# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Since it is most useful to have some kind of Fourier series representation in the present work, none of the suggestions made in [1] are useful.

The Fourier transform of the random noise signal A(t) in Equation (A. 12) is given by

OO

A (/) = £ Af/tf) *(/ - /*) (A. 15)

p~ — OO

where S(- ■ ■)' is the Dirac й-function.3 It is seen from Equation (A. 15) that the

frequency resolution in the spectrum of A(/) is given by £. This frequency resolu-

tion can be made arbitrarily small by choosing г sufficiently large. To prove that Equation (A.12) is fulfilled in the time interval -r < t < т it suffices to show that

/Iе0- 2\

\|Л(4) ~ exPb’2"P^] ) - 0 (A.16)

* — OO 1

The autocorrelation function for A(<) is expressed as a Fourier series as

R(h,h) = (Mh)y(h)) (A-17)

OO

= £ R(h,nO exp[j2Tn£<3] (A.18)

n= — OO

R(t\,n£) = £ f R(t\: t2) exp[—dt2 (A.19)

‘Two Fourier series coefficients Ai(piif) and are said to be orthogonal i?

— У ^or mteg-ilrt pi Aiul p2 *u>/H — P2-

3The Dime function is a generalized function denned bv

J j(/l) S( f — a) da - ji f )

This as a consequence implies that

J 5(a) da = 1 Л 6(ca) = -<5(a) Л g(a) S(f — a) = g(f) 6(f - a)

An extensive analysis of generalized functions has been made by Lighthill [2].

А.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 Л*(р£) exp[-j2~p(;t]')

p = —50

= J A'(ti) exp[j2-p{,t{l dty \ exp[-j2-pff ]

•X)

= Y R'it’Pi) ехр[-/2т^г] = R'(t,t) (A.21)

p=-CO

and

. CO \

(A*(f) YL A(?f) ехр^ттр^Л = R(t,i) (A.22)

' p——CO

The fourth and last term expresses the correlation between two Fourier series coefficients as

CO CO

Y Y1 (A(piO Л‘(Р2<)> esp[j2!rpiC(] ехр{-у2ягргС0

Pi = “OO p2 = -00

OO -- CO

I] W 13 <A(ti) Л'(йО) exp[-j2ffpj^] exp[—dt,

?1 = -eo J~Tp2~-co

x exp!j2;rp^{]

■X)

= Y^ R(t-PiO expljtj — n(‘-*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 [3] that if the autocorrelation function R(ti, t/) is periodic with T'eriod т then

R(h*h) = - ") 1 A.24)

'Thus the coefficients of the Fourier (‘xn^njion ar^ orrhn^onal which means

that

.. f {! AC d i 112) for pi = pi

(A(pis) A (p2s)) = ^ ; '-4'' , . 1 A.25)

'■) 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

—oo < t < oo with the autocorrelation function

R(h,t2) = cS(t2-t\) (Л.26)

where с 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 Г СО гCO

(A(/i) Л*(/2)) = (/ X(ti) exp[-j'2irfiti]dti \‘(t2) exp[j2x f2t2] dt:

W—CO J — CO

I S(U - ti) exp - f2t2)\ dti dt2

j — CO J —>0

= «(Л - /2) (A.27)

This means that the correlation of white noise at two different frequencies, fi ф 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

— т < 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

щим = {'Л-''1 ■” -r<l<T (A.28,

1 0 otherwise

The correlation between the two Fourier series coefficients is given by

(Л(Л)Л*(Л)> = jjKn) \'(t2)>

X fxli] exp[j2n/2/9] dh dh

с Г

= J exp[-j2ff(/i - /2)f2J dt2

_ с sm[7T(h - f2)r]

2 т *(h-h)r '■■■ '

The sin(a)/a function in Equation (A.29) has the following properties as т approaches infinity:

Г , r__ i

г 'oo "(/1 — }2)т [ 0 otherwise

As the factor с in Equation (A.29) generally increases with r, the ensemble average of the two Fourier series coefficients is zero for /1 ф 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[j‘2-fit] dfx

\J — oo J — CO

X J Л*(/2) exp[-j'2-/2«] df2 (A.32)

= Г (|A(/);2) df (A.33)

J — OO

Equation (A.33) is based on the assumption that A(t) is square integrable and that expL;2ir(/i — /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 £tot = oo. Usually the total energy of a random

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