Online DPD based on the FWMP model

The proposed adaptive linearization architecture is illustrated in Fig. 1. It is based on an Indirect Learning Architecture [Reference Vu and Tran36, Reference Yu, Tang and Liu37] for updating the coefficients of the MP block [Reference Bouazza, Bachir and Duvanaud35].

Figure 1. Online DPD based on the FWMP model.

Predistorter structure

As shown in Fig. 1, the proposed DPD structure is based on a combination of two blocks: a FW system that models the main behavior, that is, the interaction between nonlinearities and memory effects, and an MP model to compensate the remaining modeling errors [Reference Bouazza38]. The FW block itself is composed of two sub-blocks: a memoryless nonlinearity in the direct path and a finite impulse response (FIR) filter in the feedback path, where z ⁻¹ is the unit delay.

Each block of the DPD is described using the equivalent complex envelope model. The output v(n) of the FW system is given by the following polynomial expression:

(1)\begin{align} v(n) = \sum\limits_{p=1}^{P} c_{p} \cdot w(n)^{p} = \sum\limits_{p=1}^{P} c_{p} \cdot \left(g_0\cdot\, x(n) - d(n) \right)^{p}, \end{align}

where c _p are the complex coefficients of the nonlinear function, g ₀ is a complex gain, and P is the nonlinearity order. x(n) is the complex envelope of the input signal.

The feedback block $F(\omega)$ is implemented by an FIR filter, and its output d(n) can be formulated as

(2)\begin{align} d(n) = \sum\limits_{m=1}^{M} b_{m} \cdot v(n-m), \end{align}

where M is the memory depth of the FIR filter.

The signal v(n) is used as an input to the MP model [Reference Kim and Konstantinou8]:

(3)\begin{align} x^\prime(n) = \sum\limits_{p=0}^{P_a-1}\sum\limits_{m=0}^{M_a-1} a_{pm} \cdot v(n-m) \cdot |v(n-m)|^{p}, \end{align}

where $x^\prime(n)$ is the predistorted signal, P _a and M _a are the nonlinearity order and the memory depth, respectively.

A great advantage of the MP model in the identification is that it is linear in parameters, allowing a least squares (LS) algorithm. We can introduce its regression form, where $\underline{\hat{\theta}}_{\rm MP}$ is the vector of parameters to be estimated and $\underline{\phi}(n)$ is the regressor vector, such that

(4)\begin{align} \hat{x}^\prime(n)=\underline{\phi}(n)^{\rm T} \cdot \underline{\hat{\theta}}_{\rm MP}, \end{align}

with

(5)\begin{align} \underline{\hat{\theta}}_{\rm MP} & = \left [ a_{00} \cdots a_{\rm pm} \cdots \, a_{(P_a-1)(M_a-1)} \right ]^{\rm T} \nonumber\\ \underline{\phi}(n) & =\left[ \phi_{00}(n) \cdots \phi_{\rm pm}(n) \cdots \phi_{(P_a-1)(M_a-1)}(n) \right]^{\rm T} \end{align}

such as the regressor elements are

(6)\begin{align} \phi_{\rm pm}(n) = \left | v(n-m) \right |^{p-1} \cdot v(n-m) \end{align}

Note that, since nonlinearities and memory effects are treated separately in the FW block, the proposed model has the advantage of having an additive evolution in the number of its first block coefficients, unlike the GMP model [Reference Morgan, Ma, Kim, Zierdt and Pastalan30]. Therefore, the total number of coefficients is given by

(7)\begin{align} N_{\rm FWMP} = \overbrace{(P+M)}^{\rm FW} + \overbrace{(P_a \times M_a)}^{\rm MP}. \end{align}

In order to further reduce the complexity of the linearizer, the idea is to estimate in offline the FW sub-model with a large number of coefficients to describe the main inverse PA characteristics, while the MP structure will be minimized by reducing as much as possible the orders P _a and M _a to operate in real time. The FW system will then be fixed regardless of the application and only the MP sub-model will be updated to take into account variations and changes in the PA environment such as the waveform, temperature, bias voltage, and/or aging.

Identification algorithm

As mentioned previously, the MP model will be estimated in real time during the use of the PA. The FW model must be estimated beforehand. One of the fundamental characteristics in the FW system is its cascaded structure [Reference Wang32, Reference Wang, Abi Hussein, Venard and Baudoin34, Reference Narendra and Gallman39, Reference Zhu, Pedro and Cunha40], and here the focus will be on the unmeasured intermediate signals v(n) and w(n), required for the estimation process.

