Abstract:
A pre-distorter and a power amplifier are combined in a communication system. The purpose of the power amplifier is to provide as high a power as possible to the orthogonal frequency division multiplexing (OFDM) signal being passed by the high power amplifier to the communication system. The pre-distorter inverts the nonlinearity of the amplifier, so that the combination of pre-distorter and high power amplifier exhibit a linear characteristic beyond the normal linear range of the high power amplifier. The pre-distorter is based on exact analytic expression for the description of the input-output characteristic of the pre-distorter based on an analytic model for the power amplifier. A mixed computational-analytical approach compensates for nonlinear distortion in the high power amplifier even with time-varying characteristics. This leads to a sparse and yet accurate representation of the pre-distorter, with the capability of tracking efficiently any rapidly time-varying behavior of the power amplifier.

Description:
RELATED APPLICATIONS  
       [0001]     The present application is related to U.S. Provisional Patent Application Ser. No. 60/602,905, filed on Aug. 19, 2004, which is incorporated herein by reference and to which priority is claimed pursuant to 35 USC 119. 
     
    
     BACKGROUND OF THE INVENTION  
       [0002]     1. Field of the Invention  
         [0003]     The invention relates to the field of pre-distorters in communications systems using power amplifiers in which the signal-dependent and time-varying parameters of the power amplifier are linearized by means of the pre-distorter.  
         [0004]     2. Description of the Prior Art  
         [0005]     Orthogonal frequency-division multiplexing (OFDM) is a method of digital modulation in which a signal is split into several narrowband channels at different frequencies. The technology was first conceived in the 1960s and 1970s during research into minimizing interference among channels near each other in frequency. In some respects, OFDM is similar to conventional frequency-division multiplexing (FDM). The difference lies in the way in which the signals are modulated and demodulated. Priority is given to minimizing the interference, or crosstalk, among the channels and symbols comprising the data stream. Less importance is placed on perfecting individual channels. OFDM is used in European digital audio broadcast services. The technology lends itself to digital television, and is being considered as a method of obtaining high-speed digital data transmission over conventional telephone lines. It is also used in wireless local area networks.  
         [0006]     Orthogonal frequency division multiplexing (OFDM) has several desirable attributes, such as high immunity to inter-symbol interference, robustness with respect to multi-path fading, and ability for high data rates. These features are making OFDM to be incorporated in emerging wireless standards like IEEE 802.11a WLAN and ETSI terrestrial broadcasting. However, one of the major problems posed by OFDM is its high peak-to-average-power ratio (PAPR), which seriously limits the power efficiency of the high power amplifier (HPA) because of the nonlinear distortion caused by high peak-to-average-power ratio. This distortion constitutes a source of major concern to the RF system design community.  
         [0007]     One of the most promising approaches for the mitigation of this nonlinear distortion is to use a pre-distorter, applied to the OFDM signal prior to its entry into the high power amplifier. For the most part previous pre-distorter-based approaches consisted of: (1) using a look-up table (LUT) and updating the table via least mean square (LMS) error estimation; (2) two-stage estimation, using Wiener-type system modeling for the high power amplifier, and Hammerstein system modeling for the pre-distorter; (3) simplified Volterra-based modeling for compensation of the high power amplifier nonlinearity; and (4) polynomial approximation of this nonlinearity.  
         [0008]     However, all of these techniques are based on a general approximation form for the nonlinear system, rather than on exploiting specific forms gleaned from physical device considerations.  
         [0009]     In the case of the look-up table, it is updated by an adaptive algorithm. This has the disadvantage of inherent quantization noise caused by the limited size of look up table and a long time involved in the update of look-up table after estimating the high power amplifier.  
         [0010]     In the case of the two-stage estimation, the estimation is utilized to estimate parameters of Wiener system to first estimate high power amplifier and then to estimate parameters for pre-distorter with the information of parameters for high power amplifier. This has the disadvantage of requiring a lot of time for the convergence of parameter estimates.  
         [0011]     In the case of using a Volterra-based pre-distorter, this approach utilizes direct as well as indirect learning structure to train the coefficients more efficiently. This has the disadvantage of complexity in the modeling and estimation of Volterra series.  
         [0012]     In the case of using polynomial approximation for high power amplifier and pre-distorter, the algorithm is generic, but it has the disadvantage of complexity incurred by polynomial approximation.  
         [0013]     In the case of using an exact inverse model of traveling wave tube amplifier this has the disadvantage of not fitting time varying high power amplifier systems.  
         [0014]     All of these techniques described above are based on a general approximation form for the nonlinear system, rather than on exploiting specific forms gleaned from physical device considerations.  
       BRIEF SUMMARY OF THE INVENTION  
       [0015]     The pre-distorter of the invention can be used any kind of wireless communications, e.g. cellular phone, digital video broadcasting, digital audio broadcasting, or any kind of wireline communications, e.g., a digital subscriber line (DSL) to enhance the power transmitted by a high power amplifier with the least nonlinear distortion. The invention can have immediate future use in hand-held wireless communication devices and in digital satellite communications.  
         [0016]     The invention is a pre-distorter. The pre-distorter is an electronic nonlinear signal processing device, which is placed before the high power amplifier, which in turn is connected to the transmitting antenna of a wireless communication system. The purpose of the high power amplifier is to provide as high a power as possible to the OFDM signal being passed by the high power amplifier to the transmitting antenna. However, a large increase in power forces the signal in the high power amplifier to go beyond the linear range of the high power amplifier. In order to enable this increase in power at the output of the high power amplifier while minimizing distortion, a pre-distorter is inserted before the amplifier. The pre-distorter inverts the nonlinearity of the amplifier, so that the combination of pre-distorter and high power amplifier exhibit a linear characteristic beyond the normal linear range of the high power amplifier. This process is called linearization.  
         [0017]     The special feature of the illustrated invention is that the design of the pre-distorter is based on exact analytic expression for the description of the input-output characteristic of the pre-distorter based on an analytic model for the high power amplifier. This permits accuracy and efficiency in the performance of the above linearization task by the OFDM signal transmission system.  
         [0018]     The fundamental principle governing the application is that orthogonal frequency division multiplexing has several desirable attributes which makes it a prime candidate for a number of emerging wireless communication standards, e.g. IEEE 802.11a and g WLAM and ETSI terrestrial broadcasting. However, one of the major problems posed by the OFDM signal is its high peak-to-average-power ratio, which seriously limits the power efficiency of the high power amplifier because of the nonlinear distortion resulting from high peak-to-average-power ratio.  
         [0019]     The illustrated embodiment provides a new mixed computational-analytical approach for compensation of this nonlinear distortion for the cases in which the high power amplifier is a traveling wave tube amplifier (TWTA) or a solid state power amplifier (SSPA) with time-varying characteristic. Traveling wave tube amplifiers are used in wireless communication systems when high transmission power is required as in the case of the digital satellite channel, and solid state power amplifiers are used for land-based mobile wireless communication systems. Compared to previous pre-distorter techniques based on look-up table or adaptive schemes, the illustrated embodiment relies on the analytical inversion of the Saleh traveling wave tube amplifier model and Rapp&#39;s solid state power amplifier model in combination with a nonlinear parameter estimation algorithm. This leads to a sparse and yet accurate representation of the pre-distorter, with the capability of tracking efficiently any rapidly time-varying behavior of the high power amplifier. Computer simulations results illustrate and validate the approach presented.  
         [0020]     In the illustrated embodiment, we describe a new approach to pre-distorter for high power amplifier by using the Saleh traveling wave tube amplifier model and Rapp&#39;s solid state power amplifier model for these devices and resorting to the exact closed form expression for its inverse represented by means of only a few parameters. This approach avoids a larger number of parameters that a generic approximation expression (like the polynomial approximation) would require for accurate representation.  
         [0021]     In the illustrated approach, we capitalize on the analytical model for the solid state power amplifier and traveling wave tube amplifier to derive cogent algorithms for two pre-distorters labeled respectively pre-distorter I and pre-distorter II. The pre-distorter I algorithm applies to the solid state power amplifier and pre-distorter II to traveling wave tube amplifier.  
         [0022]     The reason we use these two types of high power amplifiers is that these two types are very important for today&#39;s wireless communication systems. traveling wave tube amplifiers are normally used for satellite communications, and solid state power amplifiers are used for mobile communication systems. Considerable work on distortion compensation has been done for the traveling wave tube amplifier, because of severe nonlinearity of this type of amplifier. However, OFDM is expected to be a standard for next generation cellular systems in a combined form with code-division multiple access (CDMA) i.e. multiple carrier code-division multiple access (MC-CDMA) or multiple carrier direct sequence code-division multiple access (MC-DS-CDMA). Code-division multiple access is a digital cellular technology that uses spread-spectrum techniques. Unlike competing systems, CDMA does not assign a specific frequency to each user. Instead, every channel uses the full available spectrum. Individual conversations are encoded with a pseudo-random digital sequence. CDMA consistently provides better capacity for voice and data communications than other commercial mobile technologies, allowing more subscribers to connect at any given time. Multi-Carrier (MC) CDMA is a combined technique of Direct Sequence (DS) CDMA (Code Division Multiple Access) and OFDM techniques. It applies spreading sequences in the frequency domain.  
