Pre-distorter for orthogonal frequency division multiplexing systems and method of operating the same

• US 20060039498A1
• Filed: 08/17/2005
• Published: 02/23/2006
• Est. Priority Date: 08/19/2004
• Status: Abandoned Application

• Current Assignee: Regents of the University of California (University of California)
• Original Assignee: Regents of the University of California (University of California)
• Inventors: de Figueiredo, Rui J. P., Lee, Byung Moo
• Application Number:
  US11/205,937
• Publication Number:
  US 20060039498A1
• Time in Patent Office: Days
• US Class Current: 375/297
• CPC Class Codes: H03F 1/3241 using predistortion circuits H03F1/3211, H03F1/3217 take precedence
• CPC Class Codes: H04L 27/2626 Arrangements specific to the transmitter only
• CPC Class Codes: H04L 27/368 adaptive predistortion

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.

Claims

1. A pre-distorter in combination with a high power amplifier in a communication system comprising a digital nonlinear signal processing device of an orthogonal frequency division multiplexing (OFDM) signal, which device is placed before the high power amplifier, which power amplifier provides as high a power as possible for the OFDM signal being passed by the high power amplifier to the communication system, where the power amplifier has a normal linear range outside of which the power amplifier is nonlinear, and where the pre-distorter inverts the nonlinearity of the power amplifier, so that the combination of the pre-distorter and high power amplifier collectively exhibit a linear characteristic beyond the normal linear range of the high power amplifier, where the pre-distorter is characterized by an 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.

2. The pre-distorter of claim 1 where the high power amplifier comprises a traveling wave tube amplifier with time-varying characteristic or a solid state power amplifier with time-varying characteristic where the pre-distorter is characterized by a mixed computational/analytical algorithm for compensation of nonlinear distortion of the power amplifier.

3. The pre-distorter of claim 2 where the analytic model for the high power amplifier is a Saleh traveling wave tube amplifier model and where the computational/analytical algorithm for compensation of nonlinear distortion comprises an algorithm for analytical inversion in combination with a nonlinear parameter estimation algorithm to provide sparse and accurate representation of the pre-distorter, with the capability of tracking efficiently any rapidly time-varying behavior of the high power amplifier.

4. The pre-distorter of claim 2 where the analytic model for the high power amplifier is a Rapp'"'"'s solid state power amplifier model and where the computational/analytical algorithm for compensation of nonlinear distortion comprises an algorithm for analytical inversion in combination with a nonlinear parameter estimation algorithm to provide sparse and accurate representation of the pre-distorter, with the capability of tracking efficiently any rapidly time-varying behavior of the high power amplifier.

5. The pre-distorter of claim 3 where the Saleh traveling wave tube amplifier model is used to provide an exact closed form expression for the inverse of the amplifier model represented by means of only a few parameters based on an analytical model for the traveling wave tube amplifier to derive a cogent algorithm for an estimated pre-distorter I.

6. The pre-distorter of claim 4 where Rapp'"'"'s solid state power amplifier model is used to provide an exact closed form expression for the inverse of the amplifier model represented by means of only a few parameters based on an analytical model for the solid state power amplifier to derive a cogent algorithm for an estimated pre-distorter II.

7. The pre-distorter of claim 1 where the pre-distorter and power amplifier are each nonlinear zero memory devices and where the pre-distorter precomputes and cancels the nonlinear distortion present in the power amplifier.

8. The pre-distorter of claim 5 where the Saleh traveling wave tube amplifier model is represented as

$u$

[ r ] = α

⁢

 

⁢

r 1 + β

⁢

 

⁢

r 2 $\mathrm{\Phi <br><br>}<br><br>\left[r\right]=\frac{\mathrm{\gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>r^2}{1+\mathrm{\varepsilon <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>r^2}$

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.

9. The pre-distorter of claim 6 where Rapp'"'"'s solid state power amplifier model is represented as

$u$

[ r ] = r ( 1 + ( r A 0 ) 2 ⁢

p ) 1 2 ⁢

p $\mathrm{\Phi <br><br>}<br><br>\left[r\right]\approx <br><br>0$

where r is input amplitude of solid state power amplifier, A₀ is the maximum output amplitude and p is the parameter which affects the smoothness of the transition.

