Module 3:Signals and Spectra

3-2

3.0 Introduction

An understanding of the different types of signals typically present in electrical systems, andthe frequency content of such signals is critical to the design of electromagnetically compatiblecircuits. Many common signals contain high frequency components which may, under certaincircumstances, act as sources of interference. Three types of signals will be examined in thismodule, which are found in many electrical systems. These include narrowband continuous wave(sinusoidal) signals, repetitive broadband signals (such as digital clock signals), and single eventbroadband signals (such as spark discharges). The emphasis of this module will be to presentmaterial required for an understanding of the frequency spectra of these types of signals. Inparticular, the relationship between rise time, wave shape and spectral content of a signal will beexamined.

3.1 Classification of signals

For the purposes of the material presented here, a signal is considered to be either a voltage orcurrent waveform that is described mathematically. Signals may be classified in many ways. Afew of the more common classifications are listed below.

• energy signals

The instantaneous power dissipated by a voltage v(t)in a resistance R is given by

p(t) � �v(t)

�2

R

and for a current i(t)

p(t) � � i(t) � 2R .

It can be seen that the power in each case is proportional to the squared magnitude of thesignal. If these signals are applied to a 1 ohm resistor, then both of the equations aboveassume the same form. For a general signal f(t)

p(t) � � f (t) � 2 .

The energy associated with this signal during a time interval from t 1 to t2 is given by

Ef��� t2

t1

�f (t)� 2dt .

A signal whose energy remains finite over an infinite time interval

3-3

Ef��� �� �

�f (t)� 2dt < �

is referred to as an energy signal.

• power signals

The average amount of power dissipated by a signal f(t) during an interval of time from t1to t2 is

P � 1(t2 t1)

t2

t1

�f (t)� 2 dt .

A signal which satisfies the relationship

0 < limT �� 1

T T/2

� T/2

�f (t)� 2dt < �

has finite average power, and is referred to as a power signal.

• deterministic signals

A signal whose behavior is precisely known is referred to as being deterministic. Theseinclude sinusoidal signals, and digital clock signals. Usually, such signals can be representedby explicit mathematical expressions.

• non-deterministic signals

A signal whose behavior is not known, and which can only be described statistically isreferred to as being non-deterministic, or random. Digital data signals are often non-deterministic.

• periodic signals

Repetitive, time-domain signals, such as the clock signals often present in digital devices,are referred to as being periodic. A function f(t) is said to be periodic if it satisfies therelationship

f ( t ) � f ( t ± nT0 ) for n � 1,2,3, ....

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for every time t, where T0 is the period of the function. The fundamental angular frequency,�o, of a periodic function is

�o � 2 � f0 � 2 �

T0

.

Periodic signals are classified as power signals because their average power is finite. Sinusoidal signals represent a class of periodic signals that are commonly used in many

analysis techniques. These techniques, such as those involving Fourier series, decomposecomplicated waveforms into a series of sinusoidal waveforms. A sinusoidal waveform f(t) isusually represented by

f (t) � Acos( � t ��� )where A is the amplitude of the signal, � is the phase, and � is the angular frequency inradians per second ( � =2� f ).• non-periodic signals

A non-periodic waveform is one that does not satisfy the criteria for a periodic waveform. Non-periodic signals are referred to as energy signals because their total energy is finite.

3.2 The electromagnetic spectrum

In the previous chapter it was seen that, in a source free region, and satisfy�

E�

Bhomogeneous wave equations. For the case of time harmonic excitation, these equations reduceto homogeneous Helmholtz equations, the solutions of which represent propagating waves. Thuselectromagnetic energy is transferred in the form of waves which propagate at velocities thatdepend on the medium of transmission ( ), and oscill ate at frequencies that depend onv � 1/ µ �the nature of the source.

Electromagnetic radiation is typically classified according to frequency or wavelength(although in the range including visible and ultraviolet light, x-rays, and � -rays, radiation issometimes classified according to photon energy). The electromagnetic spectrum, shown inFigure 1, is divided into frequency and wavelength bands. At the low frequency end lie the radiobands. Above this is the microwave region, which occupies the range from about 1 GHz to thelower infrared band, and contains the UHF, SHF, EHF, and millimeter-wave bands. The so-calledvisible spectrum extends from 4.2 x 1014 Hz (deep red, 720 nm) to 7.9 x 1014 Hz (violet, 380 nm). At higher frequencies lie ultraviolet light, x-rays, and � -rays.

It is important to note that certain sections of the electromagnetic spectrum are reserved byregulatory agencies such as the FCC. Broadcasts and emissions in these bands are subject to lawsand regulations established by such agencies.

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10 Hz25

10 Hz

0 Hz

100 Hz

1 kHz

10 kHz

100 kHz

1 MHz

10 MHz

100 MHz

1 GHz

10 GHz

100 GHz

1THz

10THz

100THz

10 Hz15

10 Hz16

10 Hz17

10 Hz18

10 Hz19

10 Hz20

10 Hz21

10 Hz22

10 Hz23

10 Hz24 3 10 nm

× -7

3 10 nm

× -8

Infinite

3 10 m

× 7

3 10 m

× 6

3 10 m

× 5

3 10 m

× 4

3 10 m× 3

3 10 m

× 2

30 m

3 m

30 cm

3 cm

3 mm

.3 mm

3 10 nm

× 3

3 10 nm

× 4

3 10 nm

× 2

3 10 nm

×

3 10 nm

× -1

3 10 nm

× -2

3 10 nm

× -3

3 10 nm

× -4

3 10 nm

× -5

3 10 nm

× -6

3 nm

Very Low Frequency (VLF)

