By W . R. BEN N ETT
I n te r c h a n n e l in te rfe re n c e c au sed b y n o n -lin e a rity of m u lti
c h a n n e l a m p lifie r c h a ra c te ris tic s is a n a ly z e d in te rm s of second a n d t h i r d o rd e r su m a n d d ifferen ce p ro d u c ts of th e b a n d s of e n erg y c o m p risin g th e v a rio u s c h a n n e ls. M e th o d s of re la tin g th e re s u ltin g d is tu r b a n c e to d is c re te fre q u e n c y m e a s u re m e n ts a re d esc rib e d a n d m e a n s fo r a rr iv in g a t m o d u la tio n re q u ire m e n ts o n in d iv id u a l a m p lifie rs th u s e sta b lish e d .
1. In t r o d u c t i o n
'\ X 7 ’H E N a repeater is used to am plify a num ber of carrier channels
’ » sim ultaneously, d epartu re from linearity in the response as a function of input am plitude tends in general to produce interference between the channels. T he non-linear com ponent of the amplifier characteristic in effect acts as a m odulator, changing the frequencies in the in p u t wave and producing com ponents which fall in bands assigned to channels other th an the original ones. This phenomenon has been called “ interchannel m o d ulatio n” or “ non-linear crosstalk.”
In form ulating the requirem ents which are imposed on a repeater to insure th a t the resulting interference between channels will not be excessive it is convenient to tre a t separately two aspects of the problem nam ely— the condition when the total load on the amplifier is w ithin the range for which the amplifier is designed and the severely overloaded condition. A ctually a transition region between these two cases m ust also exist b u t when a considerable am ount of negative feedback is used the break in th e curve of response vs. in p u t is quite sharp so th a t for practical purposes the in p u t m ay be said to be either below or above th e overload value.
Below overload, the amplifier characteristic m ay in m ost cases be represented w ith sufficient accuracy by the first few term s of a power series and the interchannel m odulation analyzed in term s of the resulting com bination tones of the frequencies present in th e different channels. T he total interference resulting from the com bination tones falling in individual channels m ust be kept below prescribed limits.
Above overload on th e other hand the resulting disturbance in all channels becomes quite large and requirem ents are based on m aking such occurrences sufficiently infrequent.
587
T h e load capacity requirem ent has been discussed in a p ap er by curren t analyzer when sinusoidal waves are im pressed upon th e system . A nother involves th e m easurem ent of noise in a narrow frequency b and when a band of noise uniform ly distrib u ted over th e transm ission range is im pressed upon the system . then be evaluated in (7) w ith th e aid of the results of preceding sections.
T he effects introduced b y m ultiple centers of d istortio n in th e am pli
fiers of a transm ission link are considered in (8). T he pap er concludes w ith a discussion of te s t m ethods presented in th e n in th section.
2 . I n t e r c h a n n e l M o d u l a t i o n a s a S o u r c e o f C r o s s t a l k
W e shall consider specifically a single sideband suppressed-carrier m ultichannel speech transm ission system of th e four-wire ty p e al
M U L T I C H A N N E L A M P L I F I E R S B E L O W O VE R LO A D 589
M ultivalued characteristics such as associated w ith ferrom agnetic m aterials and reactive characteristics in which th e coefficients v ary w ith frequency are no t included. T he mechanism by means of which th e characteristic (2.1) gives rise to interchannel interference m ay be illustrated by assuming th a t a sinusoidal signal of frequency q radians per second is im pressed on the voice frequency channel associated w ith the carrier frequency mp, where p is the base frequency in radians per second and m is an integer. T he resulting wave im pressed on the amplifier is of th e form
e„ = Q cos {mp + q)t, (2.2)
if upper sidebands are tran sm itte d ; the plus sign would be replaced by a m inus sign in a system using lower sidebands. S u bstituting the is a direct current of trivial im portance; if the system contains a tra n s
former, this com ponent is n o t tran sm itted . If q does n o t exceed p/2, am plitude is proportional to the cube of th e impressed signal am plitude.