10. The distorter of claim 1 where the power amplifier and hence the pre-distorter is characterized by parameters α

, β

, γ

, and ε

, and where q and u denote nonlinear zero memory input and output maps respectively of the pre-distorter and high power amplifier, and x_l(n), denotes the input of the pre-distorter, y_l(n) denotes the output of the pre-distorter which is also the input to the high power amplifier, and z(t) the output of the high power amplifier, such that for any given power amplifier, operation of the pre-distorter is characterized by the input-output maps

u[q(x_l(n))]=k x_l(n)

where k is a desired pre-specified linear amplification constant, and

where the power amplifier is a traveling wave tube, and where the input and output of traveling wave tube amplifier are

y(t)=q[r(t)] cos(ω

_ct+φ

(t)+θ

[r(t)])
z(t)=u[q[r(t)]] cos(ω

_ct+φ

(t)+θ

[r(t)]+Φ

[q(t)])

where

$u<br><br>\left[q<br><br>\left(r\right)\right]=\frac{\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}$$\mathrm{\Phi <br><br>}<br><br>\left[q<br><br>\left(r\right)\right]=\frac{\mathrm{\gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}{1+\mathrm{\varepsilon <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}$

where the following relationships hold

$\frac{\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}=r$$\frac{\mathrm{\gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}{1+\mathrm{\varepsilon <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}=-\mathrm{\theta <br><br>}$$r<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2-\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q+r=0$

to yield

$q<br><br>\left(r\right)=\frac{\mathrm{\alpha <br><br>}-\sqrt{{\mathrm{\alpha <br><br>}}^2-4<br><br>r^2<br><br>\mathrm{\beta <br><br>}}}{2<br><br>r<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\beta <br><br>}},r\le <br><br>1$

where parameters α

, β

, γ

, and ε

change with time so that

$\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>E<br><br>\left(\frac{q^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^3}\right)=E<br><br>\left(\frac{q^3<br><br>u}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^2}\right)$

where E is expectation with respect to β and

$A<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{q^2}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^2}\right)$$B<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{\mathrm{qu}}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}\right)$$C<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{q^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^3}\right)$$D<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{q^3<br><br>u}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^2}\right)$

so that

$\mathrm{\alpha <br><br>}=\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}$$\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}<br><br>C<br><br>\left(\mathrm{\beta <br><br>}\right)=D<br><br>\left(\mathrm{\beta <br><br>}\right)$

which is solved numerically for {circumflex over (β

, which is the estimate of β

, and then {circumflex over (β

is used in

$\mathrm{\alpha <br><br>}=\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}$

to obtain {circumflex over (α

, an estimate for a and the estimates then generated as defined by

$\begin{array}{c}\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^2}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^2}\\ \hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n<br><br>u_n}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2}\\ \hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^3}\\ \hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^3<br><br>u_n}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^2}\end{array}$

and further estimating γ and

ε

according to a similar manner,

obtaining the optimal estimation of β

, using

{circumflex over (β

_opt=min_β

|B(β

)C(β

)−

A(β

)D(β

)|²

where the optimal coefficient {circumflex over (β

_opt, satisfies {circumflex over (β

_opt=min_β

|B(β

)C(β

)−

A(β

)D(β

)|² which 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

where J is cost function to be minimized and E is expectation with respect to β

obtaining the derivative of J with respect to β

using

$\frac{\partial <br><br>J<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=<br><br>2<br><br>E<br><br>\left(\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)-\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)\right)·<br><br>\left(\frac{\partial <br><br>\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}<br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)+\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\frac{\partial <br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}-\frac{\partial <br><br>\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}<br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)-\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\frac{\partial <br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}\right)$

where

$\begin{array}{c}\frac{\partial <br><br>\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{2}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^3}\\ \frac{\partial <br><br>\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^3<br><br>u_n}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^2}\\ \frac{\partial <br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{3}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^6}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^4}\\ \frac{\partial <br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{2}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^5<br><br>u_n}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^3}\end{array}$

using a LMS (Least Mean Square) algorithm represented as

$\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n+1\right)=\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)-{\mathrm{\mu <br><br>}}_{\widehat{\mathrm{\beta <br><br>}}}·<br><br>\left(\hat{B}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\hat{C}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)-\hat{A}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\hat{D}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)\right)·<br><br>\left(\frac{\partial <br><br>\hat{B}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}<br><br>\hat{C}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)+\hat{B}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\frac{\partial <br><br>\hat{C}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}-\frac{\partial <br><br>\hat{A}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}<br><br>\hat{D}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)-\hat{A}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\frac{\partial <br><br>\hat{D}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}\right)$

after obtaining an estimation of β

, obtaining an estimation of α

from

$\mathrm{\alpha <br><br>}=\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}.$

γ and

ε

using the same sequence of above operations.