LFM

FH

FVH

FU

HF

SHF

EHF

InfraredVisible

Ultraviolet

X-R

ayG

amm

a-Ray

Cosm

ic-Ray

Audio

Range

AM Broadcast

FM Broadcast

Microw

ave

Visible

Range

InfraredU

ltraviolet

Gam

ma-R

ay

Cosm

ic-Ray

X-R

ay

3 kHz

300 GHzR

adioS

pectrum

3 kHz

300 kHz

30 kHz

300 kHz

30 MHz

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3 GHz

300 MHz

3 GHz

300 GHz

30 GHz

Not

Allocated

9 kHz

Radio

Navigation

14 kHz

190 kHz

Fixed

Maritim

e Mobile

AeronauticalR

adionavigation

AeronauticalR

adionavigation

535 kHz

1605 kHz

AM

Rad

ioB

road

casting

4 MHz

3.5 MHzAmateur

Fixed, Mobile,

Miscellaneous

54 MHz

72 MHz

108 MHz

88 MHz

Land Mobile,

Mobile,

Fixed

TV BroadcastChannels 2-4

TV BroadcastChannels 5-6

FM RadioBroadcast

136 MHz

118 MHz

AeronauticalRadionavigation

AeronauticalMobile

174 MHz

216 MHz

TV BroadcastChannels 7-13

400 MHz

Mobile,

Fixed

470 MHz

608 MHz

806 MHz

TV BroadcastCh. 21-36

TV BroadcastChannels

38-69

902 MHzLand Mobile

1215 MHz

960 MHzAeronautical

Radio-navigation

Radionavigation, R

adioAstronom

y, Fixed,M

iscellaneous

4.2 GHz4.4 GHz

Radio-

location

Radionavigation

Radionavigation, R

adiolocation, Satellite Navigation, R

adio Astronomy, Earth

Exploration Satellite, Space Research, Am

ateur, Fixed, Mobile, etc.

Activities Frequency Frequency Activities Wavelength

2.85 MHz

3.155 MHz

AeronauticalMobile

Mobile, Fixed,

Radiolocation,

Amateur

Miscellaneous

Figure 1.

The ele

ctromagnetic spe

ctrum.

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Microwave frequency band designations

Old New Frequency Range (GHz)Ka K 26.5 - 40K K 20 - 26.5K J 18 - 20Ku J 12.4 - 18X J 10 - 12.4X I 8 - 10C H 6 - 8C G 4 - 6S F 3 - 4S E 2 - 3L D 1 - 2UHF C 0.5 - 1

3.3 Series expansions and basis functions

Complex signals which are periodic can be represented as linear combinations of simplersignals known as basis functions. Thus a periodic signal f(t) with period T may be represented

f (t) �����n � 0

cn n (t)

! c0 0(t) " c1 1(t) " c2 2(t) " ...

where the functions n(t) are periodic, having the same period as f(t), and the coefficients cn arereferred to as expansion coefficients. The best choice of basis functions depends on the signal f(t)that is to be represented.

• orthogonality of basis functions

Regardless of the type of basis function selected, the process of determining the expansioncoefficients cn is greatly simplified if the basis functions possess the property

#t1 $ T

t1

%n(t)%'&

m(t) dt ( ) m for m ( n

0 for m * n

where * indicates the complex conjugate. A set of functions + n(t) having this property is saidto be orthogonal. If both sides of the signal expansion above are multiplied by , and+',m (t)then integrated over time interval T, it is seen that

3-7

-t1 . T

t1

+ ,m(t) f (t) dt /�021n 3 0

cn 4t1 5 T

t1

6'7m (t) 6 n(t)dt

8 cm 9 m

or

cn8 19 n

4t1 5 T

t1

6'7n(t) f (t)dt .

This type of signal expansion, using a series of basis functions to represent a morecomplex signal, is useful to mathematicians and engineers because the response of a linearsystem to a complex periodic signal can be determined by finding the linear superposition ofthe responses to much simpler inputs. This representation is useful to EMC engineers for aslightly different reason, however. If the basis functions are chosen correctly, this seriesexpansion corresponds to a complex signal composed of many individual single frequencysignals. Thus a square wave, like a digital clock signal, can be thought of as being composedof many components of different frequencies, each oscill ating at some integer multiple of afundamental frequency f0, any one of which may potentially “escape” and act as a source ofinterference. All that is left is to select basis functions which will form a series that properlyrepresents a particular signal.

3.4 Fourier series

The most common series representation of periodic signals is a trigonometric Fourier series. This type of series uses sinusoidal basis functions, such that:

n(t) ; 1 ... for n ; 0

and

:n(t) ; cos(n < o t)

sin(n < o t)... for n ; 1, 2,3 ,

where .< 0 ; 2 =T

Thus a periodic signal f(t) with period T may be represented as

f (t) ; a0 >@?�An B 1

an cos(n < o t) >@?�An B 1

bn sin(n < o t)

3-8

where

ao ; 1T Ct1 D T

t1

f (t)dt

an E 2T Ct1 D T

t1

f (t)cos(n F o t)dt

and

bn E 2T Ct1 D T

t1

f (t)sin(n F o t)dt .

Here a0 is the average value of the signal, the terms for which n=1 are referred to as fundamentalfrequency terms, and the terms for which n>1 are referred to as harmonic terms.It is seen that the fundamental terms have frequency f0 = 1/T, while the second harmonic termshave frequency 2f0 = 2/T, the third harmonic terms have frequency 3f0 = 3/T, and the nth harmonicterms have frequency nf0 = n/T.

The an and bn terms of the Fourier series are found using the following orthogonali ty properties:

Ct1 D T

t1

sin(n F o t ) cos(m F o t )dt E 0

.Ct1 D T

t1

sin(n F o t ) sin(m F o t )dt E Ct1 D T

t1

cos(n F o t ) cos(m F o t )dt E 0 for n G m

T2

for n E m

• complex exponential Fourier series

An equivalent and more useful form of the Fourier series discussed above can be obtainedby applying Euler’s identity

ejnH t I cos(nJ t) K j sin(nJ t)

This gives

3-9

cos(nJ t) I ejnH t K e L jnH t2

and

sin J t I ej H t M e L j H t2j

.

When expressed in complex exponential form, the basis functions are

Nn( t ) I e

j n H o tfor n IOMQP , ..., R 1,0, 1, ..., P

and have the orthogonality property

St1 T T

t1

e U jm V o te

jn V o tdt W 0 for n X m

T for n W m.

Using exponential basis functions, a periodic signal f(t) of period T may be expressed

f ( t ) WZY2[n \̂ ] [ cn e

jn _ o t.

Multiplying both sides of this expression by and integrating over a period T`'am ( t ) b e

] jm _ o t

results in

ct1 d Tt1

e e jm f o tf ( t )dt gih�j

n k e j cn

ct1 d T

t1

e e jm f o te

jn f o tdt

which after application of the orthogonality relation yields

ct1 d T

t1

e e jm f o tf ( t )dt g cmT .

