nistUNITED STATES

DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

NATL INST. OF STAND 5 TECH R.I.C.

A111D3 7L.D5TM

NIST

PUBLICATIC

-OG

NIST Technical Note 1349

Phase Characteristics and Time Responses

of Unknown Linear Systems Determined

from Measured CW Amplitude Data

M. T. Ma

J. W. Adams

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

/ W \

\

**TES O* * /

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher. Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons. Director

Issued November 1991
National Institute of Standards and Technology Technical Note 1349

Natl. Inst. Stand. Technol., Tech. Note 1349, 24 pages (November 1991)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1991

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

Page

Abstract 1

1

.

INTRODUCTION 1

2 THEORETICAL BACKGROUND 2

3 CONVENTIONAL APPROACH 3

4 AN ALTERNATIVE BUT SIMPLER APPROACH 4

5

.

CONSIDERATION OF ENERGY CONTENTS 14

6 . CONCLUSIONS 15

7 . ACKNOWLEDGEMENTS 16

8 . REFERENCES 16

m

PHASE CHARACTERISTICS AND TIME RESPONSES OF UNKNOWN LINEAR SYSTEMS

DETERMINED FROM MEASURED CW AMPLITUDE DATA

M. T. Ma and J. W. Adams

Electromagnetic Fields Division

National Institute of Standards and Technology

Boulder, CO 80303

Abstract

An alternative but simpler technique for calculating the complete

time and frequency characteristics of an unknown linear system from

the measured amplitude response to cw excitations is described. The

associated system transfer function so determined may or may not be at

minimum phase. A comparison of the time responses shows the worst

case. Results also indicate that the susceptibility of the minimumphase system to damage by pulsed excitation is the greatest during the

initial period of excitation.

Key words: all-pass function; Hilbert transform; Laplace transform;

minimum phase; non-minimum phase; system transfer function; time

response

.

1. INTRODUCTION

The time response of a linear system to a given excitation (whether it

be cw or pulse) can be uniquely determined, if the transfer function of this

system is known, by use of the inverse Laplace transform. However, for

complicated systems involved in practical applications, the system transfer

function may be unknown. The time response of such a system to pulsed

excitations can be obtained only by sophisticated time -domain measurements

or by derivations using measurements of the amplitude and phase responses to cw excitations. These measurement results can then be used to assess (a)

whether or not the system may survive an assumed pulsed excitation, and (b)

if not, what hardening is required for the system to withstand the threat.

Such time -domain or frequency- domain phase measurements typically require

expensive equipment and special considerations on radiation hazards and

regulatory compliance (if performed outdoors). On the other hand, taking cw amplitude responses of the same complicated system due to low levels of

escitation is relatively easy and less costly. If the measured cw amplitude

could be used to predict the system's phase and thus its transfer function,

the time response to a general excitation could then be determined.

In this report, we present an alternative but simpler technique to

determine the time and frequency characteristics of an unknown, linear

system, based only on a given set of cw amplitude responses. The associated

system transfer function so determined may or may not be at minimum phase.

The time responses corresponding to the minimum phase and non-minimum phases

thus obtained can then be used to assess the susceptibility of the system to

damage by pulsed excitations. The theoretical background of relationship

between the amplitude and phase of a minimum-phase transfer function is

outlined in section 2. The conventional numerical approach practiced in the

past for calculating the phase information and the time response of the

linear minimum-phase systems from the measured cw amplitude, and the

accuracy involved in this process are reviewed and discussed in section 3.
Our new approach for obtaining a set of transfer functions (both minimum and

non-minimum phases) and the corresponding impulse responses directly from

the given cw amplitudes is presented in section 4. This technique does not

require knowledge of the necessary phase information. Of course, this phase

information is automatically obtained after the system's transfer function

is determined. It can also be determined analytically, if it is required, even before the minimum-phase transfer function is obtained. Physical

meanings of the minimum-phase results are discussed. A few examples with

various orders of transfer functions and existing measured data to

demonstrate this idea are given. Energy contents associated with the system

are discussed in section 5. Some concluding remarks and extensions for

future work are presented in the final section.