11. The pre-distorter of claim 1 where the power amplifier is characterized by parameters α

, β

, γ

, and ε

, and further comprising a digital signal processor coupled between the power amplifier and the pre-distorter for generating estimated parameters {circumflex over (α

, {circumflex over (β

, {circumflex over (γ

, and {circumflex over (ε

of the power amplifier to control the pre-distorter in a time varying fashion.

12. The pre-distorter of claim 1 where the pre-distorter is characterized by at least two parameters, and further comprising a digital signal processor coupled between the power amplifier and the pre-distorter for generating at least two estimated parameters of the pre-distorter to control the pre-distorter in a time varying fashion in response to the time varying power amplifier.

13. The distorter of claim 10 where zero phase distortion is obtained

θ

(r)+Φ

(q)=0

so that

$\mathrm{\theta <br><br>}<br><br>\left(r\right)=-\mathrm{\Phi <br><br>}<br><br>\left(q\right)=-\frac{{\mathrm{\gamma <br><br>}<br><br>\left(q<br><br>\left(r\right)\right)}^2}{1+{\mathrm{\varepsilon <br><br>}<br><br>\left(q<br><br>\left(r\right)\right)}^2}.$

14. The distorter of claim 1 where q and u denote nonlinear zero memory input and output maps respectively of the pre-distorter and high power amplifier, and x_l(n), denotes the input of the pre-distorter, y_l(n) denotes the output of the pre-distorter which is also the input to the high power amplifier, and z(t) the output of the high power amplifier, such that for any given power amplifier, operation of the pre-distorter is characterized by the input-output maps

u[q(x_l(n))]=k x_l(n)

where k is a desired pre-specified linear amplification constant, and

where the power amplifier is a solid state power amplifier characterized by parameters A₀ and p which change with time,

where the input of the pre-distorter is denoted as q(n) and the output of the pre-distorter is denoted as u(n),

where during a training stage, it is assumed that pre-distorter is turned off so that the input and output of the pre-distorter is same r(n)=q(n),.

where a MSE (Mean Square Error) for LMS (Least Mean Square) algorithm is employed to generate A₀ and p in which

$A_0=\frac{q·}{}$

u(q2⁢

p-u2⁢

p)12⁢

p

so that given p, A₀ is generated as a function of time by sending two training symbols to provide a known input q to the high power amplifier and obtain an output amplitude u of the high power amplifier to generate two different estimations of A₀, namely A₀₁ and A₀₂.

$A_{01}=\frac{q_1·<br><br>u_1}{{\left(q_1^{2<br><br>p}-u_1^{2<br><br>p}\right)}^{\frac{1}{2<br><br>p}}}$$A_{02}=\frac{q_2·<br><br>u_2}{{\left(q_2^{2<br><br>p}-u_2^{2<br><br>p}\right)}^{\frac{1}{2<br><br>p}}}$

where q₁, u₁ are output amplitudes of the pre-distorter and high power amplifier each for first training symbol and q₂, u₂ are output amplitudes of the pre-distorter and high power amplifier each for second training symbol to estimate unknown A₀ and p using

{circumflex over (p_opt=min_p|A₀₁(p)−

A₀₂(p)|²
Â

₀=A₀₁({circumflex over (p_opt)—

A₀₂({circumflex over (popt)

where {circumflex over (p_opt is an optimum estimate p and an estimate of A₀ are generated so that an LMS (Least Mean Square) algorithm tracks time variation of p and an optimum coefficient {circumflex over (p_opt is determined in order to minimize the MSE (Mean Square Error) criteria defined by

J(p)=E(A₀₁(p)−

A₀₂(p))²

and the LMS algorithm to estimate p is represented as

$\hat{p}<br><br>\left(n+1\right)=<br><br>\hat{p}<br><br>\left(n\right)-{\mathrm{\mu <br><br>}}_{\hat{p}<br><br>\left(n\right)}·<br><br>\left(A_{01}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)-A_{02}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)\right)·<br><br>\left(\frac{\partial <br><br>A_{01}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)}{\partial <br><br>\hat{p}<br><br>\left(n\right)}-\frac{\partial <br><br>A_{02}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)}{\partial <br><br>\hat{p}<br><br>\left(n\right)}\right)$

where μ

_{{circumflex over (p(n)} is the step size of LMS algorithm.

