NBSIR 74-365

IMPULSE SPECTRAL INTENSITY WHAT IS IT?

M. G. Arthur

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

May 1974

I

NBSIR 74-365

IMPULSE SPECTRAL INTENSITY - WHAT IS IT?

M. G. Arthur

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

May 1974

U.S. DEPARTMENT OF COMMERCE, Frederick B Dent, Secretan/

NATIONAL BUREAU OF STANDARDS Richard W Roberts Director

CONTENTS

Page

FOREWORD-- iv

1. INTRODUCTION 1

2. FUNDAMENTALS 2

.3. APPLICATIONS 6

4 . SUMMARY 8

5. REFERENCES 9 FIGURES 10

APPENDICES 17

LIST OF FIGURES

Page

Fig. 1. Single rectangular baseband pulse. 10

Fig. 2. Three-dimensional plot of Re [y] = A cos I'nft. 11

Fig. 3. Illustrating the relationship between Re [y] and Re[Vp(£)].-- 12

Fig. 4. Plots of |Vp(£)| and 4>p(£) versus frequency. 13

Fig. 5. (a) Finite, regularly repeating rectangular baseband pulse

train. (b) Fourier transform of above pulse train. 14

Fig. 6. Three-dimensional plot of z(f,t). 15

Fig. 7. Illustrating the relationship between z(f,t) and v (t) . 16

iii
FOREWORD

The following report is the text of a paper presented at the Electromagnetic Compatibility Colloquium, U.S. Army Electronics Command, Ft.

Monmouth, New Jersey, May 1-2, 1974. Sufficient interest was expressed

by attendees at the colloquium to warrant making the text available in

printed form. In the interest of economy and prompt response to requests

for copies, the text has been kept in substantially the same form as was used in the oral presentation.

iv
IMPULSE SPECTRAL INTENSITY -- WHAT IS IT?

ABSTRACT

The term, impulse spectral intensity, is often used in discussions concerning electromagnetic interference and broadband signal processing. In these discussions, the term is not

used in a consistent manner, resulting in confusion, equivocation,

and sometimes, errors. As a step towards improving this situation,

the mathematical basis for spectral intensity is reviewed and

certain of its features are clarified. Also, misuses of spectral

intensity are discussed, along with its limitations and proper

use

Key words: Electromagnetic interference; Fourier transform;

impulse spectral intensity; spectral intensity; spectrum

amplitude; spectrum amplitude density.

1. INTRODUCTION

In EMC work, and in the EMC literature, we often encounter an electrom.agnetic quantity that has the dimensional units of volts per hertz, or more commonly, microvolts per megahertz. But as we engage in EMC work or read the literature, we sometimes are puzzled or confused by the variety

of names that this quantity is given, or we may be suspicious that the way

in which this quantity is being used is somehow different from the way that someone else has used it.

In this paper, I would like to talk about the quantity that has the

dimensional units of volts per hertz with the purpose hopefully of clarifying what some of its important characteristics are, and possibly even helping to reduce some of the large amount of confusion, half truths, and

whole untruths about it.

The most official name given to this quantity is found in the IEEE

Dictionary [1]. That name is Spectrum Amplitude. Spectrum amplitude is

defined there as 1/tt times the magnitude of the Fourier transform of a time-domain signal function. This 1/tt is an error, and should be replaced

by the numeric 2, thus:

S(f) = 2|VCf)| CI)

where

oo V(f) = / vCt)e"^^^^^ dt (2)

- cc and S[f) = Spectrum Amplitude [2]. Other names that can be found in the

EMC literature are Impulse Spectral Intensity, Spectral Intensity, Spectral

Density, Voltage Spectrum, Impulse Strength and Interference Intensity. All

of these names have been applied to quantities that have dimensional units

of volts per hertz or its mathematical equivalent, volt -seconds .
I personally am not satisfied with any o£ these names, but I do agree

with the definition as given in the proposed new IEEE standard on impulse

strength [2] . Perhaps there are others of you who have wondered what is

the theoretical or engineering basis for this quantity. In studying this

matter, the most basic starting point that I have found is^ the Fourier

transform of the time-domain signal. Therefore, I want to discuss some of the important characteristics of this Fourier transform as they pertain

to the question, "Impulse Spectral Intensity -- What Is It?".