2. THEORETICAL BACKGROUND

A well designed, stable, linear system with time- invariant and lumpedconstant elements exists only when its transfer function H(s) has no poles

in the right half of the complex frequency s-plane. In other words, H(s) is

analytic in Re(s) > 0, where Re stands for the "real part of" [1]. We

address only the stable system in this report, because otherwise the system

is not useful in application. In general, H(s) is a rational function of s or a ratio of two polynomials with the degree of the numerator polynomial

lower than the degree of the denominator polynomial. When this transfer

function is evaluated at the real radian frequency s = jw, H(jw) is a complex function of to. It then consists of a real part R(w) and an imaginary part X(w) , or a magnitude |H(w)| and a phase 6 (to) . That is,

H(jco) = R(w) + j X(w) = |H(jw)| e"J* (w)

, (1)

where the convention of assigning a minus sign to the phase is adopted. The

magnitude function |H(jw)| may also be expressed in terms of the attenuation

function a(a>) :

|H(j»)| = e- Q(w). (2)

When H(s) is analytic, the real and imaginary parts of H(jw) are related by the Hilbert transform pair [2],

X(c) = - ** r Hr^T dy,

and (3)

R(«) = 1 r f(Y)

2 dy.

In other words, if one part is given or specified, the other part can be

uniquely determined by performing one of the integrals shown in (3). The

complex function H(jw) is then completely obtained. In practice, however,

(3) is not useful because we cannot just measure R(w) or X(w)

.
If, in addition, H(s) has no zeros in Re(s) > 0, the transfer function

is said to be at minimum phase, herein denoted by H (s) . Under this

condition, the attenuation and phase functions are related by another pair

of Hilbert transforms [2, 3]:

, n ^ n l H (jy) I

-oo J -oo J

and

a( W) = a(0) - f IT y(y2

(Y)

. „*) dy. (4b)

Equation (4b) shows that the attenuation function can be determined

completely from a given phase function only when a(0) is also known. But,

for our application, only (4a) is useful because we assume that the

magnitude (or attenuation) function is given. Once & (w) is determined, the

entire complex H (jw) is obtained from (1), because |H (jo) | or a(w) is

already known. The impulse response of the system is then traditionally

calculated by the inverse Fourier transform,

h (t) = 7T r H (» ejWt

dt. (5)

-oo

The system's time response to a general excitation can subsequently be

computed by the convolution integral h (t)*e(t) , where e(t) is an arbitrary

excitation, cw or pulse, applied to the system. The success of determining

the time response in this case is based on the assumption that the system's

transfer function is at minimum phase.

3. CONVENTIONAL APPROACH

Since the type of integral in (4a) is considered extremely difficult to

calculate, the approach has been to apply another transformation known as the Wiener-Lee transform [2] to a(w) or |H(ja>)|, and then to use numerical

procedures [4] to obtain the necessary 6 (w) , H(ja>) , and h(t) .

When the Wiener-Lee transform [2],

w = - tan(5/2), (6)

is applied, we are able to transform the entire u> range, (-°°, °°) into (-n,

n) for 5. Because of this transformation, the original attenuation and

phase functions, q(w) and 9 (u>) , will become respectively a' (8) and #'(5).

Furthermore, since the attenuation function a(w) = - in|H(jw)| = - -z in

[R 2 (co) + X

2 (w)] is an even function, and the phase function 6 (u>) = tan [X(w)/R(tc>) ] is an odd function, of u> in (-«, °°) , their respective

transforms a' (5) and 0'(8) are also even and odd functions of 8 in {-n , n)

.

As such, they may be expanded into Fourier series,
a' (8) = an + a, cos 8 + ... + a cos n5 + ....

1 n

' and (7)

6' (8) = b, sin 5 + ... + b sin n5 + ... , 1 n

'

where the expansion coefficients are determined by

a„ - -^ J* a' (8) dS, a = - J

n

a' (8) cos n8 dS , (8a) 2n J n 7r -7T -7T

and

b = - J

n

6' (8) sin nS dfi . (8b) n 7T -7T

When the system under consideration is such that h(t) is causal [h(t) = when t < 0] , as is usuallyJ so in practice, r it is known that a and b are n n simply related by [2]

b = - a . (9) n n

v '

Therefore, the determination of a from (8a) automatically yields b , which

in turn gives 6' (8), 6 (w) , and thus H(jw) . The impulse response is then

obtained from (5).

