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NisrUnited States Department of Commerce

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National Institute of Standards and Technology

NIST Technical Note 1372

Asse^§ment of Data by a SecondOrder Transfer Function

D.G. Camell

M.T. Ma

QC

100

U5753

NO. 1372

199^1

NIST Technical Note 1372

Assessment of Data by a Second

Order Transfer Function

D.G. Camell

M.T. Ma

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

October 1994

*T OF

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director
i

National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1372, 72 pages (October 1994)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1994

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
Contents

page

List of Figures iv

1. Introduction 1

2. Approximation of Resonances by Second-Order Transfer Functions 2

3. Frequency Transformation 4

4. Assessment of Torpedo Measurement Data 4

4.1 Analytical Expression for Magnitude Squares 5

4.2 Extraction of Transfer Functions 8

4.3 Impulse Response 8

4.4 Phase Information 9

4.5 EMP Response 10

5. Conclusions 12

6. References 12

ni
List of Figures

page

Figure 1. The cw external excitation (magnetic field) used in the experiment 13

Figure 2. Measured C26 magnitude response to the cw excitation 14

Figures. Magnitude of the system transfer function 15

Figure 4. Possible impulse response of the system under study 16

Figures 5-11. Continuation of figure 4 17-23

Figure 12. Possible phases of the system under study 24

Figures 13 - 19. Continuation of figure 12 25-31

Figure 20. Magnitude response, as a function of normalized frequency, of the

system to the impulse excitation given in eq (36) 32

Figure 2 1

Time response to the impulse excitation given in eq (3 7) 33

Figures 22 - 52. Continuation of figure 21 34-64

Figure 53. Measured time response, after normalization, to the same impulse 65

IV
Assessment of Data by a Second-Order Transfer Function

D.G. Camell and M.T. Ma

Electromagnetic Fields Division

National Institute of Standards and Technology

Boulder, CO 80303-3328

A newly developed theory for predicting the response of a linear system to an

electromagnetic pulse, based only on the measured continuous-wave magnitude,

is applied to the problem of possible electromagnetic interferences at a sensitive

part of a torpedo. The measured magnitude representing the system's transfer

function is deduced first from the measured response at this sensitive point to a known cw source, supplied by the Naval Surface Warfare Center. We derive an

analytic expression for the magnitude square of the transfer function to

approximate the measured data and obtain a system transfer function in terms of

the complex fi"equency, from which we predict the system's cw phase

characteristics and its multiple solutions due to a given impulse source.

Key words: convolution integral; impulse response; linear system; magnitude;

phase; transfer function.

1. Introduction

A theoretical method was developed to predict the frequency and time characteristics of an unknown linear system, based only on the measured cw magnitude data, without requiring phase

information [1]. The approach is to deduce an approximate analytical expression for the overall

system magnitude square from the given measured magnitude data. The essential steps involved

in achieving this purpose [1] may be summarized as follows: (i) identify the resonant frequencies

and the maximum responses at these frequencies from the data, (ii) approximate each of the

resonant regions by a component magnitude square with a simple second-order transfer function,

(iii) sum up all of the component magnitude squares to yield an overall magnitude square to represent the unknown system, (iv) derive from it a set of multiple transfer functions in terms of

the complex frequency, (v) determine the corresponding phases and impulse functions, and (vi)

compute the time responses of the system to a known electromagnetic pulse (emp) source by

performing analytically the convolution integral.

Steps (i), (ii), and (iii) are outlined in section 4.1. The approximation for each resonant

frequency by a second-order transfer function as required in step (ii) is presented in section 2. A

short discussion on frequency transformation for avoiding manipulations with large numbers is given in section 3. The derivation of transfer functions from the approximate magnitude square

in step (iv) is detailed in section 4.2. Determination of the system impulse functions and phases

required in step (v) are given respectively in sections 4.3 and 4.4. Computation of the emp
response of the same system is derived in section 4.5. Related numerical results are presented in

figures 4 through 52.

The final fi^equency and time responses obtained from this method may then be analyzed and

compared with the measured data, if available.

2. Approximation ofResonances by Second-Order Transfer Functions

The second-order transfer function used to approximate each cw resonance displayed on

measured magnitude data takes either of the following two forms:

^ci'"> - ^^ ' (la) s ^ as ^ b

or

jj .. A{s ^ c)

s ^ as * b

where s is the complex frequency, a, b and A are all positive and real numbers with b > a^ll [1],

and c is a real number (which may be positive, 0, or negative).

For the transfer function in eq (la), the parameters a, b and A are determined by [1]

b'- <^l- \ ''

. (2a)

4

a' - 2{b - G)o) , (2b)

and

A = {(D^ - (Oj) , (2c)

where Wq is the radian resonant frequency, Wj and co, are the half-power frequencies with

Wi < Wo < 0)2 and \H^(J^)\ = WJJ^^ )\ = \HJJ^^) ^\l

^ , and |//^(/Wo )| is the maximum

magnitude at the resonant frequency.