         [0023]     Therefore, the importance of solid state power amplifier will be then much greater than now. For this reason we also use a solid state power amplifier as a high power amplifier model. While a closed form expression for the inverse of the Saleh model is known, this inverse was not used in the implementation of their pre-distorter in the illustrated embodiment in which the characteristic of the high power amplifier is time-varying. We have combined the closed form expression for the inverse of the high power amplifier characteristic with a sequential nonlinear parameter estimation algorithm, which allows sparse implementation of the pre-distorter and accurate tracking of or adaptation to the time varying behavior of the high power amplifier.  
         [0024]     Compared to the other prior art approaches mentioned above, our algorithms are fast, accurate, and of low complexity as demonstrated and verified by the computer simulations described below.  
         [0025]     While the apparatus and method has or will be described for the sake of grammatical fluidity with functional explanations, it is to be expressly understood that the claims, unless expressly formulated under 35 USC 112, are not to be construed as necessarily limited in any way by the construction of “means” or “steps” limitations, but are to be accorded the full scope of the meaning and equivalents of the definition provided by the claims under the judicial doctrine of equivalents, and in the case where the claims are expressly formulated under 35 USC 112 are to be accorded full statutory equivalents under 35 USC 112. The invention can be better visualized by turning now to the following drawings wherein like elements are referenced by like numerals. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0026]      FIG. 1  is a simplified OFDM communications transmitter with a pre-distorter and high power amplifier of the invention.  
         [0027]      FIG. 2  is a graph of the nonlinear amplitude and phase transfer function of the Saleh&#39;s traveling wave tube amplifier model showing normalized output as a function of normalized input.  
         [0028]      FIG. 3  is a graph of the nonlinear amplitude transfer function of the Rapp&#39;s solid state power amplifier model showing normalized output as a function of normalized input.  
         [0029]      FIG. 4  is a graph of the amplitude compensation effect of Saleh&#39;s traveling wave tube amplifier model with a pre-distorter showing normalized output as a function of normalized input.  
         [0030]      FIG. 5  is a simplified block diagram of a pre-distorter combined with a time varying high power amplifier.  
         [0031]      FIG. 6   a  is a graph of the compensation effect of Rapp&#39;s solid state power amplifier model using a pre-distorter showing normalized output as a function of normalized input.  
         [0032]      FIG. 6   b  is a graph of the compensation and clipping effect of Rapp&#39;s solid state power amplifier model using a pre-distorter showing normalized output as a function of normalized input.  
         [0033]      FIG. 7   a  is a graph of the received OFDM signal constellations with a traveling wave tube amplifier without a pre-distorter showing I channel vs Q channel  
         [0034]      FIG. 7   b  is a graph of the received OFDM signal constellations with a traveling wave tube amplifier with a pre-distorter showing I channel vs Q channel.  
         [0035]      FIG. 8  is a graph showing the bit error ratio (BER) output performance with and without a pre-distorter in an OFDM system with a time-invariant traveling wave tube amplifier showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio).  
         [0036]      FIG. 9   a  is a graph of the signal amplitude in the saturation condition where the normalized signal is clipped above 1 showing normalized output as a function of normalized input.  
         [0037]      FIG. 9   b  is a graph of the signal phase in the saturation condition. This figure shows normalized input amplitude vs output phase distortion, since output phase distortion is a function of normalized input amplitude  
         [0038]      FIG. 10  is a graph showing BER output performance with and without a pre-distorter in an OFDM system with a time-varying traveling wave tube amplifier with parameters are uniformly distributed with IBO (Input Back-Off)=6 dB in which the pre-distorter is provided with and without tracking showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0039]      FIG. 11  is a graph showing BER output performance with and without a pre-distorter in an OFDM system with a time-varying traveling wave tube amplifier with parameters are uniformly distributed with IBO=7 dB in which the pre-distorter is provided with and without tracking showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0040]      FIG. 12   a  is a graph of the received OFDM signal constellations with a solid state power amplifier without a pre-distorter showing I channel vs Q channel.  
         [0041]      FIG. 12   b  is a graph of the received OFDM signal constellations with a solid state power amplifier with a pre-distorter showing I channel vs Q channel.  
         [0042]      FIG. 13  is a graph of BER performance of a pre-distorter in an OFDM system with a time-invariant solid state power amplifier, when A 0 =p=1 showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR  
         [0000]     (Signal to Noise Ratio)  
         [0043]      FIG. 14  is a graph of BER performance of a pre-distorter, when the parameters are uniformly distributed in the range 1≦·A 0 ≦·1.5, 1≦·p≦·1.5, with IBO=6 dB showing BER as a function of input E b /N 0  ratio in db where E b  is the number of bit errors and N 0  the total number of input bits.  
         [0044]      FIG. 15  is a graph of BER performance of a pre-distorter, when the parameters are uniformly distributed in the range 1≦·A 0 ≦·2, 1≦··p≦··2 with IBO=6 dB showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0045]      FIG. 16  is a graph of BER performance of a pre-distorter, when the parameters are uniformly distributed in the range 1≦·A 0 ≦·2, 1≦··p≦·2 with IBO=7 dB showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0046]      FIG. 17  shows convergence of two changing parameters with Gaussian and uniformly distributed, β, ε in Saleh&#39;s TWTA model  
         [0047]      FIG. 18  is a graph showing BER output performance with and without a pre-distorter in an OFDM system with a time-varying traveling wave tube amplifier with parameters are both Gaussian and uniformly distributed with IBO (Input Back-Off)=6 dB in which the pre-distorter is provided with and without tracking showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0048]      FIG. 19  is a graph showing BER output performance with and without a pre-distorter in an OFDM system with a time-varying traveling wave tube amplifier with parameters are both Gaussian and uniformly distributed with IBO (Input Back-Off)=7 dB in which the pre-distorter is provided with and without tracking showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0049]      FIG. 20  shows convergence of two changing parameters with Gaussian distributed, A 0 , p in Rapp&#39;s SSPA model (mean=1.5, variance=0.01)  
         [0050]      FIG. 21  is a graph of BER performance of a pre-distorter, when the parameters are Gaussian distributed, variance=0.1 with IBO=6 dB showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0051]      FIG. 22  is a graph of BER performance of a pre-distorter, when the parameters are Gaussian distributed, variance=0.1 with IBO=7 dB showing BER as a function of input E b /N 0  ratio in db where E b  is the signal energy per bit and N 0  is the noise power spectral density. That is E b /N 0 =SNR (Signal to Noise Ratio)  
         [0052]     The invention and its various embodiments can now be better understood by turning to the following detailed description of the preferred embodiments which are presented as illustrated examples of the invention defined in the claims. It is expressly understood that the invention as defined by the claims may be broader than the illustrated embodiments described below. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0000]     System Description  
         [0053]      FIG. 1  is a simplified block diagram of the invention showing a system architecture, generally denoted by reference numeral  10 , for compensation of the high power amplifier nonlinearity for an OFDM system. The OFDM baseband module  12  generates an OFDM-formatted signal to pre-distorter  14 , whose digital output is converted to analog form by digital to analog converter  16  to produce phase shifted QAM outputs to multipliers  18  and  20  which are combined and summed in adder  22  and then input to power amplifier  24  for transmission to the wireless or wireline communication system. It must be understood that the hardware in  FIG. 1  can be implemented in a number of equivalent ways. For example pre-distorter  14  is a digital circuit which may be a dedicated digital signal processor using a combination of hardware and/or firmware, or may be a computer with appropriate signal interfaces which computer arranged and configured by software to process digital information as taught by the invention. There is no limitation on the specific technology by which pre-distorter  14  may be realized and all means now known or later devised are expressly contemplated as being within the scope of the invention.  
         [0054]     Typically, an OFDM signal x(t) can be analytically represented as  
               x   ⁡     (   t   )       =       1     N       ⁢       ∑     k   =   0       N   -   1       ⁢       X   ⁡     [   k   ]       ⁢     ⅇ     j2π   ⁢           ⁢     f   k     ⁢   t                     (   1   )             
        where X[k] denotes quadrature amplitude modulation (QAM) symbol, N is the number of sub-carriers, and f k  is kth sub-carrier frequency which can be represented as  
               f   k     =     k   ·     1     NT   s                 (   2   )             
    where Ts is sampling period of x(t). QAM is a method of combining two amplitude-modulated (AM) signals into a single channel, thereby doubling the effective bandwidth. QAM is used with pulse amplitude modulation (PAM) in digital systems, especially in wireless applications. In a QAM signal, there are two carriers, each having the same frequency but differing in phase by 90 degrees (one quarter of a cycle, from which the term quadrature arises). One signal is called the I signal, and the other is called the Q signal. Mathematically, one of the signals can be represented by a sine wave, and the other by a cosine wave. The two modulated carriers are combined at the source for transmission. At the destination, the carriers are separated, the data is extracted from each, and then the data is combined into the original modulating information.        
 
