Optimized Memory Polynomial with Binomial reduction in Digital Pre distortion for wireless communication systems

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OPTIMIZED MEMORY POLYNOMIAL WITH BINOMIAL

REDUCTION IN DIGITAL PRE-DISTORTION FOR

WIRELESS COMMUNICATION SYSTEMS

Hong Ning Choo

1

, Nurul Adilah Abdul Latiff

1

, Pooria Varahram

2

and Borhanuddin Mohd Ali

1

Department of Computer and Communication Systems Engineering, Faculty of Engineering, Universiti Putra Malaysia Serdang, Selangor, Malaysia

2

Department of Electronic Engineering, National University of Ireland Maynooth Maynooth, Co. Kildare, Ireland E-Mail: [email protected]

ABSTRACT

The non-linearity of the Power Amplifier (PA) causes signal amplitude and phase distortion which contributes to Adjacent Channel Interference (ACI). Moreover, today’s inevitable increasing bandwidth and transmission speed causes Memory Effects, an undesired scattering of the PA output signal. Among various PA linearization methods, Digital Pre-distortion (DPD) stands out due to its balanced advantages and trade-offs in implementation simplicity, bandwidth, efficiency, flexibility and cost. An accurate modeling of the PA is required by the DPD, where the highly popular Memory Polynomial Method (MP) is used to model the PA with Memory Effects but with reduced complexity. This project presents the Memory Polynomial with Binomial Reduction method (MPB) which is a performance optimized MP with reduced addition and multiplication operations. When compared to MP, MPB is capable of achieving improvements in Normalized Mean Square Error (NMSE) of up to 35dB. The MPB and MP methods are simulated/compared using a modeled ZVE-8G Power Amplifier and sampled 4G (LTE) signals.

Keywords: power amplifier, PA linearization, digital pre-distortion, 4G, memory polynomial.

1. INTRODUCTION

The Power Amplifier (PA) is a vital component in the transmitter design of a communication system. The ideal PA is supposed to be linear when graphed in terms of output power versus input power. However, the PA exhibits non-linearity when implemented in real world. As the PA input power increases, the output power slants away from the linear region, resulting in smaller output power than expected. The non-linearity of the PA generates spectral regrowth, which further leads to the Adjacent Channel Interference (ACI) problem [14]-[22]. The obvious solution is to back-off the PA’s operating region to ensure it only runs in its linear region. However, the solution is not efficient as limiting the PA to operate only in the linear region results in low efficiency and high cost. On the other hand, high Peak to Average Power Ratios (PAPR) in Wideband Code Division Multiple Access (WCDMA) and Orthogonal Frequency Division Multiplexing (OFDM) requires the PA to be backed off far from its saturation point. This contributes to the low efficiency of the PA.

The waste of power is a cost for telecommunication companies. In the business perspective, there is definitely a demand for a solution to improve the communication system’s efficiency. Therefore, linearizing the PA is needed to meet up the expectations of efficiency requirement from the industry.

The Digital Pre-Distortion (DPD) technique linearizes the cheaper non-linear power amplifiers,

resulting in a lower overall cost of the communication system [11], [15]-[22]. DPD offers many advantages in terms of cost, power management, reliability and handling if compared to the other linearization methods [1]-[9], [11], [13]. The Memory Polynomial (MP) is used widely for PA modelling in DPD. This paper focuses on developing a better performing MP in terms of resource optimization without compromising key performance indexes.

2. Model Description

2.1 Digital Pre-distortion

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Figure-1. Pre-distortion Linearization Block Diagram [14].

2.2 Volterra Series

Conventionally, the Volterra series is used to model the PA with considerations of the Memory Effects [8]-[9]. Volterra Series is defined as:

y(t) = ∑ ∫ ⋯ ∫ hk 2k+1(τ2k+1) ∏i=1k+1z(t − τi)∏2k+1i=k+2z∗(t −

τi)dτ2k+1 (1)

Where h2k+1(τ2k+1) = 1 22k(2k + 1_{k ) h}̃2k+1(τ2k+1)e−j2π(∑k+1i=1τi−∑2k+1i=k+2τi) 𝒇𝒐: Carrier Frequency (∙)∗_: _{Complex conjugation} Equation (1) could be rewrite in discrete-time domain: y(n) = ∑ ∑ ⋯ ∑k l1 l2k+1h2k+1(l1, l2, ⋯ , l2k+1) ∏k+1i=1 z(n − li) ∏2k+1i=k+2z∗(n − li) (2)

