II. BACKGROUND

In this paper, the nonlinear behavior of the PA is compensated using DPD. In particular, the challenge of performing adaptive DPD based on an observation pass band that is narrower than the bandwidth of the discernible IMD products is addressed.

The nonlinearity in the PA can be modeled by a polynomial function of the input signal. Consider the PA response to a Gaussian noise input signal that is band limited to fill the allocated channel. The output spectrum is the sum of the linear component and several intermodulation modes (the third- and fifth-order IMD products are shown in Fig. 1) [Reference Braithwaite and Luo3]. The linear component, also referred to as a carrier signal, remains within the allocated channel whereas the distortion spreads into the adjacent channels. The IMD bandwidth increases with the polynomial order. However, a significant portion of the distortion lies within the allocated channel.

Fig. 1.

Output spectra for the linear signal, as well as the third- and fifth-order intermodulation distortion products. Also shown are the allocated and adjacent channels. Note the increase in IMD bandwidth with the polynomial order. From [Reference Braithwaite and Luo3].

The RF transmitter may transmit several carrier signals occupying nearby channels. In such cases, the IMD comprises components originating from the intermodulation within individual carriers as well as intermodulation between carriers. The effect of the latter is to spread the distortion over a much larger bandwidth, as shown in Fig. 2.

Fig. 2.

Linear and third-order IMD spectra for a two-carrier signal. Note the increased bandwidth of the distortion spectrum arising from intermodulation between carriers.

The nonlinear behavior of a PA can be classified as memoryless or memory-based. For the former, the nonlinear modes are functions of the instantaneous input only. For the latter, the nonlinear modes are also affected by past input values. The PA memory alters the frequency response of the distortion generated and increases the number of predistortion coefficients needed for wide bandwidth IMD cancellation.

The memory effects exhibited by PAs tend to be more discernible for wide bandwidth input signals, such as those comprising multiple carrier signals. In particular, dramatic memory effects are common with the Doherty amplifier configuration, which is popular in cellular base stations due to its high efficiency when operating at power levels near saturation [Reference Cripps4].

The digitally predistorted transmitter used in this paper is shown in Fig. 3. The digital input and predistorted signals are denoted by x(k) and x _DPD(k), respectively. The RF output is

Fig. 3.

Digital transmitter with predistortion module, estimator, and narrow bandwidth output observation path. Adapted from [Reference Braithwaite2].

© 2008 IEEE. Reprinted, with permission, from [2].

(1)$$y_{RF} \lpar t\rpar =G_{PA} \left[{h\lcub x_{DPD} \lpar k\rpar \rcub \exp\! \lpar j \omega _c t\rpar } \right]\comma \eqno \lpar 1\rpar$$

where G _PA is the nonlinear gain of the PA, h{} represents the reconstruction filter used in the digital-to-analog conversion, and ω _c is the LO₁ frequency. The observation path captures a digital measurement of the output, denoted by y _obs(k), by down-converting the RF output to an intermediate frequency (IF) using LO₂, bandpass filtering, digitizing, down-converting the digital IF to baseband using a numerically controlled oscillator (NCO), then equalizing for gain flatness.

The DPD module generates several nonlinear modes derived from the input signal x(k). That is, the predistorted signal is

(2)$$x_{DPD} \lpar k\rpar =x\lpar k\rpar +\sum\limits_{i=1}^N {a_i \gamma _i } \lpar k\rpar \comma \eqno\lpar 2\rpar$$

where γ_i(k) are nonlinear functions of x(k) referred to as basis waveforms, a _i are complex coefficients, and N is the order of the DPD model. By weighting the nonlinear basis waveforms correctly, distortion is generated that cancels the IMD generated subsequently by the PA nonlinearity. To accommodate the increased bandwidth of the nonlinear basis waveforms, the sampling rate after the DPD module must be several times the Nyquist rate of the linear signal x(k).

