Wide band digital predistortion using iterative feedback decomposition

Abstract

This paper presents a new digital predistortion (DPD) technique for wide band applications. Digital predistortion is the most useful linearization technique to reduce power amplifier (PA) nonlinearity effects due to its high flexibility and low complexity. However, this technique requires high performances ADC to digitize the feedback signal whose bandwidth is equal to several times the original bandwidth due to spectral regrowth generated by the PA nonlinearity. This point represents one of the main bottlenecks for the deployment of the wideband LTE-A standard. The method proposed in this paper calculates the DPD coefficients iteratively using an ADC with a fixed bandwidth equal to the original bandwidth. The proposed method has been simulated and compared with other methods using Matlab. Simulation results show that the proposed method has almost the same performance as the other methods with an ACPR of − 60 dB. Moreover, it reduces considerably the constraints on the ADC and the power calculation resources.

Appendix: Proof of the method validity

The figures in this section have identical architecture to that shown in Sect. 3, with some modification in the view, to make explicit the proof steps. Moreover, for sake of clarity, the subband decomposition is modeled with filters though it will be implemented differently. In this section we consider a PA of nonlinearity order 7.

1.1 1st step: 7th order identification

In the first step, the BPF will select the subband holding information on the 7th order nonlinearity (2.5–3.5 BW).

By construction (according to Fig. 18), and since only inputs relevant part of the input signal is being considered:

$$\begin{aligned} {\mathbf{y}}_{{P17}} & = {\mathbf{Y}}_{{CH}}^{{1,7}} \cdot {\mathbf{c}}^{{1,7}} = \left[ {{\mathbf{Y}}_{{CH}}^{1} {\mathbf{Y}}_{{CH}}^{7} } \right] \cdot \left( {\begin{array}{*{20}l} {{\mathbf{c}}^{{{\kern 1pt} 1}} } \hfill \\ {{\mathbf{c}}^{{{\kern 1pt} 7}} } \hfill \\ \end{array} } \right) \\ & = {\mathbf{Y}}_{{CH}}^{1} \cdot {\mathbf{c}}^{{{\kern 1pt} 1}} + {\mathbf{Y}}_{{CH}}^{7} \cdot {\mathbf{c}}^{{{\kern 1pt} 7}} . \\ \end{aligned}$$

(A.1)

and

$$\begin{aligned} {\mathbf{z}}_{P7} = {\mathbf{y}}_{P17} + {\mathbf{y}}_{IM7} \end{aligned}$$

(A.2)

where using Eq. A.1 gives:

$$\begin{aligned} {\mathbf{z}}_{P7} = \left[ {\mathbf{Y}}_{CH}^{1} {\mathbf{Y}}_{CH}^{7} \right] \cdot \begin{pmatrix} {\mathbf{c}}^{\,1}\\ {\mathbf{c}}^{\,7}\end{pmatrix} + {\mathbf{y}}_{IM7}. \end{aligned}$$

(A.3)

The identification process based on minimizing the error $\left( {\mathbf{e}} \rightarrow {\mathbf{0}} \right)$ will result in having:

$$\begin{aligned} {\mathbf{u}}={\mathbf{x}}\simeq {\mathbf{z}}_{P7}. \end{aligned}$$

(A.4)

Therefore the first order and seventh order DPD coefficients are obtained by:

$$\left( {\begin{array}{*{20}l} {{\hat{\mathbf{c}}}^{{{\kern 1pt} 1}} } \\ {{\hat{\mathbf{c}}}^{{{\kern 1pt} 7}} } \\ \end{array} } \right) = \left[ {{\mathbf{Y}}_{{CH}}^{1} {\mathbf{Y}}_{{CH}}^{7} } \right]^{{ - 1}} \cdot \left( {{\mathbf{z}}_{{P7}} - {\mathbf{y}}_{{IM7}} } \right).$$

(A.5)

Once the solution is computed, the 7th order terms are computed by:

$${\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} = {\mathbf{z}}_{{P7}} - {\mathbf{y}}_{{IM7}} - {\mathbf{Y}}_{{CH}}^{1} \cdot {\hat{\mathbf{c}}}^{1} .$$