This term is of trivial in terest in the study of transm ission quality of individual channels of a well-designed m ultichannel system and is of no interest in the interference problem w ith which we are here con
cerned because it is received only in the originating channel. Finally, if q is less th an p/3, the fifth term represents interference of frequency w ith different values of Q, m and q, in th e amplifier characteristic (2.1),
we find th a t in addition to interference in channels having twice and
M U L T I C H A N N E L A M P L I F I E R S B E L O W O V E R L O A D 591
be designated by F 0p, m ay be calculated by averaging the distribution according to power, thus m odulation products by designating the m odulation product produced by 0-vu talkers as a “ zero volum e m odulation p ro d u c t” of its p a r
fundam ental appearing once in the product, two db for each db increase in volum e of a fundam ental appearing twice, etc. T hus a (2A — B )- product should increase two db for one db increase in th e volume of the A -com ponent, and one db for one db increase in the volum e of the 5-com ponent. If the fundam ental talker volumes producing a p ar
ticular product are norm ally distributed on a db scale, it follows from established relations concerning the distributions of sums 3 of norm ally d istributed quantities th a t th e volume of the product is also norm ally
(A + B — C )-type products, for example, are also norm ally dis three fundam ental channel spectra respectively which are flat from 10 per cent to 80 per cent of th e carrier spacing. A ctual speech be formed from a single band as from two equal bands. T he interfering effect of a 2T -type product from a speech channel m ay of course be
M U L T I C H A N N E L A M P L I F I E R S B E L O W O V E R L O A D 593 consider th e infrequent large bursts of m odulation from exceptionally loud talkers as a lim iting factor. T he allowance to be m ade can be estim ated by determ ining the com plete distribution curve of m odula
tion noise. C om putation of the required distribution function m ay be carried out by m ethods sim ilar to those described in the paper by Holbrook and Dixon.1 T he fact th a t the products are not independent introduces a difficulty which complicates the calculation. F or system s w ith a large num ber of channels, the requirem ents m ay be based on average values w ith a considerable resulting simplification.
If in addition to the sidebands due to speech, “ carrier leak s”
(partially suppressed carrier waves) are present, m odulation products are produced which are sums and differences of carrier frequencies and speech sidebands. P roducts of this sort m ay cause intelligible cross the level of th e speech channels. T he intelligibility tends to disappear as the num ber of channels is increased, since the num ber of super
imposed products becomes larger thereby producing m asking effects.
Carrier leak m odulation is however more serious th an m odulation from speech channels having the same power since carrier leaks are present all the tim e, while speech sidebands occur only in active channels.
Similar considerations apply to pilot and control tones.
Q uantitatively, the various frequency com ponents in m odulation noise m ust be weighted in term s of their interfering effect on reception of speech. In practice th e weighting is done by a noise m eter designed for th a t purpose. T he noise m eter readings are expressed in term s of
5 . Re l a t i o n Be t w e e n Sp e e c h a n d Si n e Wa v e Mo d u l a t i o n should be m ade on the basis of relative interfering effect. Suppose it is determ ined th a t an x-type p roduct is L x db below one 0-vu talker.
M U L T I C H A N N E L A M P L I F I E R S B E L O W O VE R LO A D 595
retically it should be possible to com pute th e S .T .M .F . for an y p a r
ticu lar p ro duct if sufficient inform ation concerning the properties of speech, th e tran sm ittin g and receiving instrum ents, th e carrier system
CARRIERS
F ig . 1— S p e c tra of second o rd e r m o d u latio n p ro d u cts fro m tw o fu n d a m e n ta l channels.
itself, and th e ear were known, b u t in practice it is found best to use experim ental determ inations. In th e case of new system s for which experim ental d a ta are no t available, estim ates based on known system s of sim ilar ty p e would be used.
6 . Nu m b e r o f Pr o d u c t s Fa l l in g i n In d i v i d u a l Ch a n n e l s
In th e appendix it is explained how the to tal num ber of possible products of each type falling in individual channels m ay be counted.
T able II shows th e result of counting all second and th ird order type pro d u cts.6 R esults for products falling both w ithin and w ith ou t the fundam ental band are given. In certain of th e (A + B — C)- and (A — B — C)-ty p e products, th e channel in which th e pro duct occurs also is th e source of one of ^he fundam entals. Since th is would give a type of interference heard only when th e disturbed channel is also carrying signal, it is n o t in general as serious a form of crosstalk as the cases of independent fundam ental and product frequencies. Therefore the num ber of these special kinds of products has also been evaluated
5 M r. J. G . K re er co llab o ra ted in th e d e riv a tio n of th e s e formulae.
POWER DENSITY
Fig. 2—Spectraofthirdorder modulationproductsfromthree fundamental channels.
Nu m b e r o f Pr o d u c t s Fa l l i n g i n ¿t h Ch a n n e l o f Mu l t i c h a n n e l Ca r r i e r Sy s t e m w i t h Ha r m o n i c Ca r r i e r Fr e q u e n c i e s n\p, (m + l ) p , . . . (wi + IV— l ) p
»i/> = Lowest C arrier Frequency. iV = N um ber of Channels.
n2p = ( » 1 + N — l) p = H ighest Carrier Frequency. I(x) = “ Largest I n t e g e r ^ . ”
kp = C arr. Freq. Associated with Mod. Product. No. of Products is 0 Outside Ranges Indicated.
