BSTJ 60: 9. November 1981: Spectral Properties and Band-Limiting Effects of Time-Compressed TV Signals in a Time-Compression Multiplexing System. (Eng, K.Y.; Yue, O.C.)

7.4 MB·English

Spectral Properties and Band-Limiting Effects
of Time-Compressed TV Signals in a Time-
Compression Multiplexing System

By K. ¥. ENG and 0-€. YUE
(Manuscript rocaved March 80, 1881)

‘Time:-compression multiplexing (row) has recently heen propose
{ar application in madtiple sv transmissions through satellites. 1 ts
‘advantageous over frequency-divisinn multiplesing bectuse of is
relive immunity to nonlinear transponder effects, Here we study
fasn important and fundamental aspects of Tes—the spectral prop

tnties and band-tiiting eects on the timeccompressed signal. We
derive te ouput spectrum of a tie-comprensed signal al shew
‘hat if the original put signal 12 assumed to be: (Band limited 10
Hs, fi) segmented ino Toned interds before time compression
by a factor of af = I), and fit) /T 2B, then eesentialy all the
spectral power inthe output time compressed signal i contoined in
‘the bamedith |f| = all Hs, Thie result applioable to the Tv ease
‘Numerical examples on various ypee of speca are also presented.
Using the rv example, sve further demonstrate that the ripples created
by low-pass fitering the time-compressed signal up to aB He are
‘small and interburet interference de Uo theve ripple can be hept
eiaible with «small guard time (about 2 percent of the burst
uration) between diferent signal bureix. We ales provide « Brief
‘discussion on some interesting spectral properties of time cumpressel
signals in spectruncexpansion applications

1. iTRODUCTION

‘Timo-compression multiplexing (70%) isa technique whereby rt
ipl signals cane maltipkexod tngetherin a common eommunicetion
hannel for tanemiasion’ A simple ilusteatin of this method. it

‘shown in Fig. | where 2(¢) a consinuous waveform intenvled for
‘transmission. Ite frst divided inco segments of 7 seconds each; and
‘ach segment is time compressed by'a facto ala =), rein in 8
ah =

se Utes

lADN

Fe 1A sing tain 0

bursty signal (0) with a bust duration of T/a seconds. A total of «
such lime-compressed signals ean then be time multiplexed together
for trantnision. In the perlicilar eave of TY transmissions through
fatelites, exis advantagcoas over fraquency-division multiplexing
(rom) becouse various degeading effees (eg, intermodulation and
Ineligible crostlk) duets Iransponder nonlinearities ean be avoided
by euploying eM, Ip 2 more general cotext, Tew more efficent
than rot whenever time division can be accomplished more eiciently
than frequency division. Tv thin paper, wo etudy wo Important and
fandemental aspects af rex—the spectral properties and the band
limiting effect of the cime-comprested signal

‘Te assume that tho orginal signal #() in ¥ig. Le band-limited to
2B Ha, then time compressing it by 8 factor of « inthe infinite time
duration, be, transforming x(@) 9 x(¢/a), would mean a Trequency
spectrums expansion by the sams faclor(e). However, as shown inthe
diagram, (0) is segmented into T-econd intervals before time com
pression on each segment. Doing 40, ics no longer obvious whal the
pectrum should Took like or what the bandwidth expansion factor
‘would be. I clear though that the spectra power in (0
beyond ad Hi due to the segmentation: and itis desirable tht this
power beyond af! Hi, le sinall to raaiain spectral efclency in Te
We derive and disci an explicit expression for the output spectrum
of the time-corapreseed aigual y(2)foee. Section TD) and show by
foumerical examples (Section 111) chat all the sialfcant power in
‘contained inthe frequency bandwidth below aB Hi, thus, confirming
the long apeculated result Unt the bandwidth expansion factor in Tcat
{athe aome asthe time-compression factor.

2168 THE BELL SYSTEM TECHNIGAL JOURNAL, NOVEMBER 1861
“To ensure compliance with che out-ntsband emission requirement,
signals are often tered before cansmiasion, Such bandclimiing
‘operation on the time-comprested signal truncates its mall bul non
sro power beyond ia paseband (a1 Hz) and ereates ipl in ite me
‘waveform. The ripples fllowing the eriling edge of each Turat are
Important hecauee they lead to interburs. interference in the system,
We demonstrate using 2 computa simulation (Section TV) that in the
specific caso of ry transmission, (7) auch a bandclimiting effect ix
‘minimal as long aa all cho spect componente below of He are
{enarited without datortion ane Gi) the interburet interference can
be Kept nelle ly introducing » small guard time in the order of
two percent of the burst duration betwoon diferent time-compressed
‘Ty signal Those encouraging eaultson both the bandwidth expansion
1nd bund miting effects ussure us of the base atractiveness of using
‘Tea to transmit TV signals in nonlinear satellice channel.

