Dual-Branch Parallel Hammerstein Model for Joint Modeling of Quadrature Imbalance and PA Distortions with Memory

The quadrature impairments in terms of gain and phase imbalance, which were modeled in Section 10.1.1 using a set of four filters applied to the in-phase and quadrature components, can also be modeled by two complex filters, G_{1} and G_{2}, where each of them takes a complex signal as input. The input to G_{1} is the original or non-conjugate complex envelope signal to be fed to the quadrature modulator and the input to G_{2} is the conjugate of this complex envelope signal [10, 11, 18,19]. It can easily be shown that the inverse model (predistorter) of this system has a structure similar to that of the forward model.

The PA nonlinearity can be corrected with any of the predistortion functions presented in the previous chapters. In [11], a Hammerstein model is used. The joint compensation of the quadrature modulator impairments and PA nonlinearity can then be obtained by cascading the Hammerstein predistorter and the quadrature impairment compensation block, where the predistorter is placed first. Figure 10.3 shows

Figure 10.3 Block diagram of a linear and nonlinear distortion compensation composed of a cascade of a Hammerstein PA predistorter and a quadrature imbalance compensator [11] the block diagram of a quadrature modulator impairment and PA nonlinear distortion compensation model as proposed in [11].

If the quadrature imbalance and the PA nonlinear distortions are known, such a model can be implemented. However, in practice, such information is not available beforehand and therefore a joint estimation is required. Since the filters of the PA predistorter and quadrature modulator impairment compensation block are in cascade, their joint estimation is not straightforward. In order to be able to achieve a joint estimation of the different coefficients of the model, the structure of Figure 10.3 can be modified from a cascade to a parallel structure, enabling one-step joint estimation of all the parameters using linear LS techniques, without any extra RF hardware.

This transformation can be achieved in different steps as described in [11]. First, for each branch of the parallel Hammerstein (PH) model, the quadrature compensation is added to the frequency response of that branch, as shown in Figure 10.4a. As shown in Figure 10.4b, the two frequency responses are merged together; H_{p}(z) and G_{1}(z) are merged in the non-conjugate path while H*(z) and G_{2}(z) are merged together in

Figure 10.4 Concept of transforming a single-branch serial structure (cascade of a Hammerstein PA predistorter and a quadrature imbalance compensator) to a parallel dual-branch parallel Hammerstein structure: (a) the original structure of the digital predistorter and quadrature imbalance compensator in one branch of the PH model; (b) the modified structure after merging the frequency dependent LTI parts; and (c) the final structure after splitting the static nonlinearity between the conjugate and non-conjugate branches

Figure 10.5 Detailed block diagram of a parallel dual-branch parallel Hammerstein structure

the conjugate path to obtain the following transfer functions:

where p e I_{P}, I_{P} = {1,2,3, ... , P} is the set of used polynomial orders, and P is the number of branches considered in the Hammerstein model.

Then, as depicted in Figure 10.4c, the final structure of the cascade obtained by splitting the static nonlinearity part of the predistorter function represented by a polynomial, y_{p}i^{x}(^{n})] = ^ a_{kp}lx(n)l^{k-}1x(n), between the non-conjugate and a conjugate

kelP

branches, resulting in a parallel connection of two PH predistorters with summed outputs. The time domain analysis of this model is explained in the following equations:

where (.)* i^the complex conjugate operator; ® is the convolution sum operator; and f_{pn} and f_{pn} are the impulse responses of the transfer functions F_{P}(z) and F_{P}(z), respectively.

Finally, the local oscillator (LO) leakage is compensated for by a constant added to the signal at the output of the joint linearization architecture. This constant is named c^{f}; and the entire block diagram of the joint compensation becomes then as depicted in Figure 10.5.

The mathematical expression for the input-output relationship of the parallel- Hammerstein-based model, assuming a finite impulse response (FIR) for the linear time invariant (LTI) blocks, can be expressed as:

where Q_{p} is the order of the FIR block in the pth non-conjugate branch, and Q_{p}' is the order of the FIR block in the pth conjugate branch, a_{pq} and a^{f}pq are the coefficients of the FIR filters F_{p}(z) and F_{p}(z), respectively; and c' is a constant term used to represent the dc offset of the modulator.