From this the expansion coefficients are found to be

cn g 1T

ct1 d T

t1

f ( t ) e e jn f o tdt .

3-10

It is often easier to compute these expansion coefficients of the complex exponential Fourierseries, than the coefficients of the trigonometric Fourier series.

It can be seen that the complex exponential Fourier series contains both positive-valuedharmonic frequencies ( l o, 2 l o, 3 l o, ...) and negative-valued harmonic frequencies (- l o, -2 l o, -3 l o, ... ). Also the expansion coefficients of the complex Fourier series may themselves becomplex, while the expansion coefficients of the trigonometric Fourier series are real valued. The expansion coefficients associated with the positive and negative harmonic frequencies areconjugates of each other

c m n n 1T ot1 p T

t1

f ( t ) ejn q o t

dt

n c rn .

The complex exponential Fourier series representation of f(t) may be written

f ( t ) n co sut�vn w 1

cn ejn x o t y{z}| v

n w | 1 cn ejn x o t

Switching the indices of the second summation to positive values then gives

f ( t ) ~ coyuz v

n w 1

cn ejn x o t yuz v

n w 1

c �n e | jn x o t

~ coy z v

n w 1

cn ejn x o t y z v

n w 1

cn ejn x o t � ~ co

y 2Re z vn w 1

cn ejn x o t

~ 2Re z vn w 0

cn ejn x o t ~ z v

n w 0

2Re (cn r

yjcn i

) (cos(n � o t ) � jsin(n � o t ) )

���2�n � 0

2 cn rcos(n � o t ) � cn i

sin(n � o t )

���2�n � 0

ancos(n � o t ) � ���n � 0

bn sin(n � o t )

3-11

where

, and .an� 2cn r

bn� � 2cn i

• justification for use of complex-exponential Fourier series

How do we know that a complex exponential Fourier series like the one presented aboveis an accurate representation of a signal? Let Sn(t) be an approximation of a periodic functionf(t) with period T. Sn(t) is a 2n+1 term sum of exponentials

Sn( t ) � c � n e� jn � o t �����

c � 2 e� j2 � o t �

c � 1 e� j � o t �

c0�

c1 e jt � c2 ej2 � o t �����

cn ejn � o t

��� nm �Q� n cm e

jm � o t.

The error of this approximation can be defined as

�n(t) � f (t) � Sn(t) .

Therefore, the approximation, Sn(t), is equal to the original function, f(t), minus an error term. The mean square error is defined as

M � 1T �T/2

� T/2

� 2n (t) dt .

To improve the accuracy of Sn(t), the coefficients cm are chosen in such a way that M is aminimum. From the expression above, it is seen that

M � 1T �T/2

� T/2

f ( t ) � Sn( t ) 2 dt

or

M � 1T �T/2

� T/2

f ( t ) ��� nm ��� n cn e

jm � o t 2dt .

3-12

The term M is minimized by setting the derivative of M with respect to cn equal to zero �

M�cl � 1

T �T/2

� T/2

2 f ( t ) �@� nm   � n cn e

jm ¡ o t ¢ � e jl ¡ o tdt � 0

where l = -n, ..., -2, -1, 0, 1, 2, ..., n. This gives

�T/2

� T/2

f ( t )ej l ¡ o t

dt � � nm   � n cn �T/2

� T/2

ejm ¡ o t

ej l ¡ o t

dt .

Using the following orthogonality relationship (where l and n are integers)

�T/2

� T/2

ejl ¡ o t

e� jm ¡ o t

dt � 0 ... l £ mT ... l � m

the Fourier coefficient cl is found to be

cl � 1T �T/2

� T/2

f ( t ) e� jl ¡ o t

dt .

It can be shown that as n approaches infinity, the error term M goes to zero. Thus, anyfunction satisfying certain conditions (to be discussed in the next subsection) can berepresented by an infinite Fourier series.

f ( t ) � �2¤n ¥�¦ ¤ cn e

jn § o t.

• existence of the Fourier series

The existance of a convergent Fourier series is guaranteed if the function f(t) satisfies the Dirichlet conditions:¨ f(t) is absolutely integrable over one period (weak Dirichlet condition)©t1 ª T

t1

| f ( t )| dt < «

3-13

¬ f(t) is finite for all t in a period T0¬ f(t) has a finite number of maxima and minima in a period T0¬ f(t) has a finite number of discontinuities in a period T0

If the requirements above are satisfied, the existence of a convergent Fourier series isguaranteed.

• properties of Fourier coeff icients

Direct computation of the expansion coefficients for the trigonometric and complexexponential Fourier series discussed in the sections above can prove difficult for certainwaveforms. For piecewise linear waveforms, certain properties of the Fourier series can beexploited to make this less difficult.¬¬ ¬¬ linearity

A waveform may be expressed as a linear combination of two or more functions

.f ( t ) ­ a1 x1( t ) ® a2 x2( t ) ® a3 x3( t ) ® ...As a result, the Fourier series representation of f(t) can be written as a linear combinationof the Fourier series representations of , etc.x1( t ), x2( t ), x3( t ), ...

.f ( t ) ­ a1 ¯2°n ±^² °

c1n ejn ³ o t ® a2 ¯�°

n ±´² °c2n e

jn ³ o t ® a3 ¯�°n ±^² °

c3n ejn ³ o t ® ...

­ ¯�°n ±^² °

[ a1 c1n ® a2 c2n ® a3 c3n ® ... ] ejn ³ o t

µ time-shifting

The Fourier coefficients of a waveform f(t) that has been shifted forward or backwardin time by an amount ¶ can be found directly from the expansion coefficients of f(t). TheFourier series representation of a time-shifted waveform is

f (t ± ¶ ) ·Z̧2¹n º̂ » ¹ cne

jn ¼ o(t ± ½ )¾Z¿ ¹

n º̂ » ¹ cne±jn ¼ o ½ ejn ¼ o t

.

