# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

**Download**(direct link)

**:**

**48**> 49 50 51 52 53 54 .. 85 >> Next

ministic excitation signals are sinusoids (not pure exponentials) then an even order contribution of order oi £ {2,4, ft. - - -} givps чоппе responses at the same frequencies as ;or lower order contributions 04 £ (2. 1,. . •. o\ — 2). Note that

6’, = ММ 1 17 .Ж)

k=ljk = l

and thus contains all the applied frequencies regardless of the ports at which they are applied. Also that ш a recursive form

160

7. Noise in non-linear systems: Theory

^o+i — + ^1,ь • ■ ■ 1 ^0,1 + ^1 ,Ji 1 • ■ •

■ • ■ I ФоЛ + фк,1, • • • , ФоЛ + <PK,Jk’.....

....., 9 о,Б, + ^1,1' • ■ •. Фо.Б., + flJi. • • •

• ■ • ) ^о,Ео + Й’Я',1, • • Ф»,е» + '/■7\,7К-} (7.37)

= {ф.+ 1,1,.-.,Фо+1,в0+1} (7.38)

and in this case £0+l < E0(JX -\----------b Jk), which means that E0 < (JL-\------------f- JK)°.

Furthermore

Фо1,е„, + 6 <Soi+o2 (7.39)

where 0i,02 t {1, 2,..., M - 1}, e0l £ (1. 2... ., Eol} and £ {1, 2,. .E:^}. Thus, u7,i,(/) in Equation (7.27) can be rewritten into the following:

A/-1 E0

“?,.,(/) = (7-40)

0 = 1 S— 1

M-l

= E °Zq,q(S0)6(f,S0) (7.41)

0=1

where

X,>,(£,) = [Ч..,(Фол).---.Ч..ч(^.е„)]Г (''.42)

<5(/,50) = [<5(/-Ф0л),-...Ч/-Ф,,Я„)]Т (7.43)

and where q £ {1,2,..., Q}, г, 6 {i, 2,..., Fq} and E,D is the number of different frequencies of order о £ {1,2,. ...M — 1}. Table 7.2 shows the upper limit for E„ versus the order о and the number of input frequencies Jl -f-• -+.//V- Ea is identical to the -upper limit listed in Table 7.2 when the input frequencies are incommensurate up to order o. Note from Table 7.2 that the upper limits in many cases are significantly lower than (J, -4- ■ • ■ 4- Jk}°- This is because the E0 < (/; 4- • • • -t- J condition does not use the fact that the addition of frequencies is associative.

A set of frequencies {u>(.... ,jg } is defined as incommensurate up to order о if

lot ли ]>j ( j- p2 a

j _ _ r n 1 лОх 1 /-..л

anu c I'-44-)

where из = [uii,.. . ,uiq\t, the vector 1-norms ||Pi,J!, [|P2,0[| £ {1,2, ...,o}, о £ {1.2...., M ]■ a.nd for

7.3. Noise sources

161

1. Exponential inputs exp[j(2Tb;it т Pi)i, • • ..expb'^jrugt + <?g)] where

ujq e TZ with la-^j ф ■ ■ ■ ji L’q[. Here the frequencies {wx,. . . ,u,'q} may be negative as well as positive.

2. Sinusoidal inputs cos(2;rc<;it + pj), ...,co$[2KWQt + &q) where 6

'R.q±. Here all the frequencies {-.-j,. . . .^'q} must be positive (zero included).

If a set of frequencies {шь .. ., o_'q} is not incommensurate up to order o, it is com-mensur ate for orders higher than or equal to v. In the literature, e.g. [8.9,10,11], the com mensur ability concept does not depend on the order, which is not in accordance with the above definition. However, the traditional definition actually assumes that, the order is infinite, which of course is not very relevant, since all practical uses of the Volterra series technique are of limited order (traditionally order 2 or 3). For a further discussion on this see [7, footnote Ij. From the definition it is seen that

1- If 0 t {cji , •. ., ljq} then the set {a-'i,... ,cjq} is incommensurate up to order o=l and commensurate for order о £ {2, 3,..., со}.

2. Any set of frequencies {u/'i,... ,^g} where uij / ^ ,jq is incommensurate

up to at least order о = 1.

The concept of commensurability is very important in the analysis and is frequently used in the following. The definition of commensurability does not depend on the number of input ports for the non-linear system but only on the overall frequencies applied. Intuitively, a set of frequencies {wj,..., wq} is incommensurate up to order

о if a frequency иj E {wj,. - -,^'q} can not be given as an intermodulation product or harmonics of the other frequencies up to order o.

Jpper limit for £

0 •Л + ■ ■ • + J к

1 2 3 4 5 6 7 Я

1 1 2 3 4 5 6 7 Я

2 1 3 6 10 15 21 28 36

3 ! 10 20 35 56 84 120

4 1 •5 15 35 70 126 2 L 330

0 1 6 21 56 126 252 162 792

6 1 - 28 8 i 210 162 02 I 1 7 1 h

Table 7.2: Upper limit for the number of frequencies E0 versus the order о and the number of applied frequencies ,/i 4- ■ ■ + J к Note that the upper limit for E0 applies to the situation

162

7. Noise in non-linear systems: Theory

The coefficient °1г,,1г1(Ф3,,.) where q e {1, 2,.. ., Q}, iq £ {1,2....,/,}, о e {1,2,...,.V/ — 1} and e £ {1, 2,..., E0) in Equation (7.4u) may be determined by comparing Equations (7.27) and (7.40). (Ф) is the Fourier series coefficient

ol order о at the frequency Ф0,. for port number (uqiiq) where q £ {1,2,..., Q} and iq & {!■ 2,..., /,}. Note that only the controlling variables

Q

.,ut{f)} - 1J{«,,!(/),..., u,,/,(/)} (7.45)

3=1

should be determined since one controlling variable may very well control more than one noise source, and thus I £ {0, 1.. .., T + • - • + Iq}. If none of the noise sources {nl(/),. .., ng(/)} are modulated then 1=0, and if none of the controlling variables are identical, i.e. n -=1 {u,,i(/)... . ,uqj4(J)} = 0 where 0 is the empty set, then I ~ I\ + ■ ■ ■ + Iq.

**48**> 49 50 51 52 53 54 .. 85 >> Next