All of the four required numbers, Wg , Oi , 0)2 , and \H^(Jwq )|, can be read directly from the

measured cw magnitude data. In this case, (Jj and Wj are not independent, but are related by

Wj + (x>2 = 2(x)q (3)

Thus, it is unimportant if either coi or 0)2 is not available in the measured curve since it can be

determined from eq (3). The dimensions of a and b in this case are respectively s'^ and s'^. The

dimension of^ depends on |//^(/a))|.
For the transfer function eq (lb), the parameters a, b, c, and A are determined by [1]

b^ = 2g)J - cOjWj , (4a)

2 ,2 _2 .2

a^ = Qj + G)2 - 4a)o + 26 , (4b)

2 2s (Wq - "l<»^2) ^ = —, -, ' (4c)

(g)i . (O2 - 2a)o)

and

A' =(2co'o -2Z) .a^)|//,0«J|^ . (4d)

Again, the required input information, cOq , coi , Wj, and \HJJix>q )|, can be read directly from the

given curve. The dimensions of a, b and c now are respectively s'\ s'^ and s"\ The dimension of

A depends on \Hi,(joiQ )|.

In eq (4c), we require that [1]

2 2 2 2 0)^ + 0) 2 > 2 G)(, and o) q > co ^ 0)2 (5)

If either of the conditions in eq (5) is not satisfied, we use the transfer function in eq (la).

Once the parameters a, b, c [under condition eq (5)], and A are determined, the magnitude square

of the component transfer function that approximately represents the resonant frequency under

consideration becomes

\H^(J^)\' - 4— , (6a)

or

I IT-/ • M2 A^(^x>^ + C^)

CO - {lb - a )(x) + b

The approximation as presented is good only for the important resonant region being studied and

is not meant to match the given measured curve everywhere.

If there are A/^ distinct resonant frequencies in the original cw data, the final approximate

magnitude square for the overall system is just the sum of all the component magnitude squares,

from which the transfer function in complex frequency H(s) is then extracted [2]. It will be

shown later that there may be multiple solutions for H(s). The corresponding solutions for

impulse transfer fijnctions h(t) can then be determined in terms of damped sine and cosine

functions.
3. Frequency Transformation

To avoid manipulations with large numbers such as the frequency in megahertz (MHz), we use

the normalized radian frequency by omitting the factor 10^. We herein designate o) as the

normalized radian frequency and to' as the actual radian frequency such that

G)' = 5(0 with B = \0^ . (7)

Substituting eq (7) in eqs (6a) and (6b), we obtain respectively the component magnitude square

in actual radian frequency.

2 d4

\G^U^')i\\2 A^B

and

(0'' - B\2b - a')(o'' . b^B*

£^

CO'' - (lb' - a'')(o'' . b'^

0)'* -B^{2b - a')(o'' . b^B'

G)'' - (lb' - a''')oi''' . *''

(8a)

(8b)

?2 AI>2 From eq (8a), we see that the parameters a\ b\ and A' will become respectively aB, bB , AB

when we convert from normalized frequency to actual frequency. For the transfer function in eq

(lb), the parameters a\ b\ c\ and A' in eq (8b) will become aB, bB^, cB, and AB for the same

conversion in frequency.

4. Assessment of Torpedo Measurement Data

The data analyzed in this report came from a test made on a special torpedo by Naval Surface

Warfare Center (NSWC). The test was instrumented with current probes in two locations,

arbitrarily called C26 and C29, where EMI problems are suspected. The external cw source used

in the test for C26 is a magnetic field in amperes/meter (input) as shown in figure 1 . The

measured probe magnitude response in amperes (output) is given in figure 2. Thus, this inputoutput configuration constitutes "the system" as far as our particular application is concerned.

Both figures 1 and 2, were originally expressed in megahertz. For our analysis, we use the

normalized frequency by omitting the factor of 10^, as discussed in section 3. The transfer

function magnitude in meters is obtained by taking the ratio of the response data (fig. 2) to the source data (fig. 1), as shown in figure 3 (solid curve), where the normalized frequency is used.

Also, we concentrate only on the portion of data up to 100 Hz for the normalized frequency or 100 MHz in real frequency. Similar analysis for C29 is not included in this report.
4.1. Analytical Expressionfor Magnitude Squares

From the measured data derived in figure 3 we see many resonant frequencies where the

measured magnitudes reach their maxima. For demonstration and considering the storage

limitation of our software, as a first step, we identify six resonant frequencies properly numbered

in the figure 3. They are : fjo = 5.593, fjo = 19.609, fjo = 53.262, f4o = 56.848, fjo = 59.373, and

fgo = 72.192 with their respective maximum values of 0.1 151 m, 0.0862 m, 0.3773 m, 0.2709 m,

0.4671m, and 0.1277 m.

For the first resonant fi^equency at fjo (cojo = 35. 143 1), we observe the half-power frequencies

fji =5.077 (g)ii = 31.9016) and fi2 =5.935 (<0i2 =37.2876). Since one of the conditions in eq

(5) is not satisfied (wn^ + Wij^ < 2a)io^ ), we use the transfer function in eq (la) and obtain a^ = 5.2870, Z>i = 1.2490(10^), mdA^ = 21.4460. Thus, the first component magnitude square

becomes

WSJ^^)? - '— , (9a)

where

Z)j(o)2) = w' - 2.4701(10^)0)2 . 1.5600(10^) . (9b)

The magnitude square in eq (9a) is used to approximate the first resonant region in figure 3.