The number of coefficients increases exponentially when non-linearity order and memory length increases. Despite being one of the most accurate PA modeling system, the Volterra Series do not perform as an optimum system, due to its heavy trade off in calculation complexity, which will be requiring more processing power and memory space, resulting in higher operational cost and total Bill of Materials (BOM). This leads to the development of a more optimized PA modeling method, lesser coefficients, but with comparable linearization performance, the Memory Polynomial (MP) method by [8]-[9]. 2.3 Memory Polynomial (MP) The MP method utilizes the diagonal kernels of the Volterra Series, resulting in a reduced number of coefficients. The MP method is shown below [8]-[9]: z(n) = ∑Kk=1 ∑Qq=0akqx(n − q)|x(n − q)|k−1 k odd (3)

and 𝑎_𝑘𝑞 is the inversed of the Power Amplifier coefficients, which is also the MP coefficients. The Least Squares (LS) method is used to obtain the MP coefficients [8]-[9]. The input signal 𝑥(𝑛) is replaced with the output signal𝑦(𝑛), yields the following 𝑧(𝑛) = ∑𝐾𝑘=1 ∑𝑄𝑞=0𝑎𝑘𝑞𝑦(𝑛 − 𝑞)|𝑦(𝑛 − 𝑞)|𝑘−1 𝑘 𝑜𝑑𝑑 (4)

(4) in matrix form: 𝑧 = 𝑌 ∙ 𝑎 (5)

Where 𝑧 = [𝑧(0), 𝑧(1), … , 𝑧(𝑁 − 1)]𝑇₍₆₎

𝑌 = [𝑦10, … , 𝑦𝐾0, … , 𝑦1𝑄, … 𝑦𝐾𝑄] (7)

𝑎 = [𝑎10, … , 𝑎𝐾0, … , 𝑎1𝑄, … 𝑎𝐾𝑄]𝑇 (8)

𝑎 = [𝑎10, … , 𝑎𝐾0, … , 𝑎1𝑄, … 𝑎𝐾𝑄]𝑇 (9)

The least square solutions in (5) could be rewrite as: 𝑎 = (𝑌𝑐𝑜𝑛𝑗_{∙ 𝑌)}−1_𝑌𝑐𝑜𝑛𝑗_𝑧₍₁₀₎

2.4 Memory Polynomial with Binomial Reduction (MPB) As in [4], the two real and imaginary forms of MPB are shown below: 𝑧(𝑛) = ∑𝐾𝑘=0∑𝑄𝑞=0𝑎𝑘𝑞𝑥(𝑛 − 𝑞)𝑥(𝑛 − 𝑞)𝑟𝑒𝑎𝑙2𝑘 (11)

z(n) = ∑Kk=0∑Qq=0akqx(n − q)x(n − q)imag2k (12)

The linearization performance of (11) and (12) are compared among each other in terms of NMSE. The graphs is shown in the NMSE graph results section, where (12) has a better NMSE value than (11). The coefficient

𝑥

𝐹(∙)

𝐺(∙)

Pre-distorter PA

𝐹(𝑥) 𝑦 = 𝐺(𝐹(𝑥))

𝑃𝑜𝑢𝑡

𝑃𝑖𝑛 𝑃𝑜𝑢𝑡

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2.5 Digital Pre-distortion Learning Architecture

High transmission bandwidth coupled with fast changes in electrical components temperatures results in Memory Effects observed in the PA, where the PA output is rapidly changing especially in high Peak to Average Power Ratio (PAPR) scenarios [8]-[9], [17]-[19]. An adaptation block is therefore justified to be required to accurately ensure the DPD block to adapt to the rapid behaviour changes in PA. The learning methodologies in the DPD blockare Direct Learning, and indirect Learning Architecture. The Direct Learning Architecture requires both the PA input and output signal, while Indirect Learning Architecture utilizes the pre-distorted signal and the PA output signal, as shown in Figure-2.

Figure-2. Direct and Indirect Learning Architectures in Digital pre-distortion [25].