The selection of the basis waveform set is important for the predistortion to achieve the required distortion cancellation. For the memoryless DPD case, the basis waveforms are

(3)$$\gamma _i \lpar k\rpar =\left\vert {x\lpar k\rpar } \right\vert ^{p - 1} x\lpar k\rpar \comma \eqno \lpar 3\rpar$$

where p is the polynomial order and the index i is a function of p. The DPD model order N tends to be small and is bounded by the highest polynomial order (denoted by P) in the basis set. In some implementations, only the odd polynomial orders are used, thereby reducing the size of N.

For the case of a DPD module with memory correction capability, the DPD model order N exceeds the highest polynomial order P. One possible basis set with memory is the Volterra series, which produces a large number of basis waveforms. The basis waveforms for the polynomial order p = 3 are

(4)$$\gamma _i \lpar k\rpar =x\lpar k - m_1 \rpar x\lpar k - m_2 \rpar x^{\ast} \lpar k - m_3 \rpar \comma \eqno \lpar 4\rpar$$

where –M ≤ m ₁, m ₂, m ₃ ≤ M and i is a function of p, m ₁, m ₂, and m ₃. The DPD model order is N = (2M + 1)³ for p = 3, which is already large, and becomes exponentially larger as higher values of p are included in the basis set. In addition, many of the basis waveforms in (4) are duplicated. As a result, pruned versions of the Volterra series are used for memory correction in practical implementations [Reference Zhu, Pedro and Cunba5].

A popular memory correction approach implements the predistortion as a nonlinear finite impulse response (FIR) structure with adaptable coefficients. The FIR can be applied to either the gain term [Reference Braithwaite6] or the distortion [Reference Ding7]. For the former case, the basis waveforms are

(5)$$\gamma _i \lpar k\rpar =\left\vert {x\lpar k - m\rpar } \right\vert ^{p - 1} x\lpar k\rpar \comma \eqno \lpar 5\rpar$$

where –M ≤ m ≤ M and i is a function of p and m. The DPD model order is N = P*(2M + 1) assuming the FIR size, 2M + 1, is the same for each polynomial order, p. The basis set (5) is referred to as a gain polynomial. An alternative basis set, referred to as a memory polynomial, uses

(6)$$\gamma _i \lpar k\rpar =\left\vert {x\lpar k - m\rpar } \right\vert ^{p - 1} x\lpar k - m\rpar \comma \eqno \lpar 6\rpar$$

as the basis waveforms.

The DPD coefficients are tuned adaptively to provide effective distortion compensation. This requires an observation of the signal present at the output of the PA. The RF observation signal is converted to the digital baseband using a receiver and compared to the original input signal x(k). Differences between the input signal x(k) and the output signal y(k) are used to estimate the coefficient adjustments required to improve the predistortion. Note that at this point, output signal y(k) and the observed signal y _obs(k) are considered to be the same signal. However, they are treated as different signals in Section III, where y _obs(k) is a band limited version of y(k).

The coefficients are adjusted to minimize the residual distortion in the output signal, which is measured using time-aligned samples of the input and output signals: x = [x(1)···x(K)]^T and y = [y(1) ···y(K)]^T, respectively, where K is the buffer size. Let the error signal and basis set be denoted by

(7)$$\varepsilon \lpar k\rpar =G_o ^{ - 1} y\lpar k\rpar - x\lpar k\rpar \comma \eqno \lpar 7\rpar$$

(8)$$\psi \lpar k\rpar =\lsqb \matrix{\! {\gamma _1 \lpar k\rpar } & \cdots & {\gamma _N \lpar k\rpar } \cr } \rsqb \comma \eqno \lpar 8\rpar$$

respectively, where G _o is the nominal gain. The nominal gain used in this paper is

(9)$$G_o=\displaystyle{{E\lsqb y\lpar k\rpar x^{\ast} \lpar k\rpar \rsqb } \over {E\lsqb \left\vert {x\lpar k\rpar } \right\vert ^2 \rsqb }}\comma \eqno \lpar 9\rpar$$

where E[ ] denotes the expected value. The estimate of the coefficient offset is

(10)$$\underline {\Delta a}=Q^{ - 1} \cdot \underline \rho\comma \eqno \lpar 10\rpar$$

where Δa = [Δa ₁,…, Δa _N]^T are the coefficient offsets,

(11)$$Q=E\lsqb \psi ^H \lpar k\rpar \psi \lpar k\rpar \rsqb =\displaystyle{1 \over K}\sum\limits_{k=1}^K {\psi ^H \lpar k\rpar \psi \lpar k\rpar }\comma \eqno \lpar 11\rpar$$