For the second resonant fi-equency at fjo = 19.609 (wjo = 123.2070), we read fji = 17.784 (oji = 1 1 1.7402), and fjj = 20.480 (0)22 = 128.6796). Again, because of 0)21^ + 022^ < 20)20^ , we use

the transfer function in eq (la) to obtain aj = 16.4905, b^ = 1.5316(10^), A^ = 1.7545(10^), and

the second component magnitude square

\H0(^)\^ = — , (10a)

i),(a>2)

where

Dl^i^) = (^' - 3.0360(10^)0)2 . 2.3458(10') . (10b)

For the third resonant frequency at fjo = 53.262 (0)30 = 334.6550), we have fji = 52.604 (0)31 = 330.5207), and f32 = 55.124 (0)32 = 346.3543). Since the second condition in eq (5) is not

satisfied, that is 0)30^ < o)3j 0)32 , we also use the transfer function in eq (la) to obtain a^ = 16.0080, Z>3 = 1.1212(10'), ^3 = 2.0218(10^), and the third component magnitude square

I^3(/'")r= ^V' Ola)

Z)3(0)2)

where

Z)3(o)2) = 0)' - 22399(lCP)o)2 . 1.2571(10'°) . (lib)
For the fourth resonant frequency at f4o = 56.848 (040 = 357. 1865), where f4i does not exist and

f42 = 58.324 (042 = 366.4605), we also use eq (la) to obtain a^ = 18.7740, b^ = 1.2776(10^),

^4 = 1.8172(10^), and the fourth component magnitude square

m<^)\'- -^, (12a)

where

i)/a)2) = co" - 2.5516(10^)0)' . 1.6322(10'°) . (12b)

Similarly, for the fifth and sixth resonance at fjo = 59.373 and f^ = 72.192, the transfer function

in eq (la) also applies. We obtain a^ = 9.6223, b^ = 1.3921(10^), A^ = 1.6769(10^); a^ = 19.2431,

b^ = 2.0593(10^), and ^6 = 1.1149(10^); which yield respectively

|//3(/G))|'= ^i—

,

(13a)

with

and

Z)/a)') = (0^ - 27833(lCP)a)' . 1.9380(10'°), (I3b)

Al

with

\H,(j^W- -^-, (14a)

D/o)') = G)' - 4.1150(10?)g)2 . 4.2409(10'°) . (Mb)

Adding all the component magnitude squares in eqs (9a) through (14a), we have the overall

system magnitude square,

6 \H(jcor- y\KU^)\' - 1.1473(1C7)^^^ , (15)

where

with Z), (g)'), i = 1, 2, ..., 6 given previously in eqs (9b) to (14b), and

iV(a)2) = 0)'° - 9.3575(10^)0)'' . 3.6725(10")o)'' - 7.8083 (10")o)"

. 9.6766(10" )o)'' - 6.9726(10'')o)'° . 2.7329(10'')o)' - 5.0434(lCP')o)'

. 4.0175(10'')o)' - 8.0753(10*^)0)' . 6.0670(10*')
= [o' - 2.4904(10')(o' + 2.1141(10')] [w' - 3.0973(10^" - 2.7749(10')]

[o)' - 23947(10')w' . 1.4446(1^°)] [ui' - 2.6894(10')co' . 1.8147(lC/° )] (17)

* [o* - 3.9387(10^)0)' . 3.9452(l(y°)] .

The final maximum value of\H(joi)\ in eq (15) at Wjo is 0.4921, somewhat larger than the actual

maximum value of 0.4671, resulting in an amplification factor of 1 .0535. This increase is due to the contributions, although small, from the other component magnitude squares. To enforce

|/f(/a)5o)| = 0.4671, we simply divide the final magnitude square in eq (15) by (1.0535)' to yield a corrected system magnitude square

|///»|' = 1.0336(10^)^^, (18)

with D(iii^) and iV(a)') given in eqs (16) and (17).

The square root of eq (18) is plotted as the broken curve in figure 3 for comparison. As far as the major regions (resonant frequencies) in figure 3 are concerned, the approximation is very

good except that the approximated value at f4o is much larger than the actual value.

According to the method known in classical network theory [2], we can extract the transfer

function in complex frequency s from the magnitude square in eq (18) by substituting s =ji^.

Thus,

H{s)H{-s) - \H(ji^)\^ {withs-ji^)

1.0336(10')^t^

where

and

with

N{-s^) - N^i^)Nl-)N^(^)Nf)N^{.)Nl-)N^{^)Nl-)N^(.)Nl-),

D{-s') - D^{.)D^i-)D2(^)Df)Dl^)D,i-)D^(.)D^i-)Dl.)Dl-)Dl.)D^i-)

N^(±) = s^ ± 20.43435 . 1.4540(10^)

Nl±) = s^ ± 48.40785 . 1.6658(10^

NJi±) = 5' ± 30.13245 . 1.2019(10')

A^/±) = 5' ± 21.92905 . 1.3471(10')

Nji±) = 5' ± 58.15255 + 1.9862(10')

(19)

(20)

(21)

(22)

and

Z).(±) = 5' ± as + b. 7= 1,2,...,6 . (23)
4. 2. Extraction of Transfer Functions

To determine a realizable transfer function H{s) from eq (19), we have to assign all D,{+) to the

denominator ofH(s) in order to yield a stable system. Then all Z),(-) automatically belong to the

denominator ofH{-s). As far as the numerator ofH(s) is concerned, we have multiple choices.