T A B L E II
an d shown in the table. T he num ber of these is to be su b trac te d from these two results should give the average to ta l m odulation of each ty pe present in a channel. A difficulty occurs however inasm uch as it is n o t certain how th e interfering effect of superim posed m odulation adds.
T h e noise caused by one m odulation p ro d u ct is of an irregular n atu re and it is probable th a t its m ost d isturb ing effect is associated w ith
* infrequent peak values. W hen two p rodu cts are superim posed their individual peaks are n o t a p t to coincide and hence th e re su lta n t dis
M U L T I C H A N N E L A M P L I F I E R S B E L O W O V E R L O A D 599
turbance m ay not be much greater th an th a t of one alone. W e shall introduce here the concept of “ plural S .T .M .F .” Suppose v products of x-type are superim posed and comparison of th e resulting noise w ith one fundam ental talker shows th a t the difference is L xv db. If in te r
fering effects add as mean power we should expect L xv to be equal to L x — 10 logio v. Hence it seems logical to write
$Xv — L xv T 10 logic ^ TLx, (7.1) where sx, is the “ plural S .T .M .F .” to be used when v products are superim posed in order th a t power addition of products m ay be valid.
Com bining (5.1) and (7.1),
L Xv = $xv s* 10 logio v} (7.2)
which shows th a t the correction to be su btracted from power addition is
Pxv = $ X V $X' ( 7 * 3 )
T he value of pxv is best determ ined by experim ent. Superposition of a large num ber of products w itho ut using an excessive num ber of talkers can be accomplished by m aking phonograph records of indi
vidual products and combining their o u tp u ts in subsequent recordings.
T he average to tal m odulation of x-type in a channel is found by m ultiplying the average value for one product by the average num ber of products, and subtracting the q u a n tity pxv, which m ay be called the
“ plural S .T .M .F . correction,” thus
Vx = VxVop + .115(X^ — i7*)<r2 + 10 logio vxh T ^ — pxv. (7.4) where Vx is th e volum e averaged on a power basis of the x-type m odulation in th e ¿-channel referred to the volum e of one x-type product from 0-vu talkers. We next wish to express Vx in term s of db above reference noise.
L et T a represent the “ noise” produced by a 0-vu talker in db above reference noise. T his is an experim entally determ inable q u an tity and is abo u t 82 db. L et T x represent th e noise from an x-type product from 0-vu talkers. T hen L x, th e q u a n tity appearing in (5.1), is given by
L x = T a - T x. (7.5)
The average total noise produced by all x-type products in db above reference noise is given by
w x = Vx + T x ~ 4 Vx
1
Ta4
5, - H x. (7.6)*
If we assum e th a t th e to tal m odulation noise allowable for an x-type p rodu ct is X db above reference noise a t zero level, we m ay eq u ate X to W x in (7.6) and solve for H x, giving
H x = Vx + T a - s x - X . (7.7) S u b stitu tin g the value of Vx from (7.4) in (7.7), we g et for th e system requirem ent in term s of allowable ratio of fu ndam ental to prod uct when there is one mw. of each fundam ental a t th e p o in t of zero tra n s
mission level:
H x = T a + n}x Vop — sx + .llStA* — t]x)a2
+ 10 logio Vxk + lO^t(x) logio T — pxv — X . (7.8) F or the convenience of th e reader, th e following recap itu lation of significance of the sym bols used in (7.8) is given:
H x = R atio in db of power of each single frequency fundam ental to power of resulting x-type p rodu ct when each fundam ental has power of one mw. a t point of zero transm ission level.
T a = R eading of 0-vu talk er on noise m eter in db above reference noise.
r)x = O rder of x-type product.
VoP = Volume in vu corresponding to th e average power of th e talk er volum e distribution a t th e point of zero transm ission level
= V 0 + .115cr2 where Vu is average talk er volum e in vu.
cr = stan d ard deviation in db of talk er volum e distribu tio n.
sx = Speech-Tone M odulation F acto r of x-type p ro d u ct as defined in Section 5.
\ x = VwF + m 22 + • • • for (m iA ± m 2B ± • • -)-ty pe product.
Vxk = to tal num ber of x-type products which can fall in channel w ith carrier frequency kp. See T able II.
ju(x) = num ber of d istinct fundam entals required to produce x-type product.
r = fraction of busiest hour th a t a channel is active.
px, = correction to be applied to S .T .M .F . when v products are superim posed. Defined in (7.1)—(7.3).
X = Allowable m odulation noise in channel in db above reference noise a t zero transm ission level point.
T he allowable noise m ay be divided equally betw een second and third order purely on a power basis by settin g the requ irem en t for each 3 db more severe th a n th e to tal value allowed, or it m ay tu rn o u t th a t th e noise from one order is m uch m ore difficult to reduce th an th a t of th e other in which the full allowance m ay be given to th e m ore
M U L T I C H A N N E L A M P L I F I E R S B E L O W O VE R LO A D 601 angles between products originating in the various repeater sections.
