t)}F{r^{5} (t)}  
 +  ã_{3} ^{2} F{r^{3} (t)}F{r^{3} (t)}+2ã_{3} ã_{5} F{r^{3} (t)}F{r^{5} (t)}  
 +  ã_{5} ^{2} F{r^{5} (t)}F{r^{5} (t)}  (17) 
where F{r(t)} = A sH(f) was obtained in Equation 5.
Let
as a result
F{r(t)} = A s P_{1} , F{r^{3} (t)} = (A s )^{3} P_{3}
and
F{r^{5} (t)} = (A s )^{5} P_{5}
where
P_{3 } = P_{1} P_{1} P_{1} , P_{5 } = P_{1} P_{1} P_{1} P_{1} P_{1}
in which * denotes a convolution operator. Also, the linear portion of the amplifier output P_{0} can be described as^{4,5}
Substituting P_{1} , P_{3} , P_{5} and P_{0} into Equation 17 produces
By the relationship between P_{y} (f) and P_{ } (f) in Equation 14, the final result of the power spectrum P_{y} (f) of y(t) in terms of G, IP_{3} , IP_{5} and P_{0} becomes
where
f_{c } = carrier center frequency
If IP_{5} is ignored, Equation 20 becomes
Thus, Equation 21 is a special case of Equation 20. It provides a simpler and easier result to use in the case where IP_{5} may be ignored.
Several observations are made by inspecting Equation 21: the first term
corresponds to the linear output power density; the remaining terms in Equation 21 are caused by the nonlinearity. In other words, these remaining terms are due to the intermodulation. For a linear amplifier, the intermodulation is usually much lower than the linear output power. Therefore, the intermodulation does not affect the passband spectrum significantly.
With the explicit power spectrum of the output TDMA signal, the outofband spurious emission power may be calculated in a particular frequency band. It is this power that is used in IS54 to specify the limit for the outofband emission level. To keep the result easy to use, only IP_{3} is considered here.
Let a frequency band be defined by f_{1} and f_{2} outside the passband. Using the results from P_{y} (f) of Equation 21, the emission power level within the band (f_{1} , f_{2} ), denoted as P_{IM3} (f_{1} , f_{2} ), can be determined easily by
Equation 22 can also be expressed as
where
In most design procedures, a designer is concerned with the required IP_{3} for a given outofband emission level. To obtain the desired IP_{3} , Equation 23 is solved for IP_{3} with given P_{IM3} (f_{1} ,f_{2} ), which yields
where C_{1} , C_{2} and C_{3} are given in Equation 24.
This result provides a direct relationship between the outofband emission power of a TDMA signal power amplifier and its IP_{3} . With a given required IP_{3} , the power amplifier design for a TDMA signal becomes a conventional RF power amplifier design.
DESIGN EXAMPLE AND COMPARISON WITH SIMULATIONS
In this example, the result shown in Equation 25 is used to design a 4 W amplifier, which complies with the outofband emission level limit requirement proposed for TDMA standard amplifiers. The outofband emission level limits required in IS54 are given as follows: the total TDMA signal bandwidth is 30 kHz. In the band of (f_{c } + 18 kHz) to (f_{c } + 47.5 kHz), the suppression level between the output power and emission power at 0.72 kHz bandwidth must be larger than 45 dB.
For this amplifier, P_{0 } = 4W and for the (f_{c } + 18 kHz) to
(f_{c } + 47.5 kHz) band, the corresponding maximum P_{IM3 } (f_{1} ,f_{2} ) is expressed as
For the worst case, f_{1} and f_{2} are assumed at the lower edge of [f_{c } + 18 kHz, f_{c } + 47.5 kHz], that is f_{1 } = f_{c } + 18 kHz = f_{c } + 0.018 MHz and f_{1 } = f_{c } + 18 kHz + 0.72 kHz = f_{c } + 0.01872 MHz.
Then, from Equation 25, the required IP_{3} becomes IP_{3 } = 48.6 dBm. For the band described above, in order to meet the IS54 requirement, the TDMA amplifier must have an IP_{3} of at least 48.6 dBm.
As mentioned before, IP_{5} is not given in the data book. Fortunately, IP_{5} could be measured through a twotone test.^{9} Therefore, without loss of generality, IP_{5} can be assumed as 45 dBm at the same output power level, and the linear gain G of the power amplifier can be normalized to 1. Thus, the nonlinearity of the power amplifier can be simulated according to Equations 9 and 13, where G = 1, IP_{3 } = 48.6 dBm and IP_{5 } = 45 dBm.
In this simulation using MATLAB software, the TDMA signals were generated in accordance with the TDMA standard IS54B.^{3 } Figure 1 shows the power spectrum predicted from this example compared to the spectrum given by simulation. The simulated RF amplifier spectrum agrees with the analytically predicted spectrum in both the passband and shoulder area.