15. The distorter of claim 1 where q and u denote nonlinear zero memory input and output maps respectively of the pre-distorter and high power amplifier, and x_l(n), denotes the input of the pre-distorter, y_l(n) denotes the output of the pre-distorter which is also the input to the high power amplifier, and z(t) the output of the high power amplifier, such that for any given power amplifier, operation of the pre-distorter is characterized by the input-output maps

u[q(x_l(n))]=k x_l(n)

where k is a desired pre-specified linear amplification constant, and

where the power amplifier is a solid state power amplifier characterized by parameters A₀ and p which change with time, where the input of the pre-distorter is denoted as q(n) and the output of the pre-distorter is denoted as u(n),

where during a training stage, it is assumed that pre-distorter is turned off so that the input and output of the pre-distorter is same r(n)=q(n),.

where a MSE (Mean Square Error) for LMS (Least Mean Square) algorithm is employed to generate A₀ and p in which

$A_0=\frac{q·}{}$

u(q2⁢

p-u2⁢

p)12⁢

p

so that for a given p, A₀ is generated, where both A₀ and p change with time

where two training symbols are sent to the distorter so that input amplitude q and the output amplitude u of the high power amplifier is known,

where corresponding to two different training symbols, two different estimations of A₀, namely A₀₁ and A₀₂ are generated, where a p is chosen which is nearly constant during the training period in the high power amplifier, the two different estimations of A₀, namely A₀₁ and A₀₂, have almost the same value or due to step size, very close values, so that a value for p can be found, which yields the smallest distance between two estimated A₀, namely D_min=A₀₁−

A₀₂|² and from the estimation of p, Â

₀=A₀₁≈

A₀₂ from the minimum distance D_min=A₀₁−

A₀₂ |² using only two training symbols and no iteration.

16. A method of operating a pre-distorter which is placed before a high power amplifier in a communication system where the power amplifier has a normal linear range outside of which the power amplifier is nonlinear comprising:

providing an orthogonal frequency division multiplexing(OFDM) signal;

pre-distorting the OFDM signal by means of the pre-distorter by inverting OFDM signal as determined by the nonlinearity of the power amplifier, where operation of the pre-distorter is characterized by an 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; and

amplifying the pre-distorted the OFDM signal with the power amplifier to as high a power as possible for the OFDM signal being passed by the high power amplifier to the communication system, so that the combination of the pre-distorter and high power amplifier collectively exhibit a linear characteristic beyond the normal linear range of the high power amplifier,.

17. The method of claim 16 where the high power amplifier comprises a traveling wave tube amplifier with time-varying characteristic a or solid state power amplifier with time-varying characteristic and where pre-distorting the OFDM signal by means of the pre-distorter comprises using a mixed computational/analytical algorithm for compensation of nonlinear distortion of the power amplifier.

18. The method of claim 17 where the analytic model for the high power amplifier is a Saleh traveling wave tube amplifier model and where using a mixed computational/analytical algorithm comprises analytical inverting and using a nonlinear parameter estimation algorithm to provide sparse and accurate representation of the pre-distorter, with the capability of tracking efficiently any rapidly time-varying behavior of the high power amplifier.

19. The method of claim 17 where the analytic model for the high power amplifier is a Rapp'"'"'s solid state power amplifier model and where using a mixed computational/analytical algorithm comprises analytically inverting and using a nonlinear parameter estimation algorithm to provide sparse and accurate representation of the pre-distorter, with the capability of tracking efficiently any rapidly time-varying behavior of the high power amplifier.

20. The method of claim 18 further comprising using the Saleh traveling wave tube amplifier model to provide an exact closed form expression for the inverse of the amplifier model represented by means of only a few parameters based on an analytical model for the traveling wave tube amplifier to derive a cogent algorithm for an estimated pre-distorter 1.

21. The method of claim 19 further comprising using Rapp'"'"'s solid state power amplifier model to provide an exact closed form expression for the inverse of the amplifier model represented by means of only a few parameters based on an analytical model for the solid state power amplifier to derive a cogent algorithm for an estimated pre-distorter II.