The justification for using the Wiener-Lee transform and the derivation

as outlined above seems reasonable and straightforward. The important

question we should ask is then: if the integral in (4a) is difficult to

compute before the Wiener-Lee transform is applied because the given |H(jw)| or a(w) are very complicated, is it much simpler to compute the Fourier

series expansion coefficients a as required in (8a) after the Wiener-Lee

transform is used? If so, what kind of penalty is paid in terms of the

final accuracy? After all, the entire procedure requires (i) conversion of

the given a(a>) data into a' (8) , (ii) numerical computation of a from a' (5) , (iii) construction of a set of 6' (5) by including a finite number of terms

in the Fourier sine series of (7) with b = - a , (iv) numerical

determination of H(jw) based on the given attenuation data and the newly

retrieved phase data, and (v) numerical computation of h (t) by performing

the discrete inverse Fourier transform. Each of these five steps involves

approximations and thus introduces inaccuracies.

4. AN ALTERNATIVE BUT SIMPLER APPROACH

Here, we review modern passive network theory together with a re- examination of the Hilbert transform (4a) to assess whether or not a simple

and analytical method may be devised to handle the problem at hand without

going through unnecessary numerical procedures. Since H(s) for the system

under study is a rational function of s, the square of the magnitude of

H(jw) , |H(jw)| 2

, is a ratio of two polynomials of even order in w. As such,
the numerator and denominator polynomials in |H(jw)| 2 may contain one or more, or any combination, of the following elementary factors:

(i) u> 2 + a

2 and (10)

(ii) w

4 + bw2 + c2

, where a, b, and c are real numbers. In addition, since |H(ju>)| 2 > for all

w, we require, for the factor (ii) in (10),

b

2 < 4c 2

. (11)

The number of the above elementary factors contained in |H(jw)| 2 depends on

the complexity of the transfer function being studied. The first-order

transfer function has just one factor in form (i) in the denominator of

|H(jw)| 2

, and its numerator is a constant. For a second-order transfer

function, the denominator of |H(jw)| 2 contains a factor in form (ii), or two

factors in form (i), and the numerator has either a factor in form (i) or a constant. For a third- order transfer function, the denominator can be

expressed as a product of form (i) and form (ii) [or as a product of three

factors in form (i)], and the numerator as a factor of form (ii), form (i), or a constant. Thus, for an nth- order transfer function, the denominator of

|H(jw)| 2 has a special form of polynomial such as w + a , u

n ~" +

an ./n - 4

. . .+ a

Q

, and the numerator is an even polynomial o/the order

of (2n - 2) or lower.

When the magnitude function |H(jw)| can be described by a mathematical

expression, H(s) can be deduced directly and exactly from it in a straightforward manner [5], without having to find the phase function first.

For example, consider the simplest, first-order case described by

|H(>| 2 = ^

2 \ a2 , a > 0. (12)

We can use the relationship, |H(jw)| 2 = H(s)H(-s)| . , to obtain

s - jw

H(s)H(-s) =

T^L_ = ___1___ , (13)

yielding

Hm(s) = l/(s + a), (14)

which has a simple pole (s = -a) on the left-hand real axis. Since it has

no zeros , the transfer function in this particular form is automatically at

minimum phase. The associated impulse function is then determined directly

by the inverse Laplace transform as h (t) = exp(-at), t > without having

to find the phase function first. It can readily be verified that (14)

indeed gives |H(jw)| 2 in (12). Note that the other factor (-s + a) in (13)

is not allowable in H(s) because the resulting system will then be unstable.
The expression in (12) may represent the output of a low-pass filter

with its maximum at u = and with its half -power (or 3 dB) point at u> = a. This same parameter "a" is also the decay rate of the time response.

Clearly, the phase function associated with the system's transfer

function in (14) is,

0(w) = tan

_1 (w/a)

.

(15)

If, however, the phase information is desired before the system transfer

function in (14) is obtained, it can also be found analytically by the

following procedure. Since

o(w) = - in |H(jw)| = - \ in |H(jw)| 2 = J

in(u> 2 + a

2 ), (16)

the required phase function for the minimum-phase case, in accordance with

(4a) , is

a r \ ^ f

00 in(y2 + a

2

) dy _ <±> r°° in(y2 + a

2 )dy H" } ~ n J 2(y2

- w

2

) " n

J y

2 - w2

= tan"V/a), (17)

which agrees with (15) . The last step in (17) is achieved by using the

definite integral [6]:

r

in(p 2 + q

2 x2

) dx n . -1, ah v , . n /1QN

g

2 x2 -h2 = ^h tan <Ji>' P>^g' h>0 - < 18 >

Another method for getting h (t) is through the inverse Fourier

integral given in (5), since we already know the complex H (jw) = |H (jw)

|

exp[-jfl(w)]

.