(12)$$\underline \rho=E\lsqb \psi ^H \lpar k\rpar \varepsilon \lpar k\rpar \rsqb =\displaystyle{1 \over K}\sum\limits_{k=1}^K {\psi ^H \lpar k\rpar \varepsilon \lpar k\rpar }\comma \eqno \lpar 12\rpar$$

and ()^H denotes conjugate transpose. The updated coefficients are

(13)$$\underline a \lpar t_{l+1} \rpar =\underline a \lpar t_l \rpar - \eta \underline {\Delta a}\comma \eqno \lpar 13\rpar$$

where a = [a ₁,…, a _N]^T, t _l indicates the iteration number, and η < 1. The process of capturing data, estimating coefficient offsets, and updating the coefficients is repeated, further reducing the output distortion.

The estimation can also be done in the frequency domain. Let the Fourier transforms of x(k), y(k), γ_i(k), ψ(k), and ɛ(k) be denoted by X(ω _k), Y(ω _k), Γ_i(ω _k), Ψ(ω _k), and E(ω _k), respectively, where

(14)$$\Psi \lpar \omega _k \rpar =\lsqb \matrix{ {\Gamma _1 \lpar \omega _k \rpar } & \cdots & {\Gamma _N \lpar \omega _k \rpar } \cr } \rsqb \eqno \lpar 14\rpar$$

and

(15)$${\rm E}\lpar \omega _k \rpar =\displaystyle{{Y\lpar \omega _k \rpar } \over {G_o }} - X\lpar \omega _k \rpar . \eqno \lpar 15\rpar$$

We have

(16)$$Q=E\lsqb \Psi ^H \lpar \omega _k \rpar \Psi \lpar \omega _k \rpar \rsqb =\displaystyle{1 \over K}\sum\limits_{k=1}^K {\Psi ^H \lpar \omega _k \rpar \Psi \lpar \omega _k \rpar } \eqno \lpar 16\rpar$$

and

(17)$$\underline \rho=E\lsqb \Psi ^H \lpar \omega _k \rpar \!\cdot \!{\rm E}\lpar \omega _k \rpar \rsqb =\displaystyle{1 \over K}\cdot \!\sum\limits_{k=1}^K {\Psi ^H \lpar \omega _k \rpar\! \cdot\! {\rm E}\lpar \omega _k \rpar } . \eqno \lpar 17\rpar$$

The Fourier-based estimation is used in the proposed approach described in Section III.

Careful selection of the basis set ψ(k) minimizes the number of basis waveforms γ_i(k) used, ensuring that the matrix Q is well conditioned. However, the selection criteria for the basis set that satisfies the DPD module's requirement for IMD reduction competes with the estimator's requirement for a well-conditioned matrix Q. A large basis set that describes accurately the physical properties of the PA provides good IMD cancellation, but only if the coefficients are selected correctly. Unfortunately, the input signal x(k) often lacks the statistical content in terms of amplitude variation and spectrum to excite fully the space spanned by the basis set ψ(k). This causes the matrix Q to be rank deficient. Thus, the appropriate model order for the coefficient estimation depends on the input signal statistics as much as the PA nonlinearity.

It is possible to reduce the model order of the estimation compared to the DPD model order. The former will be denoted by N _est and the latter by N. The difference in the model orders of the DPD module and the coefficient estimator can be resolved by using an eigenvalue decomposition of the matrix Q, as shown in the next section. When the input signal x(k) is not rich enough to provide a full rank matrix Q, the number of independent coefficient adjustments for a given iteration of (10) is reduced to equal the number of dominant (or discernible) eigenvalues available in Q. This technique is referred as singular value decomposition (SVD).

The above-mentioned estimation assumes that the bandwidth of the output observation path is larger than the bandwidth of the widest basis waveform. This is not always the case when the input signal x(k) has a wide bandwidth and the maximum polynomial order of the basis set ψ(k) is high.