If all of the Ni(+) are chosen as the numerator ofH(s), Nj(-) will be in the numerator ofH{-s).

In this case, H(s) has no zeros in the right-half s-plane. The system transfer function is said to be

at minimum phase. Other choices will produce one or more zeros in the right-half s-plane,

resulting in nonminimum-phase transfer functions. There are a total of 3 1 (from 2^ -

1)

nonminimum-phase transfer functions for the case under study. The coeflQcient in eq (19),

1.0334(10') gives a factor of 3.2146(10^) each to His) and H{-s).

For the minimum-phase case, we obtain

//i5) = 3.2146(10' )-ii

n^(-)

n^,o

i-l

where

yKs . G

L Di.)

'

Fr = 0.4305, Gi = 4.5011,

F2 = 0.8814, G2 = 46.740,

Fs = 2.5998, G3= 1196.6,

F,= -0.1656, G4 = 950.33,

Fs = -2.0624, G5 = 926.21,

Fe- -1.6837, G6 = 93.272.

(24)

(25)

and

If we replace one or more Ni(+) in eq (24) by its counterpart N,(-), the resuhing transfer function

wiU be at nonminimum phase, and the corresponding expansion coefiQcients F^ and G, will be

different from those in eq (25). The values ofF, and G, for the 3 1 nonminimum-phase cases can be computed and printed out by the software, when desired.

4.3. Impulse Response

The impulse response for the minimum-phase case in eq (24) is then

K(^) - )\,cosi^,t) . ^,sin(p,.0]^""'' , (26)

with
a, = 2.6435, P, = 35.243, c, = 0.4305, d,= 0.0954,

a? = 8.2443, P2= 123.48, C2= 0.8814, d,- 0.3197,

"3 = 8.0012, p3 = 334.75, C3= 2.5998, d,= 3.5126,

"4 = 9.3860, P4 = 357.31, C4 = -0.1656, d,= 2.6640,

a,= 4.8106, P5 = 373.08, C5 = -2.0624, ds = 2.5092, and

"6 = 9.6219, P6 = 453.70, C6 = -1.6837, de = 0.2129.

(27)

Equation (26) is plotted in figure 4 marked with '11111', because five N^ (+) are included in

the numerator of eq (24) to represent the minimum-phase case. The time scale, in seconds, used

here is based on the normalized frequency. The corresponding time scale for the actual

frequency will be microseconds. Nonminimum cases are shown in figures 4 to 1 1 for

comparison. The coefficients c, and d^ for the nonminimum-phase cases will all be different from

those in eq (27) although a, and P, will remain unchanged. The curve marked with '1 1 - 1 - 1

1'

is interpreted to indicate that A^i(+) A'2(+) A3(-) A^4(-) ^si"^) is chosen for the numerator of the

transfer function.

4.4. Phase Information

Since we define, in this report, the phase of//(/to) as

Hgco) - |//Oa>)k ^«(">

, we have for the minimum-phase case in eq (24)

0^(0)) = 0/0)) - 0„(G)) ,

where 0/w), due to the denominator factors in H^ (/co), is given by

(28)

(29)

0/0)) = > tan-^

6

»i

a 0)

0) (30)

and 0„(o)), due to the numerator factors in //,„(/o)), from eq (22), is given by

e„(a)) = tan"' 20.4343 0) 1.4540(10^) - 0)2

tan' 48.4078 0)

1.6658(10'*) - 0)2 tan 30. 1324 0) 1.2019(10^) - 0)2

+ tan" 21.9290 0) 1.3471 (10^) - 0)2

tan" 58.1525 0) 1.9862(10^) - 0)2 (31)

For the nonminimum-phase cases, one or all of the terms in eq (3 1) will change sign to make the

total phase larger than that for the minimum-phase case. Phases for the 32 cases (1 minimum

phase and 31 nonminimum phases) are presented in figures 12 through 19.
4.5. EMP Response

Once the impulse response h{t) is obtained in subsection 4.3, the response r^^Jif) at C26 due to an emp source function emp{i) can be derived analytically by performing convolution integrals.

That is, '•.JO = ^^P^^) * ^0 , (32)

where h{t) represents either minimum-phase or nonminimum-phase case.

The actual emp source function used in the test is given by

emp{t') = K

g-4(10*)r' _ g475(10«)r'

where t is the real time before frequency normalization, and the constant factor K is

J,= 52500^ 139.2573.

377

The Laplace transform of eq (33) is

(33)

EMP{s') = K

1 1 s' . 4(10^) s' . 475(10')

(34)

and its spectrum is given by

EMPiJoi') = K

1 1 4(10') .70' 475(10') . >' (35)

We now apply frequency normalization to convert eq (35) into

EMPHio) -K(m') 1 1 4 + j(x> 475 + y'o)

which yields

emp^ (0 = K{\Cr^)[e" - e''"]

(36)

(37)

Comparing eqs (33) and (37), we see that the net effects on the time excitation function from the

frequency normalization are a dropping of the (10') factor from the exponent and a multiplying

factor of (10"') to the coefficient.

Now, the frequency response to the normalized source becomes, from eq (36),

10
l^(/")i = \Hin^)\ - , (38)

v/(a)' . 4')(a)' . 475'

)

which is plotted in figure 20. Since there is only one |//c(/w)| for the minimum-phase and non- minimum-phase cases, there is only one fi-equency response as shown in figure 20.

