Spectrum Analyzer Series

APPLICATION NOTE 150-2

S P E C T R U M ANALYSIS Pulsed RF

PACKARD November 1971 HEWLETT

APPLICATION NOTE 150-2

Printed N O V E M B E R 1971

HEWLETT [hp] PACKARD

C O N T E N T S

Page

C H A P T E R 1. In t roduc t ion 1

T h e Bas ic Pu lse Spec t rum 1

C H A P T E R 2 . " L i n e " Spec t rum 5

Genera l Ru l e s and Explana t ion 5

Pulse Desens i t i za t ion ah 6

Transi t ion to the Pulse R e s p o n s e 9

C H A P T E R 3. "Pulse" Spectrum 10

Genera l Ru les and Explanat ion 10

W h y U s e a " P u l s e " Spec t rum D i s p l a y 13

Peak Pulse R e s p o n s e —Pulse Desens i t i za t ion a p 14

Very Short R F Pulses 17

C H A P T E R 4. Summary o f P u l s e d Spectra

Characterist ics 18

C o m m o n Pulse Spectra 18

A P P E N D I X A. T a b l e o f Impor tant Transforms 24

Explana t ion o f the T a b l e 24

Proper t ies o f Transforms 25

A P P E N D I X B 31

IF Ampl i f i e r R e s p o n s e 31

Dis tor t ion 32

T h e spec t rum analyzer was or ig inal ly d e s i g n e d to l o o k at the output o f radar

transmitters. A p u l s e radar signal is a train o f R F pu lses wi th a constant repet i t ion

rate, constant pu l se w i d t h and shape, and constant ampl i tude . B y l o o k i n g at the

characterist ic spectra, all important proper t ies o f the p u l s e d signal such as pu l se

wid th , o c c u p i e d b a n d w i d t h , duty c y c l e , peak and average p o w e r , etc . , can b e mea­

sured eas i ly and wi th h i g h accuracy. Perhaps an e v e n m o r e important app l ica t ion o f

the spec t rum analyzer is the de t ec t ion o f transmitter misfir ing and f r e q u e n c y pu l l ing

effects. Th i s app l ica t ion no te is i n t e n d e d as an aid for the opera t ion o f the spec t rum

analyzer and the interpretat ion of the d i s p l a y e d pu l se spectra.

T h e format ion o f a square w a v e from a fundamenta l s ine w a v e and its o d d har­

m o n i c s is a g o o d w a y to start an explanat ion o f the spectral d i sp lay for nons inuso ida l

wave fo rms . Y o u wi l l recal l perhaps at o n e t ime plot t ing a s ine w a v e and its o d d

ha rmon ics o n a shee t o f graph paper , then add ing u p all the instantaneous va lues . I f

there w e r e e n o u g h ha rmon ics p lo t ted at their co r rec t ampl i tudes and phases , the

resultant w a v e f o r m b e g a n to approach a square w a v e . T h e fundamental f r e q u e n c y

d e t e r m i n e d the square w a v e rate, and the ampl i tudes o f the ha rmon ics var ied in­

ve r se ly to their n u m b e r .

A rectangular pu l se is m e r e l y an ex tens ion o f this p r i nc ip l e , and b y c h a n g i n g

the relat ive ampl i tudes and phases of ha rmon ic s , b o t h o d d and e v e n , w e can p lo t

an infinite n u m b e r of w a v e s h a p e s . T h e spec t rum analyzer e f fec t ive ly " u n p l o t s "

w a v e f o r m s and presents the fundamental and each h a r m o n i c c o n t a i n e d in the w a v e ­

form.

C o n s i d e r a per fec t rectangular pu l se train as s h o w n in F igu re l a , pe r fec t in the

r e spec t that rise t ime is z e ro and there is n o o v e r s h o o t or o ther aberrations. Th i s

pu l se is s h o w n in the t ime d o m a i n and w e w i s h to e x a m i n e its spec t rum so it must

b e b r o k e n d o w n into its ind iv idua l f r e q u e n c y c o m p o n e n t s . F igu re l b s u p e r i m p o s e s

the fundamenta l and its s e c o n d h a r m o n i c plus a constant vo l t age to s h o w h o w the

pu l se b e g i n s to take shape as m o r e ha rmon ics are p lo t ted . If an infinite n u m b e r o f

ha rmonics w e r e p lo t ted , the resul t ing pu l se w o u l d b e per fec t ly rectangular. A spectral

p lo t o f this w o u l d b e as s h o w n in F igure 2.

Figure 1a. Periodic rectangular pulse Figure 1b. Addi t ion of a fundamenta l t ra in . cosine wave and its harmonics to fo rm

rectangular pulses.

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CHAPTER 1 INTRODUCTION

Average Value of Wave

Fundamental

Sum of Fundamental, 2nd Harmonic and

Average Value

2nd Harmonic

TIME

Spectral Lines

F R E Q U E N C Y , f

Figure 2. Spect rum of a perfect ly rectangular pulse. Ampl i tudes and phases of an inf in i te number of harmonics are plot ted, result ing in smooth envelope as shown.

T h e r e is o n e major po in t that must b e m a d e c lear b e f o r e g o i n g into the analyzer

d i sp lay further. W e have b e e n talking abou t a square w a v e and a pu l se wi thou t any

relation to a carrier or modu la t ion . Wi th this b a c k g r o u n d w e n o w app ly the pu l se

w a v e f o r m as ampl i tude m o d u l a t i o n to an R F carrier. T h i s p r o d u c e s sums and dif­

fe rences o f the carrier and all o f the h a r m o n i c c o m p o n e n t s c o n t a i n e d in the m o d u ­

lating pu lse .

W e k n o w from s ing le t o n e A M h o w the s idebands are p r o d u c e d a b o v e and b e l o w

the carrier f r equency . T h e idea is the same for a pu l se , e x c e p t that the pu l se is m a d e

u p o f m a n y tones , t he r eby p r o d u c i n g mul t ip l e s i d e b a n d s w h i c h are c o m m o n l y re­

ferred to as spectral l ines o n the analyzer display. In fact, there wi l l b e t w i c e as many

s idebands or spectral l ines as there are ha rmonics c o n t a i n e d in the modu la t ing pu l se .

F igu re 3 s h o w s the spectral p lo t resul t ing f rom rectangular ampl i tude pu l se

modu la t i on o f a carrier. T h e ind iv idua l l ines represen t the m o d u l a t i o n p r o d u c t o f the

carrier and the modu la t i ng pu l se repet i t ion f r e q u e n c y wi th its ha rmonics . T h u s , the

l ines w i l l b e s p a c e d in f r e q u e n c y b y wha teve r the pu l se repet i t ion f r e q u e n c y migh t

Figure 3. Resultant spect rum of a carrier ampl i tude modulated wi th a rectangular pulse.

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T h e e n v e l o p e o f this p lo t f o l l o w s a funct ion o f the b a s i c form: y = S1^ X

h a p p e n to b e . T h e spectral l ine f r equenc i e s m a y b e e x p r e s s e d as:

FL = Fc ± n -PRF

w h e r e Fc = carrier f r e q u e n c y

PRF = pu l se repet i t ion f r e q u e n c y

n = 0, 1, 2 , 3,

T h e " m a i n l o b e " in the cen te r and the " s i d e l o b e s " are s h o w n as groups o f spectral

l ines e x t e n d i n g a b o v e and b e l o w the base l ine . F o r per fec t ly rectangular pu lses and

o ther funct ions w h o s e der ivat ives are d i s con t inuous at s o m e poin t , the n u m b e r o f

s i d e l o b e s is infinite.

T h e m a i n l o b e contains the carrier f r e q u e n c y r ep re sen t ed b y the longes t spectral

l ine in the center . A m p l i t u d e o f the spectral l ines fo rming the l o b e s varies as a func-

T sin co —

t ion o f f r e q u e n c y a c c o r d i n g to the express ion for a per fec t ly rectangular pu l se . T

T h u s , for a g i v e n carrier f r e q u e n c y the points w h e r e these l ines g o through ze ro

ampl i tude are d e t e r m i n e d b y the modulating pulse width only. As pu l se w i d t h b e ­

c o m e s shorter, m i n i m a o f the e n v e l o p e b e c o m e further r e m o v e d in f r e q u e n c y from the

carrier, and the l o b e s b e c o m e w ide r . T h e sidelobe widths in f r e q u e n c y are re la ted to

the m o d u l a t i n g pu l se w id th b y the express ion / = 1/T. S ince the m a i n l o b e contains

the or ig in o f the spec t rum (the carrier f r e q u e n c y ) , the u p p e r and l o w e r s idebands

ex t end ing from this po in t fo rm a main l o b e 2 /T w i d e . R e m e m b e r , h o w e v e r , that the

total number o f s i d e l o b e s remains constant so l o n g as the pu l se quali ty, or shape , is un­

c h a n g e d and o n l y its repet i t ion rate is var ied. F igu re 4 c o m p a r e s the spectral plots

Figure 4a. Narrow pulse width causes wide spect rum lobes, high PRF results in low spectral line density.