         [0057]     By discretizing x(t) at t=nTs, we have the equation  
                 x   ⁡     (   n   )       ≡     x   ⁡     (     nT   s     )         =       1     N       ⁢       ∑     k   =   0       N   -   1       ⁢       X   ⁡     [   k   ]       ⁢     ⅇ       j2π   ⁢           ⁢   kn     N                     (   3   )             
 
         [0058]     The pre-distorter  14  is a nonlinear zero memory device that pre-computes and cancels the nonlinear distortion present in the zero memory high power amplifier  24  which follows the pre-distorter  14 .  
         [0000]     Traveling Wave Tube Amplifier Model  
         [0059]     As a high power amplifier model, we show Saleh&#39;s well established traveling wave tube amplifier model. In this model, AM/AM and AM/PM conversion of traveling wave tube amplifier can be represented as  
                 u   ⁡     [   r   ]       =       α   ⁢           ⁢   r       1   +     β   ⁢           ⁢     r   2             ⁢     
     ⁢       Φ   ⁡     [   r   ]       =       γ   ⁢           ⁢     r   2         1   +     ɛ   ⁢           ⁢     r   2                       (   4   )     ,     (   5   )               
        where u is amplitude response, φ is phase response, r is input amplitude of the traveling wave tube amplifier and α, β, γ, and ε are four adjustable parameters. The behavior of equations (4) and (5) is illustrated in the graph of  FIG. 2 , where normalized output of the traveling wave tube amplifier is shown as a function of normalized input. In  FIG. 2 , we use α=1.9638; β=0.9945; γ=2.5293; and ε=2.8168 as in Saleh&#39;s original work. The output z(t) of traveling wave tube amplifier  24  without pre-distorter  14  can be represented as 
 
 z ( t )= u[r ]cos(ω c   t +φ( t )+Φ[ r ])  (6) 
    where φ(t) is the phase of the input signal and ω c  is carrier frequency. 
 