The time response to the normalized source becomes

r^J) - Kim')^'' - e-'''^]. Kt) (39)

Since the source fianction is in the form of e'^' and the typical term in h{f) is

]g. cos( P . + d. sin( p. ] ^

"'

, we only need to deal with a general formula

e*' * [c.cos(p.O + J.sin(P7) V"''. One term is

t

e-'' * cos(p.O e ' = e-"^'-'^ cos(p.x) e ' dx

[{p - a^)cos(p,0 . p,sm(p7)]e-"'' - {p - a,>-''

{p - cc,T ^ P?

and the other term is

e-'' * sin(p.O e ' = e^^'-^^ sin(p.T) e ' dx

[-P,cos(P,0 . {p - a^)sin(P,0]^""'' ^ P,^"'

ip - ay . p,.^

Substituting eqs (40a) and (40b) into eq (39), we obtain

i ) l/?2. 'e

""'' +

«.'

where

, jc^p^ - g,) - ^,pjcos(p,0 . [c p, . ^,(/>, g )]sm(p,0

(/'i- a)' . P^

and

11

(40a)

(40b)

(41)

(42a)
-c(p - a) + d^

«/ = —' ' '—!- {independent of t) , /'42k\

and m" and /?," have the same expressions as w/ and n- but withpi replaced by/?2 , where/?, = 4

andpj'^^TS.

The time responses for all cases are presented in figures 21 to 52, which can be compared with

the measured emp response as shown in figure 53. The emp response r^^Jif) for the minimumphase case as shown in figure 2 1 is almost always nonnegative because the term in eq (42b) is very dominant in this case. Similar solutions for the nonminimum-phase cases can be both

positive and negative. Some of them have significant negative parts such as that shown in figure

38. Therefore, it appears that the specific input-output configuration of the torpedo under study

constitutes more likely a nonminimum-phase system.

5. Conclusions

We have applied the newly developed theory for predicting the impulse characteristics of an unknown linear system based only on available cw magnitude data to a particular system

configuration. By comparing the predicted impulse responses to the measured emp data, we

conclude that the system under study is more likely at nonminimum phase. This same procedure

can be applied to the analysis of other linear systems and to assess the meaning of the measured

data.

We thank the Naval Surface Warfare Center (NSWC) and Naval Sea Systems Command

(NAVSEA) for their partial support of this study. We also appreciate the measured data supplied

by John Bean and Rich Porter of NSWC.

6. References

[1] Ma, M.T.; Adams, J.W. Characteristics of unknown linear systems deduced from

measured cw magnitude. J. Res. Natl. Inst. Stand. Technol., 98:297-319; May-June

1993.

[2] Balabanian, N. Network Synthesis, New Jersey: Prentice-Hall; 1958.

12
20

o

(DC

O)

CO

15

E

E^

^

10

5 —

20 40 60

Frequency (MHz)

80 100

Figure 1. The cw external excitation (magnetic field) used in the experiment.

13
1.6

1.2 —

<

E

-£ 0.8

(D

o

0.4

0.0

20 40 60

Frequency (MHz)

80 100

Figure 2. Measured C26 magnitude response to the cw excitation shown in Figure 1

.

14
0.6 -n

0.4 —

CD

c

05

0.2 —

0.0

Measured Data

Approximation based on eq (18)

20 40 60 80

Normalized Frequency (Hz)

100

Figure 3. Magnitude of the system transfer function, both measured and

approximated based on eq (18).

15
Curve 1=11111

Curve 2= 1111-1

Curve 3=1 1 1-1 1

Curve 4=1 1 1-1-1

10 -n

03 o

Q.

E

<

10 - £ 5^

o

E

<

-5

-10

0.0

Curve 4

"n

\—\—\—\—\—\ \ \ \ \ \ \ \ \ \

I 0.2 0.4 0.6 0.8 1.0

Time (s)

Figure 4. Possible impulse responses of the system under study; minimum phase case and three nonminimum phase cases.

16
(U o

•^Q.

E

<

Curve 5 = 1 1 -1 1 1

Curve 6 = 1 1 -1 1 -1

Curve 7 = 1 1 -1 -1 1

Curve 8 = 1 1 -1 -1 -1

Curve 7

0) TJ3+^Q.

E

<

Curves

Figure 5. Possible impulse responses of the system under study; four nonminimum

phase cases.

17
10 -

(D

Q.

E

-5 — < _

-10

10

^i^m^ E

—

5

oa — D — a z E

-5 < —

-10

Curve 9

Curve 9 = 1 -1 1 1 1

Curve 10 = 1 -1 1 1 -1

Curve 1 1 = 1 -1 1 -1 1

Curve 12 = 1 -1 1 -1 -1

E

<

10 ^

5

^

-

-5 -

-10 -

Curve 11

D

Q.

E

<

Figure 6. Possible impulse responses of the system under study, four nonminimum

phase cases.

18
(U o

E

<

Curve 13 = 1 -1 -1 1 1

Curve 14 = 1 -1 -1 1 -1

Curve 15 = 1 -1 -1 -1 1

Curve 16 = 1 -1 -1 -1 -1

Curve 15

(D o

Q.