3. Performance Metrics

3.1 Normalized Mean Square Error (NMSE)

Normalized Mean Square Error (NMSE) is capable of properly quantify the accuracy of the improved Memory Polynomial Model with respect to the original Memory Polynomial Model.

NMSE is shown in the formula below:

𝑁𝑀𝑆𝐸(𝑑𝐵) = 10 𝑙𝑜𝑔∑𝑁𝑛=1|𝑦𝑖𝑑𝑒𝑎𝑙(𝑛)−𝑦𝑝𝑑(𝑛)|2 ∑𝑁𝑛=1|𝑦𝑖𝑑𝑒𝑎𝑙(𝑛)|2

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where𝑦_{𝑖𝑑𝑒𝑎𝑙}(𝑛) is the Ideal Power Amplifier Output, and

𝑦𝑝𝑑(𝑛) is the Pre-distorted Power Amplifier Output. A lower/smaller NMSE value is preferable because it indicates smaller error deviation from the ideal output values, therefore resulting in higher accuracy.

4. RESULTS AND DISCUSSIONS

4.1 Normalized Mean Square Error (NMSE)

The comparison in NMSE performance between (11) & (12) is shown in Tables 1-3 and Figure-3, where the non-linearity order ranges from 1 to 4 and the memory depth for the results is set to 3.(11) is represented by MPB-real-2k, and (12) is represented by MPB-imag-2k.MPB-imag-2k has relatively slightly better accuracy compared to MPB-real-2k. This is because MPB-imag-2k has lower NMSE value, indicating smaller errors against the ideal output signal.

Table-1. NMSE for MPB-imag-2k and MPB-real-2k when PAG is 2.

K MPB-imag-2k MPB-real-2k

NMSE difference between MPB-imag-2k and MPB-real-MPB-imag-2k

1 -86.197 -84.582 -1.614

2 -83.943 -85.368 1.425

3 -79.937 -83.459 3.522

4 -73.44 -66.651 -6.788

1 -71.33 -71.861 0.531

2 -71.366 -72.739 1.373

3 -69.212 -72.343 3.131

4 -67.56 -66.603 -0.956

Indirect Learning Direct Learning

𝑥

Pre-distorter

𝑃𝐴

𝑥𝑝𝑟𝑒−𝑑𝑖𝑠𝑡𝑜𝑟𝑡𝑒𝑑 _𝑦

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1 -53.47 -53.365 -0.104

2 -53.265 -53.501 0.236

3 -42.926 -46.995 4.069

4 -38.321 -24.765 -13.555

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Table-4. NMSE for MPB-imag-2k and MP when PAG is 2.

K MPB-imag-2k MP NMSE difference between

MPB-imag-2k and MP

1 -84.58229 -78.66833 5.91396

2 -85.36833 -76.51994 8.84839

3 -83.45954 -92.81956 -9.36002

4 -66.65129 -92.26313 -25.61184

Table-5. NMSE for MPB-imag-2k and MP when PAG is 3.

1 -71.86156 -64.97981 6.88175

2 -72.7396 -63.71627 9.02333

3 -72.34312 -77.03894 -4.69582

4 -66.60309 -68.11177 -1.50868

Table-6. NMSE for MPB-imag-2k and MPwhen PAG is 4.

1 -53.36592 -48.2505 5.11542

2 -53.50133 -27.1319 26.36943

3 -46.99565 -12.4829 34.51275

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[image:6.595.137.458.82.585.2]

Figure-4. NMSE for MP and MPB.

CONCLUSIONS

The Memory Polynomial with Binomial Reduction (MPB-imag-2k) Model is an improved Memory Polynomial (MP) Model, by using the Binomial Reduction Method at the MP Basis Function, with improvements in Normalized Means Square Error (NMSE) performance. Table VII shows the respective values of NMSE for non-linearity orders, k ranging from 1 to 4, with memory depth set to 3, and Pre-amplifier Gain (PAG) of 2 to 4. MPB-imag-2k shows positive improvement in NMSE performance with values up to 34.51dB when operating in

Table-7. NMSE Performance of MPB-imag-2k against MP.

K NMSE Improvement (dB)

(PAG = 2) (PAG = 3) (PAG = 4)

1 5.91396 6.88175 5.11542

2 8.84839 9.02333 26.36943

3 -9.36002 -4.69582 34.51275

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