Figure 4b. Wider pulse than 4a causes narrower lobes, but line density remains constant since PRF is unchanged.

Figure 4c. PRF lower than 4a results in higher spectral density. Lobe width is same as 4a since pulse widths are ident i­cal .

Figure 4d. Spectral density and PRF unchanged f rom 4c, but lobe widths are reduced by wider pulse.

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for t w o pu l se lengths , each at t w o repe t i t ion rates wi th carrier f r e q u e n c y h e l d c o n ­

stant.

N o t i c e in the drawings h o w the spectral l ines ex t end b e l o w the base l ine as w e l l

as a b o v e . T h i s c o r r e s p o n d s to the ha rmon ics in the modu la t ing pu l se , hav ing a phase

re la t ionship o f 180° wi th r e spec t to the fundamenta l o f the modu la t i ng w a v e f o r m .

S ince the spec t rum analyzer can o n l y d e t e c t ampl i tudes and not phase , it wi l l invert

the n e g a t i v e - g o i n g l ines and d isp lay all ampl i tudes a b o v e the base l ine .

B e c a u s e a p u l s e d R F signal has u n i q u e proper t ies w e have to b e careful to

interpret the d i sp lay on a spec t rum analyzer cor rec t ly . T h e r e s p o n s e that a spec t rum

analyzer (or any s w e p t r ece ive r ) can have to a p e r i o d i c a l l y p u l s e d R F signal can b e

o f t w o k inds , resul t ing in d isplays w h i c h are similar bu t o f c o m p l e t e l y different

s igni f icance . O n e r e sponse is ca l l ed a " l i n e s p e c t r u m " and the o ther is ca l l ed a " p u l s e

s p e c t r u m . " W e must k e e p in m i n d that these are bo th r e sponses to the same pe r iod i ­

ca l ly p u l s e d R F input signal, and the " l i n e " and " p u l s e " spec t rum refer s o l e l y to the

r e s p o n s e or d i sp lay on the spec t rum analyzer .

W e wi l l d iscuss bo th types o f r e s p o n s e to a signal wi th the bas i c appearance as

s h o w n in F igu re 5 wi th the aid o f p ic tures , and then summar ize all formulas and rules

for p r o p e r opera t ion o f the analyzer.

p(t) COS O J 0 t

T e f f = Width of Rectangular Pulse of same height and area as pulse applied to analyzer = / 0

p ^ d t

E P

Figure 5. Basic RF Pulse.

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t = o

E P

T e f f

^ = P R F

A " l i n e " spec t rum occurs w h e n the spec t rum analyzer 's 3 dB b a n d w i d t h B is

nar row c o m p a r e d to the f r e q u e n c y spac ing o f the input signal c o m p o n e n t s . S ince the

ind iv idua l spectral c o m p o n e n t s are s p a c e d at the pu l se repet i t ion f r e q u e n c y ( P R F )

o f the p e r i o d i c a l l y p u l s e d R F , w e can say:

B < P R F (1)

In this case all ind iv idua l f r e q u e n c y c o m p o n e n t s can b e r e so lved . O n l y o n e is

wi th in the b a n d w i d t h at a t ime as s h o w n in F igu re 6.

T h e d i sp lay is a f r e q u e n c y d o m a i n d i sp lay o f the actual Four i e r c o m p o n e n t s

o f the input signal. E a c h c o m p o n e n t b e h a v e s as a C W signal. T h e d i sp lay has the

normal true f r e q u e n c y d o m a i n characterist ics:

1. T h e spac ing b e t w e e n l ines o n the d i sp lay wi l l N O T c h a n g e w h e n the analyzer

scan t ime ( " s e c / D i v " ) is c h a n g e d .

2. T h e ampl i tude o f each l ine wi l l not c h a n g e w h e n the b a n d w i d t h B is c h a n g e d

as l o n g as B remains c o n s i d e r a b l y b e l o w the PRF .

W e wi l l n o w l o o k at C R T pic tures o n p a g e 7 A o f a p u l s e d R F signal to s ee h o w

different scan t imes , scan wid ths , and IF b a n d w i d t h s o f the spec t rum analyzer in­

fluence the appearance o f the signal o n the display.

A carrier signal wi th a C W ampl i tude o f — 3 0 dBm (F igure 7) is m o d u l a t e d b y a

pu l se train wi th a P R F o f 1 kHz and an ef fec t ive pu l se w i d t h r e / / o f 0.1 ms (F igu re 8 ) .

In F igu re 9 w e see the resul t ing pu l se spec t rum in a l inear display. T h e analyzer

b a n d w i d t h is 100 Hz, one- tenth o f the P R F .

T h e logar i thmic d isp lay (F igure 10) a l l o w s a m u c h bet ter evaluat ion o f the signal

spec t rum, b e c a u s e the l o w e r ampl i tudes o f the h ighe r o rder s i d e l o b e s can n o w b e

easi ly measured .

E a c h Four i e r c o m p o n e n t is r e s o l v e d and the l ine spac ing is m e a s u r e d as 1 kHz,

w h i c h is the P R F . W e can also see that the spac ing o f the s i d e l o b e m i n i m a is 10 kHz,

a c c o r d i n g to the relat ion — = ——-— = 10 kHz. TEFF 0.1 ms

B < 1 o r B < PRF

Figure 6. IF bandwidths smaller than PRF.

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W e thus can coun t ten spectral l ines in each s i d e l o b e or twen ty l ines plus the

carrier l ine in the m a i n l o b e , a c c o r d i n g to the duty c y c l e o f the p u l s e d signal.

yor r e f l • P R F = 0.1

( T h e fact that the ampl i tude o f the spectral l ines on the l o b e m i n i m a reach ze ro for

each in teger ratio o f can b e u s e d to adjust the duty c y c l e ve ry accurately.)

T h e d i sp lay in F igu re 10 d o e s not c h a n g e for different scan t imes , unless w e

se lec t a scan t ime t o o short for the g i v e n scan w i d t h and bandwid th .

T h e n e w H P Spec t rum A n a l y z e r sys tems have a buil t- in l o g i c wi th a warn ing

l ight that enab le s us to a v o i d any w r o n g c o m b i n a t i o n o f these cont ro l settings.

F o r spec t rum analyzers w i thou t this feature w e have to satisfy an addi t ional

equa t ion to a v o i d d isp lay errors:

Scan W i d t h [ H s / D i v ] < ( 2 )

Scan l i m e Lsec/UivJ

( S e e A p p e n d i x B)

In F igure 11 the b a n d w i d t h o f the analyzer has b e e n c h a n g e d to 300 H z . A l t h o u g h

the reso lu t ion o f the spectral l ines is r e d u c e d (min ima) w e still have a true Four i e r

l ine spec t rum display. F r o m this e x p e r i e n c e w e can d e r i v e a rule o f t h u m b for the

analyzer 's b a n d w i d t h to obta in a l i ne spec t rum:

B < 0.3 P R F (preferably B < 0.1 P R F ) (3)

This rule is valid for the shape factors ( 1 0 : 1 to 3 0 : 1 ) of the IF filters used in H P

Spectrum Analyzers.

In F igure 12 w e have c h a n g e d the spec t rum wid th from 100 kHz (10 kHz/Div)

to 50 kHz (5 kHzlDiv). W e see that the spec t rum e n v e l o p e and the l ine spac ing have

c h a n g e d , bu t the n u m b e r o f l ines in each l o b e remains constant.

In F igure 13 the pu l se wid th has b e e n al tered from Teff= 0.1 ms to T E / / = 0 .05 ms.

C o m p a r i n g wi th F igure 10 ( same cont ro l settings o n the analyzer) , w e find three

d i f ferences :

1. T h e s i d e l o b e m i n i m a are s p a c e d b y 20 kHz.

2. T h e n u m b e r o f l ines in e a c h s i d e l o b e is 20 . ( T h e l ine spac ing is still 1 k H z

s ince w e d id not c h a n g e the P R F . )

3. T h e ampl i tude o f the spec t rum e n v e l o p e is 6 dB l o w e r .

T h e last po in t reveals a ve ry important fact w h i c h has not b e e n m e n t i o n e d yet ,

bu t can eas i ly b e s e e n in the pic tures o f the ca l ibra ted logar i thmic d isplays on p a g e 7 A:

T h e ampl i tude o f the carrier c o m p o n e n t (h ighes t ampl i tude in the spec t rum e n v e l o p e )

o f a pu l se m o d u l a t e d signal is c o n s i d e r a b l y l o w e r than the C W ampl i t ude o f the un­

m o d u l a t e d carrier. Th i s effect is c o m m o n l y ca l l ed pu l se desens i t iza t ion .