The implementation of the observation receiver shown in Fig. 3 uses IF sampling, which has advantages in terms of cost, as well as avoiding offsets and imbalances associated with a demodulator used in zero-IF down-conversions. However, the OBW is reduced by at least a factor of two compared to a zero-IF receiver. Assume the analog-to-digital converter (ADC) and the digital-to-analog converters (DAC) within Fig. 3 have the same sampling rate, but that the transmit path uses a zero-IF up conversion (dual DACs). The observation path shown in Fig. 3 does not have enough bandwidth to measure the entire frequency content of the widest basis waveforms that can be generated by the DPD module.

It is not necessary to measure the full spectrum of the IMD waveform to adapt the DPD coefficients. Zhu [Reference Zhu8] has shown that the output of a nonlinear function in response to a band limited input signal need only be sampled at the Nyquist rate of the input signal x(k) rather than the wider bandwidth of the DPD signal x _DPD(k) if the nonlinear function is a one-to-one continuous mapping. With respect to the present application, this means that the OBW must be as wide as the linear signal bandwidth to estimate the coefficients if the DPD module is memoryless.

In general, the coefficient estimate based on (10) will have poor IMD cancellation over a wide bandwidth because the IF filter in the observation path will be interpreted incorrectly as PA memory. In such a case, any memory correction available within the DPD module will degrade the distortion cancellation compared to a memoryless model in the portion of the spectrum outside of the OBW. Thus, there is a penalty associated with over-specifying the model order of the estimation N _est when the observation path is not perfect.

It is shown in Section III that deleterious effects of the limited OBW are minimized by filtering the basis waveforms ψ(k) within the estimator so that the comparison between the observed PA output y _obs(k) and the basis waveforms is made over the identical frequency band. The filtering alters the frequency weighting of the coefficient estimate and is applied only to the estimator. The basis waveform set used by the DPD module is not altered, making it possible to extrapolate the distortion cancellation beyond the observed bandwidth (OBW). However, there is no guarantee that the unseen IMD will be reduced by the narrow bandwidth optimization. The model order of the estimation N _est affects the overall distortion cancellation performance, as shown in Section IV. The selected estimation model order N _est should improve distortion cancellation within the OBW without over-fitting so that the IMD products outside of the OBW are at least as low as those achievable using memoryless DPD.

For a DPD module with memory compensation, it is necessary to either expand the OBW or to constrain the model order of the estimator N _est so that distortion cancellation occurs outside of the OBW. ACPR provides an independent measure of the residual distortion that is used in Section IV to judge whether the selected estimation model order N _est provides adequate cancellation.

III. PROPOSED APPROACH

In this section, the estimation approach is altered so that bandwidth limitations within the observation path do not restrict the predistortion capabilities. When the output signal is measured using a small bandwidth, it is recommended that center frequency of the observation path be adjusted over time using the LO₂. Frequency-shifting the observation band allows as much distortion spectral content as possible to be measured. The estimator combines measurements from the different observation bands to improve distortion cancellation over a wider bandwidth. In addition, the model order of the estimation N _est is adjusted to obtain a low ACPR outside of the OBWs tested.

The cost function minimized by (10) is

(18)$$J=\sum\limits_{k=1}^K {\left\vert {\varepsilon \lpar k\rpar - \psi \lpar k\rpar \!\cdot\! \underline {\Delta a} } \right\vert ^2 } . \eqno \lpar18\rpar$$

In general, (10) cannot be used to estimate the coefficient offsets Δa for the system shown in Fig. 3 because the observed output y _obs(k) is band-limited. That is, y _obs(k) is available, but not y(k).

The relationship between the desired output y(k) and the observed output y _obs(k) is

(19)$$L_{IF} \left\{{y\lpar k\rpar } \right\}=y_{obs} \lpar k\rpar \comma \eqno \lpar 19\rpar$$

where L _IF{} is the frequency response of the IF filter. In most implementations, y _obs(k) would be equalized to obtain the best estimate of y(k). However, equalization can only flatten the pass band of the observation path; it will not restore the part of the spectrum for y(k) lost to band limiting.