 Solid State Power Amplifier Model 
       
 
         [0062]     For the solid state power amplifier  24 , we use normalized Rapp&#39;s model. In this model, we assume AM/PM conversion is small enough, so that it can be neglected. Then, AM/AM and AM/PM conversion of solid state power amplifier can be represented as  
                     u   ⁡     [   r   ]       =     r       (     1   +       (     r     A   0       )       2   ⁢   p         )       1     2   ⁢   p                         Φ   ⁡     [   r   ]       ≈   0                   (   7   )     ,     (   8   )               
        where r is input amplitude of solid state power amplifier  24 , A 0  is the maximum output amplitude and p is the parameter which affects the smoothness of the transition. The behavior of equation (7) is illustrated in the graph of  FIG. 3  where normalized output is shown as a function of normalized input. The output z(t) of solid state power amplifier  24  without pre-distorter  14  can be represented as 
 
 z ( t )= u[r ] cos(ω c   t +φ( t ))  (9) 
    where φ(t) is the phase of the input signal. 
 
 Pre-Distorters 
       
 
         [0065]     Now consider the pre-distorters  14  for both traveling wave tube amplifier  24  and solid state power amplifier  24  according to the invention. Let q and u denote the nonlinear zero memory input and output maps respectively of the pre-distorter  14  and high power amplifier  24 , and x l (n), the input of the pre-distorter  14 , y l (n), the output of the pre-distorter  14  which is also the input to the high power amplifier  24 , and z(t) the output of the high power amplifier  24  as shown in  FIG. 1 . Then for any given high power amplifier  24 , an ideal pre-distorter  14  according to the invention is one for which the input-output maps satisfies 
 
 u[q ( x   l ( n ))]= k x   l ( n )  (10) 
        where k is a desired pre-specified linear amplification constant. In this illustration, we assume k=1. 
 
 Pre-Distorter for Traveling Wave Tube Amplifier 
 
 Time-Invariant Case 
       
 
         [0067]     In traveling wave tube amplifier  24 , the general baseband (equivalent low pass signal) expressions for the input x l (n) and output y l (n) of the pre-distorter  14  are 
 
 x   l ( n )= r ( n ) e   jφ(n)   (11), 
 
 y   l ( n )= q[r ( n )] e   j(φ(n)+θ[r(n)])   (12) 
        where the function q and φ are to be determined by requiring that equation (10) be satisfied. According to equations (4) and (5), the input and output of traveling wave tube amplifier  24  are 
 
 y ( t )= q[r ( t )] cos(ω c   t +φ( t )+θ[ r ( t )])  (13), 
 
 z ( t )=[ q[r ( t )]]cos(ω c   t +φ( t )+θ[ r ( t )]+Φ[ q ( t )]  (14) 
    where  
                 u   ⁡     [     q   ⁡     (   r   )       ]       =       α   ⁢           ⁢   q       1   +     β   ⁢           ⁢     q   2             ⁢     
     ⁢       Φ   ⁡     [     q   ⁡     (   r   )       ]       =       γ   ⁢           ⁢     q   2         1   +     ɛ   ⁢           ⁢     q   2                       (   15   )     ,     (   16   )               
       
 
         [0070]     In order to satisfy (10), the following must hold  
                   α   ⁢           ⁢   q       1   +     β   ⁢           ⁢     q   2           =   r     ⁢     
     ⁢         γ   ⁢           ⁢     q   2         1   +     ɛ   ⁢           ⁢     q   2           =     -   θ                 (   17   )     ,     (   18   )               
 
         [0071]     From equation (17) 
 
 rβq   2   −αq+r= 0  (19) 
 
         [0072]     This equation can be solved for q to yield  
                 q   ⁡     (   r   )       =       α   -         α   2     -     4   ⁢     r   2     ⁢   β             2   ⁢   r   ⁢           ⁢   β         ,     r   ≤   1             (   20   )             
 
         [0073]     Also for zero phase distortion, we must have  
                   θ   ⁡     (   r   )       +     Φ   ⁡     (   q   )         =   0     ⁢     
     ⁢   or           (   21   )                 θ   ⁡     (   r   )       =       -     Φ   ⁡     (   q   )         =     -         γ   ⁡     (     q   ⁡     (   r   )       )       2       1   +       ɛ   ⁡     (     q   ⁡     (   r   )       )       2                     (   22   )             
 
         [0074]     If r&gt;1, equation (20) has no solution. This corresponds to the clipping of the signal according to the depiction of the graph of  FIG. 4  where the normalized output is shown as a function of the normalized input for a traveling wave tube amplifier  24  with pre-distorter  14 . This analytical solution of equations (20), (22) was previously obtained by Brajal and Chouly.  
         [0000]     Time-Varying Adaptive Case  
         [0075]     We now extend this solution to the time-varying case as follows. As a time-varying model, we assume four parameters α, β, γ, and ε change with time. We express  
               J   ⁡     (     α   ,   β     )       =       E   ⁡     (         α   ⁢           ⁢   q       1   +     β   ⁢           ⁢     q   2           -   u     )       2             (   23   )             
 
         [0076]     Where J is a cost function which should be minimized, E is expectation w.r.t α,β. Partially differentiating with respect to α and equating the result to zero, we get  
                   ∂     J   ⁡     (     α   ,   β     )           ∂   α       =       E   ⁡     [     2   ⁢     (         α   ⁢           ⁢   q       1   +     β   ⁢           ⁢     q   2           -   u     )     ⁢     q     1   +     β   ⁢           ⁢     q   2             ]       =   0       ,           (   24   )                 α   ⁢           ⁢     E   ⁡     (       q   2         (     1   +     β   ⁢           ⁢     q   2         )     2       )         =     E   ⁡     (     qu     1   +     β   ⁢           ⁢     q   2           )               (   25   )             
 
         [0077]     Proceeding similarly with respect to β, we get  
                   ∂     J   ⁡     (     α   ,   β     )           ∂   β       =       E   ⁡     [     2   ⁢     (         α   ⁢           ⁢   q       1   +     β   ⁢           ⁢     q   2           -   u     )     ⁢     (     -       α   ⁢           ⁢   q         (     1   +     β   ⁢           ⁢     q   2         )     2         )     ⁢     q   2       ]       =   0       ⁢     
     ⁢   or           (   26   )                 α   ⁢           ⁢   E   ⁢     (       q   4         (     1   +     β   ⁢           ⁢     q   2         )     3       )       =     E   ⁡     (         q   3     ⁢   u         (     1   +     β   ⁢           ⁢     q   2         )     2       )               (   27   )             
 
         [0078]     Let us define the following for the sake of simplicity.  
                 A   ⁡     (   β   )       =     E   ⁡     (       q   2         (     1   +     β   ⁢           ⁢     q   2         )     2       )         ,           (   28   )                   B   ⁡     (   β   )       =     E   ⁡     (     qu     1   +     β   ⁢           ⁢     q   2           )         ,           (   29   )                   C   ⁡     (   β   )       =     E   ⁡     (       q   4         (     1   +     β   ⁢           ⁢     q   2         )     3       )         ,           (   30   )                 D   ⁡     (   β   )       =     E   ⁡     (         q   3     ⁢   u         (     1   +     β   ⁢           ⁢     q   2         )     2       )               (   31   )             
 
