As implied by its name and defined in Table 6.1, the impulse response, h(t), is the system output, y(t), when its input, x(t), is a very narrow pulse (i. e., a unit pulse of time duration much less than the smallest important time constant of the system). The impulse response is also the Green’s function or weighting function. It is appropriate to discuss the impulse-response function in connection with transfer functions since, as shown in Table 6.1, it is the Fourier transform of the transfer function. To date the impulse-response function has not enjoyed the popularity of the transfer function as an analytical tool. Recently, however, Dorf2 pointed out that since digital computers greatly facilitate time-domain analyses of systems the impulse-response function should become more popular

Table 6.2 gives an example of the impulse response of a system with a single time constant. Evidently excitation by an input pulse, x(t), having a width much less than the system time constant stretches this pulse to a width or duration of the order of the system time constant. A physical interpretation of this is that the output, y(t), “remembers” an input pulse and shows its effect (up to about the time constant) after the input pulse.

FREQUENCY (со), radians/sec

Fig. 6.1—Magnitude and phase of zero-power reactor transfer function vs. frequency for 2 3 3U, 2 3 5 U. and 2 39 Pu. Curves are shown for various values of l, the neutron lifetime,3

Table 6 2—Illustration of Dynamic Functions for a System Having a Single Time Constant, tc, One Input, and One Output Description Frequency-domain expression Time-domain expression Differential equation of the system and its Laplace transform x(s) = (1 + itcco) y(s) x(t) = y(t) + TC Transfer function Impulse response 4 y(u>) 1 G(0J) x(w) 1 + ітсш h(t) = 0 (r < 0) e~t/Tc , h(t) =*-z—<-T> 0) Tc Spectral density of output for an arbitrary input p f Px(u>) РУ(Ш) " 1 + Tgto2 Cross spectral density of input and output (for an arbitrary input) px(m) PXy(to) = -2————- y 1 + ircOJ Autocorrelation function of output for a constant spectral-density (uncorrelated) input C(r) = e-lTl/Tc Cross-correlation function of output with a constant spectral-density input Сху(т) = 0 (r < 0) Cxy(r) = e-T/Tc (r > 0)

Finally, another insight into the nature of the impulse response comes from using it as a weighting function in relating arbitrary input and output time functions

WATTMETER о і о v (

BAND-PASS FILTER OF WIDTH df

ION CHAMBER

R = 1 .

y(t) = /0 h(t’) x(t — t’) dt’

(6 3)

3. A continuum of frequency components make up the randomly fluctuating or nonperiodic part of the signal.

The spectral-density concept applies primarily to the last. Spectral densities associated with the first two contributions are additive with the total of the third

Ptotal = (i)2 + (y + Y * ) + /-Т P<f> df <6 5)

If the signal is a current through a resistor, the three terms are, respectively, the d-c power, the a-c power of discrete frequencies, and the a-c power of random noise The usefulness of the spectral-density concept is m char­acterizing the last term, and hence P(f) is sometimes called a random-noise spectrum.

Fundamental relations associated with spectral density are given in Table 6.3. As also indicated in Table 6.1, the

Fourier integral relations between x(t) and its	x(t) = f°° X(f) exp(icot) df J-OO
transform X(f)	X(f) = *im f+T/2 x(t) exp(-icot) dt T -» °° J-T/2
Spectral density from Fourier amplitude	lim IX(f)U ‘ * T-»°° T
Total spectral power =	pt = Г P(f> df
variance = square of standard deviation = autocorrelation	= t— 7/™[x(t)]2 dt = x2
function at zero lag	II C) M II n ©

Table 6.3—Formulas Associated with Spectral-Density Analysis of a Random Signal, x(t), Having a Zero Mean Value

Description

Formula

spectral density may be obtained from Fourier amplitudes or, alternatively, by integration of an autocorrelation function. Ideally the signal duration, T of Table 6.3, would be infinite. In practice, the finite duration of the signal available for spectral analysis is an important experimental limitation (see Sec 6-7).

5- 1.5 Cross Spectral Density

Just as the spectral-density function, P(f), is used to display the relative importance of various frequency com­ponents in a single random signal, the cross spectral density, PXy(f), is used to show the joint importance of these frequency components in two related random signals, x(t) and y(t). Its definition in terms of Fourier amplitudes and in relation to the cross-correlation function is given in Table 6.1. Evidently the cross spectral density is a more general concept, which reduces to the simple spectral density, P(f), for the case x = y.

Figure 6.1 shows conceptually how one might measure the cross spectral density using a wattmeter and filters with switchable phased outputs. With the switches in the positions indicated, a quadrature spectral density is indi­cated by the time-average value of the current from one chamber and the 90° phase-shifted voltage from another, if the filters are switched in phase, the meter shows the cospectral density. In both instances the extent to which the two signals are similar in a frequency band df is being measured.

Unlike the spectral density, P(f), but like the transfer function, the cross spectral density, Pxy(f), requires two numbers at each frequency for its specification. These may be the “со” and “quadrature” spectral values or the amplitude and phase, with the relations

IPXyP = (cross-spectrum amplitude)2

= (cospectrum amplitude)2 + (quadrature spectrum amplitude)2

= (Coxy)2 + (Quxy)2 (6.6)

в = phase angle

quadrature spectrum amplitude
cospectrum amplitude

Q. uxy

Coxy

Because of the similarities in the descriptions of the transfer function and the cross spectrum, it is not surprising to find that these are related, as shown in Fig 6 3. In spite ■Ф

n 1 (t) n 2(t)

Transfer-function relation Y(s) = G(s) [ X(s) + N1 (s)] + N2(s) where (s) and N2(s) are Fourier transforms of n1 and n2

Spectral-density relation P.. = IGI2 [Pv + P_ ] + Prt

у * M2

Cross-spectral-density relation PXy = GPX

Fig. 6.3—Input—output relations of Fourier transforms and spectra in a system having uncorrelated additive noise signals, n, (t) and n2 (t), at its input and output

of additional signals (such as unwanted noise) at the input and output, a simple relation exists the transfer function, G, times the input spectral density, Px, is the cross spectral density, Pxy On the other hand, only when the unwanted signals can be neglected are the input and output spectral densities related by the square of the transfer function.

