(I)

./ .'.:" ~ AUGUST 1948

Philips .Research R~po~ts'".' ÉbITED BY 'I'HE RESEARèH LA~ORATORY. ' .

OF N. V. PHILIPS' GLOEILAlIIPENFABRIEKEN, EINDHOVEN, NETHERLANDS

R 84 Philips 1!-es. Rep. 3. 241-254, 1948

I

NOISE IN A PULSE-FREQUENCY-MODULATIONSYSTEM

by F. L. H. M. STUMPERS ,621.396.619.16:621.396.~22

Summary

The optimum filter for pulse-frequency-modulatio,p. is derived- forany given pulse form and. large signal-to-noise ratio. For some specialpulse forms the signal-to-noise ratio is calculated and it is shown thatnormal frequency modulation gives a better result. A method is givenfor the calculation of the noise spectrum, which is valid for all signal"to-noise ratios, though the intricacy of the formula restricts the possi-bilities for application. The noise threshold is estimated and thesuppression of the modulation.by noise is calculated.

Résumé

L'auteur établit le filtre optimum pour la modulation de fréquenceà impulsions (pour un grand rapport du signal au bruit de fond). Lerapport du signal au bruit de fond est calculê pour quelques formesspëciales des impulsions et on montre que la modulation de fréquencenormale donne, encore un meiUeur résultat. L'auteur indique unemëthode de calcul du spectre du bruit de fond, méthode valabIe pourtous les rapports du signal au hruit de fond mais la complexité desformules .restreint les possibilités d'application. Dans Ia dernièrepartie on considère Ia suppression de la modulation en présence debruit de fond.

1.. Introduction

Amplitude modulation, phase modulation, and frequency ~odulation.'.usually sta;t from a sinusoidal signal. Instead any other periodic signal

can be used fq,r modulation purposes, and in the last years attention has.heen drawn to the use of periodic pulses. In this case the unmodulated -signal is given by

where COo is the ca~rier-wave frequency (in radians per second) and F1{t)

a periodic function' of t, which is 'different from zero only; for, a small partof its period. We shall choose the period of F1(t) equal to T = 2[&/W1'When modulation is applied the carrier-wave frequency Wo remains

constant and F1{t) is varied in accordance with the audio input in the.. ,

Thus"'~ 1.+&1,

colLIto = - f v(.) d•. (3)

242 , F. L. H. 111. SrUMPERS. "

.'same way as is the sinusoidal signal in ordin~ry modulation systems. In

. ',this paper we shall restrict ourselves' to p~ls~-frequenéy~modulatio'n, of.. which there are two forms. Firstly, ui the periodic function' , ..

~ ~Fl(t) = ,E,. ~Ak cos kcolt +"Bk sin kcolt{ = Re 1)0 ~Clf eikQ)ll~, (2)

o I I. . 0

the' phase colt is changed into

, . .."where v(.) is proportional to the audio input. Then the form of the pulse,changes in proportion to the length of the period. In the second system theRulses remain identical and only their mutual distance is varied. We chooseour fundamental period in such a way that Fl(t) is different from zero inthe central part of it, and when modulafion is applied the centre of each

:.. period' coincides with the passing through zero in a positive direction of .1

, cos '~COlt+ f v(.) d.~ •. '

, 'As a result of the modulation the pulses shift in time. This shift is calcul- ..~ated in the following way. ÓDse~ve a zero to of cos colt. Let the shift ofthis z~ro (~nd of the attached pulse) be LIto· Then' '

"• 1.+&1,

cos ~COl~O+ colLIto + f v(.) d.~ = o., . '

. Since the effect of the modulation is continuous no extra'shift of k2n'__has to be taken into account. LIt is a function of t which so far is defined

''; _ ~ . t

.. only at the zeros to + k2n/col of cos colt. There are, however, many pulses, for.each audio-frequency period, and we can link all the ordinates of LIt. as a function of t by. a smooth -curve. Then the modulation is obtained£ron;. the shifts LIt' of the zeros by .