Before proceeding further, let me say what my major points will be.

FIRST, the Fourier Transform, in the general case, is mathematically a complex quantity . Its magnitude alone may not be adequate to meet a given

need. Phase information may also be needed.

SECOND, the Fourier transform is an amplitude and phase distribution

funct ion , and not simply an amplitude.

THIRD, the Fourier transform is a continuously defined function of

frequency, and not a discontinuous, discrete function of frequency.

FINALLY, the magnitude of the Fourier transform appears explicitly in

many equations dealing with a signal function. This leads to the identification of a spectral amplitude quantity having engineering importance to EMC

work

Having made these statements, let me show you some justifications for

and implications of these statements. Although I start out by focusing

on the Fourier transform, I'll soon shift over to a type of spectral amplitude quantity.

Every physical signal can, at least in principle if not in practice,

be represented mathemat ical ly in both the time domain and the frequency

domain. An algebraic equation representing the physical signal in the time

domain can be mathematically transformed by the Fourier transform [3], [4]

to rep resent the physical signal in the frequency domain. Conversely, the

algebraic frequency -domain representation of the physical signal can be

transformed by the inverse Fourier transform to produce an algebraic

time-domain representat ion of the physical signal. The two equations defining these two transformations are as follows:

2 . FUNDAMENTALS

OO V(f) = / v(t)e

- j2TTft dt (2)

- OO

oo v(t) = / VCf)e j2uft df (3)

- oo

2
They are called the transform pair for the physical signal which is represented by v(t) in the time domain and by V(f) in the frequency domain. Equation (2) transforms the time-domain quantity into a frequency-domain representation; eq. (3) transforms the frequency - domain quantity into a timedomain representation. Both equations represent the same physical signal,

and are simply two different ways of mathemat ical representation.

As an example, consider the single rectangular baseband pulse shown in

figure 1. This signal has an amplitude of A from t = 0 to t = t , and is zero elsewhere. Although I have selected an idealized pulse-like signal for

illustration, all of the observations and conclusions which we will discuss

generally apply to any physical signal, regardless of whether it is a single

pulse of arbitrary shape, or a series of pulses repeated at perfectly regular

intervals, or a series of pulses repeated at irregular intervals, or continuous

type signal funtions of regular or irregular shape. The Fourier transform

of our example is V^Cf) as given by the following equations (see Appendix A).

V (f) = (sin 2TTfT - j2 sin^ Trfi) (4) P 2T;f

= At sin TTfx

TTfr Equation (4) shows that Vp(f) is a complex quant ity , having both a real

and an imaginary part. This is true iil general of any V(f) , although some signal representations, depending upon their waveform and the choice of

time origin, may have only a real or an imaginary part.

Equation (5) is written in terms of the magnitude,

V (f) I = At p^ J sin 7t£t

TTfT (6)

and phase

,

(t,p(f) = -^fT (7)

of the complex frequency-domain signal representation. To repeat, the

Fourier transform is a complex quantity having both a magnitude and

phase,

A second characteristic of the Fourier transform, V(f) is that it is the

integral over time of a time function (see eq. (2)). This means that V(f)

is geometrical ly an AREA . In our example.

3
oo Vp(f) =

/ Vp(t)e-J2.ft

= / y dt (10)

where

y = Ae"j2^^^ 0 < t < T

= 0 e Is ewhere

(11)

This area is the area under the curve of eq. (11) . The function y is a function of both frequency and time, and is complex. The real part of y in our example is

Re [y] = A cos Zirft (12)

and the imaginary part is

Im[v] = -A sin 2TTft. (13)

Figure 2 shows a portion of a three-dimensional plot of the real part of y,

Tt is three dimensional because y is a function of both frequency and time.