         [0079]     According to equations (25), (28) and (29)  
             α   =       B   ⁡     (   β   )         A   ⁡     (   β   )                 (   32   )             
        and according to equations (27), (30), (31), (32)  
                   B   ⁡     (   β   )         A   ⁡     (   β   )         ⁢     C   ⁡     (   β   )         =     D   ⁡     (   β   )               (   33   )             
       
 
         [0081]     So, our approach is: Solve equation (33) in an estimator  26  shown in  FIG. 5  numerically for {circumflex over (β)}, which is the estimate of β, and then replace {circumflex over (β)} in equation (32) to obtain {circumflex over (α)} the estimate of α. The expectation in equations (28), (29), (30), (31) can be estimated using the following equations  
                   A   ^     ⁡     (   β   )       =       1   N     ⁢       ∑     n   =   1     N     ⁢       q   n   2         (     1   +     β   ⁢           ⁢     q   n   2         )     2             ,           (   34   )                     B   ^     ⁡     (   β   )       =       1   N     ⁢       ∑     n   =   1     N     ⁢         q   n     ⁢     u   n         1   +     β   ⁢           ⁢     q   n   2                 ,           (   35   )                     C   ^     ⁡     (   β   )       =       1   N     ⁢       ∑     n   =   1     N     ⁢       q   n   4         (     1   +     β   ⁢           ⁢     q   n   2         )     3             ,           (   36   )                   D   ^     ⁡     (   β   )       =       1   N     ⁢       ∑     n   =   1     N     ⁢         q   n   3     ⁢     u   n           (     1   +     β   ⁢           ⁢     q   n   2         )     2                   (   37   )             
        γ and ε also can be estimated exactly in the same way as described above. This approach is illustrated in the block diagram of  FIG. 5  which shows a pre-distorter  14  for a time varying high power amplifier where a parameter estimator  26  is provided to take parameters from high power amplifier  24  and provide them to estimator  26  to generate parameter estimates for pre-distorter  14 .        
 
         [0083]     To get the optimum estimation of 18 from (33), we use the following equation. 
 
{circumflex over (β)} opt =min  β   |B (β) C (β)− A (β) D (β)| 2   (38) 
 
         [0084]     The optimum coefficient {circumflex over (β)} opt , satisfying (38) is determined in order to minimize the MSE (Mean Square Error) defined by 
 
 J (β)= E[{circumflex over (B)} (β){circumflex over ( C )}(β)−{circumflex over ( A )}(β){circumflex over ( D )}(β)] 2   (39) 
 
 Where J is cost function to be minimized and E is expectation w.r.t β
 
         [0085]     Then, derivative J w.r.t. β 
                 ∂     J   ⁡     (   β   )           ∂   β       =       ⁢     2   ⁢       E   ⁡     (           B   ^     ⁡     (   β   )       ⁢       C   ^     ⁡     (   β   )         -         A   ^     ⁡     (   β   )       ⁢       D   ^     ⁡     (   β   )           )       ·     (           ∂       B   ^     ⁡     (   β   )           ∂   β       ⁢       C   ^     ⁡     (   β   )         +         B   ^     ⁡     (   β   )       ⁢       ∂       C   ^     ⁡     (   β   )           ∂   β         -         ∂       A   ^     ⁡     (   β   )           ∂   β       ⁢       D   ^     ⁡     (   β   )         -         A   ^     ⁡     (   β   )       ⁢       ∂       D   ^     ⁡     (   β   )           ∂   β           )                 (   40   )             
 
 Where  
                 ∂       A   ^     ⁡     (   β   )           ∂   β       =       -     2   N       ⁢       ∑     n   =   1     N     ⁢       q   n   4         (     1   +     β   ⁢           ⁢     q   n   2         )     3                   (   41   )                   ∂       B   ^     ⁡     (   β   )           ∂   β       =       -     1   N       ⁢       ∑     n   =   1     N     ⁢         q   n   3     ⁢     u   n           (     1   +     β   ⁢           ⁢     q   n   2         )     2                   (   42   )                   ∂       C   ^     ⁡     (   β   )           ∂   β       =       -     3   N       ⁢       ∑     n   =   1     N     ⁢       q   n   6         (     1   +     β   ⁢           ⁢     q   n   2         )     4                   (   43   )                   ∂       D   ^     ⁡     (   β   )           ∂   β       =       -     2   N       ⁢       ∑     n   =   1     N     ⁢         q   n   5     ⁢     u   n           (     1   +     β   ⁢           ⁢     q   n   2         )     3                   (   44   )             
 
         [0086]     After that, LMS (Least Mean Square) algorithm can be represented as  
                 β   ^     ⁡     (     n   +   1     )       =         β   ^     ⁡     (   n   )       -       μ     β   ^       ·     (           B   ^     ⁡     (       β   ^     ⁡     (   n   )       )       ⁢       C   ^     ⁡     (       β   ^     ⁡     (   n   )       )         -         A   ^     ⁡     (       β   ^     ⁡     (   n   )       )       ⁢       D   ^     ⁡     (       β   ^     ⁡     (   n   )       )           )     ·     (           ∂       B   ^     ⁡     (       β   ^     ⁡     (   n   )       )           ∂       β   ^     ⁡     (   n   )           ⁢       C   ^     ⁡     (       β   ^     ⁡     (   n   )       )         +         B   ^     ⁡     (       β   ^     ⁡     (   n   )       )       ⁢       ∂       C   ^     ⁡     (       β   ^     ⁡     (   n   )       )           ∂       β   ^     ⁡     (   n   )             -         ∂       A   ^     ⁡     (       β   ^     ⁡     (   n   )       )           ∂       β   ^     ⁡     (   n   )           ⁢       D   ^     ⁡     (       β   ^     ⁡     (   n   )       )         -         A   ^     ⁡     (       β   ^     ⁡     (   n   )       )       ⁢       ∂       D   ^     ⁡     (       β   ^     ⁡     (   n   )       )           ∂       β   ^     ⁡     (   n   )               )                 (   45   )             
 
         [0087]     Where μ {circumflex over (β)}  is the step size of LMS algorithm.  
         [0088]     Once we get estimation of β, we easily get estimation of α from (32). γ and ε can be estimated exactly same way described above.  
         [0000]     Pre-Distorter for a Solid State Power Amplifier  
         [0000]     Time-Invariant Case  
         [0089]     As in traveling wave tube amplifier  24 , the general baseband (equivalent low pass signal) expressions for the input x l (n) and output y l (n) of the pre-distorter  14  for solid state power amplifier  24  are 
 
 x   l ( n )= r ( n ) e   jφ(n)   (46), 
 
 y   l ( n )= q[r ( n )] e   jφ(n)   (47) 
        where the function q and (are to be determined by requiring that equation (10) be satisfied. As we assume phase distortion is neglected, we don&#39;t need to regard phase pre-distortion. According to equations (7) and (8), the input and output of solid state power amplifier  24  are 
 
 y   c ( t )= q[r ( t )] cos(ω c   t +φ( t ))  (48), 
 
 z ( t )= u[q[r ( t )]]cos(ω c   t +φ( t ))  (49) 
    where  
               u   ⁡     [     q   ⁡     (   r   )       ]       =       q   ⁡     (   r   )           (     1   +       (       q   ⁡     (   r   )         A   0       )       2   ⁢   p         )       1     2   ⁢   p                   (   50   )             
       
