An electronic analog computer for spectrum and gross spectrum analysis of random signals

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AICH1EF

AN ELECTRONIC ANALOG COMPUTER FOR SPECTRUM A?9D

CROSS-SPECTRUM ANALYSIS OF RANDOM SIGNALSJ) by

Wilbur Marks(2) and Paul trausser

ABSTRACT

The Taylor Model Basin spectrum analyzer is described. The extension to cross-spectrum analysis by use of matched filters is

dis-cussed. _{Some preliminary experiments, to check the method, provide} a general verification but show the need for further experimentation on

the matched filters and phase-shifting components of the system.

INTRODUCTION

The motions of a ship in a seaway, waves in a storm, and _many geophysical phenomena are classified as random _processes. _{Time hstorìes} of such processes car. be converted to _{energy spectra.} _{This permits}

statistical inferences to he made regarding the physical behavior of the system under investigation for ail time (and space) where _stationary conditions apply. If a particular event is being studied without _regard to cause or effect, its auto-spectrum will describe that _{event through} a frequency decomposition of the time history that represents it. _If, however, it is desirable to relate oria event to another, _{say the pitch} and heave of a ship, then it is necessary to calculate the phase relation-ship between the same frequency components in the two records. This Is accomplished by obtaining the cross-spectru _{cf the two records.} _The

1.ab. y.

Schesboúwknck

Tecnsch2 Lkcioa

D{L

(l)The

work reported here was carried out at the David _{Taylor Model} Basin, Washington, D.C.

(2)Head _{Ship Hydrodynamics Division,}

_Davidson

LaboraLory, Stevens Institute of Technology, Hoboken, N.J.

(3's ¼

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cross-spectrum is composed of two parts: the co-spectrum which defines the in-phase energy of the two systems and the cuad-spectrüm which defines

the 9Q0 _{out-of-phase energy in the two systems.} _{Resolution of these two}

components will yield the desired phase relationships. _{In the case of} the directional spectrum of ocean waves, one method of measurement

requires calculation of the phase differences between the wave components in the records simultaneously obtained by pairs of wave probes suitably spaced.

This report describes the Taylor Model Basin cross-spectrum analyzer which is presently capable of producing, from _{an input of two} simultaneous signals, t.he spectrum of each signal as well as the co- and quad-spectra of the pair of signals, each realization displayed _{on an} x-y recorder at one and the same time. This report will also describe some initial analysis experiments aimed at verifying the usefulness of the system as a reliable cross-spectrum analyzer.

MATCHED-FILTERS METHOD OF CROSS-SPECTRUM ANALYSIS

Consider two random functions, f(t) and g(t), measured simul-tanaously for a period T. _{The cross-spectral density is defined as}

I

c_f,g(u) + q (u)

-

Rf (-r)e" dT

f,g 2ît ,g

where Cf

_,g

is the cosine component (co-spectral density) and _qf is the sine component (quad-spectral density) of the _{cross-spectral density, and} tha cross-covariance function is defined by

R

(T)

hm

1 ( _{f(t)g(t + T)dt}

f,g - T-'

T

If the signals f(t) and g(t) are passed through _{linear filters, the} respective outputs, ft(t) and g*(t) _{are given by}

(i)

(3)

f _{( t)}g*(t)

A1A2

-ami

Ç R

(n)e

cos indn

4a

_L

-ajr - sin

f

R1g n)e

_{sn ndn]}

-ajnJ

If a is taken sufficiently small, the _{damping factor e}

has little effect on f*(t)g*(t); the result _is

References are listed at the end of the text.

3 f*(t) -

jf(x) K1(t

- x)dx (3) g*(t) -

5g(x)

K2(t - x)dx

where and _{are the impulse responses of the filters.}

The average product of the outputs of the filters is

f(t)g*(t)

jRfg(n)'(n)dn

(4)

where

jK()

+ (5)

as proven in Reference _and = t - x, and

îj -

X1 - X2.

