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

Authors: Rajendra Kumar
Journal: Journal of Applied Mathematics
 (2016)

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## 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.


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## Body

## 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 outputy(t) can be expressed in terms of its input bandpass process u(t) via its input-output characteristic function F(u) as(1)yt=Fut.With a power series expansion of F(u) about u=0, the amplifier output y(t) may be expressed in the following series form:(2)yt=K1ut+K3u3t+K5u5t+⋯,where the coefficients K1,K3,… 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 byRy(τ) may be obtained from (2) as(3)Ryτ=Eytyt+τ=EK1ut+K3u3t+⋯K1ut+τ+K3u3t+τ+⋯,where E denotes the expected value operator. Carrying out the multiplication of the two series in (3) and grouping the terms of the same order yield(4)Ryτ=K12Eutut+τ+K1K3Eutu3t+τ+K1K3Eu3tut+τ+K1K5Eutu5t+τ+K32Eu3tu3t+τ+K1K5Eu5tut+τ+⋯.Denoting by ξ(t) the normalized process u(t)/σ with σ2 denoting the variance of the process u(t), the correlation function Ry(τ) may be expressed as(5)Ryτ=σ2K12Eξξτ+σ4K1K3Eξξτ3+K1K3Eξ3ξτ+σ6K1K5Eξξτ5+K32Eξ3ξτ3+K1K5Eξ5ξτ+⋯,where ξ and ξτ denote ξ(t) and ξ(t+τ), respectively, for the convenience of notations. Assuming that u(t) is a zero mean process, the moments E[ξnξτm] for any pair of nonnegative integers may be obtained by the following integral:(6)Eξnξτm=∫-∞∞∫-∞∞ξ1nξ2mfξ1,ξ2dξ1dξ2.In (6) the function f(ξ1,ξ2) denotes the joint pdf (probability density function) of the random variables ξ and ξτ or by definition the two-dimensional pdf of the process ξ(t). For the case of Gaussian process assumed in this report, the joint pdf f(ξ1,ξ2) is given by(7)fξ1,ξ2=12π1-ρ2exp⁡-ξ12-2ρξ1ξ2+ξ221-ρ2,where ρ denotes the correlation coefficient between ξ and ξτ. In general ρ is a function of τ; however for notational convenience the argument τ of ρ has been dropped in (7). Following the approach of [18, 19], the function f(ξ1ξ2) is expressed in terms of the following series so as to evaluate the integral in (6) in closed form:(8)fξ1,ξ2=∑p=0∞gpξ1gpξ2ρpp!.In (8) the functions gp(x) are given by(9)g0x=12πexp-x22;gpx=dpg0xdxp;p=1,2,….In order to obtain closed form expressions for the desired moments, the functions gp(x) are related to the Hermite polynomials. For example, (10)dndznexpz22=Cnzexpz22;n=0,1,2,…,Cnz=zCn-1z+n-1Cn-2z;n=2,3,…,C0z=1;C1z=z.In (10), Cn(z) denotes the Hermite polynomial of degree n. Substituting z=jx with j=-1, one obtains(11)dndznexp-x22=jnCnjxexp-x22;n=0,1,2,…,(12)Cnjx=jxCn-1jx+n-1Cn-2jx;n=2,3,…;C0jx=1;C1z=jx.Multiplication on both sides of (12) by jnexp(-x2/2)/2π and application of (11) result in the following desired recursive expression for the functions gp(x) that are the derivatives of the function g0(x)=1/2πexp(-x2/2);(13)gpx=-xgp-1x-p-1gp-2x;p=1,2,…;g0x=exp-x2/22π.Now from (6) and (8), one obtains(14)Eξnξτm=∑i=0∞ηn,iηm,iρiτi!with(15)ηk,i=∫-∞∞xkgixdx;k,i=0,1,2,….Multiplying both sides of (13) by xk and integration over the interval (-∞,∞) and using (15) yield the following recursive expression for ηk,i:(16)ηk,i=-ηk+1,i-1-i-1ηk,i-2;i,k=1,2,…,(17)η0,0=1;η0,i=0fori>0.The coefficients ηk,0 which are the moments of the Gaussian distribution can be obtained by direct integration and are given by(18)ηk,0=1·3·5···k-1;keven;k>00;kodd.With the initial conditions in (17) and (18), ηk,i for i,k>0 may be computed from (16). For example, from (16) ηk,1=-ηk+1,0 and in view of (17) one obtains(19)ηk,1=-1·3·5···k;kodd;k>00;kodd.Similarly, from (16), (18), and (19)(20)ηk,2=-ηk+1,1-ηk,0ηk,2=1·3·5···k+1-1·3·5···k-1=1·3·5···k-1k+1-1=1·3·5···k-1·k;keven;k>0,ηk,2=0;kodd.In general, ηk,i have the following properties which follow from the recursion (15)–(17):(i)(21a)ηk,i=0fori>k.(ii)(21b)Fori≤ktheonlynonzeroelementsarethoseforwhichk-iisaneveninteger.(iii)
The nonzero elementsηk,i are given by(21c)ηk,i=-1kk·k-1···k-i+1·k-i-1·k-i-3···1.In (21a)–(21c) the first bracketed term has i factors and is equal to 1 for i=0, with the second bracketed term having (i-k)/2 factors. The value of ηk,i can be computed using (i)–(iii) for any integers k, i, and some of these terms are presented in Table 1.Table 1
Sample values of the coefficientsηk,i.
k i 0 1 2 3 4 5 0 1 0 0 0 0 0 1 0 - 1 0 0 0 0 2 1 0 2 ⋅ 1 0 0 0 3 0 - 3 ⋅ 1 0 - 3 ⋅ 2 ⋅ 1 0 0 4 3 ⋅ 1 0 4 ⋅ 3 ⋅ 1 0 4 ⋅ 3 ⋅ 2 ⋅ 1 0 5 0 - 5 ⋅ 3 ⋅ 1 0 - 5 ⋅ 4 ⋅ 3 ⋅ 1 0 - 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1In view of (21a)–(21c), the expression in (14) may be modified to that in (25)(22)Eξnξτm=∑i=0minn,mηn,iηm,iρiτi!.Equation (22) also shows that the expression for E[ξnξτm] is symmetric in n and m as expected from the stationarity of the process ξ(t). For the case of n=m, with n being odd, the expression for E[ξnξτm] reduces to(23)Eξnξτn=∑i=1,i:oddnηn,i2ρiτi!.For example,(24a)Eξ3ξτ3=η3,12ρτ+η3,32ρ3τ3!=9ρτ+6ρ3τ,(24b)Eξ5ξτ5=η5,12ρτ+η5,32ρ3τ3!+η5,52ρ5τ5!=225ρτ+600ρ3τ+120ρ5τ.From (22)(25)Eξ3ξτ=3ρτ;Eξ5ξτ=15ρτ.Substitution of the expressions (24a)–(25) for E[ξnξτm] in (5) yields the following expression for Ry(τ):(26)Ryτ=α1,1ρτσ2+α3,1ρτ+α3,3ρ3τσ4+α5,1ρτ+α5,3ρ3τ+α5,5ρ5τσ6+⋯,where the first few coefficients αi,j are evaluated in (27) and are given by(27)α1,1=K12,α3,1=6K1K3;α3,3=0,α5,1=30K1K5+9K32;α5,3=6K32;α5,5=0.


