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Theory of Linear and nonlinear circuits - Engberg J.

Engberg J. Theory of Linear and nonlinear circuits - Wiley & sons , 1995. - 154 p.
ISBN 0-47-94825
Download (direct link): noisetheoryoflinearandnonlinear1995.pdf
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quadratic and then the inverse does not exist. This problem can be solved by
introducing fictitious controlling or controlled variables as follows:
• If Q < R then introduce R — Q fictitious controlled sources which depend on
one or more of the controlling variables . . ,£/}(/).
• If Q > R then introduce Q — R fictitious controlling variables which control
one or more of the controlled variables y\(/)...., yq(f).
To determine the second and higher order controlling variables XfimllOhMi) where j| m j j 6 {2,3,..., oc} use Equation (9.51) to yield
QO OO OO
£•■• £ £
711=0 np(q)=0 01,1=1
- £ £
1 Qp(q),l
P(q) nr
£wiND П П '
г —1 p=t
(/-'qjn, 9i,i
• ■ ■; 'ФТЯр{я),1,- ■
Р(ч) nr
П
r=l P=1
r PM Пг
exp j j фт Y £ Яг.р
** r = l p— I
R ~o
= £ £ £ ЛЛ:!?!!)
r^i 0=1 g
Tiie coefficient oi «xpij \\ф\\\ с i f - Ijvvi) must be the same on both. sides nt bquat.ion (O.oo’l. From this, the following properties can be derived regarding which terms should be included on the left-hand side of Equation (9.55):
OO
£■■■ £ £ ■■■
<?!,! 01,n; =1 9l..n
cc>
£ £
• °p!l).nP(.,i 1
*,(IKPII)
■’■ФТЯрм,Пр(1!)
?rlP)
1 / P<'!l
*f/-*T£LO
J 4 r=l p=L 7
A.-, )3(exp! j 7i й(/ — Q )
2.32
9. Multi-port Volterra. transfer functions
Property 9.1
Р(ч) nr
E E 4r,p = 1 (9-56)
Г — 1 p=l
inhere.
1 = [1,1, ■■■■■., 1]T £ {1}II^|X* (9.57)
llroll
for any given q £ {1,2, ...,£?}.
Proof 9.1 Follows directly since {ф\.\,. ■ ■, Ф\,т1 .............., 0AMi • • ■ > Фк.тк} forms a
phase base. □
Property 9.2
qTlP.U e {0,1} (9.58)
for all r £ {1,2,..., P(q)}, p £ {1,2,..nr}, k £ {1,2,..., A'}, I £ {1,2,..., m*}.
Proof 9.2 Since {ф 1|b ..., ....., Фкл- • ■ ■ > Фк,тК} forms a phase base, then
only one of the coefficients of any where k {1,2,..., A'} and I £ {l,2,...,m*} mB5( be equal to 1 and all others must be equal to 0. □
Property 9.3
||9J| = or,p £ {1,2,..., oc} (9.59)
for all r £ {1,2,..., /’(?)} and p £ {1, 2,.. ., nr}.
Proof 9.3 Follows directly from Equation (9.55). □
Property 9.4
Pta)
r=l p—L
= IMI (9-61)
where the о-vector is defined in Equation (9.65).
9.3. Theory
233
Proof 9.4 Equation (9.60) follows directly from property 9.3 and Equation (9.61) follows directly from property 9.2. □
Property 9.5
or,pG{ 1,2------IMI-l} (9.62)
for all r £ {1, 2,. .., /-’(7)} and p £ {1,2....nT}.
Proof 9.5 From Equation (9.55) it is given that oTp £ {1.2, ...,oo} for all r £ {1,2,..., P(q)} and p £ {1, 2,. .., nT}. From properties 9.3 and 9-4 and as \\n\\ £ {2,3,...,00} then ||oj| £ {2,3,...,oo}. vis ||ojj = j|mjj from Equations (9-60) and (9.61) then or-p £ {!, 2,..., j[rn.[j - 1} for ail r £ {1, 2,. .., P{q)} and p £ {1,2, ...,nr}. □
Property 9.6
jlnj’j {2,3,..., |jmj!) (9.63)
for any q £ {1,2,...,£?}.
Proof 9.6 From Equation (9.55) it is directly seen that 'n: £ {2,3...., со} for any q £ {1,2.. ... ‘j \. Similarly, from properties 9.3 and 9-4 it follows that |!n!| .r_ {2.3,. ... j|mjj} for any q £ {1, 2,.... 0}. □
Using these properties, the second and higher order controlling variables i||m||(IIV,ll) can be determined as
234
0. Multi-port Volterra transfer functions
Wl NPU) °1,I °1,"] °P(4). 1 °FM'nP(q)
E E E E ...... E • • E
щ —0 npj?)-0 Oil—1 01 ,nj =1 0p(?)lr‘pi ,.1 ~1
E-E...................... E E
9-..1 Яр(ч ),1 Я
,P(4) nr . P(q) nr
A.coON!) л, ЕЕ Чг.р) П П ;WlP(!!?-,P!!)
V=1 р=1 ' г=1 р=1
(•07’^1,1, • • •,;................;ФтяР(я).v ■■■,ФтяР(5),-.я(1))
Л?) пг
П П (х1,,г)огМтЯг,р)
г = 1 р = 1
= Е л?.^(|1^11) Cxr )|]m.!| (ll'V’ 11J (9-64)
Г = 1
where 1 and Ai(-) are defined in Equations (9.57) and (9.34) respectively, and
O = [Ol.l, ■ • • I °l,ni ................°P(g),l----:°P(,),npif)]T 6 <2+ ‘ ' (9.65)
NT = |jm!| - («i +------------1- n,—i) (9.66)
Or,p =
llmil — (nr + • ■ ■ + nP{q)) + ^ — (°1 Д + " ■ ■ + °T-l.nr_I ) if P= 1
IMI - (nr + •—I- n p(q)) + p — (oi,i + ■■• + if P {2,3,...,oo}
(9.67)
for т 6 {1, 2,..., P(q)} and p 6 {1,2,..., nT}. Equation (9.66) is derived from
property 9.6, and Equation (9.67) is derived from properties 9.4 and 9.5.
Now Ж||т||(]|0||) is determined by using all the sets of equations for ? f {L 2,. ..,Q) in Equation (9.64) from which
,/!!./,in — 4-1/IU.Ih «... ..nUMIt
where
u||to||(I|V’||) = [ (u l) | ] m, 11 (11 "0 i 1)-,(“Q)||m||(H1/''ll)] 6 c'* (9.69)
lias elements
9.3. Theory
W||'m||(iWI)
,V, -vp (?) Oi.i
- E E E ■ • E ..................... E E
ni=0 T\pj4'—0 L>11’=1 Ol,ni— 1 ,3P<!7).!=1 °P!q),nr,(
Р{я) P(q)
E E................E • E П П ........................................9r.nr)
^L,l 'iPU)A r=l r=l
/^(7) П.- X Р[ч) Tlr
A.oodHI) лЛ]Г]Г qTA П П Л'^,р(Ii4r.p|!)
Г = 1 1 7 r= 1 p= 1
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