An alternative approach is to spectrally weight the estimation (18). That is, superposition is applied to (18), creating a new cost function based on filtered signals:

(20)$$J=\sum\limits_{k=1}^K {\left\vert {L\left\{{\varepsilon \lpar k\rpar } \right\}- \psi _L \lpar k\rpar \cdot \underline {\Delta a} } \right\vert ^2 }\comma \eqno \lpar 20\rpar$$

where L{ } is a filter operation and

(21)$$L\lcub \varepsilon \lpar k\rpar \rcub =G_o ^{ - 1} \cdot L\lcub y\lpar k\rpar \rcub - L\lcub x\lpar k\rpar \rcub \comma \eqno \lpar 21\rpar$$

(22)$$\psi _L \lpar k\rpar =\lsqb \matrix{ {L\lcub \gamma _1 \lpar k\rpar \rcub } & \cdots & {L\lcub \gamma _N \lpar k\rpar \rcub } \cr } \rsqb .\eqno \lpar 22\rpar$$

The estimate of the coefficient offsets using the band limited signals is

(23)$$\underline {\Delta a}=Q_L ^{ - 1} \cdot \underline \rho _L\comma \eqno \lpar 23\rpar$$

where

(24)$$Q_L=\displaystyle{1 \over K}\cdot \sum\limits_{k=1}^K {\psi _L ^H \lpar k\rpar \!\cdot\! \psi _L \lpar k\rpar }\comma \eqno \lpar 24\rpar$$

(25)$$\underline \rho _L=\displaystyle{1 \over K}\cdot \sum\limits_{k=1}^K {\psi _L ^H \lpar k\rpar \!\cdot\! L\lcub \varepsilon \lpar k\rpar } \rcub . \eqno \lpar 25\rpar$$

The key requirement in (20) is that the same linear operation L{} must be applied to each of x(k), y(k), and γ_i(k). We can replace y(k) with y _obs(k) as long as we account for the frequency response of the IF filter L _IF{} within (21). For example, in the simplest case when L{} is the same as L _IF{}, the filtered error signal becomes

(26)$$L_{IF} \lcub \varepsilon \lpar k\rpar \rcub =G_o ^{ - 1} \cdot y_{obs} \lpar k\rpar - L_{IF} \lcub x\lpar k\rpar \rcub . \eqno \lpar26\rpar$$

However, it is often beneficial to choose a filter response L{} that limits the OBW further. That is, if the bandwidth of L{} is smaller than the OBW of L _IF{}, so that L{y(k)} = L{y _obs(k)}, then we do not need to know the exact frequency response of the IF filter (assuming the IF filter response is flat within the frequencies of interest).

It is also beneficial to select L{} to notch the input signal x(k), so that L{x(k)} = 0. This has the effect of optimizing the coefficient estimation for the cancellation of out-of-band IMD products, which contribute to the ACPR. In particular, it removes biases in the estimate of Δa that are due to unmodeled in-band errors, which are often larger in power than the distortion generated. The benefits of notching the input signal x(k) are shown in [Reference Braithwaite9].

The digital filter L{} is implemented in the Fourier domain. The frequency response of the estimation filter L(ω _k) is the product of a notch filter L _notch(ω _k) and a band pass filter L _OBW(ω _k). The frequency response of the notched filter is

(27)$$L_{notch} \lpar \omega _k \rpar =\displaystyle{1 \over {1+\alpha \displaystyle{{\left\vert {X\lpar \omega _k \rpar } \right\vert ^2 } \over {E\lsqb \left\vert {X\lpar \omega _k \rpar } \right\vert ^2 \rsqb }}}}\comma \eqno \lpar 27\rpar$$

where X(ω _k) is the Fourier transform of x(k) and α is a constant controlling the depth of the notch. The frequency response of the band pass filter used to band limit γ_i(k) and ɛ(k) to fit within the OBW is

(28)$$L_{OBW} \lpar \omega _k \rpar =\left\{{\matrix{ 1 & {{\rm for}} & {\left\vert {\omega _k - \omega _c } \right\vert \lt \displaystyle{{\omega _o } \over 2}\comma \; } \cr 0 & {} & {{\rm otherwise\comma \; }} \cr } } \right. \eqno \lpar 28\rpar$$

where ω _o is the bandwidth of the filter and ω _c is the center frequency of the band.