Tis welnown that Lin momprision can alia be used a @ means
lovabinin spectrum expansion, eg, Henes's spectrum expander ured in
‘aio astronomy.’ In auch a case, che key concer i that of spectrum
istortion as analyzed thoroughly by Rowe We extend our resulta
‘examine this problem in an appondix, and some simple and interesting
Stet rerio nn to hs ype expan pation

IL SPECTRUM OF THE TIME-COMPRESSED SIGNAL,

Referring to Fig. 2 let x(t) be an input signal to an ideal time
oraprassor which poriorms the time eumpremion on each T-eecond
Seen of Une inj waveform lacs efor ‘Phis bs matbemat-
Teall equivalent to fst cme compressing (2) in che infiite time
Atration, resulting in x.) by the required time-compression factor
‘r= 1h and chen time-shifting aegmenta of Tya-necom duration in
‘cle to various proper time instants co amiveaty(), the desired time:
‘compressed output as shown in the diagram. We are interested in the
spectrum (or Fourier transform of yf. denoted by Y(/}

Ry the above definition, yt is related 0 x0) by

ee)
wee

29 a

and
AANA

wetiaff Sa o

‘Using the folowing,
dt) KlP “
reste ov rane ©

‘where « denotes Fourier tranetorm pai and

sine 8,

tho Fourie sent of) can win a
ven= 3 [xe enlist ~

cone gy) el-jtetF— aa Ide.
Assamng chat ehe mummation si integration can be interchange

the shave hecomes

2179 THE BELL SYSTEM TECIINICAL JOURNAL, NOVEMBER 1951
vin [ xteraineta

xexpl-forkiman + (70]de. 8)

Applying the well awown identity of

§ ewcomern = farm ®
Yip spo
ee
5 x (4) sf 24)]
Jeet n en on

sshera X1) the Romrior transform of s(t, wo got the final result of

wneh i x(r-)aode(e-£Se]. om

“The shove exprossion rslates the usta spectrum ¥(/) tothe input
samsetruen X(/) and sth hasic elon we work within the rest
fof the paper. The preceding derivation is the simples that me are
tsware of (eee Appendix A and Ref. for comparison), We now proce
te discuss some simple propertos of ¥1/)

22 Simple properties
‘We are inerested inthe properties of Yi/1 that convey informacion.
about the spectral occupancy prablem (or the bandwith expansion
factor) in ene ayers. Ta this ear, we must note that ¥(f) derived
shove is just the Fourier transform of one single time-compressed
Signal in the rest system. If there are N users for the chun iy N
timo-compreased signals for tronsmision} then the total agra in the

tranemiaion channel (without poo-timt-compreaton filtering
a= Zan, aa)

TIME-COMPESSION MULTIPLERING 2171
where uch (0) ina time-compreed signal resembling) considered
previously. ‘The power spectrum ofthe total signa inthe channel is

Kn ay

pepvelaine

wine ZIf} and Yi(/) ace the Fourier transforms of 210) and (0,
respectively ts well known thet

puy= 3 (YAN as)

‘nly when the 3:2) are all ueslated. In the particular case of 1
Unansmssion, the varius time-comprescod 1¥ signals are not totally
‘uncorrelated Tecause of the prevenee of aye pulses, color subearrier
burse, and zo on. Hosever, from the point of view of spectral vee
anoy (he the tral poorer eontined in some paschand), consideration
Of Y¢f) love sufficient, In any event, PI/} scaleable frors the
lbovo if ended

*Withoue lose of generally, we may normaliae 7 = 1, and Y¢/1
‘vecomen

r~

Sowv-ninf(e-f-a)). a

‘The simplest property observable fom the above isthe output de
samponent in YUN He

Yo Sx ante
-tx0, an

which holds for any general X(/) (se Ref. for comparison)

‘Let us now examine che bandwidth property a ye). We assume chat
the inpat signal is band-limited to B He, ie, X(j) i wero tor (f| = B.
With the normaliaation of 7 = 1, X(f) is band-imived to M = 8/7
Dropping the multiplsing constant of 1fa for convenience, nd at
Duntular frequones f= af, (call thal- is che time-compression
factor,

Yiof)= E Kafe h) sine s[fla—1)—K], 683

‘where the sin funtion provides weightings for various poins in X(/>.
Since the sine function is maximum when ils argument is ze, it i
Sensible to perform Use suanation starting with the value of & that
mnazimizes the sine function, Denoting such» value of By’ By, itis

2172 THF BELL SYSTEM TEGHINCAL JOURNAL, NOVEMBER 1901
ven hy solving

fe 6) ns

“The snaion is
iy=file— ite, el U5, 20)

whore « is necansary heeaute By i atric to bo intoger, and sis
Unique, except for the case ¢ = 208. Te xhoull bo emphasized that
depends on both f nd. Uring thie expeemion for hy Fla ean be