A quantity called coherence, cxy(f), has been defined to quantitatively assess the extent to which the presence of

Ip I2 ,r xy1 PxPy

Its square is the ratio of IgI2 (the numerator) to the spectral-density ratio (the denominator), the latter being IgI2 plus effects from extraneous signals nj and n2, according to Fig. 6.3. In frequency ranges over which cxy(f) is 1 or nearly so, the input and output can be related with negligible effects from other uncorrelated signals. Conversely, the input and output can be consideied virtually uncorrelated in frequency ranges where N[(f) and/or N2(f) are large enough to cause cxy(f) to be near zero Evidently the ease of making transfer-function mea­surements will be in proportion to how near cxy(f) is to 1.

6- 1.6 Autocorrelation

Table 6.1 shows that the function m the time domain that corresponds to the spectral density is the auto­correlation function C(t). The definition indicates that it is a measure of the amount of correlation existing at a time interval tin a signal x(t). It has its largest values at т = 0 and at other time intervals during which the signal has essentially the same value, it is smallest during time intervals over which signal values are uncorrelated. In the example shown in Table 6 2, the autocorrelation function decreases from 1 to e_1 in a time tc and approaches zero when t is large. Thus tc may be called a correlation time within which signal values are similar and beyond which they are rather unrelated.

Table 6.1 shows that the spectral density, P(f), can be obtained from either the square of the Fourier transform of x(t) or from the transform of its autocorrelation function. Conversely, the autocorrelation function can be obtained by transforming P(f). However, x(t), when random, cannot be reconstructed from either P(f) or C(r).

6- 1.7 Cross Correlation

The concept of cross correlation is more general than that of autocorrelation since the latter is a special case of the former in which the two signals are the same. The cross-correlation function defined in Table 6.1 is an applica­tion to continuous time functions of the digital concept of a correlation coefficient of statisticians if x, and y, are two time series of variable values spaced in time (x, being at the same time as yi+(T/At)) in which the degree of correlation is sought, then

N

is a measure of this. However, it is customary to define a normalized correlation coefficient in terms of fluctuations from means.

where ax and ay are the standard deviations 0^x and This is +1 or —1 for perfect correlation or anticorrelation, respectively, and is 0 if there is no correlation. The integral expression for cross correlation in Table 6.1 is evidently digitally evaluated in Eq. 6.9.

In the example in Table 6.2, Cxy is zero for t<0 because the output cannot “know” ahead of time what the perfectly random input, x(t), will be. A significant input — output correlation, however, does exist for values of r up to the order of tc, the correlation time of the system. In other more complex systems, the maximum value of Cxy might occur at some time other than zero, in which case a time-lag effect between x and у will have been identified.

Table 6.1 shows the frequency-domain function corre­sponding to Cxy(t) to be the cross spectral density, Pxy(f) These are Founer-transform pairs, and, if one is known, the other can be found from the relations shown.

5- 2 REACTOR APPLICATIONS

6- 2.1 Neutron Kinetics

For a study of the time behavior of reactors, the equations giving the time dependence of the neutron density, N + n (mean value plus deviations therefrom), and the )th group of delayed-neutron precursors, Cj + Cj, are

/-3r=[kd — (3) — 1] (N + n) + £ /Xj(Cj + Cj) + IS (611)

dc. — л r, „ ) . 0Jk<N + n>

_ Aj^j + Cj)+ і

where the sum of the delayed-neutron fractions, |3j, is the total fraction, /3, Xj is the decay constant of a precursor, S is a source, / is the prompt-neutron lifetime, and к is the effective multiplication constant that, when not unity, represents the departure of the reactor from exact criticality,

P=l-f (6.13)

к

being the excess reactivity.

The solution to these equations, under conditions of all variables undergoing small oscillations about their mean values, is the zero-power transfer function, G0, defined as

lG0l = [(amplitude of power oscillation)/(average power)] /(amplitude of reactivity oscillation) (6 14)

and having the phase

Phase angle = 360° X (fraction of a cycle that the power lags behind the reactivity oscillation) (6 15)

The zero-power transfer function can also be regarded as the quotient of the Fourier transforms of the power and reactivity divided by the average power.

Table 6.4 gives explicit formulas for this transfer function in terms of reactor constants and the frequency.

Table 6.4—Forms of the Complex Amplitude of the
Zero-Power-Reactor Transfer Function G0(u>)

Formula for complex Conditions amplitude of G0 (со)

6

1 — no 2] + loo)

No approximations CJ > 2Л Г 6 1 — к + ico I/ + к J] 1 1=1 [1 — k(l — 0) + icu/1 1 One delay group [і к + .ш(/+т+іш)] One delay group and Г. , k(3 1 1 l1-k + ,“x+,<J I I ‘|m V 3

Also, G has been tabulated in detail in Ref. 3. In essentially all but subcntical reactor applications, к may be set equal to 1 to further simplify the approximations there At mid­frequencies, where 2X < со < 0 5(3//, a very simple result, G = 1 //3, exists At these frequencies the physical inter­pretation is

Percent power oscillation about its mean

= — X (percentage reactivity amplitude) = reactivity amplitude in cents

where (3 is typically 0.007.