3-14

Thus it can be seen that the expansion coefficients of f( t ± À ) can be obtained bymultiplying the expansion coefficients of f(t) by .e

±jn Á o ÂÃ unit impulse function

A third important property of Fourier series involves the so called unit impulsefunction Ä (t). This function is defined such that

Ä ( t ) Å 0 ... for t < 0

0 ... for t > 0

or

Æ( t ) Å 0 ... t Ç 0

and

ÈÉ�ÊÌËÍ�ÎÌË^Ï ( t Ð´Ñ ) f ( t ) dt Ò f ( Ñ ) ... Ó Ô >0

where 0- and 0+ are infinitely small intervals of time just before and after t=0, respectively. This function is zero everywhere except at t=0, where it takes on an undefined value. Theunit impulse may be thought of as having zero width and infinite height, and is usuallyrepresented by a vertical arrow when plotted.

As an example, consider a signal which consists of a periodic train of impulse functions

.f ( t ) ÒÖÕ�×k ØÚÙ ×

Û( t Ü kT )

The Fourier expansion coefficients associated with this signal are

cn Ý 1T ÞT/2

Ù T/2

Û(t )e Ù jn ß o t

dt à 1T

.

If f(t) is shifted by an amount of time ±á , the expansion coefficients are

3-15

cn à 1T

e±jn ß o â .

ãã ãã differentiation property

The Fourier coefficients cn, can be found by using a property of the Fourier seriesinvolving the derivatives of f(t). The Fourier coefficient of the i th derivative of cn, has asimple relationship with cn . As stated previously, a function f(t) can be represented by aninfinite Fourier series

f ( t ) àåä�æn ç^è æ cn e

jn é 0 t.

Taking the time derivative of both sides of this expression yields

d f ( t )dt ê ä�æ

n ç^è æ ( jn ë o )cn ejn ì o t

.

From this it is seen that the i th derivative with respect to time is given by

d (i ) f ( t )

dt ( i) íZî2ïn ð^ñ ï( jn ë o ) i cn e

jn ì o t

Thus it is seen that the expansion coefficient associated with the ith derivative (c(i)) isrelated to cn by

c ( i)n í cn ( jn ë o ) i .

• Fourier coefficients of a rectangular pulse train

The properties above are often employed when determining the Fourier coefficients of aparticular waveform. To accomplish this the following general procedure is followed:ò A waveform is repeatedly differentiated until the first occurrence of an impulse

function.ò If the resulting function does not consist solely of impulse functions, then the waveformis expressed as the sum of a component consisting only of impulse functions, and aremainder.ò Expansion coefficients are determined for the component of the waveform which

3-16

consists of impulse functions.ò The remainder is differentiated, and the entire process repeated until only impulsefunctions remain.

As an example, the Fourier coefficients of a rectangular pulse train will be computed firstby direct integration, and then using the properties listed above to show that the results areequivalent. A diagram of the rectangular pulse train is shown in Figure 2.

f (t)

2

T0 T0

2

T0-T0-

2

T0- 3

2

T03

A

Figure 2: Rectangular Pulse Train

The Fourier coefficients associated with this pulse train are given by (for a period T0 andarbitrary time t1)

cn ó 1T0

ôt1 õ T0

t1

f ( t ) e ö jn ÷ o tdt .

Letting t1 equal -T0 / 2, cn becomes

cn ø 1T0

ùT0 /2

ú T0 /2

f ( t ) eú jn ÷ o t

dt .

The rectangular waveform effectively truncates the limits of integration, thus

3-17

cn ø 1T0

ùT0 /4

ú T0 /4

A eú jn ÷ o t

dt

ø AT0

eú jn ÷ o tû jn ü o

T0 /4

ú T0 /4

.

Substitution of the limits of integration gives

cn ø AT0

1û j n ü o

eú jn ÷ oT0 /4 û e

jn ÷ o T0 /4

or

cn ø AT0

1j n ü o

ejn ÷ oT0 /4 û e

ú jn ÷ o T0 /4.

As stated earlier, angular frequency is

ü 0 ø 2 ýT0

or

ü o T0 ø 2 ý .

Substituting this into the expression for cn gives

cn ø Ajn2 ý e jn þ /2 û e

ú jn þ /2

ø A

n ý e jn þ /2 û eú jn þ /2

2 j.

But

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sinx ø e jx û eú jx

2j

therefore

cn ø An ý sin(n ý / 2)

ø A2

sin( n ý /2 )n ý / 2

.

Finally

cn ø A2

sinc (n ý / 2)

where

sinc(x) ø sinxx

.

The coefficients cn will now be found by computing cn(1), the Fourier coefficients for

. A diagram of is shown in Figure 3. The Fourier coefficient cn(1) isdf ( x) /dx d f (x ) /dx

c (1)n ø 1

T0

ùT0 /2

ú T0 /2

d f ( t )dt

eú jn ÷ o t

dt .

Using the property

cn ø c (1)n

j n ü o

it is seen that

.cn ø 1j n ü o T0

ùT0/2

ú T0/2

df ( t )d t

eú jn ÷ o t

dt

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2

T0 T0

2

T0-T0-

2

T0-3

2

T03

d f(t)

d t

A

-A

Figure 3: The Derivative of the Rectangular Pulse Train

Substituting , and the values representing the derivative of the waveform T0 ÿ o � 2� df ( t ) dtinto the expression above gives

.cn � 1j n2 � �T0 /2

� T0 /2

A � ( t � T0 /4 ) � A � ( t � T0 / 4) e� jn � 0t

dt

This leads to

cn � Ajn2 e

jn � 0T0/4 e � jn � 0T0 /4

� A

n e jn � /2 e � jn � /2

2 j

� An sin n

2

� A2

sinc n2

.

3-20

This is the same result that was obtained through direct integration. :-)

3.3 Trapezoidal waveforms

Digital clock signals are an extremely important group of waveforms that need to bestudied. Clock signals are often approximated as square waves, however this is not veryaccurate. All real digital clock waveforms have a certain non-zero rise time, and a non-zero falltime. A better approximation of a digital pulse train is a trapezoidal waveform, as shown inFigure 4.

A/2

A

x(t)

rt

fT

Figure 4: Approximation of a digital clock signal.

As with the waveforms examined previously, the fundamental angular frequency � 0 of thetrapezoidal pulse train is given by

�o � 2 � f0 � 2 �

T0

.

It will be seen that rise time � r and fall time � f greatly affect the spectrum of a waveform. The rise time of a pulse train is defined to be the length of time needed for the signal totransition from 0 to A, where A is the amplitude of the signal. The fall time is then the length oftime needed for the signal to transition from A to 0. The pulse width � is the time needed forthe signal to transition from 0.5A on the rising part of the signal to 0.5A on the falling part ofthe signal.

