An Extension of Noise Figure Definition for Nonlinear Devices

IFTH-25 Noise and Distortion Figure – An Extension of Noise Figure Definition for Nonlinear Devices Pedro Miguel Lavrador, Nuno Borges de Carvalho and José Carlos Pedro Instituto de Telecomunicações, Universidade de Aveiro, 3810-193 Aveiro, Portugal. on a signal passed through that system. It is well known Abstract — This paper deals in the integration of the straightforward relation between NF and Signal to additive and nonlinear distortion noise contributions into a Noise Ratio (SNR), defined as the ratio of signal power to single new figure of merit: Noise and Distortion Figure, noise power. NF is frequently referred has the ratio NDF. In the same way as traditional noise figure, NF, was conceived to be a measure of SNR degradation, NDF is now between input and output SNRs, although the formal proposed as its extension to nonlinear systems, as a measure definition of NF is: of SINAD degradation. NDF definition is discussed and its GN + N application to SINAD evaluations in systems of practical NF = o a (1) interest to the wireless community is exemplified. GN o where, No and Na are output available noise power I. INTRODUCTION spectral densities at a given source noise temperature, as The Noise Figure (NF), is an important figure of merit seen if the system were noise free, and the system’s added for designing low-noise systems. Its only drawback is that noise, respectively. its validity is restricted to linear systems. Actually, real In a nonlinear system the approach described above is systems are not linear, and so their nonlinear distortion incomplete because the distortion noise produced by the characteristics should also be incorporated in any high nonlinearity (distortion components that have a stochastic dynamic range design. The traditional approach consists behavior relative to the signal) is not taken in account. in also taking, distortion figures as IP3, although in a Another common figure of merit, which is more useful in separate way. Beyond their separate treatment of additive the context of nonlinear systems is Signal to Noise and noise, these nonlinear distortion standards were measured Distortion Ratio (SINAD) which is defined, according to using one or two tone test signals, and this kind of signals [3], as the ratio of signal power density, to noise and does not give a complete set of all possible nonlinear distortion power densities, which can be written as: distortion effects. Although some work have already been ()ω ()ω = S done in this respect [1], it lacks generality as, again, only SINADlin (2) N()ω + D ()ω a single tone was used [2] This paper proposes Noise and Distortion Figure, a new where S, N and D, are respectively the Signal, additive Figure of Merit that simultaneously handles additive Noise and Distortion power spectral densities. Since that noise and nonlinear distortion noise, identifying how the output distortion depends on the load, delivered power these two perturbations will combine to affect the signal and not available power must be considered. Therefore processed by the nonlinear system. The differences hereinafter, power refers to power delivered to the load. between Signal to Noise Ratio, SNR, and Signal to Noise In order to evaluate and compare these two figures of and Distortion Ratio, SINAD, are pointed out first, and merit consider a nonlinear system excited by an input, x then an appropriate Noise and Distortion Figure of merit composed of a signal s and noise n, the SINADo can be is defined accordingly. Finally, some simulations are calculated if we are able to evaluate the nonlinear output made to exemplify its application and usefulness. components. These components can be separated using a consequence of Price’s Theorem [4], as was previously presented by Rowe [5], which stated that, for a II. SIGNAL TO NOISE RATIO VERSUS SIGNAL TO NOISE AND memoryless nonlinearity h(·), with input x, the output DISTORTION RATIO. z=h(x), can be decomposed in It is usual, during the design of a RF system, to have it z()t = α ⋅ x ()t + y ()t (3) characterized in terms of noise via the Noise Figure, which gives a good measure of the addictive noise impact Where y(t) is uncorrelated with x(t), and has two distinct components, the addictive noise introduced by the 2137 0-7803-7695-1/03/$17.00 © 2003 IEEE 2003 IEEE MTT-S Digest system and the produced nonlinear distortion. The origin The input signal considered in this case is composed of of these two