Ylafd = Lay ~ A sine afte — 1) — AT,

Y Kaf,~ A) sine slate 1) ~ 2

=x
+ EM et Wyrm ries en

ine at}

where X in the surmation is taken ae K-— 21, 22, and s0 on A
trephical representation ofthe above ie depicted in Fig. 3

"To got irmedite insigh ico the bandwidth property, we consider
the following eo

Case 10 = integer, f,~ integer. Under thie assumption, «
vee bieve a simple relation a

Vlas = XU ey

‘This means that ovary intoger value of fin X1/) ie mapped exsetly
‘only af in ¥¢j). Without the normalization on Z, cis i equivalent
stving that every integer mukiple of 7 in X(f) is mapped exactly
‘nto @/T in ¥if) as shown in Pig. 4. Ifthe condition that 1/7 -€
Tle iis almost contain hat Uhe spaectrum (7) ie simply the

and

TIME-COMPIESSION MULTIPLEXING 2173
ig +_Frequenerapaanm expansions prety hoagh tine compre,

frequeney-expanded version of X(/) by the faccor « with some ins
hificentsidobands for |f|> olf He, In the particular ease of reat, 7
js taken as a sean Hine dureton, yielding 1/77 = 16.73 kl With B =
42 Mile BT = 267, using a 38 the bandwidth expansion factor i
Therefore, a good sue af thumb for the TW case, We noce that the
fondition of a and f being integers is merely an artifact due to
rormalization, Therefore, We emphasize thal this case i indeed more
fneral than it appears toe, For instance, for any given set of a, we
fan always find 9 act off, sweh chat this condition holds

‘Cave 2 7 integer, fo integer. ei nonerally nonzero here, und
Yer) is given by weighted eum of X(/.—« + &) [300 (21) and Fi,
8] with X(/, ~c) a the main contributor. An alternative view that
Yar.) is the weighted average of X(f,~ 6 and its neighboring paints.
‘The output spectrum Y/} x ngsin «frequency-expanded version of
[X(j), except fr ripples created by the averaging process. 1¢ is noted
that '¢ can be zero here resulting in. Y(af.) = XU). This occurs
‘whenever fla = 1) wn integer in 20}, 08, @= 35, fe ~ 108.

"The foregoing diseusion by and larg enswors the asc question on
Dondvwidth expansion in test syacems. The result of wsing che time-
fenmpresion factor « 2 the bandwidth-expansion factor males sense
or moat cases where Ube input spectrum X(7) can be modeled us
contitwous anil batd-limited, he inelusion of pecaliar aulls ued del
fimecione in X(/) would compliace the mater, but ican be examined
‘n-decail using Ue equations provided ahove. Aa co the precze shape
fof YUP) in comparison Lo Xf), whieh is relevant in the spectrum

2174 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981
expausion applicution, some interesting discumsons are given in Ap
pens

I. NUMERICAL EXAMPLES OF OUTPUT SPECTRUM
We present in this section specific numerical examples of output
spectra calculated from (12). There are four dierent types of X(f) in
the examples:
(i) Rectangular

rie {> Wes,
wom [ Mee, a
orate te nee td Te 9/8
SeeS the a tania bein aie ate ee
Sond mace ea te Mepeit e
ioe tcl (faa eh Oe Sas os
ae tad ta 5 ron erly be te care
ral ien sera ea ae
Patelponer nace bat
eee

lien,

+ oxherwise

en

“TIME-COMPRESSION MULTIPLEXING 2175
Bis normalized to be 1 Ha, and 1° = 267/8 seconds. The results are
plot in ig. 6, Here the sidebande are oo low that they are nol
fbservable in the diagram, Of course, thie is due co tho viper
Characteristic fn X({). Some ml ripples are again present. insie che
bandoridth [/|-= 0B.

ii) Hall Covine

sn fof) ee o

o + etberwie

Bis again {Hz and 1'= 267/7 seconds, The reeuls are plotted in Fig
‘and the saz olwervations asin (2) apply here

(is) Trunonted Half-Cosine

“The equation for X(7) is che same us in (i) above, except that is
(0.90 Hs, which coneespondscoa 20-48 taper at XU as corapared to
$10), is agin tan a 267 seconde, The reals are shown in Fig. 8
‘phere the glitches ac the edge of the output pascband (Le. = @B) ae
evident, Note thatthe sidebandl outside the pascband are much lower
Computed to thee in (7 nbowe.

Iv, BAND-UIMMTING EFFECTS
‘When the original int signal x(a band-limited lo B Hs, we have
own thac most of the power in the output time-compresaed signal

Fir -Ouputnmemmot ine caneemen sg ih angling ep

2176 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981