E

<

10 -^

5 -^

-

-5 -3

-10 -

Curve 16

n—^—^\—\—\—\—\—\—\—\—\—\—\—I\—\—

r

0.0 0.2 0.4 0.6 0.8

Time (s)

1.0

Figure 7. Possible impulse responses of the system under study, four nonminimum

phase cases.

19
10 -^

Curve 17 = -1 1 1 1 1

Curve 18 = -1 1 1 1 -1

Curve 19 = -1 1 1 -1 1

Curve 20 = -1 1 1 -1 -1

10 -^

10

^

n\—\—\—i\—\—\—\—\—

0.4 0.6 0.8 1.0

Time (s)

Figure 8. Possible impulse responses of the system under study, four nonminimum

phase cases.

20
(D

Q.

e

<

Curve 21 =-1 1-1 1 1

Curve 22 = -1 1-1 1-1

Curve 23 = -1 1-1-1 1

Curve 24 = -1 1 -1 -1 -1

(U D

J

cLE

<

Curve 22

Curve 23

0) o

E

<

10

5

-5

-10

=

Curve 24

n I 1

I

I I I

I

I \ I

I

\ r

0.0 0.2 0.4 0.6

Time (s)

1—

r

0.8 1.0

Figure 9. Possible impulse responses of the system under study, four nonminimum

phase cases.

21
o

"a. E

<

Curve 25

Curve 25 = -1 -1 1 1 1

Curve 26 = -1 -1 1 1 -1

Curve 27 = -1 -1 1-1 1

Curve 28 = -1 -1 1 -1 -1

0) o

^u

Q.

E

<

10

5

-5

-10

Curve 26

10 -=,

-10 -^

Curve 27

10 -^

Curve 28

Figure 10. Possible impulse responses of the system under study, four nonminimum

phase cases.

22
<0

E

<

Curve 29 = -1 -1 -1 1 1

Curve 30 = -1 -1 -1 1 -1

Curve 31 = -1 -1 -1 -1 1

Curve 32 = -1 -1 -1 -1 -1

0) T3

Q.

E

<

Curve 30

10 -=i

-10 -^

Curve 31

Curve 32

0.4 0.6

Time (s)

1.0

Figure 1 1 . Possible impulse responses of the system under study, four nonminimum

phase cases.

23
35

30

25

Curve 11111

Curve 1111-1

Curve 111-11

Curve 1 1 1-1-1

o

(0i^ 20

(0

c

o

Q.

(0

q:: (D

0)

CO

15

10

—

-5 T

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 12. Possible phases of the system under study (C26) for one minimum phase

case and three nonminimum phase cases.

24
35

30 —

25 —

Curve 11-111

Curve 1 1-1 1-1

Curve 1 1-1-1 1 Curve 1 1-1-1 -1

T3

CO

CD

COC

o

Q.

(n

CD

20

15

10

-5

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 13. Possible phases of the system under study (C26) of four nonminimum

phase cases.

25
35

30

25 —

Curve 1-1111

Curve 1-1 1 1-1

Curve 1-1 1-1 1 Curve 1-1 1-1 -1

(D

c

o

Q.

Oi

q:

(D

cn

CD

20

15

10

5 —

/

-5

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 14. Possible phases of the system under study (C26) of four nonminimum

phase cases.

26
35

30 —

25

oCO

O)

c

o

Q.

(O

0)

q:

Q)

CO(0

CL

20

15

10

5 —

-5

/-^. .».»' -1 -1 -1 -1 ^vUl VC 1 - 1 - 1 1 1 - - - Curve 1-1-1 1 -1 Curve 1 -1 -1 -1 1 Curve 1 -1 -1 -1 -1

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 15. Possible phases of the system under study (C26) of four nonminimum

phase cases.

27
35

30

25

Curve -11111

Curve -1111-1

Curve -111-11

Curve -1 1 1-1 -1

o

(D

(0

c

o

Q.

0)

q:

(0CO

20 —

15

10 —

r

-5

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 16. Possible phases of the system under study (C26) of four nonminimum

phase cases.

28
35

30

T5

?0

(0

c

o

Q.

q:

(0

(Q

JZ

Q.

25 —

20 —

15

10 —

Curve -1 1-111

Curve -1 1-1 1-1

Curve -1 1-1-1 1 Curve -1 1 -1 -1 -1

-5

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 17. Possible phases of the system under study (C26) of four nonminimum

phase cases.

29
35

30

25 —

oCO

</) c

o

Q.

i/i (D

CD

(0

20 —

15

10 —

5 —

Curve -1-1111

Curve -1-1 1 1 -1 Curve -1 -1 1-1 1 Curve -1 -1 1 -1 -1

-5

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 18. Possible phases of the system under study (C26) of four nonminimum

phase cases.

30
35 -n

CO

(D

(0C

o

Q.

(0

0)

a:

(0

CO

30

25

i^ 20

15

10

Curve -1 -1-1 1 1 Curve -1 -1 -1 1 -1 Curve -1 -1 -1 -1 1 Curve -1 -1 -1 -1 -1

-5

200 400 600 800

Normalized Frequency (Hz)

1000

Figure 19. Possible phases of the system under study (C26) of four nonminimum

phase cases.

31
1 .2E-6

1 .OE-6

8.0E-7

(D

3 6.0E-7

Q.