 
         [0092]     According to equation (50), equation (10) implies  
                 q   ⁡     (   r   )           (     1   +       (       q   ⁡     (   r   )         A   0       )       2   ⁢   p         )       1     2   ⁢   p           =   r           (   51   )             
 
         [0093]     Then, after some algebraic manipulation, we can find the exact expression for the pre-distorter characteristic q(r):  
                 q   ⁡     (   r   )       =     r       (     1   -       (     r     A   0       )       2   ⁢   p         )       1     2   ⁢   p             ,     r   &lt;     A   0               (   52   )             
 
         [0094]     An illustration of compensation effect is shown in  FIG. 6 . When r&gt;A 0 , equation (52) has no solution. In this case, we clip the input signal as in  FIG. 6 .  
         [0000]     Time-Varying Adaptive Case  
         [0095]     Since high power amplifier  24  is time-varying system, as a time-varying model, we assume parameters A 0  and p in the solid state power amplifier model change with time. To track two parameters A 0  and p, we use training symbols. Using training symbols, we get input of pre-distorter  14 , q(n), and output of pre-distorter  14 , u(n). During the training stage, we assume pre-distorter  14  is turned off. That is, input and output of pre-distorter  14  would be same (r(n)=q(n)).  
         [0096]     To estimate parameters A 0  and p, first, we change equation (50) as  
               A   0     =       q   ·   u         (       q     2   ⁢   p       -     u     2   ⁢   p         )       1     2   ⁢   p                   (   53   )             
 
         [0097]     To summarize the algorithm, if we know p, we can get A 0  easily from equation (53). However, we assume both A 0  and p change with time. First, send two training symbols, then we know the input amplitude q and the output amplitude u of the high power amplifier  24 . Then from equation (53), corresponding to two different training symbols, we can get two different estimations of A 0 , namely A 01  and A 02  as given by equations (54) and (55) below. If we choose a correct p, which is the same for high power amplifier  24  during the training time, the two different values of A 0 , namely A 01  and A 02 , have almost the same value or due to step size, very close values. We can find p for that point, which has the smallest distance between two estimated A 0 , namely D min =|A 01 −A 02 | 2 . Then, from equation (53) and the estimation of p, we can get Â 0 =A 01 ≈A 02  from the minimum distance D min =|A 01 −A 02 | 2 . This algorithm is computationally effortless. We use only two training symbols and no iteration, hence incurring very little delay.  
                                     Brief description of the Algorithm                                1. Send two training symbols.       2. Get two estimated values of A 0 , A 01  and A 02  from equation (53).       3. Choose a step size for p and find D min  = |A 01  − A 02 |  2  to get       corresponding p which yields {circumflex over (p)}.       4. Get estimated value of A 0 , Â 0 , which is Â 0  = A 01  ≈ A 02                    
 
         [0098]     As a more practical way, if we know p, we can get A 0  easily from equation (53). However, we assume both A 0  and p change with time. In this case, we propose following algorithm. First, send two training symbols, then we know input amplitude of high power amplifier  24 , q and output amplitude of high power amplifier  24 , u. After that, from equation (53), correspond to two different training symbols, we get two different estimations of A 0 , namely A 01  and A 02 .  
                 A   01     =         q   1     ·     u   1           (       q   1     2   ⁢   p       -     u   1     2   ⁢   p         )       1     2   ⁢   p             ,           (   54   )                 A   02     =         q   2     ·     u   2           (       q   2     2   ⁢   p       -     u   2     2   ⁢   p         )       1     2   ⁢   p                   (   55   )             
        where q 1 , u 1  are output amplitudes of pre-distorter  14  and high power amplifier  24  respectively for first training symbol and q 2 , u 2  are output amplitudes of pre-distorter  14  and high power amplifier  24  respectively for the second training symbol. Training symbols are not affected by the function of pre-distorter  14  as we stated previously. During training period, we can replace q, and q 2  as r 1  and r 2  which are the original amplitudes of training symbols. We can estimate unknown A 0  and p using following equations. 
 
 {circumflex over (p)}   opt =min p   A   01 ( p )− A   02 ( p )| 2   (56), 
 
 Â   0   =A   01 ( {circumflex over (p)}   opt )≈ A   02 ( {circumflex over (p)}   opt )  (57) 
    where Â 0  is an estimator of A 0  and {circumflex over (p)}opt is the optimum {circumflex over (p)} which we can get from equation (56). 
 
 Simulation Results and Discussion 
       
 
         [0101]     Consider now a test of the illustrated pre-distortion technique for compensation of high power amplifier nonlinear distortion as demonstrated with computer simulations. The additive white gaussian noise (AWGN) channels were assumed to clearly observe the effect of nonlinearity and performance improvement by the illustrated pre-distorter  14 . An OFDM system  10  with 128 subcarriers and 16 QAMs is considered. If the input amplitude is very high, the high power amplifier  24  operates in a highly nonlinear situation. If the input amplitude is very small, the high power amplifier  24  operates with very small distortion. In the operation of high power amplifier  24 , a relative level of power back off is needed to reduce distortion. However, this power back off is not so desirable because it reduces power efficiency. In our algorithm, a compensation solution always exists in the range r&lt;A 0 , where A 0  is maximum output amplitude. So, if the input average power is same as A 0   2 , we get maximum power efficiency, but a highly nonlinear result. Thus, we need a criterion to show how much power back off from optimum power efficiency is needed. In the simulations, we define IBO (Input Back-Off) as  
             IBO   =     10   ⁢       log   10     ⁡     (       A   0   2       P     i   ⁢           ⁢   n         )                 (   58   )             
        where Pin is input average power (average power of OFDM signal). Similarly, we can also define OBO (Output Back-Off) as  
             OBO   =     10   ⁢       log   10     ⁡     (       A   0   2       P   out       )                 (   59   )             
    where P out  is output average power (average output power of high power amplifier  24 ). 
 
