Journal of Applied Mathematics

Volume 2016, Article ID 8710860, 9 pages

http://dx.doi.org/10.1155/2016/8710860

## Theoretical Analysis of the Noise Power Ratio of Nonlinear Power Amplifiers

Department of Electrical Engineering, California State University, Long Beach, CA 90840, USA

Received 29 July 2016; Revised 27 October 2016; Accepted 31 October 2016

Academic Editor: Mehmet Sezer

Copyright © 2016 Rajendra Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a theoretical analysis and derives the amplifier output noise power spectral density result in a closed form when the input to the amplifier is a band limited Gaussian noise. From the computed power spectral density the NPR is evaluated by a simple subtraction. The method can be applied to any amplifier with known input-output characteristics. The method may be applied to analyze various other important characteristics of the nonlinear amplifier such as spectral regrowth that refers to the spreading of the signal bandwidth when a band limited signal is inputted to the nonlinear amplifier. The paper presents numerical results on the NPR as a function of the noise bandwidth, depth level of the notch, and the output power back-off obtained from the analysis presented in the paper.

#### 1. Introduction

The power amplifier is one of the most important subsystems of modern communication systems [1–15]. In the case of satellite downlinks, the power efficiency of the high-power amplifiers (HPAs) is very important as the requirements on the power directly translate into the size, weight, and cost of the satellite payload. As the power efficiency is relatively high when the amplifier is operated near the saturation region, in the case of satellite links, the HPA by necessity needs to be operated in such a region. However, the nonlinearity of the amplifier in the near saturation region introduces considerable distortion in the signal to be amplified. As the output power back-off is reduced the signal to distortion power ratio at the amplifier output is correspondingly reduced. This places a serious restriction on the amount of back-off that needs to be introduced resulting in a loss of the available output power and equally importantly in a reduced power conversion efficiency thus resulting in an increased demand on the D.C. power which in case is provided by solar panels and thus has a direct implication on the size, weight, and cost of the satellite payload. In wireless communication systems any reduction in the power conversion efficiency results in a corresponding reduction in the battery life at the user terminal. It is thus extremely important to analyze the performance when the power amplifier is operated with relatively small output power back-off thereby exhibiting significant nonlinear behavior.

Among the various methods to assess the performance of an amplifier is the evaluation of the signal to distortion ratio at the amplifier output as a function of the amplifier output power back-off. In case of digitally modulated signals at the amplifier input in the digital communication systems, one of the most important performance measures is in terms of the bit error rate (BER) achieved in the presence of both the distortion introduced by the amplifier and any channel interference plus noise. However, both these measures are functions of the detailed modulation techniques, multiple accessing or multiplexing techniques, and the number of user signals at the amplifier input in addition to the amplifier input-output characteristics and the output power back-off. The various performance measures may be obtained by detailed mathematical analysis, computer simulations, and/or experimental measurements. Both the simulations and experimental measurements are, in general, very time consuming and expensive while the analysis methods are generally difficult and are for more specific cases. For example, detailed analysis has been presented by the author for the case of the code division multiple accessing (CDMA) [1–3] with an approximate analysis based on the Gaussian assumption on the nonlinear distortion appearing in earlier literature [4–6]. An infinite series result for the output of a nonlinear device in terms of the autocorrelation function of an input Gaussian process is presented in [7, 8]. An analysis for the distortion effects of the nonlinear amplifier on OFDM signals appears in [9, 10]. Extensive relatively earlier literature exists on the effects of the nonlinear amplifier on FDMA signals as in [13–17] and their references.

A performance characterization of the nonlinear power amplifiers that is independent of the specifics of the amplifier input signal, such as the multiple accessing and modulation techniques used, the number of users, and the relative power level of the various user signals, is commonly used in the practice. This measure is termed the noise power ratio (NPR). The noise power ratio is measured by inputting the amplifier with a white noise of bandwidth equal to the specified signal bandwidth. A notch in the input noise band is created with the bandwidth of the notch much smaller than the noise bandwidth. At the output of the amplifier one measures the noise power spectral density both inside and outside the notch with the ratio of the two by definition equal to the amplifier NPR that is a function of the total output noise power or the output power back-off.

While the NPR measurements are relatively less intensive compared to some other performance measurements such as the BER measurements, nevertheless these do require extensive measurements as well, in which from the measurements alone it is not possible to predict the performance for the amplifiers other than the one involved in the measurement; in other words it does not provide any measure of the sensitivity of the NPR to the amplifier input-output characteristics and does not address the validation issues of the experimental data. Thus it is of great interest to be able to evaluate the amplifier NPR by independent analytical means. This paper presents a theoretical analysis of the NPR for the nonlinear amplifier that is also applicable to any other nonlinear devices.

This paper presents a theoretical analysis and derives the amplifier output noise power spectral density result in a closed form when the input to the amplifier is a band limited noise with a notch in the spectral band. From the computed noise power spectral density (PSD) the NPR is evaluated as the ratio of the PSDs evaluated outside and inside of the notch. The method can be applied to any amplifier with known input-output characteristics. The paper presents numerical results on the NPR as a function of the noise bandwidth, depth level of the notch, and the output power back-off obtained from the analysis presented in the paper.

#### 2. Amplifier Model

It is assumed in this analysis that the amplifier output can be expressed in terms of its input bandpass process via its input-output characteristic function asWith a power series expansion of about , the amplifier output may be expressed in the following series form:where the coefficients may be obtained from the amplifier characteristics using a Taylor series expansion or by a finite degree polynomial approximation [1].

#### 3. Correlation Function of the Amplifier Output Process

The correlation function of the amplifier output process denoted by may be obtained from (2) aswhere denotes the expected value operator. Carrying out the multiplication of the two series in (3) and grouping the terms of the same order yieldDenoting by the normalized process with denoting the variance of the process , the correlation function may be expressed aswhere and denote and , respectively, for the convenience of notations. Assuming that is a zero mean process, the moments for any pair of nonnegative integers may be obtained by the following integral:In (6) the function denotes the joint pdf (probability density function) of the random variables and or by definition the two-dimensional pdf of the process . For the case of Gaussian process assumed in this report, the joint pdf is given bywhere denotes the correlation coefficient between and . In general is a function of ; however for notational convenience the argument of has been dropped in (7). Following the approach of [18, 19], the function is expressed in terms of the following series so as to evaluate the integral in (6) in closed form:In (8) the functions are given byIn order to obtain closed form expressions for the desired moments, the functions are related to the Hermite polynomials. For example, In (10), denotes the Hermite polynomial of degree . Substituting with , one obtainsMultiplication on both sides of (12) by and application of (11) result in the following desired recursive expression for the functions that are the derivatives of the function ;Now from (6) and (8), one obtainswithMultiplying both sides of (13) by and integration over the interval () and using (15) yield the following recursive expression for :The coefficients which are the moments of the Gaussian distribution can be obtained by direct integration and are given byWith the initial conditions in (17) and (18), for may be computed from (16). For example, from (16) and in view of (17) one obtainsSimilarly, from (16), (18), and (19)In general, have the following properties which follow from the recursion (15)–(17):(i) (ii) (iii)The nonzero elements are given by

In (21a)–(21c) the first bracketed term has factors and is equal to 1 for , with the second bracketed term having factors. The value of can be computed using (i)–(iii) for any integers , , and some of these terms are presented in Table 1.