The result is a wavv surface with a cosinusoidal variation in both directions,

The real part of the Fourier transform, Re[^^p(f)], at a specified frequency,

f, is the are a under the cosine curve at frequency f in the time direction.

For example, figure 3 shows four "slices" through this wavy surface parallel

to the time-axis and perpendicular to the f requency - ax i s . The value of

Re['^.^p(f)] at frequency f^ is the area marked "c." The imaginary part of

V (f) can be found in a similar wav , and then V (f) is the vector sum of

p

^ ^ ' P the real and the imaginary parts (see the vector diagram at frequency f|^) • Beinp an are a , V(f) has dimensional units h'hich are the nroduct of

the units of its constituent parts, height and base. Tf v(t) is a voltage,

then V(f) has the units of volt - s econds . From this we can see that V(f) is

not the amp 1 i tude ner se of sinusoidal signals that are components of the

physical signal in the frequency domain. These amplitudes are zero and

would have the dimensional units of volts . Rather, \'( f ) is a measure of

the amp 1 i tude di s tr i but i on of a cont inuous ly defined frequency spectrum.

It tells re lat ive amp 1 i tude s as they are distributed with frequency. Thus

it is a distribution funct ion , a type of dens i ty funct ion . Re cause of this,

V(f) is often expressed in units of volt s per hert z , which are dimens ional ly

equivalent to the units of volt - s econds

4
There is a strong potential hazard in choosing the units o£ volts per hertz

rather than volt- seconds . It's true that mathematically they are equivalent.

From a physical standpoint, volts per hertz suggests a density function whereas volt- seconds does not. And I can find no engineering justification for

using one set of units in preference to the other. However, the units, volts

per hertz, can mislead one into thinking that he can simply integrate V(f) over a finite interval of frequency and obtain a voltage that has some useful

physical meaning. Such may not be the case. What does have useful meaning, as we have seen in eq . (3), is the integral of the product of V(f) with an external exponential function, which, for the infinite integral, yields the

time-domain signal representation v(t) . Incidentally, the infinite integral

of just V(f) over the entire frequency interval is the value of the timedomain signal function at time t = 0. This is a basic theorem of Fourier

transforms. Thus the integral of V(f) over a finite interval yields the part

of vCt) at t = 0 contributed by the frequency-domain signal in that band of

frequencies. To repeat, V(f) is a spectral distribution function . Another characteristic of V(f) is that it is a continuously defined

function of frequency. Let me show you what I mean. A plot of |Vp(f)| and

( f) for our example is given in figure 4. Although there may be points

where V(f) is zero, it is generally true that there are no^ places where

V(f) does not exist. In this way the Fourier transform representation of

a physical signal is different from the Fourier Series representation [5]

in which the spectrum is a discrete spectrum, and the spectral components

are amplitudes , expressed in volts , of sinusoids at discrete frequencies . Having heard this, you may wonder why the spectrum of a so-called "periodic"

pulse train is experimentally observed to be a line spectrum. The truth

is, it is not truly a line spectrum, but only appears to be so because of

the deficiencies of our test equipment. The Fourier Transform of a regularly

repeating function, such as the one illustrated in figure 5(a), which is of

finite duration T, as all phy si cal signals are, is a sum of terms given by

the equation

where f^ is the repetition rate of the physical signal (see Appendix B)

Each term of this sum can be plotted as an individual spectral amplitude

distribution centered at a frequency f = nf^. Three such terms are shown in

figure 5(b) for n = 0 and n = ± 1. As the time interval T becomes longer

and longer, the amplitude distributions become more and more compact in the

frequency dimension, and, as T approaches infinity, the total amplitude

distribution looks like a line spectrum with vast regions of zero amplitude

V (f) = Axf T I r o _

sin u(f-nf^)T"

(14)

irnf T o TT(f-nf^)T

5
between lines. However, in these regions between the frequencies where the

amplitude distribution is bunched up in clumps, V^(£) is not zero but only

infinitesimally small. Because of this limiting form, which is what is observed in practice, many authors give the erroneous impression that the

spectrum amplitude for a finite periodic signal is_ a line spectrum. And

then, through confusion with the Fourier series line spectrum, this can mislead one into thinking that these lines represent the amplitudes of sinusoids

at specific frequencies . Such is not the case when we are dealing with the

Fourier trans form . The ordinate values still have units of vo It - seconds

and are not amplitudes in volts.