22. The method of claim 16 where the pre-distorter and power amplifier are each nonlinear zero memory devices where pre-distorting the OFDM signal by means of the pre-distorter comprises pre-computing and canceling the nonlinear distortion present in the power amplifier.

23. The method of claim 20 where using the Saleh traveling wave tube amplifier model comprises modeling the power amplifier using

$u$

[ r ] = α

⁢

 

⁢

r 1 + β

⁢

 

⁢

r 2 $\mathrm{\Phi <br><br>}<br><br>\left[r\right]=\frac{\mathrm{\gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>r^2}{1+\mathrm{\varepsilon <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>r^2}$

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.

24. The method of claim 21 where using Rapp'"'"'s solid state power amplifier model comprises modeling the power amplifier using

$u$

[ r ] = r ( 1 + ( r A 0 ) 2 ⁢

p ) 1 2 ⁢

p $\mathrm{\Phi <br><br>}<br><br>\left[r\right]\approx <br><br>0$

where r is input amplitude of solid state power amplifier, A₀ is the maximum output amplitude and p is the parameter which affects the smoothness of the transition.

25. The method of claim 16 where pre-distorting the OFDM signal by means of the pre-distorter comprises characterizing the power amplifier and hence the pr-distorter by parameters α

, β

, γ

, and ε

, and where q and u denote nonlinear zero memory input and output maps respectively of the pre-distorter and power amplifier, and x_l(n), denotes the input of the pre-distorter, y_l(n) denotes the output of the pre-distorter which is also the input to the high power amplifier, and z(t) the output of the high power amplifier, and such that for any given power amplifier, operating the pre-distorter according to the input-output maps

u[q(x_l(n))]=k x_l(n)

where k is a desired pre-specified linear amplification constant, and

where the power amplifier is a traveling wave tube, and operating the traveling wave tube amplifier so that the input and output of traveling wave tube amplifier are

y(t)=q[r(t)]cos(ω

_ct+φ

(t)+θ

[r(t)])
z(t)=u[q[r(t)]]cos(ω

_ct+φ

(t)+θ

[r(t)]+Φ

[q(t)])

where

$u<br><br>\left[q<br><br>\left(r\right)\right]=\frac{\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}$$\mathrm{\Phi <br><br>}<br><br>\left[q<br><br>\left(r\right)\right]=\frac{\mathrm{\gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}{1+\mathrm{\varepsilon <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}$

where the following relationships hold

$\frac{\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}=r$$\frac{\mathrm{\gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}{1+\mathrm{\varepsilon <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}=-\mathrm{\theta <br><br>}$$r<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2-\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q+r=0$

to yield

$q<br><br>\left(r\right)=\frac{\mathrm{\alpha <br><br>}-\sqrt{{\mathrm{\alpha <br><br>}}^2-4<br><br>r^2<br><br>\mathrm{\beta <br><br>}}}{2<br><br>r<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\beta <br><br>}},r\le <br><br>1$

where parameters α

, β

, γ

, and ε

change with time so that

$\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>E<br><br>\left(\frac{q^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^3}\right)=E<br><br>\left(\frac{q^3<br><br>u}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^2}\right)$

Where E is expectation w.r.t. β and

$A<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{q^2}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^2}\right)$$B<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{\mathrm{qu}}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2}\right)$$C<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{q^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^3}\right)$$D<br><br>\left(\mathrm{\beta <br><br>}\right)=E<br><br>\left(\frac{q^3<br><br>u}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q^2\right)}^2}\right)$

so that

$\mathrm{\alpha <br><br>}=\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}$$\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}<br><br>C<br><br>\left(\mathrm{\beta <br><br>}\right)=D<br><br>\left(\mathrm{\beta <br><br>}\right)$

solving numerically for {circumflex over (β

, which is the estimate of β

, and then using {circumflex over (β

in

$\mathrm{\alpha <br><br>}=\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}$

to to obtain {circumflex over (α

, an estimate for α

, generating the estimates as defined by

$\begin{array}{c}\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^2}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^2}\\ \hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n<br><br>u_n}{1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2}\\ \hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^3}\\ \hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)=\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^3<br><br>u_n}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^2}\end{array}$

and further estimating γ and

ε

in the same manner,

obtaining the optimum estimation of β

, using

{circumflex over (β

_opt=min_β

|B(β

)C(β

)−

A(β

)D(β

)|²

where the optimum coefficient {circumflex over (β

_opt, satisfyies {circumflex over (β

_opt=min_β

|B(β

)C(β

)−

A(β

)D(β

)|² which is determined in order to minimize the MSE (Mean Square Error) defined by