A discussion of the general problem of addition of m odulation prod
ucts from m ultiple sources is to be given in a forthcom ing paper by J. G. K reer. A point of particular interest in connection w ith broad band system s is the effect of a linear phase characteristic on the phase shift between m odulation products originating in different repeater sections. T he curve of phase shift vs. frequency throughout the frequency range occupied by a considerable num ber of adjacent channels will in general d ep a rt b u t little from a straig h t line, b u t the intercept of this straig h t line if produced to zero frequency is in general no t zero or a m ultiple of 2r. T he intercept of such a linear phase curve is effective in producing phase difference between contributions to m odulation from successive repeater sections of all the second order products and of some of th e third order products, nam ely th e types 3A, 2A + B , B — 2A, A + B + C, and A — B — C. T he phases of third order products of types 2A — B and A + B •— C however are unaffected by the value of the intercept and the contributions from the different repeaters of these types of m odulation will add in phase to give th e maxim um possible sum whenever the phase curve is linear throug hout the channels involved. T hird order m odulation require
m ents on individual repeaters of a system m ay therefore have to be based on the very severe condition of in-phase or voltage addition of separate contributions.
Experim ental verification of in-phase addition of third order m odula
tion products from th e repeaters of a 12-channel cable carrier system are included in the paper by K reer previously m entioned. C orroborat
ing d a ta obtained on an experim ental system capable of handling 480 channels are shown in Fig. 3. T he m easurem ents there shown were
m ade 7 on a loop approxim ately 50 miles inTlength w ith repeaters spaced 5 miles a p a rt. T he band tran sm itte d extended from 60-2060 kc. F un d am en tal frequencies of 920 and 840 kc. were supplied from
Fig- 3— E xperim ental d ata on addition of th ird order m odulation from a m u lti- repeater line. F undam ental te s t tones A = 920 kc, B = 840 kc. M odulation product 2A - B — 1000 kc.
tw o oscillators a t the sending end, and m easurem ents w ith a po rtable cu rren t analyzer were m ade a t each repeater to determ ine th e ra tio of 7 Messrs. M. E. Cam pbell and W. H . T idd collaborated in these m easurem ents.
M U L T I C H A N N E L A M P L I F I E R S B E L O W O VE R LO A D 603 could be m easured w ithout disturbing the operating levels throughout the system .
T he d a ta shown on Fig. 3 include m easured m odulation from indi
vidual amplifiers, and from tandem amplifiers w ith intervening cable sections. T he sum m ation of amplifiers proceeds in the sam e order as the plotted individual amplifier values. T he crosses show the calculated sum s of th e individual contributions assum ing in-phase addition. A greem ent between these values and the m easured sums is well w ithin the accuracy of the m easurem ents, considering the difficulties involved and the length of tim e required to com plete the run. T he dots show the resu ltan t m odulation which would be obtained by adding the power in th e individual com ponents instead of the voltages, which would be th e expected result for a large num ber of com ponents w ith random phase angles. T he m odulation thus calcu
lated is much smaller th an th e m easured values indicating th a t a hypothesis of random phasing is untenable for this product.
In actual system s both th e m agnitude and phase shift of m odulation products in th e different repeater sections exhibit variations because of non-uniform o u tp u t levels, differences in tubes and other amplifier p arts, and unequal repeater spacings. T he addition factor for con
verting system requirem ents to single amplifier requirem ents should therefore contain a m arginal allowance for these irregularities in phase increm ent of products from successive amplifiers throughout the system, the to tal second order m odulation m ay be much less th an calculated from addition of power. In setting th e requirem ents which each am plifier m ust m eet, m arginal allowances should be m ade for differences in lineup throughout th e system and aging effects which m ay take place afte r th e amplifier is p u t in service.
L et A x represent th e ratio expressed in db between th e to tal x-type m odulation received from th e system and the co n tribu tio n of x-type from one amplifier, assum ing th e amplifiers co n trib u te equally. F or example, if there are K am plifiers in th e system and if the contributions to th e p ro d u ct add in phase A x = 20 logio K . If power addition occurs, A x = 10 log10 K . A favorable set of phase angles m ay reduce th is factor by an am o u n t depending on th e uniform ity of th e repeaters.
If the system is divided in to K \ links having Kz am plifiers in each link, w ith phase shifts and changes of frequency allocations of individual channels present a t th e link junctions such th a t am plitud e addition oc
curs w ithin links and power addition from link to link, A x = 10 logio K \
quired to evaluate the perform ance of th e system from single frequency d ata, an overall te s t under conditions com parable to actu al operation
quired to evaluate the perform ance of th e system from single frequency d ata, an overall te s t under conditions com parable to actu al operation