That is

,

h (t) = ^ r H (J«) eJ Wt

du> = r1 T |H (J«) | J

[wt

" $ ( " } ] dc m Z7T J m

J

27r J ' m

J

1 r«o cos \cot - 6 (u>) 1 du>

'

7T J (w2 + a

2

)

1 /2

1J. r«> pco a cosos(cc)t) (cot du>cc> ip1 r<»00 cj sin(c^t) du>

" 7T i W

2 + a2

7T ^ u> 2 + a

2

1 -at 1 -at -at /ion

2 e + X e = e » ( iy )
which is the same as obtained above by a much simpler procedure (inverse

Laplace transform). The last line in (19) results from the application of

the following two integrals [6]:

r

cos (ax) dx _ _n -aB , x

2 + B2 ~ 2/3 e and r<» x sin(ax) dx

J x

2 + B

2 n -aB "2 e

a > and Re(B) > 0. (20)

This method of inverse Fourier integral is the one used in the numerical

approach discussed in the last section.

We wish to note another important point about the transfer function.

Even though the particular form in (14) is taken for analysis, it does not

necessarily imply that the system under study is minimum phase. In fact,

the original unknown system may possibly be represented by many transfer

functions with non-minimum phases, because a product of H (s) and an allpass function yields a transfer function with non-minimum phase but still

with the same I H( ico) 1 2

i \j / i . That is,' H (s) = H (s) H (s) , where H (s) is the n ma n non-minimum-phase transfer function, and H (s) is the all-pass function

a defined as

H (s) = n (s - a )/ n(s + a ), (21)

with each zero in the right-half s -plane the mirror image of the

corresponding pole in the left-half s-plane [5]. In (21), the symbol II denotes successive product, and the parameter a. is either positive real or complex. When a. is complex, (21) must also have another complex conjugate

factor. The fact that |H (ju>)| 2 = 1 explains why |H (ju))| 2 = |H (jw)| 2. 3. in m

The corresponding impulse response is then

h (t) = h (t) * h (t), (22) n m a

where h (t) is the inverse Laplace transform of H (s) , and * represents

cl 3. convolution integral. For example, consider the case with a = 1 and H (s) =

(s - 2)/(s +2). We then have h (t) = e"

t

, t > 0, and

h (t) = h (t)*h (t) = h (t)*[S(t) - 4e" 2t

] n ma m

L J

= h (t) - r(t), t > 0, (23)

where

r(t) = 4e"

t*e"2t = 4(e -t

- e"

2t

). (24)
Since r(t) > for all t > , we conclude that h (t) < h (t) . This implies

that, for this particular example, if the minimum-phase system can survive a threat from a given pulsed excitation, the non-minimum-phase system can also

successfully withstand the same threat.

Before we proceed with an example of a second-order transfer function,

we note that the impulse response h (t) given in (23) can also be obtained

by first performing a partial fraction expansion of H (s)

,

n

H (s) = H (s)H (s) = t—- -, " - rr =

~~*

. + —7S7 , (25) n m

v ' av

' (s + l)(s +2) s+1 s+2 v/

and then doing an inverse Laplace transform [7] to get

h (t) = -3e _t + 4e" 2t = h (t) - r(t)

(26) n

v ' m

v '

We will now present another example using a second-order transfer

function whose squared magnitude is given by

co 2 + 1 w

4 + w2 + 16 |H(»I ' 2 = ,,4 T ,,2 I ,c , (27)

a plot of which is shown in figure 1 with its resonant frequency at to 2 = 3 and with a half -power bandwidth of Au> 2 = 12.689. From (27), we have

H(s)H(-s) =

s4 _ s2 -16 =

(g2 + 3g + 4)(s2 _ 3g + 4) , (28)

thus yielding the minimum-phase transfer function

H (s) = 2 I

t TZ / > (29) m

v ' s

z + 3s + 4

' v '

and the simplest possible non-minimum-phase transfer function

H (s) = 2~1X

1

J. / ( 3 °) n

v / s-^ + 3s + 4

v '