 Pre-Distorter for Traveling Wave Tube Amplifier 
       
 
         [0104]     Time-Invariant Case  
         [0105]     Consider now OFDM simulation results with the assumption that parameters α, β, γ, and ε are time invariant.  FIGS. 7   a  and  7   b  are graphs which depict α as a function of I and which show the difference of signal constellation without and with pre-distorter  14  respectively. In  FIGS. 7   a  and  7   b , we use IBO=6 dB. The bit error rate or bit error ratio (BER) performance curve, shown in the graph of  FIG. 8 , shows BER as a function of Eb/N 0  where Eb is the signal energy per bit and N 0  is noise power spectral density, and shows that the pre-distorter  14  can significantly reduce nonlinear distortion in an OFDM system  10 . BER is the number of erroneous bits divided by the total number of bits transmitted, received, or processed over some stipulated period. Examples of bit error ratio are (a) transmission BER, i.e., the number of erroneous bits received divided by the total number of bits transmitted; and (b) information BER, i.e., the number of erroneous decoded (corrected) bits divided by the total number of decoded (corrected) bits. The BER is usually expressed as a coefficient and a power of 10; for example, 2.5 erroneous bits out of 100,000 bits transmitted would be  2 . 5  out of 10 5  or 2.5×10 −5 .  
         [0000]     Time-Varying Adaptive Case with Uniform Distribution  
         [0106]     As mentioned previously, high power amplifier  24  is a time varying system. Assume the four parameters α, β, γ, and ε are now time-varying, thus we should track the variations of α, β, γ, and ε. We assume that these four parameters change with uniform distribution according to the following conditions.  
         [0107]     (1) The four parameters change in the following ranges 
 
1.01≦α0≦2  (60) 
 
0.01≦β≦1  (61) 
 
1.5&lt;γ, ε·≦3  (62) 
 
         [0108]     (2) Input and output normalization condition, β=α−1.  
         [0109]     (3) Saturation condition, signal is clipped above 1, as shown in the graph of  FIGS. 9   a  and  9   b.    
         [0110]     The reason why we choose the above conditions on amplitude and phase is to maintain normalization constraints in both input and output and the saturation condition in the above range (r&gt;A 0 ), even if the amplitude is changed. These restrictions are just for convenience of representation, so in a real system, even if the above condition does not hold, our algorithm works well. Table 1 below shows errors after tracking α, β, γ, and ε using our algorithm. We used the following equations to get the results of Table 1.  
                 Error   ⁡     (   α   )       =            α   -     α   ^                     α   max     -     α   min                ,           (   63   )                   Error   ⁡     (   β   )       =            β   -     β   ^                     β   max     -     β   min                ,           (   64   )                   Error   ⁡     (   γ   )       =            γ   -     γ   ^                     γ   max     -     γ   min                ,           (   65   )                 Error   ⁡     (   ɛ   )       =            ɛ   -     ɛ   ^                     ɛ   max     -     ɛ   min                      (   66   )             
 
         [0111]     We get the results of Table 1, using only two training symbols, calculating 1000 times and averaging the results.  
                                                           TABLE 1                           Error of parameters            Step size   Error (α)   Error (β)   Error (γ)   Error (ε)                    0.1   1.02 × 10 −2     2.74 × 10 −2      6.3 × 10 −3     1.72 × 10 −2         0.01   9.67 × 10 −4      2.5 × 10 −3     6.04 × 10 −4      1.7 × 10 −3         0.001   9.49 × 10 −5     2.54 × 10 −4     6.18 × 10 −5     1.69 × 10 −4                    
 
         [0112]     The results of Table 1 show that only two training symbols are enough for our algorithm. This indicates that our algorithm is very fast and has little delay. The BER performance of pre-distorter  14  in OFDM  10  with time-varying high power amplifier  24  is shown in the graphs of  FIG. 10  and  FIG. 11 . In these curves, we assume step size=0.01. As is clear from  FIG. 10  and  FIG. 11 , if the variation of high power amplifier  24  is not tracked, the performance is much worse compared with the case of tracking. The simulation results thus show that this ability to track changes in parameters adds value to system performance.  
         [0000]     Time-Varying Adaptive Case with Gaussian Distribution and LMS Algorithm  
         [0113]     We simulate our PD again, but different parameter distribution. We assume 4 parameters α, β, γ, ε are time-varying with both Gaussian and uniform distribution and track the variation of parameters using LMS (Least Mean Square) algorithm. First we show convergence of our algorithm in  FIG. 17 . The reason why we show only two parameters β and ε is that, as we show in previously, once we get both β and ε, other parameters α and γ can be achieve easily. In this simulation, we assume β is uniformly distributed and ε is Gaussian distribution with mean E(ε)=2.8168 as in Saleh&#39;s original model and variance 0.01. We use step size μ β =6000000 and μ ε =600000000000 for fast convergence.  
         [0114]     Now we show comparison of BER performance between with and without tracking. In these simulations, we assume that four parameters change according to the following conditions.  
         [0115]     (1) The two parameters change in the following ranges 
 
1.01≦α≦2  (67) 
 
0.01≦α≦1  (68) 
 
         [0116]     (2) Phase parameters γ and ε change with Gaussian distribution with averages E(γ)=2.5293, E(ε)=2.8168 and variance σ=0.1 each.  
         [0117]     (3) Input and output normalization condition, β=α−1.  
         [0118]     (4) Saturation condition, signal is clipped above 1, as shown in the graph of  FIGS. 9   a  and  9   b.    
         [0119]     As we explained in previous section, these restrictions are only for convenience of representation. The BER performance of PD in OFDM with time-varying HPA is shown in  FIG. 18  (IBO=6 dB) and  FIG. 19  (IBO=6 dB). In these BER performance simulation, we assume step sizes μ β =50000000=and μ ε =10000000000. We use two training symbols and iterate 1000 times. Even usually PD needs much less iteration, we use enough number of iteration to make sure all of parameters are converge. As is clear from  FIG. 18  and  FIG. 19 , if the variation of HPA is not tracked, the performance is much worse compare with the case of tracking. The simulation results thus show that this ability to track changes in parameters adds value to system performance.  
         [0000]     Pre-Distorter for Solid State Power Amplifier  
         [0000]     Time-Invariant Case  
         [0120]     Consider OFDM simulation results with the assumption that solid state power amplifier  24  is time invariant system. In this simulation, 16 QAMs were employed as modulation scheme and used 128 sub-carriers. Because of high peak to average power ratio, OFDM needs much more IBO than single carrier system.  FIGS. 12   a  and  12   b  show the signal constellation output without and with pre-distorter  14  respectively. In comparison with the traveling wave tube amplifier case, amplitude distortion is not so severe and no phase distortion exists. However, without pre-distorter  14 , even if IBO=6 dB, amplitude distortion is high. In  FIG. 13 , the BER performance curves show that our pre-distorter  14  can significantly reduce the effect of nonlinear distortion in OFDM system  10 . In  FIG. 13 , we use A 0 =p=1.  
         [0000]     Time-Varying Adaptive Case with Uniform Distribution  
         [0121]     As we mentioned previously, high power amplifier  14  is time-varying system. Assume the two parameters A 0  and p are time-varying, thus we should track the variation of A 0  and p. As in the case of traveling wave tube amplifier  24 , two parameters A 0  and p have uniform distribution. The simulations used a simple search algorithm. Table 2 shows errors after track A 0  and p using our algorithm. We used following 
        equations to get the results of Table 2.  
                 Error   ⁡     (     A   0     )       =              A   0     -       A   0     ^                     A   max     -     A   min                ,           (   69   )                 Error   ⁡     (   p   )       =            p   -     p   ^                     p   max     -     p   min                      (   70   )             
       
