# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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The contribution of order 0 for n„[J) in Equation (7.19). which is identical to the response when no signals (including dc) are applied, is excluded in the systems in Figures 7.2 and 7.3. It is very important to emphasize that the dc sources can not bp removed as sources bv introducing a zeroth order term in Equation (7.19). This

-A',1, ■ • -, —K,o[г ,

li=l i JC = L

ok (pu,{2i,i—..................; 'Pk,!K{-k. i, • • •

7.3. iNoise sources

155

is because the zeroth order contribution, identical to (G7)c,o,. „ of), in this case can not give information on che non-linear interference between dc and other applied deterministic signals. Actually, (G7)o,o„..,o() is a system contribution which is there but it can not interfere with other signals (it may be calied a 'mathematical offset’). It lies in the modelling of non-linear devices that the zeroth order contribution is not of interest (it does not describe any physical behaviour of the circuit). Note that there may very well be interference between dc and other deterministic signals and noise in the system in Figure 7.2. In much work on Volterra series analysis it

is stated that the zeroth order contribution (G7)o,o.........o() represents the internal dc

sources in the system. However, this is generally not correct for the above reasons.

The upper limit for the order m0 4- mi 4- ■ ■ • 4- miq is the maximum order M. Actually, M —> со but for practical reasons M must be chosen to have some finite value. Of course, M must be chosen sufficiently large so as not to give incorrect results because of too low a maximum order. For a given M it is sometimes useful to determine limits for the amplitudes of the applied deterministic signals to ensure that the residual components to nq(f) in Equation (7.19) of orders higher than M are insignificant. This problem is also closely related to the functional modelling of the non-linear elements in the system. The model of a non-linear element is valid for the controlling variable in some finite interval. Therefore, it is a good idea to calculate the deterministic contributions at the controlling variables for the non-linear elements to assure that the elements are not too strongly/weakly excited.

The response for the modulated noise source nq(f) where q £ {1,2,....Q} is limited to exclude zeroth order noise contributions — actually, this is no limitation as in this case the modulated noise source n,(/) is just a controlled non-linearity, and this type of non-linear element (subsystem) is already included in the non-linear noise free system in Figure 7.2. Using the above assumptions and treating only low-level noise, then

•Vf-1 M-1

CO /*CC ,*cc rc о rc

/ ............................./ •••/

• CO %/ — СО •/ — CO j — '>0 ^ — '30

1=0 m f, = о

4- - ■ • -f mh\

(Cj) 1,-n, m,„ .I .I

U\.(fio.l ) tl?ti(fiM) •••K7,1(ni.7?,.1)....Ub^(Qfvi) ■ )

off- Пол-fi...---------------.................-4,1-------------nv,lv>

-ifto.i •••</«,.,*,............i 7,24'.

The multi-purt 'v - -! ■ ■ ■ j: : ti angler function :: . !-) in liquation ( ,'.24 i i

referred to as the modulating hinctiori for the modulated noise source , where q 6 {1,2,...,Q}. Note from Equation (7.24) that the contribution corresponding to mi = ■ ■■ = mj - 0 is simply a linear filtering of the fundamental (unmodulated) noise source u>,(/), n.,(f) = ! Gqi i /)- and corresponds to the

156

7. Noise in non-linear systems: Theory

response when (i) all the controlling variables Ji7,i(/) = = u?,/,(/) = 0 where

q 6 {1,2,. .., Q}, or (ii) when M = 1 meaning that the system is linear. Similarly, if Iq = 0 then nq(f) = (Gq)i(f)wq(f) in which case nq(f) is an unmodulated noise source given as a linear filtering of the fundamental (unmodulated) noise source wq(f) where q £ {1,2,...,Q}. Using the formulation in Equation (7.24) it is possible to introduce very complicated modulating functions in a simple way. For example, the modulated noise source nq(f) where q £ {1,2,... ,Q] may depend on several deterministic signals in the non-linear network, and the non-linear network in Figure 7.3. In Table 7.1 some examples of modulated noise sources and the corresponding modulating functions are shown.

7.3.2 Controlling variables

Since low level noise is assumed, which implies that the noise is a small perturbation of the deterministic signal regime, the controlling variables are determined as the contributions to {u,hi(/),..., ”iq.iq{{)} due to the deterministic signals {•Si(/)’ • ■ • > sk(I)} only. Therefore the modulated noise sources {n\(f), ■ ■., riq(f)} do not have any feedback impact on the controlling variables, which would be the case for high level modulated noise sources. Thus u9,; (7) where q £ {1,2,... ,Q} and iq £ {1,2,...,/,} can be determined as

M 1 M l fOO /СО Г СО ГО

w/) = £-£/ ••■/ ............................../ •••/

Ol =0 oK=0 J,-co J~co

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