odv(t) = dt (colLlt) . (4)

r ~

In addition to the shift c~used by modulation, there is a shift caused .by disturbances and noise. In the receiver we cannot distinguish betweenthese two. Therefor~ the shifts caused by noise, too, are translated into anaudio-output, which we shall now investigate.. Tn the mixer stage of the receiver the high frequency 0)0 is changed Iptothe mtermediate frequency CO2 and, mainly hy the intermediate-frequencyfil,t!;lrs,F1(t) is changed into F(t). Apart from the signaL

Re ~F(t) eiw,lt,

and in the same way r(w) is defined. Then,'", +'" w,f2 " ..,.g(t) = Re f r(w)f(w) eiOJ' dw = Re i» f t(kWl +.~)f(kwl + s) ei(kOJ'+s), d~. (8)

-.co . .... -eo -(J)./2 t .'

The function .g(t) defines the time shift at points a distance' k2n/wl apart,and at all those points eikOJ1' has the jsame value. Thus for' the differen-tiation it can be regarded as a constant,

+cowl/2. .g'(t) = Re .Ek f isf(kWl + s) r(kwl + s) eikOJl'. e:" ds. .:

-'" --oJl/2 '

, , .,.

NOIS;E IN A PULSE.FREQUENCY.M,O~ULATION SYSTEM'

) t ' .. , -. . , , "

we' receive noise, which ~fter the filters ,may be given by. '

, , Re ~.g(t) ~iW,.~ ;

.A Iinear' rectifi~r followèd by a low-pass filter gives a lo,;-f~e~'ency output", proportional to

IF(t) + g(t)l. . ,(5) ,, .'Let the noise be small compared to the signal. Without noise F(to) = A.

Let, L1to he the shift' ca~sed by' the noise; then ' ,_"

F(to) + L1to~~'(to) = -g(t) + A , e ,

g(t)L1t =.~ -r--' --.

, 0 " F' (to)-_

(6),

Comparing this result with formula(4) we see that the noise-is received as 'a ~:spurious modulation ' ':- ,

, • o'(t), vn(t) = -Wl _0__ •

, F' (to)'(7)

2. The optimum filter characteristic for pulee-fcequency-modulation

Let the complex filter ch~racteristic be given by ft (w ). This means thata signal Re ~Aei(OJ'+'I')~ before the filter is changed into Re ~fl(W) A ei(w.+9')~

. after it. At the input of the receiver the noise is given by

Re ~j rl(w) iw' dw~.

o -. The mean-square amplitude of all noise frequencies is the same. About the.noise spectrum more will be said in section 4. After it has passed the filter _the .noise is defined by ,

~Re ~r rl(w)ft(w) iOJ' dw(= Re ~iw" j r1(w + w2)fl(W'+ w2) i;'" q_cót.

I •• 0 ' . -CO2 •

c;:;, Here W2 is the intermediate frequency.• Now we introduce a new function,

f(w) , fl (w + W2)f(w) = 0

(__:w2 ~ W < 00),(-00 < w < ~2)'

t-" .'

" '

"

.'

/,

"

244' F, I!. H. 1.!. STUII.IPERS

The energy of the noise for the frequency s is

. W12T2s2 .E~I!(kwl + S)12+ 1!(kwl-S)12~ , '(9)

E(s) = . 21F'(to)l2 •

';l'he original signal is"

Re ~FI(t) iw,,~= Re ~.ECkikw,l iw,,~.

~fter passing the intermediate-frequency. filters, this beco~es

Re wet) eiw,l~= Re~.E Ck!I(W2 + kWI) ikw,'+iw.'~?