T h e express ion " p u l s e desens i t i za t ion" is qui te m i s l ead ing s ince the sensi t ivi ty o f

the spec t rum analyzer is not r e d u c e d b y a pu l se m o d u l a t e d signal. T h e apparent re­

duc t i on in peak ampl i tude can b e e x p l a i n e d in the f o l l o w i n g manner : Pu ls ing a C W

carrier results in its p o w e r b e i n g d is t r ibuted o v e r a n u m b e r o f spectral c o m p o n e n t s

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(carrier and s idebands ) . E a c h o f these spectral c o m p o n e n t s then conta ins o n l y a frac­

t ion o f the total p o w e r .

In F igu re 10, w h e r e w e have a duty c y c l e — o f 0 .1 , w e measu re a d i sp lay ampl i ­

tude w h i c h has a d i f fe rence o f —20 dB c o m p a r e d to the C W ampl i t ude o f the carrier.

In F igu re 13, wi th a duty c y c l e o f 0 .05 , w e measu re —26 dB. T h i s leads to the equa t ion

for the l ine spec t rum pu l se desens i t i za t ion factor a,;.

aL [dB] = 20 \ogw

T-f

= 20 l o g 1 0 Teff • P R F (4)

T h i s relat ion is o n l y va l id for a true Four i e r l ine spec t rum (B < 0.3 P R F ) . W e can s e e

that aL is o n l y d e p e n d e n t on the duty c y c l e - ^ f o f the p u l s e d signal.

T h e average p o w e r P a v g o f the signal is a lso d e p e n d e n t o n the duty c y c l e :

^avg Ppeak ' rp Or Pavg Ppeak ' Teff ' P R F

Wri t ten as a ratio in d B :

| ^ [dB] = 10 l o g 1 0 reff • P R F (4a) ^peak

F i g u r e 14 represents equa t ions (4) and (4a) in a d iagram.

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^yJL [ d B ] Ppeak

dL [dB]

Y OR r. PRF

Figure 14. Pulse desensi t izat ion aL (l ine spect rum).

LINE SPECTRA OF A PULSED MODULATED 50 MHz CARRIER

Figure 7. CW signal 50 MHz, - 3 0 dBm, scan width 10/rHz/Div, bandwidth 100 Hz Log Ref. - 2 0 cff im, 10 offi/Div.

Figure 9. Line spect rum of the pulsed 50 MHz signal. Linear display 100 fiVI Div, scan 10 kHz/D'w.

Figure 10. Same spect rum in logarith­mic display scan wid th 10 /cHz/Div, band­wid th 100 Hz, Log Ref. - 2 0 dBm, 10 c/fi/Div.

Figure 11. Same spect rum wi th 300 Hz analyzer bandwidth . Scan width 10 kHz/ Div, Log Ref. - 2 0 dBm, 10 c/B/Div.

Figure 13. Carrier now modulated with a pulse w id th of r e f f = 0.05 ms (PRF = 1 kHz) scan w id th 10 kHz/D'w, bandwidth 100 Hz, Log Ref. - 2 0 dBm, 10 df i /Div.

7 A

Figure 8. T ime domain display of the 50 MHz signal pulse modulated wi th reff = 0.1 ms and PRF = 1 kHz (0.5 ms/Div).

Figure 12. Same signal but scan width changed to 5 /c/Vz/Div. Bandwidth 100 Hz, Log Ref. - 2 0 dBm, 10 dS/Div.

W e read from the d iagram that for a du ty c y c l e o f 0.1 w e wi l l get a d i sp lay d e ­sensi t izat ion o f — 2 0 dB, and for a ratio o f 0.05 w e ge t —26 dB as s h o w n in F igu re 10 and F igu re 13. T h e diagram also s h o w s that the desens i t iza t ion factor aL b e c o m e s ve ry large for l o w duty c y c l e s . In this case , the sensi t ivi ty o f the analyzer and the m a x i m u m signal l e v e l at the b r o a d b a n d front e n d m i x e r b e c o m e important factors. W e shall d e s c r i b e the necessa ry cons idera t ions for these analyzer proper t ies in the next chapter abou t the m o r e important " P u l s e " spec t rum display.

I f w e increase the IF b a n d w i d t h in our e x a m p l e further to 1 kHz, w e ge t the dis­p lay s h o w n in F igu re 15. W e no t i ce that the analyzer has lost the abil i ty to r e s o l v e the spectral l ines s ince B = P R F . T h e l ines n o w d i s p l a y e d are gene ra t ed in the t ime d o m a i n b y the s ingle pu lses o f the signal. W e also see that the d i s p l a y e d ampl i t ude o f the spec t rum e n v e l o p e has increased . Th i s is d u e to the fact that the IF filter is n o w sampl ing a b r o a d e r part o f the spec t rum at a t ime , thus c o l l e c t i n g the p o w e r o f several spectral l ines .

A pu l se repet i t ion rate equa l to the reso lu t ion b a n d w i d t h is the demarca t ion l ine

b e t w e e n a true Four ier -ser ies spec t rum, w h e r e each l ine is a r e sponse represen t ing

the e n e r g y c o n t a i n e d in that ha rmon ic , and a " p u l s e " or Fourier- transform r e sponse .

Figure 15. Bandwidth 1 kHz = PRF.

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CHAPTER 3 "PULSE" SPECTRUM

G E N E R A L R U L E S A N D E X P L A N A T I O N

A " p u l s e " spec t rum occu r s w h e n the b a n d w i d t h B o f the spec t rum analyzer is

equa l to /or greater than the P R F . T h e spec t rum analyzer in this case canno t r e s o l v e the

actual ind iv idua l Four ie r f r e q u e n c y d o m a i n c o m p o n e n t s , s ince several l ines are

wi th in its b a n d w i d t h . H o w e v e r , if the b a n d w i d t h is nar row c o m p a r e d to the spec t rum

e n v e l o p e , then the e n v e l o p e can b e r e s o l v e d (F igure 16). T h e resultant d i sp lay is

not a true f r e q u e n c y d o m a i n d i sp lay , bu t a c o m b i n a t i o n o f t ime and f r e q u e n c y display.

It is a t ime d o m a i n d i sp lay o f the pu l se l ines , s ince each l ine is d i s p l a y e d w h e n a

pu l se occu r s , regardless o f the f r e q u e n c y wi th in the pu l se spec t rum to w h i c h the

analyzer is t uned at that m o m e n t . It is a f r e q u e n c y d o m a i n d isp lay o f the spec t rum

e n v e l o p e . T h e d isp lay has three d i s t inguish ing characterist ics:

1. T h e spac ing b e t w e e n the p u l s e l ines and their n u m b e r w i l l c h a n g e w h e n the

scan t ime o f the analyzer is c h a n g e d . T h e l ines are s p a c e d in real t ime b y

1/PRF. T h e shape o f the spec t rum e n v e l o p e wi l l not c h a n g e with the scan

t ime.

2. T h e spac ing b e t w e e n the l ines wi l l not c h a n g e w h e n the scan w id th ("MHzl

D i v " or "kHz/Div") is c h a n g e d . T h e spec t rum e n v e l o p e wi l l c h a n g e hori­

zontal ly as w e w o u l d expec t .

3. T h e ampl i tude o f the d i sp lay e n v e l o p e wil l increase l inearly as the band­

wid th B is increased . T h i s means an ampl i tude increase o f 6 dB for d o u b l i n g B. 0.2 1

Th is is true as l o n g as B d o e s not e x c e e d ——. W h e n the b a n d w i d t h equals — Teff Teff

(or V2 o f the m a i n l o b e w i d t h ) , the d i sp lay ampl i t ude is pract ical ly the peak

ampl i tude o f the signal. At this po in t the IF filter c o v e r s near ly all significant

spectral c o m p o n e n t s . But then w e have lost the abil i ty to r e so lve the spec t rum

e n v e l o p e .

W e s h o w these characterist ics in the f o l l o w i n g p ic tures :

In F igure 17 w e modu la t e the —30 dBm C W carrier b y a pu l se train with a P R F o f

100 Hz and Ten~TKTTT = ' u " s ' a n a i y z e r s IF b a n d w i d t h is 1 kHz; i .e., B

Figure 16. If bandwidth greater than PRF.

10

B >̂ r or B > P R F

W e can s ee the spec t rum e n v e l o p e wi th the m a i n l o b e and s i d e l o b e s and the m i n i m a

in b e t w e e n . T h e l ines w h i c h form the e n v e l o p e are no t spectral l ines bu t pu l se l ines

in the t ime d o ma in .

W e can verify this b y c h a n g i n g the scan t ime (F igu re 18). If w e r e d u c e the scan

t ime further, w e l o s e the informat ion abou t the shape o f the spec t rum e n v e l o p e ; i .e. ,

the f r e q u e n c y d o m a i n information. But w e n o w can eas i ly m e a s u r e the P R F in the

t ime d o m a i n (F igure 19 and F igu re 20 ) .

