22.08.2015   Frequency Response Testing. in Nuclear Reactors

Power Spectra

In analogy with the power dissipated in a resistor in an electric circuit (P = I2R), the generalized power in any kind of signal is given by

image071 Подпись: (2.6.1)

P = x2(t)

The total energy contained in this signal is

E = f 1 x2{t)dt = PT J — 77 2

(2.6.2)

The energy may be related to the Fourier coefficients for a periodic signal of period T by making the substitution

00

x{t) = Z Ck export)

к — — oo

(2.6.3)

to obtain

oo /*77 2

E = Z exP(MO dt

к =-00 j-ТІ2

(2.6.4)

The integrand is identified as T times the Fourier coefficient complex conjugate of Ck). That is,

С-к (the

00

E = 7^ Z СкС-к

(2.6.5)

The average power is

oo

p= z скС-к

(2.6.6)

к = — oo

The summations give the energy or average power in all harmonics. The

energy and average power

in the /cth harmonic are

given by

E0 = TCqCq

, Ek = 2 TCkC-k

for

к Ф 0

(2.6.7)

and

Po = CqCq,

Pk = 2 CkC-k

for

к Ф 0

(2.6.8)

The factor 2 appears for к Ф 0 because the summation extends from — со to oo j and each term appears twice in the series. Similar results using the trigo-

nometric form of the Fourier coefficient are

E0 = T(A02 + B02)/4,

Ek = TU,2 + B,2)/2

for

к Ф 0

(2.6.9)

and

P0 = (V + B02)/4,

Pk = (Ak2 + B2)/ 2

for

к Ф 0

(2.6.10)

The energy or average power as a function of harmonic number (or equiva­lently, frequency) is called the energy spectrum or power spectrum of the

signal. The relation

/* T/2 00

xt)dt=T £ C, C_, (2.6.11)

4-772 k= — 00

is called Parseval’s relation for periodic signals.

Cross spectra are defined for pairs of signals. Two related periodic signals with the same period may be used to give

№12)0 = TC0D0, (E2)k ~ 2TCkD_k for к # 0 (2.6.12)

and

(P12)о = C0D0, (P12)k = 2CkD„k for к Ф 0 (2.6.13)

where (£12)* is the component of the cross-energy spectrum at harmonic k, (ЛА the component of the cross-power spectrum at harmonic k, Ck the Fourier coefficient of signal 1 at harmonic k, and Dk the Fourier coefficient of signal 2 at harmonic k. The power spectrum of a single signal is always real. The cross-power spectrum of two different signals may be complex.

Similar results may be obtained for nonperiodic signals. Let y(t) be a general, nonperiodic signal. Then

/* 00

y(t) = (1/2я) Ytiabe^dto (2.6.14)

v — 00

Подпись: /» 00 /» c image074

and

Подпись: /* 00 Y(jw)dw
image33

Interchanging the order of integration gives

or

E = (1/2я) Y(jco) Y( —jco) dco (2.6.16)

** — GO

This expression gives the total energy in the signal. Since this is obtained by integration over frequency, (jln)Y{j(a)Y(—jw) is interpreted as an energy density (energy per unit frequency). Thus, in a nonperiodic signal, there is a continuous energy density spectrum. The relation

Ґ y2(t)dt = (1/2тг) f°° Y(jco)Y(-jco)dco (2.6.17)

J — 00 v — 00

is called Parseval’s relation for nonperiodic signals.