As a check, both (29) and (30) give the same |H(jw)| 2 in (27). Of course,

the other possible non-minimum-phase transfer functions having the same

|H(jw)| 2 in (27) are the product of H (s) in (29) [or H (s) in (30)] and the

all-pass function H (s) in (21).

cl

The phases associated with the transfer functions given in (29) and

(30) are respectively

(w) = tan"

1 [3w/(4 - w

2

)

] - tan"

1 (w), (31)
and

9 (w) = tan"

1 [3w/(4 - w

2 )] + tan'^u). (32)

The impulse responses of (29) and (30) can be determined, respectively,

to be [7]

h (t) = e'

L5t (cos 1.32288t - 0.37796 sin 1.32288t), (33) m

and

h (t) = e~ 1,5t

(- cos 1.32288t + 1.88982 sin 1.32288t) n

= hm

(t) - r(t), (34)

where

r(t) = 2e"

1 - 5t (cos 1.32288t - 1.13389 sin 1.32288t) >

in < t < 0.54634. This means that h (t) < h (t) shortly after an n

v ' m

v ' J excitation is applied to the system. The minimum-phase response will not

underestimate the system behavior.

Thus, we have shown once again that (i) the minimum -phase transfer function

for an unknown system and the resultant impulse response can be determined

directly from the given |H(jw)| 2 without having to obtain the associated

phase function, and (ii) the minimum-phase time response h (t) so determined

is the most conservative estimate as far as the initial threat to the system

by an excitation is concerned.

The phase function corresponding to the minimum phase can also be

determined from (4a) before its transfer function in (29) is obtained. We

begin with the denominator of (27)

,

w

4 + w2 + 16 = (w 2 + af)(w2 + a

2

,) , (35)

where a\ and a\ are a complex conjugate pair. Even though it is

straightforward to determine a\ and a

2

, [in this case, a\ = (1 + jj63)/2, and

a 2 = (1 - j J 63)/2 ], we only need to know, as shown later, a x a 2 and a x + a 2

. We know from (35) that a\ + a\ = 1 and a\a\ = 16. We obtain a x a 2 = 4, and

(a x + a 2 ) 2 = a\ + a% + 2a x a 2 =1+8=9, ora 1 +a2 =3. Then we have

in|H(jw)| = - | in(w2 + a?) - | in(w2 + a|) + |

in(w2 + 1). (36)

Applying (4a) we obtain

0(w) = 6 1 (o)) + 6 2 (u) - 8 3 (u>), (37)
where, with the aid of (18),

in(y2 + a\) . M«>) = *Z r —^2 ^2— dy = tan" (w/ai ), (38) n

Q y w

in(y2 + a

2

,) , M") = ^ r y2 . ^2 dy = tan" (»/a2 ), (39)

and

a / n w f°° ln(v2 + 1) ^_ -1. . ..... M") = ~ J y2 . w

2 dy = tan (w)

.

(40)

However

,

! (w) + # 2 (w) = tan (u)/a 1

) + tan (w/a2 )

tan [ (a2 + a 2 )w/(a 1 a2

- o> 2

)

]

= tan"

1 [3w/(4 - u> 2 )] . (41)

Thus , we have

0(w) = tan"

1 [3w/(4 - w

2 )] - tan'V), (42)

which checks with (31)

.

The impulse response given in (33) can also be obtained by the inverse

Fourier integral method, as we demonstrated in the first example for the

first-order transfer function, with a much more involved procedure.

We make two comments about the examples presented above. First, in

obtaining (38) and (39), we have extended the application of (18) to the case with a complex number for p. Evidently, (18) is still valid as long as

a complex conjugate pair appears together. This is certainly satisfied in our study since the given |H(jw)| 2 is a rational function of w

2

.

Second, the variable o> has been used loosely as frequency. In (27) we have identified u> 2 = 3 (or u> = J 3) as the resonant frequency for the secondorder case. Strictly speaking, u> should be treated as the normalized radian

frequency. It can be translated into any frequency of interest by a simple

frequency transformation [5],

w' = Aw, (43)

where A is a normalization constant.

10
If, for example, the actual resonant frequency occurs at J 3 MHz, we then use A = 2tt(10) 6 to translate u> = J 3 to u>' = J 3 x 27r(10) 6

. Substituting

u> = u>' /A into (27) gives a new squared magnitude function,

A

2

(cj' 2 + A

2

) , (44)

I g O')| 2 - ,y 4 + A

2

u>' 2 + 16A4

yielding the |G(jw')| 2 vs w' 2 curve identical to that shown in figure 1.