In F igure 21 w e c h a n g e d the scan /wid th to 5 kHz/Div. T h e scan t ime is the same

as in F i g u r e 18. W e can s ee that the spec t rum e n v e l o p e c h a n g e d ( f r e q u e n c y d o m a i n ) ,

bu t the l ine spac ing remains constant ( t ime d o m a i n ) .

In F igure 22 w e use an IF b a n d w i d t h o f 3 0 0 Hz. W e can measu re an ampl i t ude

d e c r e a s e o f approx imate ly 10 dB c o m p a r e d to F igu re 17, w h i c h s h o w s the l inear

re la t ionship b e t w e e n IF b a n d w i d t h and d i sp lay ampl i tude . W e also can s ee that the

m i n i m a are bet ter r e s o l v e d than in F igu re 17. In F i g u r e 2 3 the b a n d w i d t h is i nc r ea sed

to 3 kHz. T h e d i sp lay ampl i t ude increase c o m p a r e d to F i g u r e 2 2 is not 20 dB bu t

o n l y 18 dB. W e lost the l inear re la t ionship b e t w e e n b a n d w i d t h and d i sp lay ampl i t ude

0 2 b e c a u s e B is greater than —— in this case . A l s o the r e so lu t ion o f the s i d e l o b e s is lost

Teff to a great extent.

If w e increase B to 10 kHz ( w h i c h is e q u a l to — ) , w e ge t a d i sp lay wi th an ampl i -V TeffJ

tude pract ical ly equa l to the peak amp l i t ude o f the p u l s e d signal (F igu re 24) .

S o m e addi t ional rules o f t h u m b are o f impor t ance :

1. F o r a sufficient reso lu t ion o f the spec t rum e n v e l o p e the b a n d w i d t h s h o u l d b e

less than 5 % o f the m a i n l o b e w id th or:

B < — (5) Teff

F o r h i g h e r reso lu t ion into the l o b e m i n i m a (20 to 30 d B ) w e s h o u l d use :

B < ^ (6) Teff

2. T h e sys tem must r e s p o n d to e a c h p u l s e i n d e p e n d e n t l y . T h e effects o f o n e

pu l se must d e c a y out b e f o r e the next p u l s e o c c u r s . T h e IF amplif ier d e c a y t ime

constant is approx ima te ly 0 .3/B. A d e c a y o f the p u l s e effect d o w n to 1%

(—40 dB) requi res five t ime constants . T h i s leads to the rule :

B > 1.5 P R F (7)

H o w e v e r , w e ge t less than 1 dB error if B = P R F , w h e r e the ba se l i ne is o n l y

20 to 25 dB b e l o w the spec t rum e n v e l o p e ( see F i g u r e 15) .

T h e range b e t w e e n B < 0.3 P R F ( l ine spec t rum) and B > P R F (pu l se s p e c -

*trum) s h o w s proper t ies o f b o t h r e s p o n s e types and s h o u l d b e a v o i d e d .

3. T h e n u m b e r o f pu l se l ines w h i c h form the spec t rum e n v e l o p e d i sp lay is

d e t e r m i n e d b y the P R F and the scan t ime. F o r a d i sp lay wi th useful reso lu t ion ,

i .e. , a sufficient n u m b e r o f l ines , the scan t ime s h o u l d b e s e l e c t e d to:

scan t ime [ s / D i v ] 3* p R - p ^ j ^ ] ( 8 )

W e then have m o r e than 100 l ines fo rming the spec t rum e n v e l o p e , thus assuring

that the m a i n l o b e peak is d i s p l a y e d o n each scan ( see F i g u r e 18 and F igure 22) .

11

Figure 17. Signal (peak ampl i tude - 3 0 dBm) pulsed wi th PRF = 100 Hz, T E F F =

1/10 kHz = 100 / is . Scan width 10 kHz) Div, 6 = 1 /(Hz scan t ime 0.5 s/Div.

Figure 19. Same signal wi th a scan t ime of 20 ms/Div.

Figure 21. Same signal wi th scan width 5 kHzlD'n, B = 1 kHz, scan t ime 0.1 s/Div.

Figure 18. Same signal, but scan t ime changed to 0.1 s/Div.

Figure 20. Same signal, but B = 300 /(Hz and scan t ime 2 ms/Div. The PRF can be measured to 1/10 ms = 100 Hz.

Figure 22. Same signal w i th B = 300 Hz scan width 10 AHz/Div, scan t ime 0.2 s/ Div. *

Figure 23. Same signal wi th B = 3 /(Hz. Figure 24. Same signal w i th B = 10 /(Hz.

11A

PULSED RF SIGNAL IN "PULSED" SPECTRUM DISPLAY * i pi 11 how the same Log Ref of 20 dBm).

T h e signal in the cen te r of the displays (base l ine lift) is the residual carrier w h i c h

is still p resen t dur ing the " o f f " p e r i o d s o f the p u l s e d signal. B e c a u s e it is essent ia l ly

a C W signal, the on-off ratio o f the p u l s e modu la to r can o n l y b e m e a s u r e d d i rec t ly if

B g — as s h o w n in F igu re 24 . W e measure an on-of f ratio o f 38 dB. Teff

In the other displays w h e r e B < — w e again have to c o n s i d e r a " p u l s e de sen -Teff

s i t izat ion" factor s ince w e c o m p a r e a C W signal wi th a p u l s e d signal. T h i s factor wi l l

b e ex t ens ive ly d i s c u s s e d later.

WHY U S E A "PULSE" SPECTRUM DISPLAY? In m a n y instances , it is ne i ther p o s s i b l e nor des i rab le to make a fine grain l ine-

by - l i ne analysis o f a spec t rum. A g o o d e x a m p l e o f such a case is a train o f short R F

pu lses at a l o w repet i t ion f r e q u e n c y as normal ly u s e d in radar transmitters. N o t o n l y

must the IF b a n d w i d t h b e c o m e i n c o n v e n i e n t l y narrow, bu t often the f r e q u e n c y m o d u ­

lation on the p u l s e d carrier is so e x c e s s i v e that the resul t ing d i sp lay is confus ing .

In the " p u l s e " spec t rum m o d e w e can ge t all informat ion w e n e e d : the spec t rum

e n v e l o p e and ampl i tude in the f r e q u e n c y d o m a i n and the P R F in the t ime doma in .

W e also have t w o advantages o v e r the " l i n e " spec t rum display:

1. W e can use shorter scan t imes b e c a u s e o f the greater b a n d w i d t h .

2. W e can increase the d i sp lay ampl i tude o f the p u l s e d signal b y c h o o s i n g a

b r o a d e r bandwid th . W e k n o w that the d i sp lay ampl i tude increases l inearly

wi th the b a n d w i d t h B. T h e no i s e l e v e l o f the analyzer increases o n l y p ropor ­

t ional to V B . S O w e can increase the s ignal- to-noise ratio p ropor t iona l to V B .

This is o p p o s i t e to the C W and " l i n e " case w h e r e w e h a v e to use nar rower

b a n d w i d t h s to d e c r e a s e the no i s e l e v e l , thus increas ing the s ignal- to-noise

ratio. F igu re 25 and F igure 26 s h o w these effects clearly.

F r o m the p r e c e d i n g d i s cus s ion abou t the " p u l s e " spec t rum r e s p o n s e w e can find

another important fact: T h e spec t rum analyzer mus t p r o v i d e i n d e p e n d e n t controls

for b a n d w i d t h , scan wid th , and scan t ime to o p t i m i z e the d i sp lay a c c o r d i n g to the

rules o f t h u m b g i v e n for this t y p e o f r e sponse . A l s o the var iable pe r s i s t ence C R T

offers a great advantage if w e want to have a flicker-free d i sp lay o f p u l s e d signals wi th

l o w P R F .

Figure 25. A carrier wi th - 5 0 dBm am­pl i tude is modulated by a pulse t ra in wi th PRF = 400 Hz, Teff = 3 ^ s . The band­width B = 3 kHz, scan width = 0.5 MHz/ Div, scan t ime = 0.1 s/Div. Only the low order sidelobes can be seen but not measured accurately. Log Ref.—40 dBm.

Figure 26. The same signal displayed wi th 30 kHz bandwidth . The noise level increased by 10 dB, but the signal level by 20 dB. The lobes and minima can be measured easily. Log Ref. - 4 0 dBm.

13

In the " p u l s e " spec t rum just d e s c r i b e d , the r e s p o n s e o f the spec t rum analyzer to

each R F input pu l se is in e s s e n c e the p u l s e r e s p o n s e o f the analyzer 's IF amplifier.

T h e peak pu l se r e s p o n s e o f the H P Spec t rum Analyzers has b e e n es tab l i shed and

is re la t ive ly i n d e p e n d e n t o f p u l s e shape and pu l se repet i t ion f r e q u e n c y (for B > P R F ) .

T h e exp res s ion relating the peak pu l se r e s p o n s e to a C W signal r e sponse is the pu l se

desens i t iza t ion factor a„.