(A.6)

Fig. 18

Block diagram of the system at the 1st step

1.2 2nd step—5th order identification

In the second step (iteration), the 1st and 7th order coefficients are already estimated, then both could be used in the estimation of the 5th order coefficients as following:

By construction (according to Fig. 19):

$$\begin{aligned} {\mathbf{y}}_{P15} &= {\mathbf{Y}}_{CH}^{1,5} \cdot {\mathbf{c}}^{1,5} = \left[ {\mathbf{Y}}_{CH}^{1} {\mathbf{Y}}_{CH}^{5} \right] \cdot \begin{pmatrix} {\mathbf{c}}^{\,1}\\ {\mathbf{c}}^{\,5} \end{pmatrix}\\ &= {\mathbf{Y}}_{CH}^{1} \cdot {\mathbf{c}}^{\,1} + {\mathbf{Y}}_{CH}^{5} \cdot {\mathbf{c}}^{\,5}. \end{aligned}$$

(A.7)

and

$$\begin{aligned} {\mathbf{z}}_{P5} = {\mathbf{y}}_{P15} + {\mathbf{y}}_{IM5} + {\mathbf{y}}_{P7} \end{aligned}$$

(A.8)

where using Eq. A.7 gives:

$$\begin{aligned} {\mathbf {z}}_{P5} = \left[ {\mathbf {Y}}_{CH}^{1} {\mathbf {Y}}_{CH}^{5} \right] \cdot \begin{pmatrix} {\mathbf {c}}^{\,1}\\ {\mathbf {c}}^{\,5}\end{pmatrix} + {\mathbf {y}}_{IM5} + {\mathbf {Y}}_{CH}^{7} \cdot {\hat {\mathbf{c}}}^{\,7}. \end{aligned}$$

(A.9)

The identification process based on minimizing the error $\left( {\mathbf{e}} \rightarrow {\mathbf{0}} \right)$ will result in having:

$$\begin{aligned} {\mathbf{u}}={\mathbf{x}}\simeq {\mathbf{z}}_{P5}. \end{aligned}$$

(A.10)

Therefore the 5th order coefficients are obtained by:

$$\left( {\begin{array}{*{20}c} {{\hat{\mathbf{c}}}^{1} } \\ {{\hat{\mathbf{c}}}^{5} } \\ \end{array} } \right) = \left[ {{\mathbf{Y}}_{{CH}}^{1} {\mathbf{Y}}_{{CH}}^{5} } \right]^{{ - 1}} \cdot \left( {{\mathbf{z}}_{{P5}} - {\mathbf{y}}_{{IM5}} - {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} } \right),$$

(A.11)

and the 5th order terms are computed by:

$${\mathbf{Y}}_{{CH}}^{5} \cdot {\hat{\mathbf{c}}}^{5} = {\mathbf{z}}_{{P5}} - {\mathbf{y}}_{{IM5}} - {\mathbf{Y}}_{{CH}}^{1} \cdot {\hat{\mathbf{c}}}^{1} - {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} .$$

(A.12)

Fig. 19

Block diagram of the system at the 2nd step

1.3 3rd step: 1st and 3rd order identification

Now, after 1st, 5th, 7th coefficients are estimated, 3rd order coefficients could be estimated, after applying a BPF on the out of band feedback path, allowing the band ranging from $0.5\,$BW to $1.5\,$BW as following:

By construction (according to Fig. 20):

$$\begin{aligned} {\mathbf{y}}_{P13} &= {\mathbf{Y}}_{CH}^{1,3} \cdot {\mathbf{c}}^{1,3} = \left[ {\mathbf{Y}}_{CH}^{1} {\mathbf{Y}}_{CH}^{3} \right] \cdot \begin{pmatrix} {\mathbf{c}}^{\,1}\\ {\mathbf{c}}^{\,3} \end{pmatrix}\\ &= {\mathbf{Y}}_{CH}^{1} \cdot {\mathbf{c}}^{\,1} + {\mathbf{Y}}_{CH}^{3} \cdot {\mathbf{c}}^{\,3}. \end{aligned}$$