The corresponding minimum-phase transfer function, impulse response, and

phase function then become respectively:

A(s' + A)

nT ' s' G-(s') - .?'; 3L-+ 4A» • <45)

-1 SAr

g (t) = A e • (cos 1.32288At - 0.37796 sin 1.32288At), (46)

and

0(w') = tan'^Aw'/OA2 . w

' 2 )] - tan'V' /A) . (47)

The same principle and procedures apply to higher-order cases. The

results on extracting H (s) , h (t) , and 6 (w) from a given |H(jw)| 2 and the

procedures involved, as demonstrated in the two examples, are exact, and no approximations have been exercised. Approximations are needed only when an analytical expression for |H(jw)| 2 is unknown. In practice, when only the

measured data of |H(jw)| are available, all we have to do is to obtain an experimental curve for |H(jw)| 2 and then to approximate it with a realizable, mathematical expression. By realizability , we mean (i) |H(jw)| 2 = N(c<; 2 )/D(w2

) > for all w, (ii) the numerator and denominator polynomials,

N(w2

) and D(w2) are in even orders of to with some restrictions on their real

coefficients, and (iii) the degree of N(w2

) is lower than the degree of

D(w2

) by at least 2. This can be achieved by the available approximation

theory and curve -fitting techniques. In fact, this is the only step

involving approximations. The remaining procedures will be exact.

The last example to demonstrate the applicability of the approach

presented in this section is now given, and is based on the measured data

shown in figure 2 (solid curve) . The data represent normalized, vertically

polarized electric fields reflected from a helicopter when it is irradiated

by an impulse. By examining the solid curve in figure 2 more carefully, we notice at least three resonant frequencies at 16.5 MHz, 25.6 MHz, and 54.3

MHz. Obviously, this system is too complicated to be represented by a simple second-order transfer function. However, for demonstration purpose,

we only consider the most important resonant frequency at 25.6 MHz. This means that we treat the dashed curve in figure 2 as "the given" one, and

then try to approximate it by a second-order transfer function with minimum

phase in the form of

H (s) = B/(s 2 + as + b), (48)

where B, a, and b are three unknown constants to be determined.

11
From (48) , we have

" Hm

( J w) l 2 = w * - (2b ^a2

) u> 2 + b

2 • < 49 >

which has the resonant frequency at u>% = b - a

2 /2.

Expressing (49) in terms of u> gives

l Hm(»l 2 ~ u< - ill a, 2 + b2 ' < 50 >

Two essential conditions are then imposed on (50)

<4 = b - a

2 /2 = (2tt x 25.6 x 10 6

) 2

, (51)

and

|H (jw )l

2 = 532 (° r 34.48 dB as given in fig. 2) (52)

A third condition may be imposed at either (i) 20 MHz or (ii) 30 MHz.

Indeed, any other condition based on a physical reasoning may be used. If we choose (i) with wa = 2n x 20 x (10) 6 = 1.2566 x 10 8

, and use

|Hm

(jWi)| 2 = 13 2 (or 22.28 dB as indicated in fig. 2), (53)

we have the solutions from (51) , (52) , and (53)

:

B = 13.5185 x 10 16

, a = 0.1584 x 10 8

, b = 2.5998 x 10 16

. (54)

The |H (jw)| 2 in (50) with the values of B, a, and b given in (54) is

plotted in figure 2 as the curve marked with dots. Not surprisingly, the

frequency region from 10 MHz to 25.6 MHz is very accurately matched to the

dashed curve, while the frequency region greater than 25.6 MHz is not.

If we choose (ii) for matching at 30 MHz (which is a 3-dB point), we have

w 2 = 1.8850 x 10 8

. The condition

|Hm

(j^ 2 )

|

2 = 53 2 /2 (or 31.48 dB), (55)

with (51) and (52), yields

B = 51.1874 x 10 16

, a = 0.5906 x 10 8, b = 2.7616 x 10 16 . (56)

The |H (jw)| 2 in (50) with this new set of B, a, and b values is presented

in figure 2 as the curve marked with crosses. Here we see a better

approximation for frequencies higher than the resonant frequency, but a poorer approximation for lower frequencies.