If the impulse responses of the filters and

5)

are properly selected, fr(r) can be made proportional _{to either sin}

or cos q with the

* *

result that the average products of _{the outputs of the filters (f (t)g (t))} yields c or q . For the second order-linear filters used in

the f,g

Taylor Model Basin system, it is shown (Reference _{1) that}

-a t K1 - A1e

_cos(t

+ -at K2 A2e

cos(t

+

Substitution of equations (5)and

(6) in equation (4) yields

(o)

(4)

and (0V R

()e

cos c

()

) f,g

fg

-a I p (r)e sin q

()

10V f,g f,g

AA r

* * f (t)g (t) lcos c

() sinØqg(

L f,g

If the phase angle (Ø) is zero, the resulting average product of the out-puts of the filters is the co-spectrum (Cf _). If the phase angle is

o ,g

90 , the result is the quad-spectrum (qf _). Consequently, matched filters combined with a multiplier and 900 phase-shifter can produce the desired cross-spectral densities of two simultaneous random signals.

The phase angle between the same components is then given by

c(w) - tan

(a))

g

The same ends can be accomplished by a single filter, but corn-putaticn of the quad-spectrum requires differentiation of one of the signals, before multiplication, and it is usually difficult to obtain accurate differentiation over the entire frequency range of interest. The matched filters method has convenience of operation, involves minimum design and construction of special components, and is highly accurate, in principle.

GENERAL DESCRIPTION OF CROSS-SPECTRUM ANALYZER

The Taylor Model Basin analog spectrum analyzer was first

assembled in 1959 at

which

_{time its purpose was to convert time histories} of ship motions and waves into energy spectra. While cross-spectrum analysis was not considered at that time, it was appreciated that such computations would eventually be required (References 2 and 3). Accord-ingly, the assembled system comprised two analyzers driven by _{a common}

4

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oscillator as well as two square-low integrators and two x-y plotters. The net result was the simultaneous generation of two spectra, one for each of two input signals. _{The unique feature of the original system} was the installation of specal1y m3tched filters such that the same out-put resulted from passing the saine signal through both filters. _{This is} critical to cross-spectrum analysis, because the product of the outputs of the matched filters is the co-spectrum or in-phase component of the two signals while the product of the output of _{one filter and the output} of the other filter, phase-shifted 900, _{is the quad-spectrum or 90°} out-of-phase component of the two signals, as has been shown.

To complete the cross-spectrum analysis system, a multiplier was added, 90° phase-shifting circuitry was installed, and two-additional x-y recorders were included.

Figure 1 is a block diagram describing the basic operation of the analyzer system and its options. _{Each of two FM signals,} represent-ing the random signals to be analyzed, are fed simultaneously _{to analyzers} A and B respectively. _{The analyzers contain the filters which separate}

the frequency components in the random signals. The central oscillator selects the center frequency to be analyzed and the filters determine the frequency band to be passed. There is a choice of four filters, 2, 5, 10, and 20 cps. _{Reference 3 explains in detail the operation of the} oscillator, amplitude modulation, behavior of filters, and analysis technique; these matters will not be discussed. _{Suffice to say, the} selected filter should be narrow enough to give adequate _{resolution of} the spectral density being measured. _{For cross-spectral analysis, even} greater resolution is required (Reference 4).

The outputs of the analyzers are voltages proportional to _the amplitudes of the frequency being examined. _{These outputs are fed cithar} to the spectral density integrator or total _{energy integrator and then} to the associated x-y plotters. _{These same outputs are also fed to the} co-spectrum multiplier and the resulting outputs _{are displayed cri the} third x-y recorder. _{Lastly, the output of analyzer A is fed to the 90°} phase-shifter and

thence

_{to the quad-spectrum multiplier where it} _is combined with the output of analyzer B; the resulting output is displayed

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on the rernaining x-y plotter. The end product of analyzing a pair of random signals is four displays on x-y plotters comprising

Spectral density (or cumulative density) of signal 1, Spectral density (or cumulative density) of sional 2, Co-spectrum of signal 1 and signal 2, and

Quad-spectrum of signal i and signal 2.