(A.13)

and

$$\begin{aligned} {\mathbf{z}}_{P3} = {\mathbf{y}}_{P13} + {\mathbf{y}}_{IM3} + {\mathbf{y}}_{P5} + {\mathbf{y}}_{P7} \end{aligned}$$

(A.14)

where using Eq. A.13 gives:

$${\mathbf{z}}_{{P3}} = \left[ {{\mathbf{Y}}_{{CH}}^{1} {\mathbf{Y}}_{{CH}}^{3} } \right] \cdot \left( {\begin{array}{*{20}c} {{\mathbf{c}}^{1} } \\ {{\mathbf{c}}^{3} } \\ \end{array} } \right) + {\mathbf{y}}_{{IM3}} + \left[ {{\mathbf{Y}}_{{CH}}^{5} {\mathbf{Y}}_{{CH}}^{7} } \right] \cdot \left( {\begin{array}{*{20}c} {{\hat{\mathbf{c}}}^{5} } \\ {{\hat{\mathbf{c}}}^{7} } \\ \end{array} } \right).$$

(A.15)

The identification process based on minimizing the error $\left( {\mathbf{e}} \rightarrow {\mathbf{0}} \right)$ will result in having:

$$\begin{aligned} {\mathbf{u}}={\mathbf{x}}\simeq {\mathbf{z}}_{P1}. \end{aligned}$$

(A.16)

Therefore the 3rd order coefficients are obtained by:

$$\left( {\begin{array}{*{20}c} {{\hat{\mathbf{c}}}^{1} } \\ {{\hat{\mathbf{c}}}^{3} } \\ \end{array} } \right) = \left[ {{\mathbf{Y}}_{{CH}}^{1} {\mathbf{Y}}_{{CH}}^{3} } \right]^{{ - 1}} \cdot \left( {{\mathbf{z}}_{{P3}} - {\mathbf{y}}_{{IM3}} - {\mathbf{Y}}_{{CH}}^{5} \cdot {\hat{\mathbf{c}}}^{5} - {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} } \right),$$

(A.17)

and the 3rd order terms are computed by:

$$\begin{aligned} \mathbf{Y}_{CH}^{3} \cdot \hat{\mathbf{c}}^{\,3} = \mathbf{z}_{P3} - \mathbf{y}_{IM3} - \mathbf{Y}_{CH}^{1} \cdot \hat{\mathbf{c}}^{\,1} -\mathbf{Y}_{CH}^{5} \cdot \hat{\mathbf{c}}^{\,5} -\mathbf{Y}_{CH}^{7} \cdot \hat{\mathbf{c}}^{\,7}. \end{aligned}$$

(A.18)

Fig. 20

Block diagram of the system at the 3rd step

1.4 4th step—1st order identification

Finally the 1st order coefficients are to be estimated, by taking only the in-band part of the feedback signal.

By construction (according to Fig. 21):

$$\begin{aligned} {\mathbf{y}}_{P1} = {\mathbf{Y}}_{CH}^{1} \cdot {\mathbf{c}}^{1} \end{aligned}$$

(A.19)

and

$$\begin{aligned} {\mathbf{z}}_{P1} = {\mathbf{y}}_{P1} + {\mathbf{y}}_{P3} + {\mathbf{y}}_{P5} + {\mathbf{y}}_{P7} \end{aligned}$$

(A.20)

which can be written using the estimated parameters:

$${\mathbf{z}}_{{P1}} = {\mathbf{Y}}_{{CH}}^{1} \cdot {\mathbf{c}}^{1} + \left[ {{\mathbf{Y}}_{{CH}}^{3} {\mathbf{Y}}_{{CH}}^{5} {\mathbf{Y}}_{{CH}}^{7} } \right] \cdot \left( {\begin{array}{*{20}c} {{\hat{\mathbf{c}}}^{3} } \\ {{\hat{\mathbf{c}}}^{5} } \\ {{\hat{\mathbf{c}}}^{7} } \\ \end{array} } \right).$$