J(β

)=E[{circumflex over (B(β

)Ĉ

(β

)−

Â

(β

){circumflex over (D(β

)]²

where J is cost function to be minimized and E is expectation with respect to β

obtaining the derivative of J with respect to β

$\frac{\partial <br><br>J<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=<br><br>2<br><br>E<br><br>\left(\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)-\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)\right)·<br><br>\left(\frac{\partial <br><br>\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}<br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)+\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\frac{\partial <br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}-\frac{\partial <br><br>\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}<br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)-\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)<br><br>\frac{\partial <br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}\right)$

where

$\begin{array}{c}\frac{\partial <br><br>\hat{A}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{2}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^4}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^3}\\ \frac{\partial <br><br>\hat{B}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{1}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^3<br><br>u_n}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^2}\\ \frac{\partial <br><br>\hat{C}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{3}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^6}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^4}\\ \frac{\partial <br><br>\hat{D}<br><br>\left(\mathrm{\beta <br><br>}\right)}{\partial <br><br>\mathrm{\beta <br><br>}}=-\frac{2}{N}<br><br>\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\frac{q_n^5<br><br>u_n}{{\left(1+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q_n^2\right)}^3}\end{array}$

using a LMS (Least Mean Square) algorithm represented as

$\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n+1\right)=\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)-{\mathrm{\mu <br><br>}}_{\widehat{\mathrm{\beta <br><br>}}}·<br><br>\left(\hat{B}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\hat{C}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)-\hat{A}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\hat{D}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)\right)·<br><br>\left(\frac{\partial <br><br>\hat{B}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}<br><br>\hat{C}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)+\hat{B}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\frac{\partial <br><br>\hat{C}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}-\frac{\partial <br><br>\hat{A}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}<br><br>\hat{D}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)-\hat{A}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)<br><br>\frac{\partial <br><br>\hat{D}<br><br>\left(\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)\right)}{\partial <br><br>\widehat{\mathrm{\beta <br><br>}}<br><br>\left(n\right)}\right)$

to obtain an estimation of β

,

obtaining an estimation of α

from

$\mathrm{\alpha <br><br>}=\frac{B<br><br>\left(\mathrm{\beta <br><br>}\right)}{A<br><br>\left(\mathrm{\beta <br><br>}\right)}.$

and estimating γ and

ε

using the same approach.

26. The method of claim 16 where pre-distorting the OFDM signal by means of the pre-distorter comprises characterizing the power amplifier by time varying parameters α

, β

, γ

, and ε

, and generating estimated parameters {circumflex over (α

, {circumflex over (β

, {circumflex over (γ

, and {circumflex over (ε

of the power amplifier to control the pre-distorter in a time varying fashion.

27. The method of claim 16 where pre-distorting the OFDM signal by means of the pre-distorter comprises characterizing the power amplifier by at least two time varying parameters, and generating at least two estimated parameters of the power amplifier to control the pre-distorter in a time varying fashion.

28. The method of claim 25 where pre-distorting the OFDM signal by means of the pre-distorter comprises providing for zero phase distortion so that

$\theta$

⁡

( r ) + Φ

⁡

( q ) = 0 $\mathrm{and}$ $\mathrm{\theta <br><br>}<br><br>\left(r\right)=-\mathrm{\Phi <br><br>}<br><br>\left(q\right)=-\frac{{\mathrm{\gamma <br><br>}<br><br>\left(q<br><br>\left(r\right)\right)}^2}{1+{\mathrm{\varepsilon <br><br>}<br><br>\left(q<br><br>\left(r\right)\right)}^2}.$

29. The method of claim 16 where pre-distorting the OFDM signal by means of the pre-distorter comprises using q and u to denote nonlinear zero memory input and output maps respectively of the pre-distorter and high power amplifier, and x_l(n), to denote the input of the pre-distorter, y_l(n) to denote the output of the pre-distorter which is also the input to the high power amplifier, and z(t) the output of the high power amplifier, such that for any given power amplifier, operating the pre-distorter according to the input-output maps

u[q(x_l(n))]=k x_l(n)

where k is a desired pre-specified linear amplification constant, and characterizing the power amplifier as a solid state power amplifier by parameters A₀ and p which change with time,

where the input of the pre-distorter is denoted as q(n) and the output of the pre-distorter is denoted as u(n), providing a training stage, during which it is assumed that pre-distorter is turned off so that the input and output of the pre-distorter is same r(n)=q(n),

generating A₀ and p using a MSE (Mean Square Error) for LMS (Least Mean Square) algorithm in which