F(t) = R~~.ECk!(kwI) eik(IJ,,~,

F'(t) = Re~.E Ck ikwI!(kwI) eihw,lt· (10)_ and

In equation (9) we can approximate\

. very w~~lby 2 1!(kwI) 12.The energy for the noise frequency s then is+(X) l

wb2j:2 s» 1!(kwI)l2

E(s) = 127 Ck ik~:.f(kwI) ikw,I'I2 . (ll)

. .The best choice for the filter characteristic is now deduced by the use of the"Séhwarz's inequality". In our case this inequality yields for the quotientin (11) , . ' ,

co 2S2,2E(s):;;::: I ,- 1.'1Ck kwl2 ~

and the equality sign holds only when'

f* (kWI) = c; ikw1 eikw,l,

(f* (kWI) is the conjugate~complex of fl{kwl»· ,Using a property of Fourier series we can also write

2 2 2E(s) :;;::: :n;s WITI

- nkoJ'IFIÎ(t) 12 dt-nlw,

(12)

(13) q;

(12a)

T!ms for the, best o?tainable signal-to-noise ratio the mean- squareof the derivative of the pulse with regard to time is conclusive. The opti-mum filter characteristic is - apart from a phase factor - the complexconjugated of the Fourier transfórm of the time derivative of the pulse ... Tills result is at variance with that given by Van Vleck andMiddleton I), where the "si~ple criterion ,- óf1:he highest ratio- of tb.è signal.. ..) ~. .,

j.

NOISE IN A PULS!,-FREQPENCY-MODULATION SYSTEA~

amplitude to the rms value of the noi~e amplitude is postulated, For a'. , ' I

large signal-to-noise ratio this result is easily derived, as after ,the filterthe signal is given by

... \ "

and the ~eán square of the noise amplitude is proportional to

•

+'"

;2 .r If(w)l2 dco ~ COl ~ El',lf(kcolW.-:0

Then:, according to the "S~hwartz's inequality", the optimum' filter charac-,, teristic is the complex conjugated of the Fourier transform of the pulse.Van V1eck and Middleton have proved that this result is also true forlarger noise values; Are this "matched filter" and the 'optimum filter forpulse-frequency-modulation realizable? The answer is that they can beapproximated ~s accurately as necessary. For if the periodic pulse is givenby F1(t) cos wot and we de:fine' , '

(- n/w1 < t < n/~1)'(-00 < t < - n/wl, 7&/W1 < t <: 00) ,

h(t) = Fl(t)h(t) = 0

" (16)

then the optimum filter for pulse-frequency-modulation is the filter withtransfer time impedance h(to-t). See fig. 1._This~mea'ns' that such a :filtergives an output h(to-t) for the input unit fun~tion U(t). Th~ unit functionis defined by' '" " _ . ' "

U(t) = 0, - oo < t < 0,U(O) = i, -1J(t) =.1, 0 < t <: 00.

Choose toin such a way that h(to-t) = 0 for t <O.Then, for instance, Lee's 2)-, method for synthesis of electrical networks by means of the Fourier .trans-

\ forms of Laguerre functions can be applied to approximate the desiredfilter characteristic. Another meth?d is discussed by Lévy 3). • "

•SJ6S2

Fig. 1. Left: Pulse form. Right:' Time impedance of the "optimum filter" for, pulse- I

frequency-modulation. The filter gives an output voltage proportional to the given formfor the input current function U(t).

245

(14),

(15), -Ór

, ,

"

",

. ,

.I

,

.,

246 F. L. H. M. STUMPERS

'.In the same way' the "matched filter" is the filter that gives the

response,'h(to-t) to anInput a-function:The best result for pulse modulation is evidently reached also when the

"matched filter" of Van Vleck ana Middleton is applied first, and the low-frequency output' given by a linear rectifier is differentiated. An objectionto this procedure may be 'made, as the largest value of F'(t) is. reacliéd

l when F(t) is zero. Of course we cannot define the modulation by the num-.<: ber of times F(t) pa~ses through zero. It is, however, possible to count only'those zero~ coming after a large deviation, We can, for,instance, applypulse renewal. Normally this means that a new ideal pulse is given eachtime the received signal passes a value A in thè positive direction (fig. ,2).