T h i s factor ap for the " p u l s e " r e s p o n s e d e p e n d s o n different phys ica l cond i t i ons

c o m p a r e d to aL in the " l i n e " spec t rum:

ap = 20 l o g 1 ( ) • ' Bimp [dB]

In this equa t ion w e find a n e w express ion : the Effect ive I m p u l s e Bandwid th ,

B i m p . Th i s can b e v i sua l i zed as the b a n d w i d t h o f an idea l , rectangular ly shaped filter

wi th a pu l se r e sponse equ iva l en t to the actual filter wi th the 3 dB b a n d w i d t h B (F igure 27) . S i n c e the i m p u l s e b a n d w i d t h B i m p o f the IF amplif ier is not the same as

its 3 dB b a n d w i d t h B, a co r r ec t ion factor K has b e e n in t roduced . Th i s factor K repre­

sents an empi r i ca l approach def in ing Bimp relat ive to B:

K can b e determined with a pulsed signal with known properties ( A N 142, " E M I

Measurement Procedure," p . 1 2 5 ) .

F o r s y n c h r o n u o u s l y t u n e d filters as u s e d in the H P IF sec t ions , the va lue o f K has

b e e n m e a s u r e d as approx imate ly 1.5.* T h e error i n t r o d u c e d b y different shape factors

o f the IF filter and different pu l se shapes is normal ly less than 1 dB.

W e can n o w wri te :

a p [dB] = 20 l o g 1 0 • Teff • K - B; K = 1.5 (10)

"If the 300 kHz filter is used in the 8552A/B IF sections, a measurement of its impulse bandwidth is recom­mended, since a number of these plug-ins have a B i m p smaller than 450 kHz (1.5 X 300 kHz). This results in an additional desensitization of 1 to 2 dB. A quick and simple check can be made by measuring the 6 dB bandwidth, which is approximately equal to BimB.

Figure 27. Equivalent B i m p of Gaussian f i l ter.

14

T h e r e are several cond i t i ons w h i c h must b e satisfied if E q . (10) is to b e va l id :

1. T h e IF b a n d w i d t h - p u l s e w id th p r o d u c t must b e less than two-tenths :

B • < 0.2 or B < — Teff

2. T h e n o r m a l i z e d scan rate ( N S R ) o f the analyzer must b e less than o n e :

Scan W i d t h [ f f z / D i v ]

Scan T i m e [ s / D i v ] • (B[Hz])2 K

3. T h e IF b a n d w i d t h must b e greater than the P R F : B > P R F

T h e c o n d i t i o n s in 1 to 3 are automat ical ly a c c o m p l i s h e d if the equa t ions (5) , (8) , and

(7) are satisfied.

4. T h e peak pu l se ampl i tude at the b r o a d b a n d input mixe r o f the analyzer mus t

stay b e l o w the saturation p o i n t (1 dB c o m p r e s s i o n ) . T h e typical saturation

p o i n t for H P spec t rum analyzers is b e t w e e n —10 d B m and —5 d B m :

Ppeak ^ - 1 0 dBm (11)

F igu re 28 is a d iagram s h o w i n g the p u l s e desens i t iza t ion ap in relat ion to IF

b a n d w i d t h B and pu l se w id th r e / / . W e see that the P R F d o e s not appear , s ince it is o f

n o s igni f icance for the d i sp lay ampl i tude as l o n g as B > PRF . T h e shaded area b e -

1 r, 0 - ° 3 , „ 0.1 . . i J • , , r t w e e n the B = and B = represents the o p t i m u m b a n d w i d t h range tor an

~rff ~ eff

analysis o f a p u l s e d signal. T h e r e are a lso three do t t ed l ines w h i c h s h o w different

no i s e l e v e l s o f an analyzer for a fast de te rmina t ion o f the d y n a m i c range.

dp [dB] N [ dB] I

Figure 28. Pulse desensit izat ion ap (pulse spectrum).

15

Noise Level N of Analyzer (B = 1 kHz)

B > 1.7 PRF

B = M3: T e f f

'eff

M A I N LOBE W I D T H

7"eff

W e wi l l n o w take a f e w example s to s h o w h o w the diagram is u sed :

1. W e assume a p u l s e d signal wi th the f o l l o w i n g characterist ics: Ppeak = ~~ 30

dBm, Tejj = 1 [JLS, P R F = 1 kHz. T h e no i s e l e v e l o f the analyzer is IV = —100

dBm for 1 kHz b a n d w i d t h . W e find o n the diagram for T E F F = 1 fis an o p t i m u m

b a n d w i d t h o f 100 kHz (—>B>PRF). W e then can read a pu l se desens i t iza t ion

o f av ~ —16 dB. T h e d i s p l a y e d ampl i tude o f the spec t rum e n v e l o p e wi l l b e

~ —46 dBm. W e also read f rom the c ross ing po in t o f the l ine for IV = —100 dBm

and the l ine for B = 100 kHz a resultant no i s e l e v e l o f—80 dBm. W e thus ge t a

usab le d i sp lay range (S/IV ratio) o f o n l y 34 dB. A l t h o u g h this range is sufficient

in m o s t cases for evaluat ion o f the p u l s e spec t rum, this e x a m p l e s h o w s h o w

important a spec t rum analyzer wi th a l o w no i s e l e v e l is.

2 . P U L S E P O W E R M E A S U R E M E N T S : W e see o n the spec t rum analyzer dis­

p lay the spec t rum e n v e l o p e o f a p u l s e d signal wi th the f o l l o w i n g charac­

teristics: the d i sp lay ampl i tude is —50 dBm, the m a i n l o b e wid th is 10 MHz.

T h e analyzer 's b a n d w i d t h is 300 kHz. W h a t are the peak and the average

p o w e r s o f the s igna l?

T h e ef fect ive p u l s e durat ion T E / / is ca lcu la ted from the l o b e wid th or read

from the diagram:

2 T e f f = 1 Q M H z = 0.2 fis or 2 0 0 ns

In the diagram, w e find a pu l se desens i t iza t ion of—21 dB for T E F F = 200 ns and

B = 300 kHz. T h e peak p o w e r is 21 dB greater than the d i s p l a y e d ampl i t ude ,

and w e can ca lcula te the peak p o w e r to Ppeak — — 2 9 dBm.

T o find the average p o w e r , w e first have to measure the P R F . Th is is d o n e

b y r e d u c i n g the scan t ime until w e can easi ly measure the pu l se l ine spac ing

in t ime doma in . A s s u m e w e m e a s u r e d the l ine spac ing to 1 ms w h i c h equa l s a

P R F o f 1 kHz, w e then can ca lcula te the average p o w e r P a v g = Ppeak " T E F F • P R F .

Teff • P R F = 2 x 10~ 7 sec x 1 0 3 H z = 2 x 10~ 4

P a v

Using the diagram for aL, F igu re 14 o n page 7, w e find a factor — o f

* peak —37 dB. T h u s , w i th the peak p o w e r Ppeak o f — 2 9 dBm and the factor o f — 3 7

dB, w e can calcula te the average p o w e r P a v g = —66 dBm.

3. W e want to calcula te the peak p o w e r o f a signal d i s p l a y e d wi th an ampl i t ude

o f—30 dBm and a m a i n l o b e w id th o f 100 MHz. T h e analyzer b a n d w i d t h is 3 0 0 2

kHz. T h e signal has a pu l se durat ion TPff — , ^ , , „ — 20 ns. W e find a de -100 MHz

sensi t izat ion factor of—41 dB.

This w o u l d y i e l d a signal peak p o w e r o f + 1 1 dBm, far b e y o n d the satura­

t ion l e v e l o f — 1 0 dBm. T h u s , the ca lcu la t ion is no t val id. W e have to insert at

least 20 dB at tenuation b e f o r e the input mixer .

T o c h e c k that the input signal l e v e l at the front e n d mixe r is b e l o w the

saturation point , w e have to o b s e r v e that for a 10 dB s tep o f the input attenuator

the d i sp lay ampl i tude must also c h a n g e b y exact ly 10 dB.

1 6

V E R Y S H O R T R F P U L S E S

W e k n o w from the diagram for ap (F igure 28 ) that the desens i t iza t ion o f the an­

alyzer d i sp lay b e c o m e s ve ry h igh for ve ry short R F pu l ses , e v e n wi th the w i d e s t

b a n d w i d t h . If w e assume that w e can p r o v i d e the m a x i m u m usab le input signal l e v e l

o f —10 dBm ( w h i c h is normal ly p o s s i b l e w h e n w e measure in the proximi ty o f the

radar transmitter to b e inves t iga ted) , w e are then l imi t ed o n l y b y the sensi t ivi ty o f

the analyzer. F o r a sufficient evaluat ion o f a p u l s e d R F signal w e shou ld have a dis­

p lay range o f at least 30 dB a b o v e the no i s e l eve l . F igu re 29 is a d iagram w h i c h s h o w s

the m a x i m u m usable d i sp lay range as a funct ion o f p u l s e wid th and analyzer sensi­

tivity for a m a x i m u m input l e v e l o f—10 dBm and a b a n d w i d t h o f 300 kHz.®

W e can easily see that for a pulse width of, for example, 1 ns, an analyzer must

have a sensitivity of —110 dBm (specified for B = 300 kHz) or better to yield a usable

display. It is not possible to improve the signal-to-noise ratio with a l o w noise pre­

amplifier, since w e are already limited b y the saturation level of the input mixer. The

n e w generation of H P spectrum analyzers offers exceptionally high sensitivities wh ich

al low measurements of extremely short R F pulses.

e S e e note on page 14.