## 4. Power Spectral Density of the Amplifier Output Process

The power spectral densityPy(f) of the amplifier output process is obtained by taking the Fourier transform on both sides of (26) which results in(28)Pyf=α1,1Pξfσ2+α3,1Pξf+α3,3Pξ3fσ4+α5,1Pξf+α5,3Pξ3f+α5,5Pξ5fσ6+⋯.In (28) Pξk(f) for any positive integer k denotes the k times convolution of Pξ(f) with itself k times. Thus(29)Pξ2f=Pξf⊗Pξf;Pξ3f=Pξf⊗Pξf⊗Pξf;⋮where ⊗ denotes convolution. Direct convolution of the PSD of the input bandpass process Pξ(f) with itself multiple times results in spectral zones centered around frequencies that are multiples of the input signal center frequency fc. In most applications including the NPR evaluation, only the PSD in the fundamental zone is of interest. In this case the convolution process can be significantly simplified using the following approach. The correlation function of the input bandpass process ρ(τ) is written as(30)ρτ=ρbτcos⁡2πfcτ,where ρb(τ) denotes the autocorrelation function of the baseband process ξb(t) corresponding to the input process ξ(t) by(31)ξt=2ξbtcos⁡2πfct.From (30) the PSD of ξb(t) denoted by Pξb(f) is related to Pξ(f) by(32)Pξf=12Pξbf-fc+12Pξbf+fc.It is also apparent from (30) or (32) that E[ξ2]=E[ξb2].It follows from (30) that(33)ρ3τ=ρb3τcos32πfcτ=ρb3τ0.75cos⁡2πfcτ+0.25cos⁡6πfcτ.Taking Fourier transform on both sides of (33) it follows that the one-sided fundamental zone component of the PSD Pξs3(f) is given by(34)Pξ,fzs3f=0.75Pξb3f-fc.Thus except for the constant 0.75, the one-sided fundamental zone component of the PSD Pξs3(f) is obtained by simply shifting the corresponding (two-sided) baseband PSD Pξb3(f), where(35)Pξb2f=Pξbf⊗Pξbf;Pξb3f=Pξbf⊗Pξbf⊗Pξbf;⋮In general the one-sided fundamental zone component of the PSD Pξsk(f) is obtained by simply shifting the corresponding (two-sided) baseband PSD Pξbk(f) by(36)Pξ,fzskf=ak,1Pξbkf-fc;k=1,3,5,…,where ak,1 is the coefficient of cos⁡(2πfct) in the trigonometric expansion of cosk⁡(2πfct) in (37)cosk2πfct=ak,1cos⁡2πfct+ak,3cos⁡6πfct+⋯.These coefficients ak,j for any odd integer can be obtained by the following recursion [1]:(38)ak+2,1=0.75ak,1+0.25ak,3;ak+2,j=0.5ak,j+0.25ak,j-2;1<j≤k;j:odd;ak+2,k+2=0.25ak,kwith a1,1=1 and ak,j=0 for j>k.In summary the PSD of the amplifier output fundamental zone component of the bandpass power spectral densityPξsk(f) is obtained by first obtaining the corresponding baseband (two-sided) power spectral density Pξb(f) by shifting the one-sided PSD 2Pξ(f) to f equal to 0, convolving Pξb(f) with itself k times, and scaling the result of convolution by the constant ak,1 and finally shifting the result to the input signal center frequency fc. This process also results in a considerable simplification in the actual numerical evaluation of the result.