"

' ' 3. APPLICATIONS OF VCf)

Thus far, we have talked about the nature of V(f) . Now let us look at

how V(f) is used, and this will lead us to identify a spectral ampl itude . One basic mathematical use of V(f) is to obtain the time-domain equation of the physical signal by taking the inverse Fourier transform of V(f).

Here we can write the inverse transform equation in a form that shows the

amplitude and phase of V(f). For our example, v^Ct) is given by the equation

( see Appendix C)

no sin TTfi V (t) = At /

TTf T cos 2TTf (t-T/2) df . (15)

Again we can see that the value of v(t) at any time t is an intergral over frequency of a function of frequency. Thus, v(t) is an AREA which, in our example, has the dimensional units of vo 1 1

. Figure 6 shows a portion of a three-dimensional plot of the integrand,

z(f,t), of eq . (15), where

z(f,t) = At

sin Tift

TTft cos ZTTf (t-T/2) . (16)

This wavy surface has a "sine x over x" shape in the frequency direction

at time t = t/2 and a cosine shape in the time direction. The surface

extends to plus and minus infinity in both the time and frequency domains.

The area under the function z (f,t) , at a specified time t, is the value

of Vp(t) at that time (see fig. 7).

Because v(t) for every physical real-world signal is a purely real

function of time, V(f) is Hermitian, mathematically speaking, which simply

means that its real part is always an even function, its imaginary part is

always odd , and

V(f) = v*(-£) • (17)

6
It is this mathematical property that allows us, and this is important in

a practical sense, to write

00 v(t) = 2 / I V(f) I cos [2^£t + <i)(t)]d£ (18)

o

for ALL real-world signals, not just for our example. This form of mathematical representation has the engineering virtue that the integral is over

the pos it i ve frequency domain only.

The spectrum amplitude

S(f) = 2|VCf) I (1)

thus appears explicitly in eq. (18).

The quantity 2|V(f)| also appears in most other working equations where

a spectrum amplitude is involved, some of which I will show in a moment. I believe, therefore, that this quantity has utility as a measurand of the

physical signal, and should be adopted by the EMC community as the quantity

which is identified by the units of volts per hertz. I would suggest that it

be named Spectrum Amplitude Density , or perhaps Ampl itude Density . The name,

spectrum amplitude density, would be in keeping with the frequency- domain

characteristics that we have discussed. But the name is a matter for standards committee to decide.

S(f) appears in other applications, some of which are as follows:

1. If v(t) is applied to a two-port network which has an impulse response

h(t), the response, r(t), is given by the equation

00 r(t) = / S(f) • |H(f)| • cos[2TTft + (l)(f) + e(f)]df (19)

o where |H(f)| and 9(f) are the magnitude and phase of the transfer function

of the network. It is this application for which the knowledge of spectrum

amplitude has its greatest practical application, namely, for the calibration of Field Intensity Meters and Noise Meters.

2. .Another application is the measurement of the impulse bandwidth, IB,

of an idealized b andpass f i Iter [2] using the equation

IB = (20)

S(f)G

where Rj-jj^g^-j^-j is the maximum value of the envelope of the network response

r(t) , and G is the network maximum gain.