• Fourier series of a trapezoidal waveform

The spectrum of a trapezoidal waveform will be determined to demonstrate the effects ofvarious rise times, and duty cycles. To accomplish this, the waveform shown above is firstexpanded in a Fourier series. The expansion coefficients are given by

3-21

cn � 1T

t0 � Tt0

f ( t ) e � jn � o tdt .

Due to the nature of the waveform, two derivatives with respect to time are required toproduce a train of impulse functions

c (2)n � 1

T

t0 � Tt0

d 2 f ( t )

dt2e � jn � o t

dt .

which becomes

c (2 )n � 1

T

0 �0 � A�

r � ( t ) e � jn � o tdt � � �1� �

1

A�r � ( t � 1 ) e � jn � o t

dt

� � �2� �2

A�f � ( t � 2 )e � jn � o t

dt � � �3� �3

A�f � ( t � 3 )e � jn � o t

dt

where

�1 � � r

�2 � � � � r

�f

2

and �3 � � � � r � � f

2.

Performing the various integrations gives

3-22

c (2 )n � A

T1�r

e 0 1�r

e � jn � o�

1 1�f

e � jn � o�

2 � 1�f

e � jn � o�

3

and substitution of the expressions for � 1, �

2, and � 3 given above yields

c (2 )n � A

T1�r

1�r

e � jn � o�

r 1�f

e � jn � o� � � r � � f2 � 1�

f

e � jn � o� � � r � � f2 .

If the rise time of the clock signal is equal to its fall time, which is often a goodapproximation, this expression can be simplified. Let

�r � � f

and then

c (2)n � A

T1�r

1 e � jn � o�

r e � jn � o� � e � jn � o ( � � � r )

.

Factoring out

e � jn � o

� � � r2

and grouping terms gives

c (2)n � A

T1�r

e � jn � o

� � � r2 e

jn � o

�2 e

jn � o

�r

2 e � jn � o

�r

2 e � jn � o

�2 e

jn � o

�r

2 e � jn � o

�r

2 .

Now

sin n ! o

�r

2 � ejn � o

�r

2 e � jn � o

�r

2

2j

3-23

A/2

A ""

rt

"f

T

x t( )

A ""

rt

"f

T

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"f

-

""

rt

"f

T

d x t

d t

( )

d x t

d t

2

2

( )

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"f

A

"r

A

"r

A

"r

A

"r

A

"r

A

"r

A

"f

- -

Figure 5: Diagram of the Trapezoidal Waveform and Its Derivatives

3-24

therefore

c (2)n # 2 j

AT

1$r

e % jn & o ')(*' r2 sin n + o , r

2e

jn & o '2 - e . jn & o '2and

c (2)n / - 4

AT

1, r

e . jn & o ')(*' r2 sin n + o , r

2sin n + o ,2 .

Applying the property

c (2)n / cn ( jn + o )2

gives

.cn / 4

n 2 + 2o

AT

1, r

e . jn & o '0(1' r2 sin n + o , r

2sin n + o ,2

This may be expressed as

cn / A ,T sin n + o ,212

n + o ,sin n + o , r

212

n + o , r

e . jn & o ')(*' r2

or

cn / A ,T sinc n + o ,2 sinc n + o , r

2e . jn & o '0(1' r2 .

It must be remembered that this expression for cn is only valid for the special case of ., r / , f

3-25

• magnitude spectrum envelope

The ampli tudes of the expansion coefficients determined for a particular waveform fallwithin a certain envelope. For the trapezoidal waveform examined above, this envelope is givenby

envtrap. / upper bound of: 2A ,T sinc( 2 , f ) sinc( 2 , r f )

where f = n/T. In order to quickly deduce the significance that high frequency spectralcomponents associated with a waveform such as this may have, upper bounds can be establishedon this envelope. These upper bounds represent “worst case”, or maximum possible expansioncoefficient amplitudes for a given frequency range.

The spectral bounds are established by first taking the logarithm of both sides of thewaveform envelope expression

.20log10 (envtrap.) / 20log10 2A ,T 3 20log10 sinc( 2 , f ) 3 20log10 sinc( 2 , r f )

It is seen that the first term

20log10 2A ,Thas a slope of 0 dB per decade, and a level of 2A 4 /T =2A 4 fo. The second term and third termshave the form . For small arguments , thereforesinx x sinx 5 x

67sinx

x

6718 1 ... for small x

19x9 ... for largex

which can be drawn as two linear asymptotes, the first with a slope of 0 dB/decade, and thesecond with a slope of -20 dB/decade.

Thus, the term , has an asymptote with a slope of 0 dB/decade and an20log10 sinc( :<; f )asymptote with a slope of -20 dB/decade. These asymptotes meet at . The thirdf = 1/ :<;term, , also has an asymptote with a slope of 0 dB/decade and an20log10 sinc( :>; r f )asymptote with a slope of -20 dB per decade, which meet at . The spectral boundsf = 1/ :>; rof the trapezoidal waveform therefore consist of a three segment line with two breakpoints. Thefirst segment has a slope of 0 dB/decade. The second segment, starting at , has af = 1/ :<;slope of -20 dB/decade. The third segment, starting at , has a slope of -40f = 1/ :>; rdB/decade.