The Fourier-based coefficient estimation is described below. Time-aligned samples of the input signal x = [x(1),…, x(K)]^T, observed signal y_obs = [y _obs(1),…, y _obs(K)]^T, and basis waveforms γ_i = [γ_i(1),…, γ_i(K)]^T are multiplied by Hann windows, then transformed into the Fourier domain using fast Fourier transforms (FFTs) to obtain X(ω _k), Y _obs(ω _k), and Γ_i(ω _k). The filtered values of the basis set and error signal are

(29)$$\Psi _L \lpar \omega _k \rpar =\lsqb \matrix{ {L\lpar \omega _k \rpar \Gamma _1 \lpar \omega _k \rpar } & \cdots & {L\lpar \omega _k \rpar \Gamma _1 \lpar \omega _k \rpar } \cr } \rsqb \comma \eqno \lpar 29\rpar$$

(30)$${\rm E}_L \lpar \omega _k \rpar =G_o ^{ - 1} \cdot L\lpar \omega _k \rpar \!\cdot \!Y_{obs} \lpar \omega _k \rpar - L\lpar \omega _k \rpar \!\cdot \!X\lpar \omega _k \rpar \comma \eqno \lpar 30\rpar$$

where

(31)$$L\lpar \omega _k \rpar =L_{OBW} \lpar \omega _k \rpar \cdot L_{notch} \lpar \omega _k \rpar . \eqno \lpar 31\rpar$$

It is assumed that L(ω _k) is band limited enough that L(ω _k) Y _obs(ω _k) = L(ω_k) Y(ω _k). If not, one must multiply L(ω _k) Γ_i(ω _k) and L(ω _k) X(ω _k) by L _IF(ω _k) to compensate. The Fourier-based estimates of matrix Q _L and vector ρ _L become

(32)$$Q_L=\displaystyle{1 \over K}\sum\limits_{k=1}^K {\cdot \Psi _L ^H \lpar \omega _k \rpar \!\cdot\! \Psi _L \lpar \omega _k \rpar }\comma \eqno \lpar 32\rpar$$

(33)$$\underline \rho=\displaystyle{1 \over K}\sum\limits_{k=1}^K {\cdot \Psi _L ^H \lpar \omega _k \rpar \!\cdot\! {\rm E}_L \lpar \omega _k \rpar }. \eqno \lpar 33\rpar$$

The length of the batches of x, y_obs, and γ_i should be short to reduce the complexity of the FFT computations, preferably K = 2048 samples. In general, at least 16 000 samples are needed for good steady-state results [Reference Zhu, Pedro and Cunba5]. The integration of several 2048 blocks to obtain Δa is performed as follows:

(34)$$\underline {\Delta a}=\left[{\sum\limits_{l=1}^8 {B_l \left\{{Q_L } \right\}} } \right]^{ - 1} \cdot \sum\limits_{l=1}^8 {B_l \lcub \underline \rho _L \rcub }\comma \eqno \lpar 34\rpar$$

where B _l{} indicates samples from block l. An advantage of (34) is that the blocks from different filters L{} may be integrated to extend the effective bandwidth of the observation path.

OBW extension is achieved using three LO₂ settings to provide spectral measurements above, below, and surrounding the allocated channels containing the input signal x(k). Changing the frequency of LO₂ has the effect of moving the center frequency of the observation band. Combining measurements from the three observation bands using (34) provides an improved estimate of the coefficient offset Δa. Note that it is necessary for every observation band to maintain partial overlap with the input spectrum to provide a reference for time-alignment of the input and output signals.

The estimation of Δa using (23) or (34) can be ill-conditioned because Q _L often has an insufficient rank. This occurs for two reasons: the signal x(k) does not stimulate all the nonlinear modes spanned by the selected basis set ψ(k) or the OBW is not measuring the modes. The narrow OBW problem is made less severe by using (34) to combine data obtained from different observation bands. The under-stimulated modes problem is due to poor selection of the basis waveform set for the input signal x(k).