12
Even though these two approximations may not be wholly satisfactory,

they are still considered effective because the complicated experimental

curve is approximated by a simple second-order transfer function. A higherorder (4th or 6th) transfer function, or even a different kind of a secondorder transfer function such as H (s) = B(s + c)/(s 2 + as + b) with an additional control parameter c will certainly give a better approximation.

The use of higher-order transfer functions and additional control parameters

should be topics for future work. The objective of this report is primarily

to demonstrate the feasibility of solving this kind of problem with a simpler approach.

The impulse responses for the two approximations under discussion are:

(i) with the matching point at 20 MHz,

h ,(t) = 8.4347 x 10 8 e"

ait sin ^t, t > 0, (57) ml

with ai = 0.0792 x 10 8

, and fi 1 = 1.6104 x 10 8

and

(ii) with the matching point at 30 MHz,

h

m2 (t) = 31.0755 x 10 8 e"

a2<: sin /3 2 t , t > , (58)

with a 2 = 0.2953 x 10 8

, and 2 = 1.6354 x 10 8

.

The normalized time responses are shown in figure 3 with curves marked

respectively with dots and crosses, along with the normalized measured time

response (the unmarked curve). On the first look, the predicted minimumphase time responses clearly do not agree well with the measured curve. This implies that the actual system (helicopter plus its attached

instruments and equipment) may have a non- minimum -phase transfer function

with one or more zeros in the right half s -plane. This fact is also

indicated by the initial negative response appearing in the experimental

curve. On the other hand, the general shape of a damped sinusoid with a

_ Q _ O period between 3 . 84 x (10) and 3.90 x (10) seconds agrees well with the

shape of the experimental curve. In addition, the rate of decay for h «(t)

with a 2 = 0.2953 x 10 8 also approaches the experimental curve. This

indicates that matching at 30 MHz not only improves |H (jw)| 2 at higher

frequencies, but also yields a better approximation to the measured time

response. This is desirable because the receiving antenna used in the

actual measurement also works more accurately in the upper frequency range

.

The corresponding phases are:

e 1 (u>) = tan"

1 [0.1584 x 10 8 w/(2.5998 x 10 16 - u> 2

) ] , (59)

and

13
B 2

(u>) = tan

_1 [0.5906 x 10 8 w/(2.7616 x 10 16 - u> 2

) ] . (60)

5. CONSIDERATION OF ENERGY CONTENTS

To assess the ability of a system to withstand damage from an excitation, it is often useful to compute the energy associated with an impulse response. Indeed, if h(t) represents a voltage waveform across a 1 Q resistor, the quantity

E = J°°h 2 (t) dt (61)

equals the total energy delivered to the resistor by the excitation [2]

Equation (61) also represents the area under the curve h

2 (t).

The energy E may also be computed, in view of Parseval's theorem [2], by

E = ^ r |H(j")| 2 dw. (62)

-00

Since the minimum-phase impulse response h (t) and the associated non- minimum-phase impulse response h (t) = h (t)*h (t) have the same |H(jw)| 2

, XT m a. their respective total energies [in < t < °°] are equal even though h (t) <

h (t) during the initial period near t = as demonstrated in section 4.

For the first example considered in section 4, we have

IH (jw)| 2 = l/(w2 + a

2 ), and h (t) = e"

at

.

i m

\j /i / / m

The energy is then, from (62),

E - ^ r -a^-2 = i r -TW = V(2a) ; (63)

-00

or, from (61)

,

E = /VCt) dt = f°eZat dt = l/(2a). (64) m

Referring to the example with a non-minimum-phase impulse response presented

in (23), we have h (t) = e _t (with a = 1) , h (t) = 6(t) - 4e" 2t

, and h (t) =

\ / . m a n

-t -2t h (t) * h (t) = -3e + 4e . The energy is then m

v ' a

v bJ

2t nl -3t ,, -4t,

E = J

N°°h 2 (t) dt = f°(9e'

Zt

- 24e"-

3t + 16e C

) dt n

14
= 1/2 = l/(2a) with a = 1, (65)

which is indeed the same as in (63) and (64)

.