7
3. The total energy per unit resistance, , contained in the physical

signal is given by the equations

oo oo E = / v^(t)dt = / |VC£)r d£ (21)

- 00 - oo

- 00 = \ j S^(£)d£. (22)

0

The part, E^ , o£ E^, that lies in the £requency band £rom f-^ to is given

by the equation

E^ = ^ / SHf)df. (23)

£i

The total energy per unit resistance, E^ , contained in the output signal

£rom a two-port network is given by the equation

oo

/ . E^ = i- / S^(£) • |H(f) I ' d£. (24)

o Finally, i£ we were to de£ine a root mean square spectrum amplitude, S^^^ [2],

as

S = = /2 I V(£) I , (25) rms ^2

then the square o£ S^^^ is 2|V(£)|^, and the last three energy equations

become

E^ = / d£ (26)

t rms ^ o

£2

E, = / d£ (27) b ir rms ^

^ 1

E = /

. |H(£) I 2 d£. (28) n rms \ ^ > i ^ o

4 . SUMMARY

In summary, we have seen that:

First, the frequency-domain representation o£ a physical signal is a complex mathematical quantity with both a magnitude and a phase.

Second, this complex quantity, the Fourier transform, is an ampl itude

distribution funct ion -- a type of dens ity function -- and is not simply the

amplitudes of individual spectral components. It must be treated as a density function and not as an amplitude.

8
Third, the Fourier transform o£ any and every physical signal is

continuously defined throughout the entire frequency domain . It is not

a discrete or line function of frequency.

Fourth, twice the magnitude of this Fourier transform is a useful

engineering quantity because it appears in this form in many useful algebraic relationships. However, one may also need to know the phase of V(f).

Fifth, we should give this quantity a descriptive name, such as Spectrum

Ampl itude Dens ity , and should agree upon one definition to the exclusion

of all others. Such a definition should be in terms of signal parameters

and not in terms of measurement network parameters as is found in parts

of the literature.

Sixth, the preferable dimensional units of spectrum amplitude may

be volt -seconds , although there are valid arguments for both volt-seconds

and volts per hertz.

Finally, a root -mean-square spectrum amplitude density can be defined

which may have utility for engineering purposes, especially when dealing with

signal power . Although I have limited this review to a few of the basic matters, I hope it has been helpful in providing the basis for a better understanding

of spectrum amplitude, and for a more careful use of it in the future.

5. REFERENCES

[1] IEEE Standard Dictionary of Electrical and Electronics Terms, IEEE

Std. 100-1972, John Wiley § Sons, Inc., New York, 1972, p. 549.

[2] The Measurement of Impulse Strength and Impulse Bandwidth, proposed

standard by IEEE Subcommittee 27.1, Dr. R.M. Showers, chairman (in

final stages of approval)

[3] Westman, H.P. (editor), "Reference Data for Radio Engineers," Fifth

Edition, Howard W. Sams 5 Co., Inc., New York, 1968, Chapter 42.

[4] Bracewell, R. , "The Fourier Transform and its Applications," McGrawHill Book Company, New York, 1965.

[5] Lathi, B.P., "Signals, Systems and Communication," John Wiley § Sons,

Inc., New York, 1965, Chapter 3.

[6] Lathi, ibid. , p. 130.

9
10
11
Q.

I I /

/

<D 4-

12
/

13
f

1

H| C\J O

I ^

k J; X!

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in

14
15
4-> M-N

I

16
APPENDIX A

For the baseband pulse, figure 1,

Vp(t) = A, 0 < t < T

= 0 elsewhere

The Fourier transform, Vp(f), of v^Ct) is

oo (f) =

/ V

(t)e"^2TTft

loo P

= A /^e-j2^ft dt

(29)

(8)

(9)

^_^^-j2Trft^

j2^f

A

[j(e-j2-£- - 1)]

[sin 2Ti-fT + j (cos 2TrfT - 1)]

27Tf A

2'fff A

2uf

[sin 27TfT - j2 sin^ Trfi]

Writing Vp(f) in polar form.

where

Vp(f) A

V (f) =

I V (f) I e P

[(sin 27TfT)2 + (-2 sin^ ufx)^]^^^

= At

2TTf

A

2^f

A

2^f

sin TTfT[cos^ irfr + sin^ irfx]

sin TTfx

[sin^ 2TrfT + 4 sin"* nfx]^^^

[4 sin^ TTfT cos^ TTfT + 4 sin"* -nfj]^^^

1/2

irf T

(4)

(30)

(6)

17
and

The re fore

,

(J)^(£) = tan'

= tan

-1

-2 sin^ TTfx

^ sin 2tt£t

-2 sin^ TTfT

(I sin irfT cos tt£t

= tan '"(-tan irfr)

= -tt£t.

