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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 Previous << 1 .. 71 72 73 74 75 76 < 77 > 78 79 80 81 82 83 .. 85 >> Next Since it is most useful to have some kind of Fourier series representation in the present work, none of the suggestions made in  are useful.
The Fourier transform of the random noise signal A(t) in Equation (A. 12) is given by
OO
A (/) = ┬ú Af/tf) *(/ - /*) (A. 15)
p~ ÔÇö OO
where S(- Ôûá Ôûá)' is the Dirac ð╣-function.3 It is seen from Equation (A. 15) that the
frequency resolution in the spectrum of A(/) is given by ┬ú. This frequency resolu-
tion can be made arbitrarily small by choosing ð│ sufficiently large. To prove that Equation (A.12) is fulfilled in the time interval -r < t < Ðé it suffices to show that
/IðÁ0- 2\
\|ðø(4) ~ exPbÔÇÖ2"P^] ) - 0 (A.16)
* ÔÇö OO 1
The autocorrelation function for A(<) is expressed as a Fourier series as
R(h,h) = (Mh)y(h)) (A-17)
OO
= ┬ú R(h,nO exp[j2Tn┬ú<3] (A.18)
n= ÔÇö OO
R(t\,n┬ú) = ┬ú f R(t\: t2) exp[ÔÇödt2 (A.19)
ÔÇÿTwo Fourier series coefficients Ai(piif) and are said to be orthogonal i?
ÔÇö ðú ^or mteg-ilrt pi Aiul p2 *u>/H ÔÇö P2-
3The Dime ð function is a generalized function denned bv
J j(/l) S( f ÔÇö a) da - ji f )
This as a consequence implies that
J 5(a) da = 1 ðø 6(ca) = -<5(a) ðø g(a) S(f ÔÇö a) = g(f) 6(f - a)
An extensive analysis of generalized functions has been made by Lighthill .
ðÉ.2. Fourier series representation
261
Using the fact that |o|2 = a a" in Equation (A. 16) gives four terms. The first term can be determined as
(|A(i)|2) = R(t,t) (A.20)
The second and third terms express the correlation between the time domain signal and a Fourier series coefficient as
!\{t) Y ðø*(ÐÇ┬ú) exp[-j2~p(;t]')
p = ÔÇö50
= J A'(ti) exp[j2-p{,t{l dty \ exp[-j2-pff ]
ÔÇóX)
= Y R'itÔÇÖPi) ðÁÐàÐÇ[-/2Ðé^ð│] = R'(t,t) (A.21)
p=-CO
and
. CO \
(A*(f) YL A(?f) ðÁÐàÐÇ^ÐéÐéÐÇ^ðø = R(t,i) (A.22)
' pÔÇöÔÇöCO
The fourth and last term expresses the correlation between two Fourier series coefficients as
CO CO
Y Y1 (A(piO ðøÔÇÿ(ðá2<)> esp[j2!rpiC(] ðÁÐàÐÇ{-Ðâ2ÐÅð│ÐÇð│ðí0
Pi = ÔÇ£OO p2 = -00
OO -- CO
I] W 13 <A(ti) ðø'(ð╣ð×) exp[-j2ffpj^] exp[ÔÇödt,
?1 = -eo J~Tp2~-co
x exp!j2;rp^{]
ÔûáX)
= Y^ R(t-PiO expljtj ÔÇö n(ÔÇÿ-*L} (A.23)
Pl = -^=
Thus insertion of Equations (A.20) - (A.23) into liquation (A.16) using R(t,t) = R"(t.t) since A(<) is a real valued signal completes the proof.
It can be shown  that if the autocorrelation function R(ti, t/) is periodic with T'eriod Ðé then
R(h*h) = - ") 1 A.24)
'Thus the coefficients of the Fourier (ÔÇÿxn^njion ar^ orrhn^onal which means
that
.. f {! AC d i 112) for pi = pi
(A(pis) A (p2s)) = ^ ; '-4'' , . 1 A.25)
'Ôûá) utueiWiii-
262
A. Mathematical concepts
However, in general this property is not valid since the autocorrelation function R(ti,t2) is generally not periodic.
Example A.l Consider a white noise random variable A(f) extending over all time
ÔÇöoo < t < oo with the autocorrelation function
R(h,t2) = cS(t2-t\) (ðø.26)
where Ðü is a positive real constant. The correlation between the frequency domain representation of A(t) at two arbitrary frequencies f\ and /2 can be determined as
I ðô ðíð× ð│CO
(A(/i) ðø*(/2)) = (/ X(ti) exp[-j'2irfiti]dti \ÔÇÿ(t2) exp[j2x f2t2] dt:
WÔÇöCO J ÔÇö CO
I S(U - ti) exp - f2t2)\ dti dt2
j ÔÇö CO J ÔÇö>0
= ┬½(ðø - /2) (A.27)
This means that the correlation of white noise at two different frequencies, fi Ðä f2, is zero and that there is correlation not equal to zero only when the two frequencies are identical, fi = f2.
Example A.2 Consider a white noise random variable A(/) in a finite time interval
ÔÇö Ðé < t < t where the correlation between two Fourier coefficients of arbitrary frequencies f\ and /> is to be determined. The autocorrelation function for the time domain random variable A(t) is
Ðëð©ð╝ = {'ðø-''1 ÔûáÔÇØ -r<l<T (A.28,
1 0 otherwise
The correlation between the two Fourier series coefficients is given by
(ðø(ðø)ðø*(ðø)> = jjKn) \'(t2)>
X fxli] exp[j2n/2/9] dh dh
Ðü ðô
= J exp[-j2ff(/i - /2)f2J dt2
_ Ðü sm[7T(h - f2)r]
2 Ðé *(h-h)r 'ÔûáÔûáÔûá '
The sin(a)/a function in Equation (A.29) has the following properties as Ðé approaches infinity:
ðô , r__ i
ð│ 'oo "(/1 ÔÇö }2)Ðé [ 0 otherwise
As the factor Ðü in Equation (A.29) generally increases with r, the ensemble average of the two Fourier series coefficients is zero for /1 Ðä f2 and some non-zero quantity for
/1 = /2.
A.3. Signal energy and average power
263
A.3 Signal energy and average power
The total energy Stot of a random noise signal X(t) averaged over the ensemble of realizations is given by
rco \
I 1-4012 dl ) (A.3i)
J ~ <:o /
= (f f MJi) exp[jÔÇÿ2-fit] dfx
\J ÔÇö oo J ÔÇö CO
X J ðø*(/2) exp[-j'2-/2┬½] df2 (A.32)
= ðô (|A(/);2) df (A.33)
J ÔÇö OO
Equation (A.33) is based on the assumption that A(t) is square integrable and that expL;2ir(/i ÔÇö /j )t] dt = 6(J\ - f2). In Equation (A.31) the quantity t',0, can be interpreted as the total energy in joules if the right-hand side of Equation (A.31) is divided by 1 where \(t) is a voltage or multiplied by 1 S where A(f) is a current. The case for A(/) in Equation (A.33) is similar. Note from Equation (A.31) that if A(t) is not square integrable then ┬útot = oo. Usually the total energy of a random Previous << 1 .. 71 72 73 74 75 76 < 77 > 78 79 80 81 82 83 .. 85 >> Next 