-r ~. . •

n . n n

"

__jJ c=J c=J lI I I

: i - :I I

53653

Fig. 2:Above:' Output of the linear rectifier fu 'the receiver. Below: A new pU:lseis given,'each timethis output passes the value A in the positive direction •

, '

In the revised form, the passing through D in the posrtrve direction ,ofF'(t) 'operates a relay circuit and a new pulse is given at the first passingof a zero. This puts

lthe relay circuit out of action. until the next passing

of D (fig. 3). . , '

,.

I .:.I :! II I I.·

, ~J6~

'Fig. 3. AhOY~:The differentiated output of fig. 2. Below: The first zero aftér each passingof the value D in the positive direction coincides with a new pulse. '

3. Application of the 'theory to special pulse forms

As an example 'we shall apply -the theory to the pulse form defined by ,

h(t) = «r",

.... ,._,'I ~

NOISE'IN A PULSE_FR~QUENCY_MODULATION SYSTEM 2417 ' -.: _-I '

" .to \ • • ~ I'

,~vhereh(t) is "definedas in equation (16). The F~urièr transferm is

f("')" +J'" -a';' :iOJld' t' " in -<IJ~/~Izw = ',e e -_ -. - e •,',-co' a,,'.

'/

. -..(It is assumed here that e-a'I' is near enough to zero for t = nlwi to permitthe use of the liillits ,± 00.) .' ,The optimum filter characteristic for pulse-frequency-modulation.is pro-<portional to '"

f~(w) " w e-OJ'/4a'. ," .

The characteristic of the :'inat~he~" filter is proportional to

,I' ( ) _ -(J)'/4alJ4 W - e • " .

By application of the equatióhs (11) to (15), we see thàrthe noise energy.for the filter f3 and the frequency s is

.', . " "WI, -E3 S = 2 'n • - S2 r2,', a

I,vor for the band from °to Wa:

, 2'I'n'l·w '-E3(O, wa) = _'_' __!. wa3 r2•_ 3 a •

. ,

For the second filter f4 the noise-energy values areI I "\ '

, 'I "I WI - 2'1. 1 WI -E4 S = 2 • n • e - S272 and E4(O, wa) =_ nl. e - wa3 r2'a 3 a .

respectively .Here e is the base of natural logarithms. On the other hand the rat~o

of the noise energy to the squared maximum signal amplitude for -the"matched" filter f4 is . _ '

I ,

..

"I and after differentiation, or for the filter f3' A (f3) = eaT2 i2n•

Now we shall investigate whether a better result for pulse-modulation,noise can be reached when a filter ischosen with a characteristic similar'. to the one of the matched filter but ~ith a different constant,

fs(w) = e:,,'/4P'.: ,

The ratio of the noise energy to the squared maximum signal amplitude• for the filter fs is calculated in the same way: '

I

"

248 F •.L. H. M. STIDIPERS

The noise energy for the frequency s after this filter is

(a2 + {J2)2. _ __E5 S = -'---,-- OOI S2 r2e ,1n/2 ,

>. a2 p3 ,rand for the hand (0, Wa)

The lowest noise energy is obtained for fJ = a {3.Then the noise energy is0'7698 times the value for the matched filter, hut still2'093 times the hestattainahle value. The value of A(fs) then is>t {3 =.1'155 times its

- minimum value.We shall now compare the, signal-to-no_i!e ratio for pulse modulation

with the same ratio for,ordinl_lry frequency modulation. For pulse modul-ation the maximum shift is PWI' where fJ > 1, for instance P ,= 0'8.,The maximum audio-signal energy is' proportional to fJ2w1

2j2. This has to ,he compared 'with the maximum audio-signal energy for ordinary frequencymodulation, Llw2/2. The ratio of the maximum audio-signal energy to thenoise energy in the hand (0, wa) is for ordinary frequency modulation