VpC£) = At

sin irfx

TT £t

-

j 7t£t

(7)

(5)

18
APPENDIX B

For the finite regularly repeating function shown in figure 5(a), v^(t)

is the convolution of an eternal periodic function, Vg(t), and a gate

function , g(t)

,

where

v^(t) = Vg(t) * g(t) (31)

Vg(t) = A, 1 < t < -

1 <t <^o

T

"

2 < t < -2

= A,

1 0

< t <

< t <

= 0

^ +l<t< 2 _T

o o

O 0

etc (32)

and

^ < t < TT- 2 2 g(t) = 1,

= 0 elsewhere (33)

From the t ime - convo lut ion theorem,

Vg(t) * g(t) VJf)G(f) (34)

where V (f) and G(f) are the Fourier transforms of v (t) and g(t) , respecp e tively. The symbol (-^-*-) reads "is the transform of" in both directions.

The Fourier transform of v^(t) is [6]

n = oo sin irnf x V (f) = ATTf y ^ • 6(f-nf ) e ^ 0 ^ r ^0^

n=-°° TTnf T o

(35)

where 6 is the unit impulse function. The Fourier transform of g(t) is

^ sin irfT

G(f)

TTfT (36)

19
which is similar to eq. (4) except for the zero imaginary part which results

from the choice of time origin. Therefore, the Fourier transform of v (t) is

V^(f) = V^(f)G(f)

n=°° sin Trnf t

= Axf T I 5_ . 6(£-nf ) TTfT n=-°° TTnf T 0

Axf T

o

n = ~ sin irnf t sin TT(f-nf )T

'l_n=-o° TTnf^T wCf-nf^)! J (14)

20
APPENDIX C

For the baseband pulse, figure 1, we use eqs . (3) and (5) to obtain

oo = ^ V (£)e^^''^^ df

^ ~ CO ^

= At /

- c

c = At /

- C

c = At /

sin irfT

tt£t

sin 7t£t irfT

sin TrfT

Trfx

J(2^£t-^£T)

[cos 2u£(t-T/2) - j sin 27r£(t -t/2) ]d£

cos 27T£(t-T/2) d£ (15)

since

sin tt£t

Tr£T sin 27T£(t-T/2)d£ = 0.

21
-

I, VI NBS-1 14A (1-71

)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 74-3652. Gov't Accession No.

3. Recipient's Accession No.

4. TH I.I-: AND SUHT1T1T-: IMPULSE SPECTRAL INTENSITY — WHAT IS IT?

5. Publication Date May 1974

6. Perfonning Organization Code

7. MiTHORCS)

M. G. Arthur

8. Performing Organization

9. P1:K1- ORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OE STANDARDS

DEPARTMENT OE COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No. 2728281

11. Contract/Grant No.

12, Sponsoring Orgtin iz3t it^n Ntirnc tinJ Address

Same as Item 9.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The term, impulse spectral intensity, is often used in

discussions concerning electromagnetic interference and

broadband signal processing. In these discussions, the

term is not used in a consistent manner, resulting in

confusion, equivocation, and sometimes, errors. As a step towards improving this situation, the mathematical

basis for spectral intensity is reviewed and certain of

its features are clarified. Also, misuses of spectral

intensity are discussed, along with its limitations and

proper use.

17. KEY WORDS (Alphabetical order, separated by semicolons) ^-i t j j_ • • j

' f

" Electromagnetic interference;

Fourier transform; impulse spectral intensity; spectral intensity;

spectrum ampl i tude ; spectrum amplitude den.si

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