T e f f [ n s ]

Figure 29. Display range vs. sensit ivi ty.

17

DIS

PL

AY

RA

NG

E [

dB

]

Ppeak = - 1 ° d B m

B = 3 0 0 kHz

C O M M O N P U L S E S P E C T R A

F igu re 30 s h o w s s o m e example s o f typical spec t rum displays for pu l se signals

wi th different pu l se shapes and with the p r e s e n c e o f A M and F M . A n ex tens ive

mathemat ica l treatment o f different pu l se forms and their spectra can b e f o u n d in

A p p e n d i x A.

T h e ideal rectangular R F pu l se free o f F M wi l l p r o d u c e a symmetr ica l pu l se

spec t rum as s h o w n in (a). W h e n the pu l se is c h a n g e d to a triangular shape , the s p e c ­

trum remains symmetr ica l wi th d e c r e a s e d ampl i tude o f the s i d e l o b e s (b) . T h e pu l se

18

CHAPTER 4 SUMMARY OF PULSE SPECTRA CHARACTERISTICS

p(t) cos co0t

T y p e o f R e s p o n s e " L i n e " Spec t rum

(Four ie r Ser ies )

" P u l s e " Spec t rum

(Four ie r Transform)

R e q u i r e m e n t s for each

T y p e o f Spec t rum:

B a n d w i d t h

Scan T i m e

Peak Input P o w e r

B < 0 . 3 P R F F

Ppeak ^ - 1 0 d B m

B > 1 . 7 P R F

B < — 7eff

T, ,>10 /PRF

Ppeak ^ - 1 0 dBm

Desens i t i za t ion Factor aL = 20 l o g 1 0 - ^ f ap = 20 l o g 1 0 • T E F F • K - B

A m p l i t u d e o f Spec t rum

D i s p l a y at to = <u„

A = E'p • T-f

= E'p • T E F F • P R F

A = E'p • Teff • K • B

T y p e o f D i s p l a y U s e d

for D u t y C y c l e -j?- o f

Four ie r or spectral

l ines

> 0.05

Pulse repet i t ion rate

l ines

< 0.05

N u m b e r o f L ines /

D i v i s i o n

C h a n g e s wi th scan wid th

no t scan t ime

C h a n g e s wi th scan t ime

not scan wid th

E'„ = response on CRT due to C W signal B = IF bandwidth (3 dB) E„ cos <ji„t

K = constant of IF amplifier (K ~ 1.5) T, = scan time in sec/Div

T e S S = width of rectangular pulse of same F., = scan width in Hz/Div h e i g h t a n d a r e a a s p i u s e a p pi i e ci t o

2 analyzer = p(t) dt PRF = f = P u l s e repetition frequency in H z E'„

spec t rum wi l l remain symmetr ica l e v e n if the pu l se shape is dis tor ted or unsym-

metr ical .

P U L S E S P E C T R U M IN T H E P R E S E N C E O F F M : A symmetr ica l pu l se wi th

l inear c o h e r e n t F M wi l l p r o d u c e a symmetr ica l spec t rum wi th inc reased s i d e l o b e

ampl i t ude and m i n i m a no t reach ing ze ro , ( c ) , (d) .

If inc iden ta l F M ( F M d u e to ampl i tude modu la t i on ) or c o h e r e n t F M is i n t r o d u c e d

toge ther wi th an unsymmet r ica l pu l se , an unsymmet r i ca l pu l se spec t rum wi th the

m i n i m a not r each ing ze ro wi l l b e p r o d u c e d , ( e ) , ( f ) . Th i s is a lso true for a symmetr ica l

p u l s e wi th non l inea r c o h e r e n t F M .

Pictures o f the p u l s e spectra p r o d u c e d b y actual radar transmitters can b e f o u n d

on pages 20 to 23 .

Spect rum of rectangular pulse wi thout AM or FM occurr ing dur ing pulse. Shape

sin o j i is that of func t ion . T Clip

Spect rum of rectangular pulse with linear FM result ing in increased sidelobe ampl i ­tude and minimas not reaching zero.

(e) Effect of linear AM and FM dur ing pulse. Note loss in symmetry due to pulse ampl i ­tude slope.

(b) Triangular pulse spectrum wi thout FM dur ing pulse. Effect ive pulse wid th is shorter than (a) causing minimas to occur at wider intervals of f requency.

(d) Same pulse spectrum as (c) wi th more severe FM.

(f) More severe case of FM and AM occurring dur ing pulse.

Figure 30. Common pulse spectra.

19

SP

EC

TR

UM

A

MP

LIT

UD

E A m p l .

1/ns Freq. A m p l . Freq.

AF = 2Mc/Sec A F = 4Mc/Sec

AF = 2Mc/Sec

AF = 6Mc/Sec

30 MHz/Div 10 MHzjD'w

3 MHz/Div 1 MHz/D'w

T h e a b o v e is a typical spec t rum signature o f the fundamental f r equenc i e s o f an

" L " b a n d radar wi th an approximate 1.0 /jusec pu l se wid th . T h e a b o v e pic tures w e r e

m a d e wi th a 10 kHz b andwid th . N o t e transients from magne t ron a long wi th ex t reme

F M .

Th i s is the third ha rmon ic output o f a radar opera t ing at a pu l se w id th o f 4.5 fxsec

and fc o f 1300 MHz. A n interdigital filter (2-4 GHz) was u s e d to trap out the funda­

mental . T h e s w e e p wid th is 3 M H z / D i v .

20

V 1 • i.

.:-T. r r~i r w

* ! • 1 )_ j {. _ j:-\„j _ i J : : 1 i [ *o

Ik -•' — L Band Radar 300 pps/4.5 ^sec. 3 MHz/ Div spect rum width log display

Same condi t ions as at left except 1 MHzl Div spect rum width

H e r e is a spec t rum display o f o n e o f the c leanes t radars no ted . No te the a b s e n c e o f

m o d i n g . T h e F M presen t s e e m s to b e normal for c o n v e n t i o n a l magnet ron .

21

10 MWz/Div 3 MHzlDlv

1 Mtfz /Div

H e r e is a radar w h e r e the magne t ron is m o d i n g rather badly . N o t e the transients

occur r ing o n the h igh s ide o f the carrier f r equency . O n l y a h igh s p e e d p h o t o g r a p h i c

r eco rd w o u l d s h o w this.

Channel A; 1 MHz/Div Channel B; 1 MHz/Div

lagnetron output ; 100 MHz/Div Duplexer ou tpu t ; 100 MHz/Div

This is another radar w h i c h e m p l o y e d t w o channe ls . C h a n n e l " A " opera ted nor­

mally, bu t C h a n n e l "B" had marginal F M . T h e magne t ron output had c o n s i d e r a b l e

no i s e and spikes s h o w i n g u p w h i c h d i d not s h o w u p o n the d u p l e x e r output . N o t e the

o n e p ic ture w h e n the base - l ine d i m m e r was no t used . W h e n base- l ine d i m m e r is

u sed , true base l e v e l remains at zero .

4.5 /xsec 1 usee

T h e s e are spectra o f a radar w h e n the pu l se wid th is c h a n g e d from 4.5 /u-sec to

1.0 /usee. N o t e the c h a n g e in spec t rum. A n IF b a n d w i d t h o f 10 kHz and s w e e p w id th

o f 10 MHz/Div was used . N o t e the inc reased transient effects o n narrow pu l se ope r ­

ations.

22

This is a d i sp lay o f a 1 megawat t radar at m i d - L b a n d s h o w i n g spur ious radiation

o v e r 300 MHz b e l o w the main carrier and interfering in T A C A N channe l s . Hor izonta l

100 M H z / D i v , vert ical 10 dB/Div. S o m e emi t t ed b r o a d b a n d output is o n l y 20 dB

d o w n from carrier (10 kW).