## 5. Numerical Results

This section presents some numerical results obtained by using the analysis presented in the paper. The analytical results have been encoded in the form of a relatively simple MATLAB program that can provide the result for any specified amplifier input-output characteristics, notch bandwidth and depth, and the output power back-off. The MATLAB program then generates the desired numerical result almost instantaneously.The first example generates the various termsPk(ξ) appearing in the expression for the amplifier output power spectral density Py(f) in (28) for various values of the integer k for the case wherein the input is a band limited white noise with a specified notch in the spectral band of the input noise. Thus the result of Example 1 may be applied to any specific nonlinear amplifier. The second example considers an example amplifier input-output characteristics and evaluates the noise power ratio (NPR) as a function of the output back-off using the result of Example 1. The NPR ratio is the ratio of the maximum PSD outside and in the vicinity of the notch band and the PSD in the center of the notch band at the output of the nonlinear amplifier.Example 1.
The processξ(t) is a band limited white noise with power σξ2 equal to 1 W and bandwidth BT equal to 100 MHz and has a notch of bandwidth 2 MHz and 80 dB depth. In the example, the power level is adjusted to 1 W after the notch. Figure 1 shows the one-sided PSD 2Pξ(f) shifted to f equal to 0. The figure also plots the corresponding fundamental zone component of the bandpass power spectral density Pξk(f) for the order k equal to 3, 5, and 7. All the PSDs in the figure are the one-sided PSDs shifted to f equal to 0. Figure 2 shows the corresponding result in dB scale over a 4 MHz band around the notch.
It may be observed from Figure1 that there is only a small difference in the power spectral density inside the notch and at the border of the notch for the third-order term and a negligible difference in the higher order terms. Figure 3 shows the result of Figure 2 over a bandwidth of 200 MHz that is two times the input noise bandwidth. Figure 2 shows the details of the PSD of various terms in the immediate vicinity of the notch.Figure 1
Power spectral density (W/Hz) of various order convolution terms (BT=100 MHz, Bnotch = 2 MHz, and σξ2 = 1 W).Figure 2
Power spectral density (dBW/Hz) of various order convolution terms (BT = 100 MHz, Bnotch = 2 MHz, and σξ2 = 1 W).Figure 3
Power spectral density (dBW/Hz) of various order convolution terms over a bandwidth of 200 MHz (BT = 100 MHz, Bnotch = 2 MHz, and σξ2 = 1 W).Example 2.
As an example of the application of the theory presented in this paper, the nonlinear amplifier is approximated by a soft limiter model with its input-output characteristic described by the following nonlinear functionF(u) given by(39)Fu=erf⁡u2≡2π∫0uexp⁡-x22dx.Figure 4 plots the nonlinearity F(u) depicting good linearity over the input range of (–1, 1). The function F(u) may be approximated by a polynomial of sufficiently high order such that the approximation error is negligible over the specified range of u. When the input to the amplifier is a sine wave of constant amplitude A, the amplifier output contains the fundamental zone and various harmonic zones. The fundamental zone output is just a sine wave of amplitude g(A), which is given by the Chebyshev transform [16] of F(u) and for the specific case of (39) is given by(40)gA=2πAexp⁡-A24I0A24+I1A24.In (40) I0 and I1 are the modified Bessel functions of zero and first order, respectively. Figure 5 plots the function g(A) versus A for the soft limiter. In addition, Figure 6 plots the input signal power level Pi equal to A2/2 versus the amplifier output signal power equal to gA2/2 in dB scale.