The ccmponents comprising the cross-spectrum _{analyzer are all} commercially available (References 5-7) with the exception of the constant voltage phase shifter which was develcped at the Taylor _{Model Basin. A} schematic drawing of the 900 phase-shifter

_iS

_{shown in Figure 2.} _The important feature of this component is the application _{of a variable} resistor ganged to the tuning condenser of the _{oscillator that permits}

accurate 900 _{phase shiftinQ for all frequency components.}

INITIAL ANALYSIS EXPERIMENTS

It now remains to be established that supplementina _{the original} system with a multiplier and phase-shifter, in conjunction _{with the}

matched filters, will yield reliable estimates of the co- and quad-spectra. To this end, several experiments were devised using _{existing FM tape}

records of a full-scale seakeeping trial.

Figure 3 shows the output of the cross-spectrum _{analyzer for} simultaneous inputs of wave height and roll angle. _{The upper figure shows} a composite of the wave height and roll auto-spectra while _{the lower}

figure shows a composite of the _{co- and quad-spectra of the randcm} signals.

To verify the computed co-spectrum, a simple summing _circuit was installed in the system between the readout head arid wave analyzer A. Therefore, the records f(t) and g(t.) _{are added together and fed through} one analyzer system. _{The result graphed on the x-y plotter is the} spectrum of the signai. f(t) + g(t). _{The components of this combined} spectrum S can be evaluated by considering the autocorrelaton function of the sum of the two records

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f(t) + g(t)][f(t + -r) + g(t + T)J f(t) f(t + -t) + g(t) g(t

't)

+ f(t) (t + -t) + a(t) «t -r) (il)

If the Fourier cosine transform of equation lì is taken

termwise, the

result is

f+g

5f + S

2c

g

Cg

where the co-spectrum is defined in equations 1 and 2, The spectrum of the sum of the wave and t-ail records appears in the top part of

figure 4.

If the spectra of roll (s ) arid of wave S are subtracted frcm S

f g

f+g

and the remainder halved, the result is an estimate of the co-spectrum using a single filter. The co-spectra obtained by matched filters and by a single filter are plotted in the bottom graph of Figure 4 where

it

is seen that the two curves are in fair agreement. The single-filter method is by no means a standard of comparison and the fair agreement

i

quite satisfactory for a first attempt at verification. Please note that the single-filter method used here is not the same as the usual single-filter method which requires consecutively passing the sum and difference of the two signals through the filter and then taking the difference between the mean squared outputs. The result is 4cf,g

directly. A little computational labor is added by the method applied here, but additional temporary electronic circuitry is bypassed. The

single-filter method is not used for evaluating the quad-spectrum estimate because of the differentiation problem mentioned earlier. Instead two signals with a known quad-spectrum are treated.

Figures SA , and 5E show the roll amplitude and roll velocity

spectra corresponding to the simultaneous recording of roll amplitude and roll velocity. Figure 50 shows the co- and quad-spectra of the two

signals. It was expected that the co-spectrum would contain none of the spectral energy and the quad-spectrum all but there is some energy in the co-spectrum which suggests that there may be some phase-shifting differences in the matched filters. If all the spectral density were

concentrated in the quad-spectrum, the phase-relationship between the o

two signals should be 90 for all frequencies . Instead trie phase angles

s

(12)

(8)

computed by equation 10 and shown in figure SD indicate a phase difference between O and 5°. This

not too

serious but it implies that further

experiment with the matched filters is desirable.

Another test involved the treatment of the simultaneous recording of wave height and heave (acceleration). _{The spectra of the} two input, appear in figure 6A and the co- and quad-spectra of the two signals appear in figure 6B. _{Comparison of the co-spectrum obtained by} the single filter and matched-filters methods is shown

in

_{figure 6C and} again fair agreement is found. A quad-spectrum is generated by treating only the wave height record. The output of analyzer A is phase-shifted 90° and multiplied by the output of analyzer A. _{The result should be}

zero but is actually very small as seen in figure 6D. _{The quad-spectrum} of waveandwave is plotted against the wave spectrum (which is also the co-spectrum of waveandwave) but is ten times larger than it really is; otherwise it would not be seen.