2 3

APPENDIX A

E X P L A N A T I O N O F T H E T A B L E

T h e t ime funct ions and c o r r e s p o n d i n g f r e q u e n c y funct ions in this table are

related b y the f o l l o w i n g express ions :

F(cu) = jf(t) e~ i c u t dt ( D i r e c t transform)

f(t) = TT" J f ( C O ) e i f t J t dco ( Inverse transform)

T h e 1/2-77 mul t ip l i e r in the inverse transform arises m e r e l y b e c a u s e the integrat ion is

writ ten wi th r e spec t to o>, rather than c y c l i c f r equency . O t h e r w i s e the express ions are

ident ica l e x c e p t for the d i f ference o f s ign in the e x p o n e n t . As a result, funct ions and

their transforms can b e in t e rchanged wi th o n l y sl ight modi f ica t ion . T h u s , if F(co) is the

d i rect transform o f / ( f ) , it is also true that 2irf(—a)) is the d i rec t transform o f F(t). F o r

e x a m p l e , the spec t rum o f a S 1 ^ X pu l se is rectangular (pair 6) w h i l e the spec t rum o f a

rectangular pu l se is o f the form S 1 " X (pair 7) . L i k e w i s e pair IS is the counterpar t o f the

w e l l - k n o w n fact that the spec t rum o f a constant (d-c) is a sp ike at z e ro f r equency .

T h e f r e q u e n c y funct ions in the table are in m a n y cases l is ted bo th as funct ions

o f co and also o f p. Th i s is d o n e m e r e l y for c o n v e n i e n c e . F(p) in all cases is f o u n d b y

substi tuting p for ito in F(a>). (No t s imp ly p for a> as the notat ion w o u l d ordinar i ly

indicate . That is, in the usual mathemat ical c o n v e n t i o n o n e w o u l d wri te F(co) =

= G(p) w h e r e the c h a n g e in letter indica tes the resul t ing c h a n g e in funct ional form.

T h e notat ion u s e d a b o v e has g r o w n through usage and causes n o confus ion , o n c e

under s tood . ) T h u s , in the p-notat ion

F (p ) = jf(t) e-»< dt f(t) = ^ JF(p) e>" dp

T h e latter integral is c o n v e n i e n t l y eva lua ted as a c o n t o u r integral in the p -p lane ,

lett ing p a s sume c o m p l e x values .

T h e f r e q u e n c y funct ions have b e e n p lo t t ed on l inear ampl i tude and f r e q u e n c y

scales , and w h e r e c o n v e n i e n t , also on logar i thmic scales . T h e latter scales often br ing

out characterist ics no t e v i d e n t in the l inear plot . T h u s , m a n y o f the spectra are a symp­

totic to first or s e c o n d d e g r e e hype rbo l a s o n a l inear plot . O n a l o g p lo t these a symp­

totes b e c o m e straight l ines o f s l ope —1 or —2 (i .e . , —6 or —12 ciB/octave) .

T h e t ime funct ions in the table have all b e e n n o r m a l i z e d to c o n v e n i e n t peak

ampl i tudes , areas or s lopes . F o r any o ther ampl i t ude , mul t ip ly b o t h s ides b y the

appropriate factor. T h u s , the spec t rum o f a rectangular pu l se 10 volts in ampl i tude

and 2 s e c o n d s l o n g is (from pair 7) 20 S m 0 0 v o l t - s e c o n d s .

Aga in , u p o n mul t ip l ica t ion b y a constant hav ing appropr ia te d i m e n s i o n s , the fre­

q u e n c y funct ions b e c o m e filter t ransmissions. T h u s , if pair 1 is mu l t i p l i ed b y a, the

COPYRIGHT, 1954, HEWLETT-PACKARD CO.

24

f r e q u e n c y funct ion represents a s i m p l e R C cutoff. A o n e c o u l o m b i m p u l s e (pair IS )

a p p l i e d to this filter w o u l d p r o d u c e an output ( impu l se r e sponse ) wi th the spec t rum

- — — X 1 c o u l o m b , represen t ing the t ime funct ion ae~aT c o u l o m b s ( w h i c h has the

d i m e n s i o n s o f amperes ) . O r a 1 vo l t s tep funct ion (pair 2S) w o u l d p r o d u c e the output

a 1 spec t rum X — vol ts , w h i c h represents the t ime funct ion (1—e m) vol ts (pair 4S) .

p + a p

T h e entries IS through 6S in the table are singular funct ions for w h i c h the trans­

forms as de f ined a b o v e exist o n l y as a limit. F o r e x a m p l e , IS may b e thought o f as the

l imit o f pair 7 ^mul t ip l ied b y as r > 0.

T h e r e are a n u m b e r o f important relations w h i c h d e s c r i b e wha t h a p p e n s to the

transforms o f funct ions w h e n the funct ions t h e m s e l v e s are a d d e d , mu l t ip l i ed , c o n ­

v o l v e d , etc . T h e s e relat ions state mathemat ica l ly m a n y o f the opera t ions e n c o u n t e r e d

in c o m m u n i c a t i o n s sys tems: opera t ions such as l inear ampli f icat ion, mix ing , m o d u ­

lation, filtering, sampl ing , etc. T h e s e relat ions are all readi ly d e d u c i b l e f rom the

def in ing equa t ions a b o v e ; bu t for ready r e fe rence s o m e o f the m o r e impor tant o n e s are

l is ted in the Proper t ies o f Transforms on the last page o f this a p p e n d i x .

Aga in , b e c a u s e o f the similarity o f the d i rec t and inverse transforms, a symmet ry

exists in these proper t ies . T h u s , d e l a y i n g a funct ion mul t ip l ies its spec t rum b y a

c o m p l e x exponent ia l ; w h i l e mu l t ip ly ing the funct ion b y a c o m p l e x exponen t i a l

de lays its spec t rum. M u l t i p l y i n g any t w o funct ions is equ iva l en t to c o n v o l v i n g their

spectra; mul t ip ly ing their spectra is equ iva l en t to c o n v o l v i n g the funct ions ; e tc .

M a n y o f the pairs l is ted in the T a b l e o f Transforms can b e o b t a i n e d f rom others

b y us ing o n e or m o r e o f the rules o f manipu la t ion l is ted in the Proper t ies o f Trans­

forms. F o r e x a m p l e , the t ime func t ion in pair 8 is — t imes the convolution o f that in

pair 7 wi th itself. T h e spec t rum s h o u l d therefore b e — t imes the product o f that in pair T

7 wi th itself, as it i n d e e d is. Further, b y us ing these proper t ies , m a n y pairs not in the

table can b e o b t a i n e d from those g iven . F o r e x a m p l e , the spec t rum o£f(t) = (1 — ar)

1 ex p e~m is (by the addi t ion proper ty ) F(p) = - - — = —.

p + a (p + ay (p + a)z

2 5

2 . R: . . . . :

r o, t < o f(t)=

[e"a\ t>o 1

F(P)= pi*

f 0 ) t <° f(t)=

[ctte"<tl,t>o 2

F ( P , a (pfd). F H = ( a + /aJ)2

,,,, -«ltl f(t)= e

3 F(P) = i-/ v 2a!

1 0 ,t<0

f ( t ) = e^siri/st; t>o 4 F ( P ) = ( P + d ) « . / . «

n - ) * W J (a2

+ Js2)-w2

+;2dw

0 , t < 0 f ( t ) = ' e" a t (cosM-|sinm),

t>0 5

F ( p ) = lp .« )» . / . ' l"'~(d2+j8-2)-wz + «2(lw

fm = sin(̂ T) U t ) ( 1 ) 6

fr, lul < f

TIME FUNCTIONS NO. FREQUENCY FUNCTIONS 1 (LINEAR SCALES) i (LOG AMPL. - LOG FREQ.)

26

27

TIME FUNCTIONS NO. FREQUENCY FUNCTIONS , (LINEAR SCALES) i (LOG AMPL. - LOG FREQ.)

f l , l t l < i f(0=

I o, ltl>|-7 F(a>)-r

s i n t 2 )

( ^ )

f l - ' ^ . U K r f ( t ) =

1 o , l t l > r 8

• 2 / T L L T \

F M = T S I N (

2

2 }

f j l - ( f ) 2 J t l < r

f(t)= [_ 0 , l t l > T

9 F (w) = 2 T

f(t)= e z W ) 10 ,_ -i(rw)2

F(w) = T{ZW e 2

0 ,lt|<o

'<«>•«-- «•* , l t l > ( , 11 F ( P ) _ (ptc*np+0)

COS w 0 t , It' < J § f ( t ) \ o ,m>x 12 F M = f

s i n ( ^ ) r s i n ( ^ ) T

-

TIME FUNCTIONS NO. FREQUENCY FUNCTIONS ( (LINEAR SCALES) ( (LOG AMPL. - LOG FREQ.)

f ^ = r ^ [ 0 , | t l > l

= S ( t ) ( FUNCTION )

IS Rp) = F(u,)=l

r>t |o , t<o f ( t )= / a (X)dx= | , ) t > 0

= u(t)(s

u

T^) 2S

pt [ o , t < o

•sit)(sr„;E) 3S F(P)«-£»

0 , t < 0

f(tH [i-e- a t,t>o

4S F ( P ) = p(?.«)

f (t)= COS cuct 5S F(OJ)= JTJS (w+tij0)+8(w-wo)j

f(t) = Z S(t-nr) 6S F{u*) = ^_|8(o>-n 2 -f)

28

• : . O F . . .

to CD

TIME O P E R A T I O N F R E Q . O P E R A T I O N S I G N I F I C A N C E

LINEAR ADDITION a f ( t ) + b g ( t )

LINEAR ADDITION a F ( u ) + b G { « )

Linearity and superposition apply in both domains. The spectrum of a linear sum regarded as a sum of component parts and the spectrum is the sum of the compo-of functions is the same linear sum of their spectra (if spectra are complex, usual nent spectra, rules of addition of complex quantities apply). Further, any function may be

SCALE CHANGE f ( k t )

INVERSE SCALE CHANGE

Ik l \ k /

Time—Bandwidth invariance. Compressing a time function expands its spectrum in as for k = 1, multiply both functions by \ / \ k |. The case where k = — 1 reverses frequency and reduces it in amplitude by the same factor. The amplitude reduces the function in time. This merely interchanges positive and negative frequencies; so because less energy is spread over a greater bandwidth. For same energy pulse for real time functions, reverses the phase.