From Eqs. (1) and (2), we can write the FW output by separating the first term such as

(8)\begin{align} v(n) &= c_1 \left ( g_0 \, x(n) - \sum_{m=1}^{M} b_m \, v(n-m) \right ) \nonumber \\ & \quad + \sum_{p=2}^{P} c_p \, w(n)^p. \end{align}

Noted that the gains g ₀ and c ₁ are correlated and to avoid the overparametrization problem [Reference Zhu, Pedro and Cunha40], $c_1\,$is set to one and only g ₀ is estimated. Thus, for parameters estimation, the FW block can be expressed in linear regression system:

(9)\begin{align} v(n) &= g_0 x(n) - \sum_{m=1}^{M} b_m \, v(n-m) + \sum_{p=2}^{P} c_p w(n)^p \nonumber \\ &= \underline{\varphi}(n)^{\rm T} \cdot \underline{\hat{\theta}}_{\rm FW}, \end{align}

where $\underline{\varphi}_n$ is the regressor vector and $\underline{\hat{\theta}}_{\rm FW}$ is the vector of coefficients to be estimated:

(10)\begin{align} \underline{\hat{\theta}}_{\rm FW} = \left [ g_0 \quad b_1 \cdots b_M\quad c_2 \cdots c_P \right ]^{\rm T} \end{align}

\begin{align*} \underline{\varphi}(n) = \left[ x(n) \quad -v(n-1) \cdots -v(n-M) \quad w(n)^2 \, \cdots \, w(n)^P \right]^{\rm T}. \end{align*}

Noted that in the linearization scheme, the aim is to modelize the inverse PA characteristic. Thus, v(n) is replaced by the PA input, x(n) by the normalized output $\hat{y}(n)$, $v(n-m)$ and w(n) by their simulation during the identification process. The parameter vector $\underline{\hat{\theta}}_{\rm FW}$ is obtained by minimizing the quadratic criterion J based on N measured samples such as

(11)\begin{align} J = \sum_{n=1}^{N} \varepsilon(n)^2, \end{align}

where $\varepsilon(n)$ is the complex–envelope error between measured input $x^\prime(n)$ and its estimation $\hat{x}^\prime(n)$.

Once the FW block has been identified, its coefficients are maintained constant during operation. The aim is now to find an MP model that describes the modeling error, remaining after the FW block, as accurately as possible. We propose then to use the recursive weighted least squares (RWLS) algorithm for the real-time identification. The RWLS algorithm [Reference Haykin41] is a recursive form of the ordinary LS method [Reference Zhou and Raich42], which is performed sample by sample to identify the parameters that are subject to change to maintain the tracking ability. The main principle of the RWLS technique is to introduce a forgetting factor λ that gives greater importance to the recent measurements [Reference Li and Zhang43]. The parameter vector $\underline{\hat{\theta}}_{{\rm MP}}(n)$ is updated at each iteration n such as

(12)\begin{align} \underline{\hat{\theta}}_{\rm MP}(n) = \underline{\hat{\theta}}_{\rm MP}(n-1) + K(n) \left ( x^\prime(n) - \hat{x}^\prime(n) \right ), \end{align}

where K(n) is the gain matrix depending on the coefficients covariance matrix C(n), such as

(13)\begin{align} K(n)=\frac{C(n-1) \underline{\phi}^{*}(n)}{\lambda + \underline{\phi}^{H}(n) C(n-1) \phi(n)}, \end{align}

(14)\begin{align} C(n) = \lambda^{-1} \left( C(n-1)- K(n) \underline{\phi}^{*}(n) C(n-1) \right), \end{align}

where $\underline{\phi}(n)$ is the MP regressor vector defined in Eq. (5), $(\cdot)^*$ and $(\cdot)^H$ represent, respectively, the conjugate operation and the transposition–conjugate transform.

Experimental results

The proposed structure has been used for the linearization of a 75 W UHF/VHF PA designed by Telerad Society for aeronautical communications. The test bench used in this study is shown in Fig. 2. The PA input is a Quadrature Phase Shift Keying modulated signal with 16.8 kHz bandwidth at the carrier frequency of 127.5 MHz. The output is recovered via a directional coupler and then digitally converted by a fast ADC converter. The input and output signals are synchronized using a cross-correlation technique.

Figure 2. Amplifier to be linearized and digital boards.

For the measurements, different carrier frequencies, different PA bias voltages, and different operating temperatures were tested. In our targeted airborne VHF applications, the VDL2 standard imposes strong constraints on the ACPR values. The out-of-band spurious emissions are specified by the first- and second-order ACPR such as

• • ACPR$_{\rm L1}$, ACPR$_{\rm U1} \lt -65\,$dB

• • ACPR$_{\rm L2}$, ACPR$_{\rm U2} \lt -75\,$dB

The first adjacent band is located at 25 kHz from the main band and with 16.8 kHz of width (ACPR$_{\rm L1}$ and ACPR$_{\rm U1}$) when the second adjacent band is located at 50 kHz from the main band and with 25 kHz of width (ACPR$_{\rm L2}$ and ACPR$_{\rm U2}$). L and U denote the lower and upper bands, respectively, which are located in the left and right of the main band.

The first step is to determine the minimum structure of the MP model needed to guarantee the specifications. Then, we look for the optimal structure of the FWMP model with a minimum number of coefficients.