The basis set ψ(k) is selected, typically, for the worst-case input signal x(k). For other input signals having less bandwidth or lower magnitude, the basis set ψ(k) will be over-specified. Singular value decomposition (SVD) is used to improve the conditioning of the estimation and avoid the need to change the basis set as a function of the input signal x(k). In such cases, Q _L⁻¹ is replaced in (23) by a pseudo-inverse S whose rank is lower.

Assume the eigenvectors of Q _L are denoted by v_j, where the index j is listed from the largest to smallest eigenvalue λ _j. SVD reduces the rank of S from N to N _est:

(35)$$S=V\!\cdot\! D\!\cdot\! V^H \approx Q_L ^{ - 1}\comma \eqno \lpar 35\rpar$$

where V = [v₁,…, v _N] and D = diag{λ₁⁻¹ … λ_Nest⁻¹ 0 … 0}. Since the degrees of freedom in the estimation are reduced, condition problems are avoided. Note that the basis set ψ(k) used in the DPD module remains unchanged.

The accuracy of the coefficient estimation is affected by the bandwidth of the observation path (OBW) and the model order of the estimator (N _est). In Section IV, various combinations of OBW and N _est are tested and compared. It is shown that the estimation model order N _est should be reduced when OBW is small to prevent spectral growth outside of the OBW. That is, the excess degrees of freedom in the estimator model should be constrained using SVD (or other regularization method) to avoid coefficient biases and unwanted spectral growth.

V. FUTURE RESEARCH

The reduction of the model order of the estimation using SVD (see (35)) is done because the observation measurement (ρ_L) does not see the nonlinear modes associated with the smaller eigenvectors, [v_Nest,…, v_N]. However, these discarded modes affect the ACPR measurements outside of the OBW. Once the reduced order coefficient set has converged, it is possible to search the sub-space of coefficients spanned by the discarded eigenvectors [v _Nest,…, v _N] for the minimum ACPR.

The coefficient offset vector, Δa, can be written as a weighted sum of eigenvectors of Q _L: that is,

(36)$$\underline {\Delta a}=\sum\limits_{j=1}^N {\Delta b_j \cdot \underline {v} _j }\comma \eqno \lpar 36\rpar$$

where Δb _j are complex weights. Similarly, the vector ρ_L becomes

(37)$$\underline \rho _L=\sum\limits_{j=1}^N {\lambda _j\! \cdot\! \Delta } b_j\! \cdot\! \underline v _j . \eqno \lpar 37\rpar$$

Since SVD approximates (37) by using only the dominant eigenvectors, its application reduces the upper bound of the summation from N to N _est. The approximation error is modest as long as the ignored eigenvalues, [λ_Nest, …, λ_N], are small. That is, the narrow OBW estimator cannot see the nonlinear modes associated with the discarded eigenvectors [v_Nest,…, v_N].

The steady-state DPD coefficients become

(38)$$\underline a=\sum\limits_{j=1}^N {b_j \!\cdot\! \underline v _j }\comma \eqno \lpar 38\rpar$$

where b _j are complex weights. SVD ensures that [b _Nest … b _N] = [0 … 0]. However, these discarded modes can be adjusted to reduce the ACPR outside of the OBW.

A descent-based search for the minimum ACPR from the sub-space spanned by discarded eigenvectors [v_Nest … v_N] begins with the selection of an orientation, Δθ = [0 … 0 Δb _Nest … Δb _N]. The orientation Δθ is chosen such that the nonlinear mode δ(k) = ψ(k)*Δθ has energy in the ACPR channels being measured. The coefficient vector b is stepped along this orientation,

(39)$$\underline b \lpar t_1 \rpar =\underline b \lpar t_0 \rpar - \beta \!\cdot \!\underline {\Delta \theta }\comma \eqno \lpar 39\rpar$$

where β is a step size. Measurements made before and after the step are used to determine the direction (positive or negative) along Δθ that reduces the ACPR. Additional orientations are tested to extract (partial) gradient information, after which a standard line search is performed in the direction having the steepest descent. This process is repeated until no improvement in the ACPR is found. Alternative descent approaches are discussed in [Reference Luenberger10].