EVEN AND ODD PARTITION

y [ f ( t ) ± f ( - t )

EVEN AND ODD PARTITION

y [ F ( « ) ± « - « ) ]

Any real function f(t) may be separated into an even part — £ f(t) + f(—t) J The transform of the odd part is — F(«) — F(—a;) J which is purely imaginary

1 f" ~] and involves only odd powers of co. Note: for f(t) real, F(—a>) = F(<*>). and an odd part — 1 f(t) — f(—t) J . The transform of the even part is

— |^ F(a>) + F(—<ii) J which is purely real and involves only even powers of oi.

DELAY f ( t - t o )

LINEAR ADDED PHASE

e _ i u , ° F ( c o )

. . d9 Delaying a function by a time to multiplies its spectrum by e ° , thus adding a produces a delay of — = to-linear phase 6 = —uto to the original phase. Conversely a linear phase filter

COMPLEX MODULATION

• " f ( t )

SHIFT OF SPECTRUM

F ( w — wo)

Multiplying a time function by e ' w o * "delays" its spectrum, i.e., shifts it to center say—produces the time function — ( e ! u o ' + e l w o , ) f ( t ) with the spectrum about CJO rather than zero frequency. Ordinary real modulation—by cos wot ^

CONVOLUTION

f f ( r ) g ( t - r ) d r

MULTIPLICATION (FILTERING)

F ( w ) G ( a > )

The spectrum of the convolution of two time functions is the product of their spectra. ning is equivalent to filtering the signal with a filter whose transmission is the trans-In convolution one of the two functions to be convolved is reversed left-to-right and form of the scanning function (reversed in time). Conversely, the effect of an displaced. The integral of the product is then evaluated and is a new function of electrical filter is equivalent to a convolution of the input with a time function which the displacement. Convolution occurs whenever a signal is obtained which is pro- is the transform of filter characteristic. This function, the so-called "memory curve" portional to the integral of the product of two functions as they slide past each of the filter, is identical with the filter impulse response, aside from dimensions, other—in other words, in any scanning operation such as in optical or magnetic (Note: the convolution of a time function with a unit impulse gives the same function recording or picture scanning in television. Transform theory states that such scan- times the dimensions of the impulse.)

MULTIPLICATION

f ( t ) g ( t )

CONVOLUTION

1 F(S)G(OJ - s ) d s 2ir J

The spectrum of t i e product of two time functions is the convolution of their spectra. about each component of the (line) spectrum of the train of impulses (see pair 6S). This is the more general statement of the modulation property. For example, sam- For no overlap, highest frequency in signal to be sampled must be less than half pling a signal is equivalent to multiplying it by a regular train of unit area impulses. sampling frequency. If this is true original signal spectrum (hence signal) can be The spectrum of the sampled signal consists of the original signal spectrum repeated recovered by low pass filter (Sampling theorem).

TIME OPERATION FREQ. OPERATION SIGNIFICANCE

DIFFERENTIATION

d"f(t)

dt"

MULTIPLICATION BY p

p " F ( p )

P The spectrum of the nth derivative of a function is ( iu) n times the spectrum of the a transmission K — where K is dimensionless or has the dimensions of impedance function. A "differentiating network" has (over the appropriate frequency range)

or admittance. Thus the output wave is proportional to the derivative of the input.

I N T E G R A T I O N f t

f Jf(r)(dr)n

MULTIPL ICATION

BY — P

-Vf(p) p

The spectrum of the nth integral of a function is (ia>) " times the spectrum of the range) a transmission K — , where K is dimensionless or has the dimensions of im-function. Thus, the response of any filter to a step function is the integral of its

. , ... , . , „ . , . . , pedance or admittance. Thus, the output is proportional to the integral of the past impulse response. An integrating network has (over the appropriate frequency of the input.

30

I F A M P L I F I E R R E S P O N S E

M e n t i o n was m a d e in the test o f the p h e n o m e n o n o f d e c r e a s e d sensi t ivi ty and

reso lu t ion that results w h e n a C W signal is s w e p t b y the IF amplif ier at a h igh rate

c o m p a r e d to the b a n d w i d t h squared. A s s u m i n g a Gaussian r e sponse for the amplif ier ,

the result ing transient can b e d e t e r m i n e d as f o l l o w s :

Figure B-1.

A s w e e p f r e q u e n c y signal as i l lustrated in F igure B - l can b e r ep re sen t ed b y

s(t) = e m F ^ H \ ( B - l )

us ing pair 10 o f A p p e n d i x A

S(co) = TV2^ e~mi™)2 (B-2)

w h e r e T = V(jT s ) / (27rF s .

If w e assume a Gaussian r e s p o n s e ,

H (w) = e~mi»IS)2, (B-3)

the p r o d u c t o f S(a>) H(a>) g ives

Y((o) = S(to) H((o) = T V 2 ^ e x p - - ( r 2 + ~j co2 (B-4)

T h e output transient is the inve r se transform o f this funct ion, again us ing pair 10

Substi tuting b a c k for T and s impl i fy ing

1 . 5 2 T S ~ m 1 J 2 7 r F s 5 2 * 2

^ ) = [ , , W 6 X P ,(T£\* T - ( B " 6 )

J T S S 2 J | _ \2TTF S / _

31

APPENDIX B

F s Sweep width SLOPE = Ts

= Sweep t ime

Figure B-2. Sensit ivi ty loss and normalized effect ive bandwidth vs. normalized sweep rate.

32

T h e e n v e l o p e o f y(t) is then

SH2

"""Fmr" N o t e that for l o w s w e e p rates

T 1

2TTFs S 2

y(*) = e x p [ - | ( ^ ) % 2 ] . (B-8)

Th i s , as was stated earlier , is a p lo t o f the f r e q u e n c y r e s p o n s e o f the IF amplifier.

D I S T O R T I O N

If the c o n d i t i o n on (B-8) is not satisfied, the result ing transient wi l l b e al tered in

bo th wid th ( t ime durat ion) and ampl i tude . T h e reduc t ion in ampl i tude wi l l b e

No t ing that 8 = ( W V l n 2 ) Bf w h e r e Bf is the 3 dB b a n d w i d t h ,

A plot o f this funct ion in dB versus —FJ(TSB'~) is i n c l u d e d as F igure B-2.

LOS

S

IN A

MP

LIT

UD

E

AN

D S

EN

SIT

IVIT

Y

a (d

B)

Be

ff

NO

RM

AL

IZE

D

EF

FE

CT

IVE

B

AN

DW

IDT

H

B

F s N O R M A L I Z E D SWEEP R A T E

T s B 2

33

If w e so lve for the 3 dB t ime durat ion At f rom equa t ion (B-8) b y setting the func­

t ion to 1/ V 2 and so lv ing for the appropr ia te At, w e ge t

2Vb?sr / r t „ . A i = - ^ F T ~ ( B - n >

In a l ike manner , the 3 dB b a n d w i d t h o f the funct ion (B-l) is

2VL¥r (jiyf2 (B-i2)

T h e ratio o f these t imes is

At' r / 2 7 r F s \ H 1 / 2

i H 1 + M J • ( B - l 3 )

This is the ratio o f the effect ive r e so lv ing b a n d w i d t h o f a spec t rum analyzer to the

b a n d w i d t h o f the IF amplif ier as a func t ion o f s w e e p rate. Rewr i t t en in terms o f 3 dB b a n d w i d t h B.

T h i s funct ion is p lo t ted in F igure B-2.

Fs = S w e e p W i d t h

Ts = S w e e p T i m e

B = 3 dB IF B a n d w i d t h

Beff= Ef fec t ive B a n d w i d t h

PRINTED IN U.S.A. 5952-1039

HEWLETT PACKARD