--'-.-_-'2 wa3 r2

- Iand for'pulse~frequen,cy-modulation

The" energy per second necessary to produce this signal-to-noise ratio, '. +~

is t for frequency modulation and (w1/,2n) f e-2at,t C~~2 wot dt ' wl/4ay2n,-,for pulse modulation. ' -co

, For the s~mé transmitter energy the signal-to-n:oise ratio for ordinary"frequency modulation is LJw2/a2fJ2 times its value for pulse modulation..The bandwidth of the frequency spectrum is 2L1w for frequency modulation

. +~(if the sweep is large compared to theaudio frequencies) and ·re-wt/2a' dw =.., ai2n for, pulse modulation. The latter value is flattered. If we ask forthe bandwidth containing 95% of the energy we get 3'92a, and for 99~'o~ven,5'15 a. In any case it is ,9uite clear t~at for the sam~ transmitter energyand the same audio signal-to-noise ratio, pulse modulation needs a widerband of the spectrum. We have proved this only for pulses of the form e-<1"':

_' \ .. NOISE IN A PULSE-FREQUENCY-lIIqnULATION SYSTEM ,. 249

, ,For a pulse form F1(t) corresponding to a transmitter signal Fl(t) coswot,

\ the audio signal-to~~oise ~àtio for pulse frequency-modulation is . .

TJ2 '

p2 I IFl'(t)l2 dt .

(~jN)p = ~~~2 • '(SjNh.m .•Llw2 I IF1(t)12 dt

-TJ2

(17)

where (SjN)f.m. is the value obtainable with ordinary frèquency modulationof the same transmitter energy, and WI T =,2:n;:

As (2)

we can also writep2 _L' ICk kwl2 ,

(SjN)p = Llw2 L' ICkl2 (SjNkm.·

We shall apply formula (14) tó triangular and trapezoidal pulse f~rms . - ~.I (see fig. :4». '

" (a) Triangular pulse form:3p2y2 '

(SjN)p = Llw2 (SjN)f,m.' (I9a)

'(I8a)'

I , .

The bandwidth containing 95% of the transmitter energy is 3'14 J', andfor,99% it is 4y. .'(b) For the trapezoidal pulse form,

. 3 R2y2(SjN) -- , /' (SjN)'

- p - Llw2 (1+ 3 yj(5) f.m.· - (19h)

T:g.ebandwidth containing 95% of the transmitter energy is 2y whenIj , 2 r. and 1'48 v when Ij :..- r- For 99% the bandwidths are 3 y and 1'6 y

_respectively., It is seen. that for these examples, too, the signal-to-noise ratio for'ordinary frequency modulation is greater .. Moreover it may he moredifficult to reach the theoretical ratio in actual pulse-modulation systems. '

c »

....

'_

S:J6'!>b

Fig. '4. Left: Idealized tri~lDguIarpnlse form. Right: Idealized trapezoidalpulse form..

. , _ ~

'" ~

, I

, \

250 F. L. H. 111. STUMPERS

"

4. The calculation of signal-to-noise ratio "in general

So far we have assumed that the noise energy is small compared 'to thesignal energy. Elsewhere we have already given a calculation of signal-,to-noise energy for frequency' modulation that does not depend o.n thisassumption 4;). Now we shall try to' outline such "a calculation methodfor puls~ modulation, without, however, entering into det~iled applications.

The pulse signal is passed through an intermediate-frequency filter andrectified by means of a linear rectifier. Then pulse renewal is applied and~ new pulse is given each time the signal obtained from the rectifier pa~sesthe value A in the positive direction. The new pulses are congruent. Thoughit is :not necessary to do so, we shall give the new pulses in our calculationthe form of a Dirac Cl-function. When the new system of pulses passes anaudio-frequency filter the output is directly proportional to the desiredaudio signal. This demodulation system is~not the only one possible, but

- .we have chosen it as it is sure to give an undistorted output and to make acalculatiön of the results readily possible., '.Then, let input signal and noise be 'given by

.'