E

<

4.0E-7

2.0E-7

O.OE+0

- - - based on the measured data in fig (3)

based on the approximation in eq (18)

-^-4-

20 40 60

Normalized Frequency (Hz)

80 100

Figure 20. Magnitude response, as a function of normalized frequency, of the system

to the impulse excitation given in eq (36).

32
curve = 1 (11111)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.43048

0.88139

2.59983

-0.16558

-2.0624

-1.68371

d =

0.09543

0.31967

3.5126

2.66402

2.50921

0.24129

0)3

Time (s)

Figure 21. Time response of the system to the impulse excitation given in eq (37),

for the minimum phase case.

33
curve =2 (1 1 1 1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

0.42908 0.10451

0.85894 0.3899

0.88812

-1.6675

d =

4.44494

2.30787

-3.34611 0.62801

2.83748 1.38157

-2*10 " -

Time (s)

Figure 22. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

34
curve = 3 (1 1 1-1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.42972

0.86933

0.21726

-6.01609

6.09814

-1.59836

d =

0.1005

0.36057

5.05079

1.49411

1.57604

0.70862

0)3

Time (s)

Figure 23. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

35
curve =4 (1 1 1 -1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.42822

0.84363

-2.02735

-6.27638

3.83916

3.19272

d =

0.10957

0.42997

4.83595

-2.0812

5.37101

0.57488

0) •o3

Time (s)

Figure 24. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

36
curve = 5 (1 1-1 1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.42929

0.86184

-6.85277

5.79528

1.33149

-1.56512

d =

0.10324

0.38348

4.6978

-0.22031

3.53812

0.76908

3

Time (s)

Figure 25. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

37
curve =6 (1 1-1 1 -1 )

2.64345 35.2426 0.42773 0.11231

8.24428 123.482 0.83432 0.45237

8.00117 334.749 -8.50636 1.38143 a =

9.38598

4.81063

p=

357.312

373.079

c =

5.35135

-1.31161

d =

3.104

3.73917

- 9.62187 - 453.699 3.20458 0.4474

Q*1/\ O

6- 10"^ - »

(I

1 1 1 1 1 1 1 1 1

4'10~^ M _

_-> 2M0~^ "

1

1 1f\ V/^ X_-—-^-.^^ -^

-2-10~^ -

-4M0~* 1 1 1 1 1 1 1 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

Figure 26. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

38
curve =7 (1 1-1-1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.42844

0.84682

-9.61129

4.53606

5.1337

-1.33374

d =

0.1083

0.42367

0.06512

12.6818

-5.23318

1.19767

V

T3

3

-2*10 "-

-4'10

Time (s)

Figure 27. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

39
curve =8 (1 1 -1 -1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

0.42678

"

0.8161

-9.04859

-3.13691

7.64746

3.29517

d =

0.11736

0.49149

-4.18477

14.0207

-0.74951

-0.45175

a>3

Time (s)

Figure 28. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

40
curve = 9 (1-1 1 1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.40692

-1.86731

3.65158

0.65855

-1.25693

-1.59282

d =

0.19072

0.28103

2.48475

2.61536

3.01165

0.62392

3

-2*10 " -

-4*10

Time (s)

Figure 29. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

41
curve = 10 (1-111-1)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.40351

-1.89505

2.3292

-0.89662

-3.03709

3.09605

d =

0.19944

0.13515

3.94496

2.73366

1.57389

0.69922

4>3

1*10 ^

8*10~^

Time (s)

Figure 30. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

42
curve = 11 (1-1 1-1 1 )

a =

2.64345 35.2426

8.24428 123.482

8.00117

9.38598

P =

334.749

357.312

4.81063 373.079

9.62187 453.699

c =

0.40504 0.19559

-1.88524 0.19705

1.8911 -5.33264

d =

4.74668

3.28547

6.32339 -0.25106

-1.40165 1.06178

• in 6 8*10

I

Time (s)

Figure 3 1 . Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

43
curve 12 (1-1 1 -1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.40153

-1.90644

-0.32217

-6.68097

5.24997

3.25809

d =

0.20428

0.04951

5.28998

-0.07639

4.05451

-0.17234

• ift 6

8*10

0) 2

Time (s)

Figure 32. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

44
curve = 13 (1-1-1 1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.404

-1.89401

-4.9723

5.51118

2.3063

-1.35518

d =

0.19822

0.14929

6.76684

-1.99145

3.01794

1.11332

a> 5

Time (s)

Figure 33. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

45
curve = 14 (1-1-11-1)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

0.40043

-1.91148

-7.65563

6.10452

-0.17819

3.24034

d =

0.20689

9.1858M0

4.15386

1.345 3.97734

-0.29987

-4

3

Time (s)

Figure 34. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

46
curve = 15 (1-1-1-1 1 )

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.40204

-1.90604

-9.14869

8.26044

3.42222

-1.02997

d =

0.20307

0.06374

3.26628

10.8154

-6.52229

1.47964

-2*10 " -

Time (s)

Figure 35. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

47
curve = 16 (1-1-1 -1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

0.39836

-1.91682

-10.0286

1.28513

7.13995

3.122

d =

0.2117

-0.08584

-0.97624

14.4602

-2.93732

-1.20102

• in 6 8*10

3

Time (s)