The predicted result using only IP_{3} vs. both IP_{3} and IP_{5} is shown in Figure 2. It can be seen clearly that a better fit exists when both IP_{3} and IP_{5} are used vs. IP_{3} only.


CONCLUSION
It was assumed traditionally that the effects of the fifth or higher order intermodulation could be ignored. However, if the fifthorder intermodulation is relatively high compared to the thirdorder intermodulation, the outofband emission power levels caused by fifthorder intermodulation can be significant.
In this article, a theoretical method is proposed to predict the output power spectrum of a TDMA standard RF power amplifier so that the traditional nonlinearity parameter IP_{3} and additional parameter IP_{5} are linked directly with outofband emission levels. This analysis makes it possible for RF power amplifier designers to use a conventional approach to design RF power amplifiers for TDMA signals. In addition to the results presented in this article, this derivation approach can be applied to outofband emission level analysis for other communication standards.
References
1. David D. Falconer, Fumiyuki Adachi and Bjorn Gudmundson, "Time Division Multiple Access Methods for Wireless Personal Communications," IEEE Communications Magazine, January 1995, Vol. 33, No. 1,
pp. 5057.
2. Jay E. Padgett, Christopher G. Günter and Takeshi Hattori, "Overview of Wireless Personal Communications," IEEE Communications Magazine, January 1995, Vol. 33, No. 1, pp. 2841.
3. EIA/TIA Interim Standard IS54B, Cellular System Dual Mode Mobile Station  Basestation Compatibility Standard, April 1992.
4. Leon W. Couch II, Digital and Analog Communication Systems, PrenticeHall Inc., New Jersey, 1996.
5. Theodore S. Rappaport, Wireless Communication Principle and Practice, PrenticeHall Inc., New Jersey, 1996.
6. Qiang Wu, Heng Xiao and Fu Li, "Linear RF Power Amplifier Design for CDMA Signals: A Spectrum Analysis Approach," Microwave Journal, December 1998, Vol. 41, No. 12, pp. 2240.
7. Heng Xiao, Qiang Wu and Fu Li, "Nonlinear Distortion Analysis on CDMA Communication Systems," IEE Electronics Letters, April 1998, Vol. 34, No. 8, pp. 730731.
8. John Minkoff, "The Role of AM/PM Conversion in Memoryless Nonlinear Systems," IEEE Transactions on Communications, February 1985, Vol. COM33, No. 2, pp. 139144.
9. Heng Xiao, Qiang Wu and Fu Li, "Measure A Power Amplifier's Fifthorder Interception Point," RF Design, April 1999, pp. 5456.
Chunming Liu received his BS and MS degrees in electrical engineering from Northwestern Polytechnic University, China, in 1995 and 1998, respectively. He is currently a PhD candidate in the department of electrical and computer engineering at Portland State University. His research interests include wireless communications, signal spectral analysis, and image and video signal processing.
Heng Xiao received his BS and MS degrees in computer and system engineering from Xiamen University, China, in 1982 and 1987, respectively. He also received his MS and PhD degrees in electrical and computer engineering from Portland State University in 1994 and 1999, respectively. Dr. Xiao is currently a system engineer with Lucent Technologies. Prior to that, he worked for Atlas Communications Inc. and Metro One Telecommunications Inc. His research interests include wireless communication and multichannel signal processing.
Fu Li received his BS and MS degrees in physics from Sichuan University, China, in 1982 and 1985, respectively. He received his PhD degree in electrical engineering from the University of Rhode Island in 1990. He is a professional engineer licensed in the state of Oregon. Since 1990 Dr. Li has been with Portland State University, where he is currently a professor of electrical engineering with indefinite tenure. He worked at Philips Laboratories and Prime Computer prior to 1990. He has been consulting for a number of companies, including Intel Corp., which he visited fulltime during his sabbatical leave. His research interests include wireless and network communications, and signal, image and video processing. He has published over seventy papers. Dr. Li received a Pew Teaching Leadership Award at the Second National Conference on Training and Employment of Teaching Assistants in 1989, and is also a recipient of several IEEE awards.
Qiang (Chung) Wu received his BSc and MSc degrees from Shanghai Jiao Tong University, Shanghai, China, in 1982 and 1984, respectively, and his DSc degree from Washington University, St. Louis, MO, in 1989, all in electrical engineering. From 1989 to 1994 he was with Communication Research Laboratory, McMaster University, Hamilton, Canada, as a senior research staff and adjunct professor. From 1994 to 1997 he worked at Celwave, a division of Alcatel North America, as an engineering manager for RF repeater and digital subsystem. Currently, as a senior staff system architect, he is working at Platform Networking Group of Intel. His research interests include RF and wireless system design, DSP for broadband access and array signal processing.