,

-vo(t) = Re ~F(t) + g(t)( iOJo'.

After passing the'filter the voltage is 111(t) cos (w2t + cp). Newpulees aregiven when v1(t) passes +A in the positive direction or -A in the negativedirection, viz.

"', ' v2(t) = Cl~V1(t) --:A~v1' U(v1') + Cl~V1(t) +A~ v1' U(-v').

" " N~,vintro'duce the Laplace transforms and make a Fourier spectrum'

v2(t) = E.fm eim,;

n co-ic co-ic

: '~i f~; .- 4~2'I e-imi dtf.d;lf dz2 Z2-2~ei'1(Vl:_A)+~..:" + ei~l(V'+A)-i"Vl'~. '(20)

-n --0:1 -ic --co -ic

_~ Thé 'output energy for the frequency ut is given by 2 fmfm *, and for' direct,', current by j~fo *.

The Fourier spectrum of the receiver input-noise' isee

g(t) = :E (an cos nt - bn sin nt) .o .The probabilitv distrihutions of an and bn are- '. ..

W(a )da =.~ e-an'IC dati n -VnC ' n,

.- 1 -bn'IC 'W(bn) dbn = -r:» e dbn •

~' ynç

(21)

(21a)

'." I'

'" NOISE IN A PULSE-FREQUENCY-lIlODUL~TION 'SYSTÈM~ ., .. 251", ,

, (F~r more ~formation about 'noise see Ric'; 5». ,Let the frequency characteristio of the intermediate-frequency filter be

.given b'y f (iQ), where, as 'in section 1, 'w, is r~ckoned frOin' the ce'nt~alfrequency. Afterthis filterwe thus have " , 1,1

, I

(22a)

g(t) = f'(an + ibn)f(n) inl,

v1(t) = F(t) + g(t).

(22b)

(23). ' .Introduce (22) in (23)' and then (23) in (20). Write down also fm": in,

analogous form. Then" 2 fm fm * is a function of the coefficients an~ b~. We'have to take the average 'of this function for all vallfes ~f an, bn: '

+co ,2 fmfm* = J J J ... J {IW(un) W(bn) 2fmfm* dan dbri·

-co

-./ ... .'. ' -. (24) .,Tins integration gives the result

. '

" , +co -ic ,I .

2fmfm* = ~ffe-im{t.-;.) dt1dt2ffrf·e-i~. H(ZlZ~tl~ H;Zaz4t2) dz1d:z2dzadz.1',8n ~, Z2 Z4 . •-1f ~ -«I-ic \ .. '

, whereX = .E If(n)l2 ~Z12+ n2z22+Za2+ n2Z'12+ 2n(zlz4-Z2Za) sinn(t~~t2) ++ 2(ZlZa + n2Z2z.1)COSn(t1 - t2)~'" , '- .

"'- '> ,

(25)

and

H(z Z t ) - ei=.{F-A)+i=.F' + ei=.{F+A)-iz,F'1 21 - . . ,

F = F(t1), F' == F'(h).We take g(n) = If(n)l2, 'andlof the exponential form we develop

" ,'exp [- ~ Eg(n) )2n(.,z, - """; :.. ~u + 2(z,z, + n'.,z,) CO,"U!]:' ,(26)"., .

.where u = ~- t2, into. a power series and integrate term-by-term. Inthe linear term we introduce n = kw + s, where - 00 < ,k < 00 and .-W1/2 < s <'w1/2. ' . .

In this case the result of the integrations with respect to Zl' Z2' za' Z4 is: ' I

,2 fsfs_~ - - W~~S2If dt du e(t) e(t + u) ~2 a(u) +, '':+ '~15;,2.rf d~du}e(t) 1/(t + ,u) - e(t + u) 1/(t)~ u a(u) +.+ W

1CS.2 If dt du.1](t) 1/(t + u) a(u) +

4n ..