(A.21)

The identification process based on minimizing the error $\left( {\mathbf{e}} \rightarrow {\mathbf{0}} \right)$ will result in having:

$$\begin{aligned} {\mathbf{u}}={\mathbf{x}}\simeq {\mathbf{z}}_{P1}, \end{aligned}$$

(A.22)

Therefore the 1st order coefficients can be obtained by:

$${\hat{\mathbf{c}}}^{1} = {\mathbf{Y}}_{{CH}}^{1} {\mkern 1mu} _{{}}^{{ - 1}} \cdot \left( {{\mathbf{z}}_{{P1}} - {\mathbf{Y}}_{{CH}}^{3} \cdot {\hat{\mathbf{c}}}^{3} - {\mathbf{Y}}_{{CH}}^{5} \cdot {\hat{\mathbf{c}}}^{5} - {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} } \right),$$

(A.23)

and the 1st order terms are computed by:

$${\mathbf{Y}}_{{CH}}^{1} \cdot {\hat{\mathbf{c}}}^{1} = {\mathbf{z}}_{{P1}} - {\mathbf{Y}}_{{CH}}^{3} \cdot {\hat{\mathbf{c}}}^{3} - {\mathbf{Y}}_{{CH}}^{5} \cdot {\hat{\mathbf{c}}}^{5} - {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} .$$

(A.24)

Fig. 21

Block diagram of the system at the extra step

1.5 Verification

We get the complete predistorter by putting all the estimated coefficients together:

$$\begin{aligned} {\mathbf{y}}_{P} & = \left[ {{\mathbf{Y}}_{{CH}}^{1} {\mathbf{Y}}_{{CH}}^{3} {\mathbf{Y}}_{{CH}}^{5} {\mathbf{Y}}_{{CH}}^{7} } \right] \cdot \left( {\begin{array}{*{20}c} {{\hat{\mathbf{c}}}^{1} } \\ {{\hat{\mathbf{c}}}^{3} } \\ {{\hat{\mathbf{c}}}^{5} } \\ {{\hat{\mathbf{c}}}^{7} } \\ \end{array} } \right) \\ {\mathbf{y}}_{P} & = {\mathbf{Y}}_{{CH}}^{1} \cdot {\hat{\mathbf{c}}}^{1} + {\mathbf{Y}}_{{CH}}^{3} \cdot {\hat{\mathbf{c}}}^{3} + {\mathbf{Y}}_{{CH}}^{5} \cdot {\hat{\mathbf{c}}}^{5} + {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} . \\ \end{aligned}$$

(A.25)

Let us replace the first order term by Eq. (A.24) with the assumption ${\mathbf{x}}\simeq {\mathbf{z}}_{P1}$ (Eq. A.22):

$${\mathbf{y}}_{P} = {\mathbf{x}} - {\mathbf{Y}}_{{CH}}^{3} \cdot {\hat{\mathbf{c}}}^{3} - {\mathbf{Y}}_{{CH}}^{5} \cdot {\hat{\mathbf{c}}}^{5} - {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} + {\mathbf{Y}}_{{CH}}^{3} \cdot {\hat{\mathbf{c}}}^{3} + {\mathbf{Y}}_{{CH}}^{5} \cdot {\hat{\mathbf{c}}}^{5} + {\mathbf{Y}}_{{CH}}^{7} \cdot {\hat{\mathbf{c}}}^{7} ,$$

(A.26)

which gives:

$$\begin{aligned} {\mathbf{y}}_P = {\mathbf{x}}. \end{aligned}$$

(A.27)

Therefore, the identified parameters ${\hat{\mathbf{c}}}^{1} ,{\hat{\mathbf{c}}}^{3} ,{\hat{\mathbf{c}}}^{5} ,{\hat{\mathbf{c}}}^{7}$ ideally yield to the ideal linearizer.