$A_0=\frac{q·}{}$

u(q2⁢

p-u2⁢

p)12⁢

p

so that given p, A₀ is generated as a function of time by sending two training symbols to provide a known input q to the power amplifier and obtaining an output amplitude u of the power amplifier to generate two different estimations of A₀, namely A₀₁ and A₀₂.

$A_{01}=\frac{q_1·<br><br>u_1}{{\left(q_1^{2<br><br>p}-u_1^{2<br><br>p}\right)}^{\frac{1}{2<br><br>p}}}$$A_{02}=\frac{q_2·<br><br>u_2}{{\left(q_2^{2<br><br>p}-u_2^{2<br><br>p}\right)}^{\frac{1}{2<br><br>p}}}$

where q₁, u₁ are output amplitudes of the pre-distorter and power amplifier each for first a training symbol and q₂, u₂ are output amplitudes of the pre-distorter and power amplifier each for a second training symbol,

estimating unknown A₀ and p using

{circumflex over (p_opt=min_p|A₀₁(p)−

A₀₂(p)|²
Â

₀=A₀₁({circumflex over (p_opt)≈

A₀₂({circumflex over (p_opt)

where {circumflex over (p_opt is an optimum estimate p and generating an estimate of A₀′

tracking time variation of p using an LMS (Least Mean Square) algorithm and determining an optimum coefficient {circumflex over (p_opt in order to minimize the MSE (Mean Square Error) criteria defined by

J(p)=E(A₀₁(p)−

A₀₂(p))²

and estimating p using the LMS algorithm with

$\hat{p}<br><br>\left(n+1\right)=\hat{p}<br><br>\left(n\right)-{\mathrm{\mu <br><br>}}_{\hat{p}<br><br>\left(n\right)}·<br><br>\left(A_{01}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)-A_{02}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)\right)·<br><br>\left(\frac{\partial <br><br>A_{01}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)}{\partial <br><br>\hat{p}<br><br>\left(n\right)}-\frac{\partial <br><br>A_{02}<br><br>\left(\hat{p}<br><br>\left(n\right)\right)}{\partial <br><br>\hat{p}<br><br>\left(n\right)}\right)$

where μ

_{{circumflex over (p(n)} is the step size of LMS algorithm.

30. The method of claim 16 where pre-distorting the OFDM signal by means of the pre-distorter comprises using q and u to denote nonlinear zero memory input and output maps respectively of the pre-distorter and high power amplifier, and x_l(n), to denote the input of the pre-distorter, y_l(n) to denote the output of the pre-distorter which is also the input to the high power amplifier, and z(t) the output of the high power amplifier, such that for any given power amplifier, operating the pre-distorter according to the input-output maps

u[q(x_l(n))]=k x_l(n)

where k is a desired pre-specified linear amplification constant, and characterizing the power amplifier as a solid state power amplifier with parameters A₀ and p which change with time,

where the input of the pre-distorter is denoted as q(n) and the output of the pre-distorter is denoted as u(n), providing a training stage, during which it is assumed that pre-distorter is turned off so that the input and output of the pre-distorter is same r(n)=q(n),. generating A₀ and p using a MSE (Mean Square Error) for LMS (Least Mean Square) algorithm in which

$A_0=\frac{q·}{}$

u(q2⁢

p-u2⁢

p)12⁢

p

so that for a given p, A₀ is generated, where both A₀ and p change with time

sending two training symbols to the distorter so that input amplitude q and the output amplitude u of the high power amplifier is known,

generating two different estimations of A₀, namely A₀₁ and A₀₂ corresponding to two different training symbols,

choosing a p which is nearly constant during the training period in the high power amplifier, the two different estimations of A₀, namely A₀₁ and A₀₂, having almost the same value or due to step size, very close values, and

finding a value for p, which yields the smallest distance between two estimated A₀, namely D_min=|A₀₁−

A₀₂|² and from the estimation of p, Â

₀=A₀₁≈

A₀₂ from the minimum distance D_min=|A01−

A₀₂|² using only two training symbols and no iteration.