+w~;.'.r.rdt d~ e(t) e(t + ft) a(u) ,

, ,.

r :

I"

(27)

\,

) '_.\

252- ,

F. L_ H. M. STUMPERS

where

~nd

go~g2 e(t) = e-(~-A)'ig"-F"18" )(F- .4)822 +'F" 802~4-. + e-(F+Al'18"-F"IS")(F+ A)822 _ F" 802~_ t

-2 yng2(F+A)F' é-(F+Al'18"~1 _:. (jj (Fjlg2)~. ',.

go 'YJ(t) , e-(F-Al'IS")l + (jj(F'lg2)~ + e-(F+Al'18")1-.(jj(F'lg2)~'. +'" ,

, a(u) = f g(n) einudn.f _'"

In these formsx

802 = C £ 8(n), g22= C £ n2 g(n) , (jj(x) = (21{ii) f e-Y' dy.o

The limits of mtegration in (27) are -nlwl and nlwl. It is assumed that ,the bandwidth of the filters is large compared to Wl'

, For small noise energy the third of the four integrals in equation (27)is. dominant; it is equivalent to the approximation (ll).' The first twointegrals in (27) give only a smallcorrection of the .proportionality factorof the triangular noise .spectrum. The fourth' integral is more important.since it indicates' the presence of a n?ise background independent offrequency., IÏLaddition to the linear term of the power-series development offormula

(26) we have also taken into account the quadratic term. This giv.e~againa frequency-independent term and a term proportional to S2. As the

: resultingformulae, are rather long, we shall omit them here." . We have applied the. theory to a pulse form e-a.:"and used an inter-. mediate-frequency filter e-w'/12a' as suggested iii section 3. -4 rough approxim- ', ation of the first-order term for the noise spectrum, corresponding to thélinear term in the development (If (22), gives the spectrum

\0'026 Wl e-0'23/8" , 8 e Wl g02 S2 .

n ~23 804 + 1'2802 + 0·014f/. + 9 ft + 2'67 eg02f/.;u2'In the same way the second-orde! term for the noise spectrum is

_" 3 (g 2+ 0'06)3 W e-O'23/8" 32 e2p 4 W S2o . 1 +. 00 1 _.

, ~4n802 )(0'46802 + 0'0075) (6804 + 0'84 8u2 + Q'0075)f" 27 (1 +5'33 eg02)"/. u2

It is tempting to try these first- and second-order terms as anapproximate solution, for the output noise spectrum as a function of the. input noise-to-signal ratio. The experience with analogous calculations forordinary frequency modulation has shown' that the convergence of theseries, of which we now have only the first and second terms, is rather bad(nth term proportional to n-s/.).

, ,

, ,

NOISE .IN A PULSE-FREQUENCY-MODULATION SYSTEM

'" .... ~ .If the audio-frequency bandwidth Wa is of the order of 0'05 to 0'1 'a,

then, if go2' 0'025, the energy of the frequency-independent noise isalready of' the same order" as that' of the triangular noise spectrum. Thismeans that for' this value of go we are at the other side of the noise 'thres-hold.' (The noise threshold corresponds to the value of the input-noise-to-signal ratio, where the deviation of the simplified analysis 'becomesapparent. It is not exactly defined.) It seems that the noise threshold willhe nearer to go2= 0'01. Since t go{2' is the rms v;Uue of the noise voltageafter the filter, and 'the top value of the puls'e is 1 before and ti'S' afterthe filter, a ratio between the rms value of the noise and the top-value ofthe pulse of 1 : 8 is insufficient, and 1 : 12 seems permissilile, if we desirethat the noise en~rgy shall confirm the simplified analysis. The signal energycalcul:tted in section 3 was w1/4a i27ï. Since a is from 10 to 20 Wa and WIis of the order of 2 to 3 Wa this makes the input-noise-to-signal-energy ratioat t~e threshold about 0'4, whereas for ordinary frequency modulationit is about 0'1. This results in a slightly more favourable behaviour- ofpulse modulation for large noise values.