If the duty cycle is short enough that the frequency is greater than the fundamental1/ ( :<; )frequency,

3-26

,1:<; > 2 :

T( ; < T

2)

then the spectrum’s envelope will have a slope of 0 dB per decade from the fundamental to thatfrequency. If the frequency occurs before the fundamental frequency, then the envelope1/ ( :<; )begins with a slope of -20 dB per decade. The envelope then begins a much sharper descent of -40 dB per decade at .f = 1/:?; r

Figure 6: Spectral Bounds on the Trapezoidal Waveform (Paul; p. 365)

• comparison of trapezoidal waveform spectra

@ relationship between risetime and signal spectrum

In Figure 7, the magnitude spectra of two trapezoidal waveforms with different risetimes areplotted as a function of frequency. The risetime of waveform 1 is 0.1 ns and the risetime ofwaveform 2 is 0.5 ns. Both waveforms have 5V amplitudes, with expansion coefficientmagnitudes plotted in dBµV. The fundamental frequencies of both waveforms are 100 MHz,and the duty cycle for both waveforms is 50% .( ACB 0.5T )

As shown above, the envelope of the magnitude spectrum has a slope of 0 dB per decadeuntil i t reaches the frequency

.1D A B 1D .5×1/ (100MHz)B 63.66MHz

3-27

This frequency, however, is lower than the fundamental frequency itself. Therefore, theenvelopes are plotted beginning with a slope of -20 dB per decade at the fundamental frequencyof 100 MHz. Both envelopes decrease at a rate of -20 dB per decade until

20

30

40

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80

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100

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108

109

10102 3 4 5 6 7 8 2 3 4 5 7 86 2 3

Cm( )dB Vµ

130

( )Hzf

Trapezoidalwaveform with 100MHz fundamentaland .1 ns rise time

Trapezoidalwaveform with 100MHz fundamentaland .5 ns rise time

Fundamentalfrequency

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-20 dB perdecade

-40 dB perdecade

1

1πτr

1

2πτ r

636.6M H z

3.18GHz

Figure 7: Magnitude spectra of trapezoidal waveforms with different rise times

3-28

envelope 2 reaches its second corner frequency at

1D A r2

B 1D ( .5×10E 9)B 636.6MHz .

After this, envelope 2 decreases at a rate of -40 dB per decade while envelope 1 continues todecrease at -20 dB per decade. The second corner frequency for envelope 1 occurs at

1D A r1

B 1D ( .1×10E 9)B 3.183GHz .

At this point, envelope 1 begins to decrease at -40 dB per decade. Above this frequency, noticethat envelope 1 is about 13 dBµV higher than envelope 2.

It is seen in this figure that the energy of a signal such as the trapezoidal clock waveform isspread over a broad range of frequencies, and that the high frequency content of such awaveform is impacted significantly by the rise time of the signal.@ relationship between duty cycle and signal spectrum

Now trapezoidal waveform with various duty cycles will be compared. In Figure 8, thespectra of two trapezoidal waveforms is plotted as a function of frequency. Waveform 1 has aduty cycle of 50%, while waveform 2 has a duty cycle of 25%. All other quantities of thewaveforms are the same (fundamental frequency is 10 MHz, rise time is 1 ns, and the ampli tudeis 5 V). As with the previous example, since the corner frequency for envelope 1

1D A 1 B 1D .5×1/ (10MHz)B 6.366MHz

is below the fundamental frequency, the plot of envelope 1 begins at the fundamental frequencyand decreases at -20 dB/decade. The first corner frequency for envelope 2 occurs at

1D A 2 B 1D .25×1/(10MHz)

B 12.73MHz .

This frequency is above the 10 MHz fundamental frequency. Envelope 2 will then have a slopeof 0 dB per decade until i t reaches 12.73 MHz, where it will begin to decrease at -20 dB perdecade. Since 12.73 MHz is so close to the fundamental, the eff ect of the 0 dB per decadeportion is diff icult to detect.

3-29

The second corner frequency for both envelopes occurs at

1D A r

B 1D (1×10E 9)B 318.3MHz .

This frequency is the same for both waveforms because they have the same rise time. Bothenvelopes begin to decrease at 40 dB per decade at this point.

Notice that although the shapes of both spectra are similar, envelope 1 is about 3 dB higherthan envelope 2 over the entire range of frequencies. This is the expected result because the25% duty cycle waveform should carry about half of the energy (-3 dB) that the 50% duty cyclewaveform carries.

The important observation here is that the high-frequency content of the signals examinedwas affected significantly by changing the risetime of the waveform. The length of thewaveform duty cycle had almost no effect on the high frequency components. Only a 3 dBdecrease of the magnitudes of all spectral components occurred, and this is because the overallenergy of the 25% duty cycle waveform is half that of the 50% duty cycle waveform.

The spectra of additional waveforms is shown in Figure 9. This figure shows the spectra oftrapezoidal waveforms with three different duty cycles (50%, 65%, and 10%). All otherquantities for the signals are identical (fundamental frequency is 100 MHz, rise time is 0.2 ns,and the ampli tude is 5 V). Let the magnitude spectra of the 50%, 65%, 10% duty cyclewaveforms be described by envelope 1, envelope 2 and envelope 3, respectively.

The first corner frequency for envelope 1 (upper left hand corner) occurs at

1D A 1 B 1D .5×1/(100MHz)B 63.66MHz .

This is below the fundamental frequency of the waveform, so envelope 1 begins to decrease at arate of -20 dB/decade. The second corner frequency of envelope 1 occurs at

.1D A r

B 1D ( .2 ns)B 1.592GHz

This frequency is the same for all three spectra, since the waveforms all have the same rise time. Envelope 1 begins to decrease at -40 dB/ decade above this frequency.

The spectrum of the 65% duty cycle waveform is shown in the upper right hand corner ofthe figure. The first corner frequency for the envelope of this spectrum (envelope 2) occurs at

.1D A 2 B 1D .65×1/ (100MHz)

B 48.97MHz

This is also below the fundamental frequency, so envelope 2 also is plotted with an initial slopeof -20 dB per decade. The second corner frequency for envelope 2 occurs at 1.592 GHz, which

3-30

was the same frequency for envelope 1. Therefore, the envelopes of the magnitude spectra forwaveforms having 50% and 65% duty cycles are the same. The

10 7 10 8 10 92 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3

20

30

40

50

60

70

80

90

100

110

120

130

Cm( )dB Vµ

( )Hzf

Fundamentalfrequency

Trapezoidal waveformwith a 10 MHz

fundamental frequencyand a 50% duty cycle

Trapezoidal waveformwith a 10 MHz

fundamental frequencyand a 25% duty cycle

The rise time is 1 ns forboth waveforms

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1

πτ

318.3MHz

-40 dB perdecade

-20 dB perdecade

r

Figure 8: Magnitude spectra of a 50% and a 25% duty cycle trapezoidal waveform

3-31

108 109 10 1 02 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3

20

30

40

50

60

70

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110

120

130

10 8 10 9 10 1 02 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3

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110

120

130

10 8 10 9 10 1 02 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3

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100

110

120

Cm( )dB Vµ

10 8 10 9 10 102 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3

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( )Hzf

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Earth Exploration Satell i te,Space Research,

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1.592GHz

1πτr

F u n d a m e n t a lf requency

Fundamentalf requency

Fundamentalf requency

Fundamentalf requency

318 .3MHz

-40 dB perdecade

-20 dB perdecade-20 dB per

decade

-40 dB perdecade

0 dB perdecade

-20 dB perdecade

-40 dB perdecade

τ = ⋅.5 T τ = ⋅.65 T

τ = ⋅.1 T

|Cm | (dBµV)

|Cm | (dBµV) |Cm | (dBµV)

1 .592GHz

1πτr

1πτr

1πτr

1 .592GHz

1 .592GHz

1πτ

f (Hz) f (Hz)

f (Hz)

Figure 9: Spectra of Trapezoidal Signals with Random Duty Cycles

3-32

individual spectral components are different, but the bounds on the magnitude spectra take thesame shape and ampli tude.