 
         [0123]     where Â 0  and {circumflex over (p)} are tracked parameters using simple search algorithm and |A max −A min | and |p max −p min | variation ranges. We calculate equations (69) and (70) 1000 times and average each error. According to Table 2, even step size is 0.1, the errors are very small.  
                                                                                             TABLE 2                           Error of Â 0 , and {circumflex over (p)}            Step   1 ≦ A 0 , p ≦ 1.5   1 ≦ A 0 , p ≦ 2   1 ≦ A 0 , p ≦ 3            size   Error (A 0 )   Error (p)   Error (A 0 )   Error (p)   Error (A 0 )   Error (p)                    0.1   3.86 × 10 −2     5.1 × 10 −2     2.29 × 10 −2     2.56 × 10 −2     1.40 × 10 −2     1.23 × 10 −2         0.01    3.7 × 10 −3     5.0 × 10 −3     2.22 × 10 −3      2.5 × 10 −3      1.5 × 10 −3      1.3 × 10 −3         0.001   3.66 × 10 −4     4.8875 × 10 −4      2.1718 × 10 −1     2.4870 × 10 −1     1.4934 × 10 −4     1.2852 × 10 −4                    
 
         [0124]     We now show BER performance of pre-distorter  14  for time-varying solid state power amplifier  24 . We use a step size 0.01 in the following BER performance simulations. In  FIG. 14 , we assume two parameters are uniform distribution in the range 1≦A 0 , p&lt;1.5 with mean=1.25 each, IBO=6 dB. In the case of without tracking, we use mean value 1.25 for both parameters. In  FIG. 15  and  FIG. 16 , we show BER performance of pre-distorter  14  for time-varying solid state power amplifier  24  when two parameters are uniform distribution in the wider range 1≦A 0 , p&lt;2 with mean=1.5, IBO=6 dB and 7 dB each. In the case of without tracking, we use mean value 1.5 for both parameters.  
         [0000]     Time-Varying Adaptive Case with Gaussian Distribution  
         [0125]     Now, we assume both parameters A 0  and p are time-varying with Gaussian distribution and track the variation using LMS algorithm. First, we simulate convergence of our algorithm in  FIG. 20 . In this simulation, we assume two parameters A 0  and p are change continuously with Gaussian distribution (Mean E(A 0 )=1.5, E(p)=1.5 and variance σ A     0   =0.01, σ p =0.01. We use step size μ {circumflex over (p(n) =10000 for fast convergence. As a MSE (Mean Square Error), we calculate error 100 times each and average them. Since MSE of A 0  depends on MSE of p, their MSE show similar characteristic. In the  FIG. 21  (IBO=6 dB) and  FIG. 22  (IBO=7 dB), we compare the case of tracking the variation of parameters p and A 0 , and without tracking the variation of parameters p and A 0 . In these simulations, we assumed two parameters p and A 0  are Gaussian distribution with variance 0.1. Since, in real system, the characteristic of HPA is not change so rapidly, we assume the two parameters p and A 0  change every 768 symbols and we know when the parameters may change. If the parameters change faster, then we just reduce the period of training stage to track the variance of two parameters timely. We use step size μ {circumflex over (p(n) =5000 for fast convergence. In the case of without tracking, we use average values of two parameters p and A 0  which 1.5 each. One more thing, we should mention is that regarding choose training symbols, we should choose symbols from nonlinear enough place in the HPA function. If input is very small, HPA operates in very close to linear situation. That is to say, this case is input=output. Then from equation (53), A 0  goes to infinity and we can&#39;t find two parameters p and A 0 . However, HPA has always nonlinear region, (If it doesn&#39;t have nonlinear part, we don&#39;t need to use pre-distorter), we can always find two appropriate parameters p and A 0 .  
         [0126]     The advantages of the model-based pre-distortion approach described above for eliminating or mitigating nonlinear distortion in time-varying high power amplifier amplifiers  24  used in OFDM-based wireless communications  10  can now be appreciated. The approach uses closed form inverses of the Saleh model of traveling wave tube amplifier and the Rapp&#39;s model of solid state power amplifier, with very few parameters required in the representation of the inverse. This sparse and yet accurate representation enables the rapid tracking of the time-varying behavior of the high power amplifier  24 . These properties have been verified by simple computer simulations.  
         [0127]     Many alterations and modifications may be made by those having ordinary skill in the art without departing from the spirit and scope of the invention. Therefore, it must be understood that the illustrated embodiment has been set forth only for the purposes of example and that it should not be taken as limiting the invention as defined by the following invention and its various embodiments.  
         [0128]     Therefore, it must be understood that the illustrated embodiment has been set forth only for the purposes of example and that it should not be taken as limiting the invention as defined by the following claims. For example, notwithstanding the fact that the elements of a claim are set forth below in a certain combination, it must be expressly understood that the invention includes other combinations of fewer, more or different elements, which are disclosed in above even when not initially claimed in such combinations. A teaching that two elements are combined in a claimed combination is further to be understood as also allowing for a claimed combination in which the two elements are not combined with each other, but may be used alone or combined in other combinations. The excision of any disclosed element of the invention is explicitly contemplated as within the scope of the invention.  
         [0129]     The words used in this specification to describe the invention and its various embodiments are to be understood not only in the sense of their commonly defined meanings, but to include by special definition in this specification structure, material or acts beyond the scope of the commonly defined meanings. Thus if an element can be understood in the context of this specification as including more than one meaning, then its use in a claim must be understood as being generic to all possible meanings supported by the specification and by the word itself.  
         [0130]     The definitions of the words or elements of the following claims are, therefore, defined in this specification to include not only the combination of elements which are literally set forth, but all equivalent structure, material or acts for performing substantially the same function in substantially the same way to obtain substantially the same result. In this sense it is therefore contemplated that an equivalent substitution of two or more elements may be made for any one of the elements in the claims below or that a single element may be substituted for two or more elements in a claim. Although elements may be described above as acting in certain combinations and even initially claimed as such, it is to be expressly understood that one or more elements from a claimed combination can in some cases be excised from the combination and that the claimed combination may be directed to a subcombination or variation of a subcombination.  
         [0131]     Insubstantial changes from the claimed subject matter as viewed by a person with ordinary skill in the art, now known or later devised, are expressly contemplated as being equivalently within the scope of the claims. Therefore, obvious substitutions now or later known to one with ordinary skill in the art are defined to be within the scope of the defined elements.  
         [0132]     The claims are thus to be understood to include what is specifically illustrated and described above, what is conceptionally equivalent, what can be obviously substituted and also what essentially incorporates the essential idea of the invention.