" .5. Suppression of the modulation hy noise

I

So far it did not matter whether the first or the second system of pulse-frequency-modulation was chosen since no modulation was present,' Nowwe shall only consider the system with identical pulses. We .as~ume thatthe modulation is .dw cos pt and that the centres of the pulses coincidewith the passage through zero in the positive direction of cos ~Wlt +(.dw/p) sinpt~. Tlieformula (25) can be used directly for the calculatiorrofthe modulation energy, as only the repetition frequency of the functions F .changes. This leads to the result that in the presence of noise the amplitudeof the demodulated-signal' frequency pis: "

I

, ,

nfro'

Ap = .dW! [~é e-(F-A)'fllo' ~1+ cp (F'jg2H-. 2 gol'n •

-;rfro,I '

-~ F'e-(F+A)'fll,' ~1-<p(F:'jg2H +2 goin .

+ ~ ~~-iF-A)'fll"-F~·fg,· + e-(F+A);·fllo'-F';ig.·~_2ngo ' .

_ g2 e-A'!go' dt] .go

. '

\.

It is easily seen that the integral has the limit 1 when no noise is present.The integral' can he calculated sinc~ the pulse fo'i-m after the filter. F( t)and the filter characteristic (determining go and g2) are known. We' have

'.

" .

....

(28)

\ -",

254 F. ~, H. 111.STUMPERS

\. . .' "

, again applied the formula to the pulse form. e-a"~ and the filter charac-teri~tic e-oJ'/12a'. The result is shown in fig. 5. The abscissa indicates go' theratio of the rIDS value of the noise after ,the filter, multiplied by {2", to

A,~

fOO

40

20

oO' 0,1 0.2 0,3 0.4 O,S 0,6 0.7 0,8 0,9

.-, -,

.'-I\.

\"'-

I'..r-

f>.... l- f-"

80

'50

f,' 1,2 ',3 1,4 1,5go53657

,Fig. 5. Suppression of the modulation by noise. The received level of modulation as afunction ofthe ratio hetween the rms value of the noise after the :filter,multiplied hij ]12:to the top value of the impulse before the filter.

the top value of the pulse before the filter (to be multiplied by 2/l'J, if the,r,atio to th_e'top value of the pulse after the filter is required). As the input

" noise-to-signal ratio is a simple function of go' it is easy to calculate the. curve for the suppression of the' modulation as a function of the inputnoise-to-signal ratio from fig. 5. 'In most cases the integral' (28) lias to becalculated by graphical methods.

I wish to express my jhanke to ,my colleague Mr Tj. Douma for anadvanced copy of his report on "Methods of modulation and noise-signalratio", inwhich he t~ckles similar problems along different lines, and, forinstance, re;ches results very similar to' my formulae (19) in the case' of

, ":triimgular and trapezoidal pulses.Eindhoven, December 1947

REFERENCES

1)' J. H. van Vleck and D. Middleton,'A theoretical .comparison of the visual,aural and meter reception of pulsed signals in the presence of noise. J. appl. Phys. 17,940-971, 1946.' ,

.2) Y. W. Lee, Synthesis of electric.networks hy means of the Foutier transforms ofLaguerre's functioris. 'J. Math. 'Phys. 11, 83-113, 1932. '

3) M. Lëvy, Etude des propriëtës des quadripêles pur la rëponse impulsiaIe. Ondeëlect. 27,261-275,1947. ' I

4) F. L. H. M. St ump er s, Theory of frequency modulation noise, Proc. Inst. RadioEngrs 36, 194!l. .

5) S. O. Rice, Mathematical theory of random noise, Bell. Syst. tech. J. 24, 282-332,1944; 25, 46-156, 1945.