S1ARY

The preliminary experiments described in this report indicate that the Taylor Model Basin cross-spectrum analyzer is capable _of

pro-ducing co-

and quad-spectra with some measure of accuracy. There is some evidence that tò. filters are not perfectly matched (figure 5c). _Although the 900 phase-shifter was not isolated for testing _{its use with the filters} did not show an increase in error over that experienced _{by the filters}

above (figure 6D).

The system as it presently exists is probably adequate for estimating cross-spectra. _{However, additional tests should be made for}

further verification. In particular, it Is recommended that simple experiments with known signals, such as _{square waves, should be} under-taken. _{Further, comparisons of the type made previously for}

auto-spectra (References 2 and 3) should be made between the _{cross-spectra computed} for this paper and cross-spectra of the _{same Inputs obtained on digital} computers. _{The warning in Reference 4 regarding the high}

resolution

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required for cross-spectral analysis _{should be heeded.} _{The narrowest} pair of matched filters in the system Is 5 cps and this may not be narrow enough. _{Speeding up the magnetic tape by}

a factor of 2 will effectively double the resolution which should be adequate for most recorded random signals.

REFERENCES

Uberol, M. S. and Gilbert, E. G.: _{"Technique for Measurement of}

Cross-Spectral Density of Two Random Functions," Review of Scientific Instruments, V. 30, No. 3, 1959. _{pp. 176-180.}

Marks, Wilbur and Strausser, P. _E.: _{"Data Reduction Methods at the} Taylor Model Basin," D#1B Report _{1361, Transactions Tw1fth American} Towing Tank Conference, September 1959.

Marks, Wilbur and Strausser, _{P. E.:}

"SEADAC--The Taylor Model BasIn Seakeeping Data Analysis Center," _{TMB Report 1353, July 1960.}

Pierson, W. J., Jr. and Dalzell, _{J. F.} _{'The Apparent Loss of}

Coherency in Vector Gaussian Processes Due to Computational Procedures With Applications to Ship Motions and Random Seas," NYU College _of Engineering, Research Division, September 1960.

"Instruction Booklet for TP-25 Wave Analyzer System," Technical Products Corp. Los Angeles, _California.

'%iodel MU/DV Duplex Multiplier/Divider

and MU/t-l-659" G.A, Philbrick Researchers, Inc. Boston, Massachusetts.

"Instruction and Operating Manual _{for Model 3 Autograf x-y Recorder,"} F.L. Moseley Co.,Pasadena, California.

An electronic analog computer for spectrum and gross spectrum analysis of random signals

AICH1EF

1.ab. y.

Schesboúwknck

Tecnsch2 Lkcioa

D{L

Davidson

-

where Cf

,g

(T)

hm

T

-ami

(n)e

cos indn

L

f

R1g n)e

sn ndn]

jf(x) K1(t

5g(x)

jRfg(n)'(n)dn

(4)

jK()

îj -

5)

cos(t

cos(t

(6) in equation (4) yields

()e

()

fg

()

AA r

() sinØqg(

which

thence

iS

't)

f+g

2c

Cg

f+g

not too

in

S1ARY

pro-ducing co-

above (figure 6D).

i

2-2g, 000cps

t

2-25,000cp3

L

SPEETRAL DESIIY

LIN

Fig. i

Block Diagram of IMB Cros

Spectrtm .naly'zer

x-y PLO1ER

1TT

i

/

/

/ /

/

FIG. 2 CIUIT DIAÛRM OF CONS

-+ 010

000

--OIo

.4

V.Iave

Heih4-WE - Rod/Sec.

0301

020

.010

000

(

Wove Hek1

plus Roll

_Davidson

_,g

_L

_{sn ndn]}

_cos(t

_iS

_/

_/

_{Corut.ed by}

_{Spectra, Co- mid Quad-Spectra, and Phase Angles of Roll}

_{Angle mid}

_{Spectra of Wave Height and Heave (Acceleration)}

_{d Quad-Spectra of Wave Height and Heave}

_{Ccriparison of Co-Spectra of Wave Height}

_{and Heave}

_{Different Methods}