Figure 36. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

48
curve = 17 (-11111)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

35.2426

123.482

334.749

357.312

373.079

453.699

-0.69547

0.96452

3.02172

0.14293

-1.77551

-1.65819

d =

0.27288

-0.01112

3.17528

2.67362

2.72541

0.39343

I

Time (s)

Figure 37. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

49
curve = 18 (-1111-1)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.70209

0.96832

1.43306

-1.39478

-3.26112

2.95661

d =

0.25879

0.06463

4.31475

2.49234

0.99484

1.12132

B«in ^

3

Time (s)

Figure 38. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

50
curve = 19 (-1 1 1-1 1)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.6992

0.96771

0.84013

-5.82185

6.24387

-1.53066

d =

0.26504

0.03263

5.00068

2.18413

0.89483

0.85201

•in ^

8*10

3

I

Time (s)

Figure 39. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

51
curve = 20 (-1 1 1 -1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.70566

0.96808

-1.42067

-^.49447

4.41478

3.23794

d =

0.25087

0.10877

5.06408

-1.3483

4.92173

0.28416

•1f> 6

8'10

<L> 3

0.1

Time (s)

Figure 40. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

52
curve =21 (-1 1-1 1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.70119

0.96879

-6.24092

5.74907

1.71625

-1.492

d =

0.26078

0.05741

5.5226

-0.88936

3.37449

0.90932

• in 6 8*10

3

Time (s)

Figure 41 . Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

53
curve = 22 (-1 1-1 1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.70757

0.96721

-8.29652

5.69096

-0.8923

3.23821

d =

0.24655

0.13371

2.42576

2.47436

3.86668

0.15589

a

Time (s)

Figure 42. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

54
curve = 23 (-1 1-1-1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

"

123.482

334.749

357.312

373.079

453.699

c =

-0.70478

0.96887

-9.55895

5.98564

4.53149

-1.22226

0.25287

0.10154

d =

1.252

12.1121

-5.77608

1.31601

3

Time (s)

Figure 43. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

55
curve = 24 (-1 1 -1 -1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.711

0.96383

-9.52379

-1.50512

7.52899

3.24709

d =

0.23855

0.17798

-3.04781

14.3331

-1.59108

-0.74954

3

Time (s)

Figure 44. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

56
curve =25 (-1-1 1 1 1)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

p =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.75246

-1.69662

3.94166

0.95849

-0.91827

-1.5328

d =

0.11664

0.94645

2.02226

2.52988

3.1365

0.76698

9> 2

I

Time (s)

Figure 45. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

57
curve = 26 (-1 -1 1 1 -1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.75591

-1.77613

2.80575

-0.57744

-2.84901

3.15275

d =

0.10115

0.816

3.63908

2.8275

1.90216

0.417

3

Time (s)

Figure 46. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

58
curve = 27 (-1 -1 1-1 1 )

a =

2.64345

'

8.24428

8.00117

9.38598

4.81063

9.62187

p =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.75442

-1.74429

2.46868

-4.9338

6.26617

-1.30235

d =

0.10801

0.87207

4.49121

3.89008

-0.94864

1.18657

B»in ^

4>3

-2*10 " -

Time (s)

Figure 47. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

59
curve =28 (-1-1 1-1-1)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.7577

-1.8181

0.33273

-6.66588

5.67353

3.23541

d =

0.09247

0.73765

5.30539

0.6961

3.45547

-0.46737

4> I

I

Time (s)

Figure 48. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

60
curve = 29 (-1 -1-1 1 1 )

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.75545

-1.77001

-4.11355

5.26127

2.62905

-1.25131

d =

0.10333

0.82925

7.34982

-2.62133

2.74901

1.23379

8* 10 ^

3

Time (s)

Figure 49. Time response ofthe system to the impulse excitation given in eq (37),

for a nonminimum phase case.

61
curve = 30 (-1 -1 -1 1 -1 )

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

35.2426

123.482

334.749

357.312

373.079

453.699

-0.75863

-1.84053

-7.10726

6.23813

0.26215

3.20615

d =

0.08776

"

0.69268

5.08033

0.6346

3.97849

-0.59301

B«in ^

2

Time (s)

Figure 50. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

62
curve = 3

1

(-1 -1 -1-1 1 )

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.75727

-1.81256

-8.70308

9.48096

2.68553

-0.89359

d =

0.09466

0.7512

4.38127

9.82194

-6.87006

1.56982

3

-2*10 " -

Time (s)

Figure 5 1 . Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

63
curve =32 (-1-1-1-1-1)

a =

2.64345

8.24428

8.00117

9.38598

4.81063

9.62187

P =

35.2426

123.482

334.749

357.312

373.079

453.699

c =

-0.76027

-1.87708

-10.103

2.95189

6.78208

3.00635

d =

0.07904

0.61105

0.26698

14.2599

-3.71262

-1.48148

• in *

8* 10

a

Time (s)

Figure 52. Time response of the system to the impulse excitation given in eq (37),

for a nonminimum phase case.

64
2.0

1.5

1.0

0.5 - <

3

(U o13 1= 0.0 -

Q.

E

< -0.5 -

-1.0

-1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time (s)

Figure 53. Measured time response of C26, after normalization, to the same impulse

excitation.

*U.S. GOVERNMENT PRINTING OFFICE: 1994-673-018/00054

65



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