Now the spectral envelope of the 10% duty cycle waveform will be examined. (Thisspectrum can be found in the middle of the figure). The first corner frequency for envelope 3 is

1D A 3 B 1D .1×1/(100MHz)B 318.3MHz .

Therefore, this envelope has a slope of 0 dB/decade from the fundamental frequency until i treaches about 318 MHz. Then it decreases at -20 dB/decade. Like the other two envelopes,envelope 3 decreased at -40 dB/decade after 1.592 GHz.

Notice that although the individual spectral components of the various waveforms aredifferent at most frequencies, the envelopes are nearly identical for the high frequency portion ofthe spectrum. Thus, duty cycle does not play as important a role in determining the highfrequency components of a trapezoidal waveform as does the rise time of the signal.

3.4 Fourier Transforms and Non-Periodic Waveforms

Just as a periodic waveform can be represented by an equivalent series (Fourier series), anaperiodic waveform can be expressed by an equivalent representation known as the inverseFourier Transform. Let fp(t) represent a periodic function

.fp ( t ) BGF*Hm IKJ H cn e

jn L o t

This can be rewritten as

.fp ( t ) MNF*Hn IOJ H T0 cn e

jn L o t 1T0

By using , this becomesT0 M 2 PQo

fp ( t ) R 12 PTS*Un VOW U

T0 cn ejn X o t Y

o .

3-33

If T0 the period of fp(t) is extended to infinity, the resulting aperiodic function can be representedby

.limT0 Z\[ fp ( t ) ] f ( t ) ] 1

2 ^ limT0 Z\[ _ [n `Ka [ T0 cn e

jn X o t Yo

The term can be written asT0 cn

.T0 cn ] T0 b 1T0

T0 /2

c T0 /2

fp ( t )ec jn d o t

dt e T0/2

c T0 /2

fp ( t )ec jn d o t

dt

As T0 approaches infinity, fp(t) approaches f(t) and n f o becomes a new variable g lim

T0 h\i T0 cn j ik i f ( t ) ek j l t dt j ik i f ( m ) e

k j l?n d m .

Recalling that

,limT0 h\i fp ( t ) j f ( t ) j 1

2 o limT0 h\i prqn sKt q

T0 cn ejn u o t v

o

and applying the new variable , the function becomesv

f ( t )

f ( t ) w 12 x lim

T0 y\z { zn |O} z T0 cn e j ~ t ���where . The infinite sum now becomes an integral from over the variable . �����C�

0 ( �O� , � ) �Using the representation derived above for , f(t) , can now be written aslim

T0 �\� T0 cn

.f ( t ) � 12 � �

� ��� � f ( � ) e

� j �O� d � e j � t d �Let

.F( � ) � �� � f ( � ) e� j �?� d �

3-34

known is the Fourier transform of . Now can be expressedF( � ) f ( � ) f ( t )

f ( t ) � 12 � �� � F ( � ) e j � t d �

where the function is the inverse Fourier transform of .f ( t ) F( � )This result indicates that the function can be represented by a continuous superpositionf ( t )

of exponentials weighted by a function for each frequency . If the frequency variableF( � ) � is to be used instead of the angular frequency variable , the Fourier transformf �C� / 2 � �

pairs become

F( f ) � �� � f ( t ) e� j2 � f t dt

and

f ( t ) � �� � F( f ) e j2 � f td f .

• frequency content of signals

The Fourier transform is used to examine the frequency content of aperiodic signals. In thefollowing example the Fourier transform will be used to examine the frequency spectrum of ashort pulse (broadband signal). This pulse can be used to represent an electrostatic discharge,an electromagnetic pulse, or a lightning event. An approximation of a short pulse is a Gaussianfunction

x( t ) � e �1� (at )2

A plot of can be found in Figure 10. The Fourier transform of (using the frequencyx( t ) x( t )variable f ) is

.X ( f ) � �� �x ( t ) e � j2 � f t dt

Using Fourier transform pairs and the scaling property, the Fourier transform of isx( t )found to be

X ( f ) � 1|a|

e �*� ( f /a )2.

3-35

-2 -1 0 1 2

Time (seconds)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Am

plitu

de (

dim

ensi

onle

ss)

Figure 10: Amplitude of Pulse vs. Time

-5 -4 -3 -2 -1 -0 1 2 3 4 5

Time (seconds) or Frequency (Hertz)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Am

plitu

de (

dim

ensi

onle

ss)

X ( f ) (Hertz)x ( t ) (seconds)

Figure 11: Illustration of the Frequency Content of a Narrow Pulse

3-36

As the parameter a increases, the frequency spectrum of widens out. This effect isx( t )shown in Figure 11. In this plot, a was chosen to be 4.

This demonstrates that pulses or signals which are narrow in time have broad frequencyspectra. Thus short duration pulse signals, such as an electrostatic discharge, anelectromagnetic pulse, or a lightning event are typically rich in spectral content. These signalscan then interfere with many electronic systems operating at a broad range of frequencies.