For the second example as given in (27) , we have

do a*dw

„ _ _1 f<o (u> 2 + l)do) 1 r«>

1 1 r<»

1 = 2, J m w* + u>* + 16 " * J 2(«» + ^) * J 2(w2 + ^

= | {aj/J^! + at/J/8t) = 5/24, (66)

where

a, - LLi/lM _ and /Ji _ Lij!63 (6?)

The same result can be obtained by computing

E = fV(t) dt or E = fV(t) dt, (68)

where h (t) and h (t) are given respectively in (33) and (34).

For our third example involving experimental data, the energies are:

(i) when the frequency point is matched at 20 MHz,

E, = f° h

2

1 (t) dt = 2.229 x (10) 101 , £ ml ' \ / .

and (ii) when the frequency point is matched at 30 MHz,

E 2 = rV2 (t) dt = 7.978 x (10) 10

,

where h -, (t) and h

? (t) are given respectively in (57) and (58).

6. CONCLUSIONS

We have shown an alternative and simpler method to determine the

complete transfer function (including the associated phase information) and

the impulse response for an unknown linear system from a given cw amplitude

response, without having to retrieve the phase function first. The

solutions (both minimum and non-minimum phases) are exact only if an analytical expression for the cw amplitude response is known. If not,

approximations can be used to derive a realizable squared magnitude

describing the cw system response. The remaining procedures for obtaining

the complete system characteristics are exact. Three examples, using known

analytical expressions and measured data for the cw magnitude, have been

examined and shown to validate the proposed technique. We also have shown

that the minimum-phase time response represents the most conservative

15
estimate as far as the response to an initial threat to the system by an excitation is concerned. Examination of the basic properties of transfer

functions with higher orders and with different forms for each order, and

study of accuracies involved in the approximation (when the measured data

are available) to cover more complicated cases should be topics for future

research.

7. ACKNOWLEDGEMENTS

We acknowledge with gratitude the partial support provided by the Naval

Sea Systems Command (NAVSEA) , and appreciate the interest and discussions of

the problem under study with Vincent Paschke of NAVSEA and Robert Amadori of

Naval Surface Warfare Center (NSWC)

8. REFERENCES

[1] Reza, F. M. ; Seely, S. Modern Network Analysis, New York, NY:

McGraw-Hill Co. ; 1959.

[2] Papoulis, A. The Fourier Integral and its Applications, New York,

NY: McGraw-Hill Co.; 1962.

[3] Bode, H. W. Network Analysis and Feedback Amplifier Design,

Princeton, N. J.: Van Nostrand; 1945.

[4] Berkhout, A. J. On the minimum phase criterion of sampled signals,

Brunei Shell Petroleum Co., State of Brunei; 1987.

[5] Balabanian, N. Network Synthesis, Englewood Cliffs, N. J.: PrenticeHall, Inc. ; 1958.

[6] Gradshteyn, I. S.; Ryzhik, I. M. Tables of Integrals, Series, and

Products, New York, N. Y. : Academic Press; 1965.

[7] Gardner, M. F. ; Barnes, J. L. Transients in Linear Systems, Vol. I,

New York, N. Y. : John Wiley and Sons, Inc., 1956.

16
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19

.'. US GOVERNMENT PRINTING OFFICE 1991—673-614 65.006 REGION NO 8
BL-114A

(5-90) U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

PUBLICATION OR REPORT NUMBER

NT ST TN 134Q

2. PERFORMING ORGANIZATION REPORT NUMBER

B91-0285

PUBLICATION DATE

4. TITLE AND SUBTITLE Phase characteristics and time responses of unknown linear systems

determined from measured cw amplitude data

5. AUTHOR(S)

M. T. Ma and J. W. Adams

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BOULDER, COLORADO 80303-3328

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

10. SUPPLEMENTARY NOTES

11. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBLIOGRAPHY OR

UTERATURE SURVEY, MENTION IT HERE.)

An alternative but simpler technique for calculating the complete time and frequency

characteristics of an unknown linear system from the measured amplitude response to cw excitations is described. The associated system transfer function so determined

may or may not be at minimum phase. A comparison of the time responses shows the

worst case. Results also indicate that the susceptibility of the minimum-phase system

to damage by pulsed excitation is the greatest during the initial period of excitation.

12. KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

all-pass function; Hilbert transform; Laplace transform; minimum phase; non-minimum

phase; system transfer function; time response.

13. AVAILABILITY UNUMITED

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS). JORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE, WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

24

15. PRICE

ELECTRONIC FORM
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