components is physically distinct so they are two independent Gaussian variables: one representing an uncorrelated with each other and so they add in power. arbitrary modulated signal, and another one representing According to this formulation we have found our output the noise, as indicated in (6), where s(t) and n(t) , stands signal component: a.x(t), that is, we considered as signal for the signal and noise components respectively. every component that can be obtained by a gain factor x()t =s ()t + n ()t (6) from the input signal. The value a, is also calculated in [5], and can be interpreted as a cross-correlation of the Applying this signal to the nonlinearity represented in output with the input signal. With all these statements we Fig. 1 will lead us two the output signal y(t): can write the SINADo. 2 y()t = α ()()()s ()t + n ()t + α ()()()s t + n t 2 + α ()s t + n t 3 (7) α ⋅ S ()ω 1 2 3 SINAD ()ω = i (4) o α 2 ⋅ ()ω + + ()ω N i N a IMD Since when using random signals we must represent them with its autocorrelation function, we assume that, In this expression S , and N stand for the input signal i i the input signal x(t), has an autocorrelation function and input noise power spectral densities respectively, a2 is Rxx(t ) = Rss(t)+Rnn(t). This is due to the uncorrelated the equivalent linear power gain of the system, Na the behavior of n and s. power density of addictive noise and IMD the power Using the result [6], which states that the output density of stochastic nonlinear Intermodulation autocorrelation of a Gaussian signal passed through a Distortion. third degree nonlinearity is given by: ()τ = α 2 ()2 + []α 2 + α α ()+ α 2 ()2 Ryy 2 Rxx 0 1 6 1 3 Rxx 0 9 3 Rxx 0 III. – NOISE AND DISTORTION FIGURE. (8) ⋅ R ()τ + 2α 2 R ()τ 2 + 6α 2 R ()τ 3 It was already referred above that NF can represent the xx 2 xx 3 xx ratio of the input SNR (SNRi) to the output SNR (SNRo). If In the frequency domain: the same ratio is evaluated using SINAD, a figure 2 S (ω )= α 2 ()()R ()0 + R ()0 δ ω identical to NF will be found except that it will now also yy 2 ss nn + []α + α ()()+ ()2 ⋅ ()()ω + ()ω include the distortion impact. Accordingly, we will call it 1 3 3 Rss 0 Rnn 0 Sss Snn Noise & Distortion Figure (NDF): + α 2 []()ω ∗ ()ω + ()ω ∗ ()ω + ()ω ∗ ()ω 2 2 Sss Sss Snn Snn 2Snn Sss ()ω + 6α 2 [S ()ω ∗ S ()ω ∗ S ()ω + S ()ω ∗ S ()ω ∗ S ()ω Si 3 ss ss ss nn nn nn + ⋅ ()ω ∗ ()ω ∗ ()ω + ⋅ ()ω ∗ ()ω ∗ ()ω ] SINAD ()ω N ()ω 3 Snn Snn Sss 3 Snn Sss Sss NDF(ω) ≡ i = i ()ω α 2 ⋅ ()ω (9) SINADo Si α 2 ⋅ ()ω + + ()ω N i N a IMD In this expression it can be seen that there is a α 2 ⋅ N ()ω + N + IMD ()ω component of the input signal and noise that emerges at = i a α 2 ⋅ ()ω the output simply affected by a gain factor which depends N i on a 1,a3 and on the total input power. Note that the (5) second order components appear clearly out of band: one In (5) it can be seen that NDF does not depend only on component is DC and the other component is at the the linear gain but also on the distortion produced. The second harmonic zone. The extra perturbation to the main advantage of this new definition is that it allows signal is produced by the third order component which simultaneous study of noise and distortion. This new combines to appear at the fundamental output zone and at figure of merit can be used without many effort compared the third harmonic zone. with the usual NF designs. For a particular case of the input, where the signals are Let us know obtain NDF for a typical third order flat over a bandwidth B, with power Ps and Pn. polynomial nonlinearity, with Gaussian random inputs. It P / 2B, −ω ≤ ω ≤ −ω ,ω ≤ ω ≤ ω is already well known that the response of a polynomial ()ω = s H L L H Sss nonlinearity similar to the one represented in Fig. 1 can 0 ,elsewhere be easily calculated [6]. − ω ≤ ω ≤ −ω ω ≤ ω ≤ ω Pn / 2B, H L , L H x(t) 2 3 y(t) ()ω = y()t = α x ()t + α x ()t + α x ()t Snn 1 2 3 0 ,elsewhere Fig. 1 The nonlinear model used. (10) 2138 2 P []α + α ()+ s 1 3 3 Ps Pn SINAD (ω) = 2B o 2 3 (12) 2 B ⋅ ()+ []α + α ()+ Pn + α 2 − ω 2 + ()ω + ω ω + − ω ω ⋅ 3 Ps Pn + 1 3 3 Ps Pn 6 3 L H L H 3 N a 2B 2 8B 2 3 2 9 B ()+ []α + α ()+ + α 2 − ω 2 + ()ω + ω ω + − ω ω Ps Pn + 1 3 3 Ps Pn Pn 3 L H L H 2 N a 2 2 B NDF(ω) = (13) []α + α ()+ 2 1 3 3 Ps Pn Pn The output power spectral density in the fundamental The power densities of the three components are zone may be written as: measured in the mid band, and SINADo calculated.

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