Another example involving the frequency content of aperiodic signals will now be presentedby examining a trapezoidal pulse. A diagram of a trapezoidal pulse is shown in Figure 12. TheFourier transform of this waveform will be computed by using the first and second derivatives ofx(t). Let

x � ( t ) � dx ( t )dt

and

.x � � ( t ) � d 2 x ( t )

dt 2

τr τr

τA

t

x t( )

Figure 12: Aperiodic Trapezoidal Pulse

The second derivative of x(t), as shown in Figure 13, can be written as

x � � ( t ) � A�r � t � �

2� � r � � t � �

2 � � t � � r � � t � �2 � � r

Let

3-37

X � � ( � ) ��� �� �x � � ( t ) e

� j   t dt

and let . Then¡1 ¢ ¡ / 2 £ ¡ r

X ¤ ¤ ( ¥ ) ¢ A¡r

ej  K¦ 1 § e j  K¦ /2 § e ¨ j  O¦ /2 £ e ¨ j  O¦ 1

.X ¤ ¤ ( ¥ ) ¢ 2A¡r

cos ¥ ¡2£ ¡ r

§ cos ¥ ¡2

Now

X ¤ ¤ ( ¥ ) ¢ ( j ¥ )2 X ( ¥ )

where

.X ( ¥ ) ¢�© ª« ªx ( t ) e

« j ¬ t dt

Then

X ( ­ ) ® 2A

( j ­ )2 ¯r

cos ­2̄ ° ¯ r ± cos ­

2̄

and since ,­�® 2 ² f

.X ( f ) ® 2A¯r ( 2 ² f )2

cos 2 ² f2̄ ± cos 2 ² f

2̄ ° ¯ r

3-38

τ r

τ r

τ t

dx t

dt

( )

A rτ

− A rτ

τ r τrτ

A rτ

t

d x t

dt

2

2

( )

− A rτ− A rτ

A rτ

Figure 13: First and second derivatives of the trapezoidal pulse

³ relationship between rise time on pulse spectrum

The spectra of two waveforms with the same pulse width, ´ , but different rise times areshown in Figure 14 and Figure 15. Both spectral envelopes have a slope of 0 dB/decade from 0Hz to the first corner frequency, which is the same for both waveforms. This frequency is

3-39

107 108 109 10102 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 9

-200

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-100

-80

-60

-40

-20

0 dB perdecade

X f( ) ( )dB Vµ

-20 dB perdecade

1

1πτ r

1

πτ

Spectrum ofwaveform with.1 ns rise t ime

and 10 ns pulsewidth

-40 dB perdecade

( )Hzf

3 1 . 8 3M H z

3 . 1 8 3G H z

Figure 14: Magnitude spectrum of pulse with a .1 ns rise time and a 10 ns pulse width

.1µ ´·¶ 1µ (10ns) ¶ 31.83 MHz

Above this frequency, both envelopes decrease at a rate of -20 dB/ decade.The second corner frequency occurs at a lower frequency for the spectrum waveform having

a 1 ns risetime (Figure 15). This second corner frequency occurs at

.1µ ´ r2¶ 1µ (1ns) ¶ 318.3 MHz

Above this frequency, the spectral envelope of the waveform decreases at -40 dB/decade, whilethe spectral envelope of the pulse having a .1 ns risetime continues to decrease at a rate of only -20 dB/decade. At the frequency

,1µ ´ r1¶ 1µ ( .1ns) ¶ 3.183 GHz

3-40

107 108 109 10102 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 9

-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0 dB perdecade

X f( ) ( )dB Vµ

-20 dB perdecade

1

2πτr

1πτ

Spectrum ofwaveform with1 ns rise time

and 10 ns pulsewidth

-40 dB perdecade

( )Hzf

3 1 . 8 3M H z

3 1 8 . 3M H z

Figure 15: Magnitude spectrum of pulse with a 1 ns rise time and a 10 ns pulse width

the spectral envelope of the pulse with the .1 ns rise time begins to decrease at a rate of -40 dB/decade.

A direct comparison of the envelopes of the spectra of these two waveforms can be seen inFigure 16 . In the frequency range above 3.183 Ghz it is noted that the spectral envelope of thewaveform with a .1 ns rise time is 20 dB higher than that of the pulse with the 1 ns rise time. Thus even for an aperiodic trapezoidal pulse, the high frequency content is greatly affected bythe rise time of the pulse.

³ relationship between pulse width and spectral content

Now the effect of the pulse width, ́ , on the high frequency components of the magnitudespectra of a trapezoidal pulse will be examined. Figure 17 shows the magnitude spectrum of atrapezoidal pulse with a rise time of .1 ns and a pulse width of 10 ns. Figure 18 displays thespectrum of another trapezoidal pulse with the same rise time (.1 ns) but a shorter pulse width(1 ns). At first glance, it appears that the spectral envelope of the shorter pulse would be muchhigher than that of the wider pulse at high frequencies, because its first corner frequency occursat a higher frequency (318.3 MHz instead of 31.83 MHz). However, this

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10102 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9

( )Hzf

X f( ) ( )dB Vµ

-40 dB perdecade

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0 dB perdecade

1

1πτr

1

2πτr

1πτ

3 1 . 8 3M H z

3 1 8 . 3M H z

3 . 1 8 3G H z

Envelope forwaveform with1 ns rise time

Envelope forwaveform with.1 ns rise time

Both waveformshave a pulse

width of 10 ns

Figure 16: Envelopes of magnitude spectra of pulses with different rise times

the spectral envelope of the short pulse starts at about 20 dB below that of the 10 ns pulse. Theenvelopes of the magnitude spectra of both pulses meet at 318.3 MHz, as shown in Figure 19. Above 318.3 MHz, the spectral envelopes of both signals are very similar. Therefore, the highfrequency content of a trapezoidal pulse is determined more by its rise and fall times than by itspulse width.

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Spectrum ofwaveform with.1 ns rise t ime

and 10 ns pulsewidth

-40 dB perdecade

( )Hzf

31.83MHz

3.183GHz

Figure 17: Magnitude spectrum of pulse with a .1 ns rise time and 10 ns pulse width

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Spect rum ofwaveform wi th.1 ns rise t imeand 1 ns pulse

width

-40 dB perdecade

( )Hzf

3 1 8 . 3M H z

3 . 1 8 3G H z

Figure 18: Magnitude spectrum of pulse with a .1 ns rise time and 1 ns pulse width

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3.183G H z

Envelope forwaveform with

1 ns pulse width

Envelope forwaveform with

10 ns pulse width

Both waveformshave a rise time

of .1 ns

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Figure 19: Envelopes of magnitude spectra of pulses with different pulse widths