Table 1 gives the obtained ACPR values for different structures of MP and FW blocks. For better readability and unlike the values shown in bold italic, which shows the respect for the standard, the other values in plain format indicate that the specifications are not reached. The analysis of these results shows that it is impossible to reduce the number of coefficients in the MP block beyond 8 (see the three last lines). The introduction of the FW block improves the results on the L2 and U2 bands, for which the constraints are difficult to meet. Although a reduced two-coefficient FW block allows the respect of the specification (model in line 3), for a safety margin, we used the FWMP model with an MP block of eight coefficients and an FW block of three coefficients (model in line 4).

Table 1. Optimal MP and FWMP structures

Considering the optimized FWMP structure with 11 coefficients (P = 3, M = 0, $P_a=8$, and $M_a=1$) with 3 fixed coefficients for FW block and 8 estimated coefficients for the MP block, the ACPR results without and with DPD (in offline and online versions), using a modulated signal transposed on a carrier frequency of 127.5 MHz, are presented in Table 2 for a symbol rate of 10 kS/s. This structure is compared to an MP model with nine coefficients ($P_a=9$ and $M_a=1$).

Table 2. ACPR without and with DPD

Table 2 is decomposed into three parts: the first one gives the results without DPD, followed by the results of the offline linearization with the MP models alone and the proposed model. The last part shows the results of both models in the online version.

Here, offline DPD consists of applying a static predistorter, estimated from measurements at 127.5 MHz. The estimation of the offline DPD models (MP and FWMP) is carried out by the least squares algorithm [Reference Wang32, Reference Bouazza, Bachir and Duvanaud35]. There is no update of the DPD coefficients in this case. The online DPD architecture is the one described in Fig. 1, using the recursive algorithm described in Eqs. (12)–(14).

From this table, it can be seen that, contrary to the case without DPD, the offline DPD functions based on the MP model (9 coefficients) and the FWMP model allow to achieve the specifications (values in bold italic). However, switching to the online mode deteriorates the performance of the MP and FWMP model on the U ₂ and L ₂ side bands, but the DPD based on the proposed FWMP structure maintains the respect of the specifications. Also, note that only eight tunable coefficients are updated in the case of the FWMP model.

Output spectra with and without DPD based on the two models MP and FWMP are shown in Fig. 3. The real-time adjustment of the DPD parameters allows an improvement of about 40 dB in the lower and upper first side bands.

Figure 3. Output spectra with and without DPD.

Table 3 presents a comparison of the estimated vector $\underline{\hat{\theta}}_{\rm MP}$ for both MP and FWMP models, resulting from the two versions of identifications (offline and online). This allows to see how the parameters of the MP block evolve between the two versions for each model.

Table 3. Evolution of $\underline{\hat{\theta}}_{\rm MP}$ for MP and FWMP models (in offline and online versions)

We can see from Table 3 that the FWMP model obtained in offline version is very close to that obtained in real time with the RLWS algorithm. This is not the case for the MP model where the values of the coefficients are different. This result is predictable because the MP structure is a black-box model without any physical significance, contrary to the FWMP model where FW block is close to the behavior of PA [Reference Bouazza38]. In the case of the FWMP model, use of the FW block helps to make the variations in the MP block less fluctuating.

In general case, the complexity of the RLS algorithm is dominated by the complexity of updating the gain matrix K(n) in Eqs. (13) and (14), which requires a computational load by $\mathcal{O}(N^2)$ operations per iteration, where N is the number of coefficients. The other steps of the algorithm have a complexity at most linear in $\mathcal{O}(N)$.

In the time domain, the AM/AM curves with and without linearization using the adaptive DPD, with a fixed FW block and only the MP block coefficients to be updated, are presented in Fig. 4. We can see that the proposed structure improves the linearity of the communication system.

Figure 4. AM/AM curves with and without linearization.

In order to test the robustness of the proposed model, we study the effect of the temperature changes at 26, 31, and 36^∘C by switching the PA with the previously identified models at each temperature. Figure 5 shows the convergence of the coefficients during operation, while Fig. 6 gives the linearization error between the PA input x(n) and the normalized PA output $\hat{y}(n)$. Note that the used FW structure is the one obtained at 36^∘C.

Figure 5. Variation of MP coefficients versus temperature changes.

Figure 6. Linearization error versus temperature changes.

The obtained results (Fig. 5) show that the algorithm based on the FWMP model adapts to temperature changes. Also, we can see from the comparison between the input and the linearized output (Fig. 6) that the difference is small, which means that during temperature changes, the system is able to adapt in a robust way. In practice, the algorithm converges after the temperature change in about 1 ms.

The value of the forgetting factor $\lambda \in \left]0, 1\right]$ has a significant influence on the performance of the algorithm. When it is close to unity, the online algorithm achieves good robustness, but the convergence becomes slower, like the Gradient algorithm. A smaller value improves the speed convergence but it increases the probability of being unstable. In this case, fluctuations appear on the estimates because the new measured samples are more weighted in the algorithm.

In our case, we choose λ = 0.95, which achieves a good tradeoff between the stability and the tracking ability, with a time convergence of about 1 ms.