The amplifier characteristicsF(u) are approximated by a polynomial of sufficiently high degree over the range of input signal which is taken equal to 3σ, where σ2 is the variance or power Pi of the zero mean Gaussian input noise process u(t). Figure 7 shows the power spectral density of the amplifier output y(t) computed from (28) with Pi=0 dBW. The coefficients K1,K3,… are obtained by a degree 9 polynomial approximation of F(u) over the interval (−3, 3) and are equal to 0.7962-0.12840.0165-0.00134.12×10-5. The RMS approximation error in the polynomial approximation is equal to 3.38×10-4 thus achieving very good curve fit.
Figures8 and 9 plot the corresponding results for the case of Pi equal to −3 dBW and −4 dBW, respectively. The amplifier coefficients K1,K3,… for the two cases are equal to 0.7978-0.13240.0191-0.00199.26×10-5 and 0.7978-0.13270.0195-0.00201.11×10-4, respectively. The corresponding RMS approximation error is equal to 3.38×10-4 and 1.76×10-5, respectively, for the two cases. Figure 10 plots the corresponding result for the case of notch bandwidth equal to 8 MHz.
The NPR values obtained for the three cases shown in Figures7–9 are equal to 6.09, 15.38, and 18.49 dB, respectively. Figure 10 shows the output noise power spectral density ratio when the notch bandwidth is increased to 8 MHz for the case of Pi=-3 dBW depicting an NPR of 15.96 dB. Comparing the result of Figure 8 for the case of Bnotch=2 MHz shows that the difference in the NPR values for the two cases is about 0.58 dB.
Figure11 plots the NPR versus the input power level for the three cases of Bnotch equal to 1 MHz, 2 MHz, and 8 MHz, respectively. There is no significant difference in the NPR between the two cases of 1 MHz and 2 MHz notch bandwidth while for the case of 8 MHz bandwidth the NPR is about 1 dB higher. As may be inferred from Figure 11, the plots of NPR versus Pi have a slope of about −3.1 dB/dB for power levels higher than −5 dBW and a slope of about −2.16 dB/dB for lower values of Pi.
Figure12 plots the NPR versus the output power back-off BO for the three cases of Bnotch equal to 1 MHz, 2 MHz, and 8 MHz, respectively. The output power back-off BO is defined as the amplifier output signal power level relative to the fundamental zone power of a hard limiter with sine wave input which is equal to 8/π2=-0.9 dB. Figure 12 also plots the result for the amplifier output signal to distortion power ratio for the case of M equal to 256 CDMA signals with QPSK modulation obtained from the analysis in [1]. The result for CDMA is close to that of the NPR result of the paper providing a cross-validation of the result as for relatively higher value of M the CDMA signal may be approximated by a Gaussian distribution.Figure 4
Soft limiter input-output characteristics.Figure 5
Plot of the Chebyshev transformg(A).Figure 6
Amplifier output versus input power level.Figure 7
Amplifier output noise power spectral density withPi = 0 dBW, Bnotch = 2 MHz, and notch depth = 80 dB.Figure 8
Amplifier output noise power spectral density withPi=-3 dBW, Bnotch = 2 MHz, and notch depth = 80 dB.Figure 9
Amplifier output noise power spectral density withPi=-4 dBW, Bnotch = 2 MHz, and notch depth = 80 dB.Figure 10
Amplifier output noise power spectral density withPi=-3 dBW, Bnotch = 8 MHz, and notch depth = 80 dB.Figure 11
Plot of NPR versus the input power level.Figure 12
Plot of NPR versus the output power back-off BO.


## 6. Conclusions

This paper has presented a theoretical analysis of the amplifier output noise power spectral density result in a closed form from which the NPR is evaluated by a simple subtraction. The method can be applied to any amplifier with known input-output characteristics. The paper has presented 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. The analytical results of the paper have been encoded in the form of a MATLAB program that can provide the result for any specified amplifier input-output characteristics, notch bandwidth and depth, and the output power back-off. The MATLAB program generates the desired numerical result almost instantaneously compared to extensive efforts and time required for obtaining experimental and simulation results for any power amplifier.



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