compression, filtering, and signal-to-noise ratio in a pulse-modulated system

Compression, Filtering, andSignal-to-Noise Ratio in a

Pulse-Modulated SystemBy E. A. MARCATILI

(Manuscript received April 19, 1961)

Some aspects of the transmission of Gaussian pulses in a frequency-divi-sion multiplex system have been calculated. It is assumed that each channeltransmitter includes an amplifier, which may be nonlinear, with input andoutput filters, and that each channel receiver includes a linear amplifierwith a thermal noise source at its input plus input and output filters. Theinfluence that (a) transmitting amplifier compression, (b) maximally flatchannel-dropping filters of first or second order, and (c) distribution offiltering in the system have on the signal-to-noise ratio, time crosstalk, andfrequency crosstalk is calculated.

I. INTRODUCTION

A proposed long distance waveguide system' will transmit a largenumber of wideband PCM channels via a single waveguide by means ofa frequency multiplexing arrangement. In previous papers2 ,3 an idealizedfrequency multiplex system was analyzed. These papers considered asystem in which each channel signal, consisting of on-off carrier pulses,arrives at a detector together with three unwanted signals: (a) time cross-talk, due to leading and trailing edges of neighboring pulses; (b) fre-quency crosstalk, which is the interference from neighboring channels;and (c) thermal noise" generated essentially in the first amplifier of thereceiver. Since these unwanted signals cause errors in the reading of thepulses, the filtering characteristics are closely related to the probabilityof errors. It has been shown that for any given probability of error it ispossible to design the filters in such a way that a very desirable result isachieved-namely, the simultaneous minimization of time spacing be-tween successive pulses, of frequency separation between neighboringchannels, and of signal-to-noise ratio.

Throughout the paper, when we talk about "noise" we mean thermal noise.1421

1422 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1961

These conclusions were derived by analyzing a linear system consist-ing of a transmitting filter and a receiving filter. Such a scheme does notgive quantitative answers to three practical questions:

i. Because RF power is expensive it is desirable to drive the trans-mitting amplifiers as hard as possible, and consequently they mustoperate in the nonlinear portion of their characteristic. What is the in-fluence of nonlinearity on the system's performance?

ii. Filters between transmitting and receiving amplifiers reduce thesignal without changing the noise. Consequently, in order to increasesignal-to-noise ratio it would be desirable to have no filter between thoseamplifiers. But then the frequency crosstalk would be prohibitively high.How much filtering before and after each amplifier makes a reasonablecompromise?

iii. All of the immediately available channel-dropping filters thatconnect transmitters and receivers with the transmission media4 6 6 areapproximately Butterworth filters of first or second order. Pulses belong-ing to one channel will lose part of their power in passing through neigh-boring channel-dropping filters. How is the frequency spacing of channelsaffected by this loss?

The object of this paper is to answer those questions. More specifically,we want to find out the influences that compression, distribution of filter-ing with respect to the amplifiers, and the use of the available droppingfilters have on time crosstalk, frequency crosstalk, and signal-to-noiseratio. The results extend those already derived previously.s-"

In Section II the system to be analyzed is described and the funda-mental formulas are introduced. The influences of neighboring channel-dropping filters and nonlinearity of amplifiers are described in SectionsIII and IV. Signal-to-noise ratio is evaluated in Section V, and some de-sign examples are given in Section VI. Section VII contains a resumeof results and conclusions.

II. SYSTEM DESCRIPTION AND ANALYSIS

The system to be analyzed, Fig. 1, is ideally phase-equalized. The ef-fect of imperfect equalization can be derived? as a perturbation of theresults obtained in this paper.

Table I shows that the transfer characteristic of each transmitter,excluding the amplifier is Gaussian and that that of each receiver isgiven by the sum of two slightly displaced Gaussian functions, whichapproximates the characteristic of a maximally flat Butterworth filterof third order and is easy to handle mathematically. Quantitative rea-sons for the selection of these filters can be found in a previous paper.!

COMPRESSION, FILTERING, SIGNAL-TO-NOISE RATIO 1423

INPUT PULSE I

~ /\ t[I+(1TT~tlX)2r

t-I II II IL J

RECEIVERFILTER 2

PORT 3

PORTt..~_TRANSMISSION

MEDIA

CHANNEL-DROPPING E(t)tFILTERS

~...L-~

IIIII-.,.-III

1-------- ----,II

Fig. 1 - Block diugram or system (see definitions or filters in Tuble I).

Qualitative reasons are that these filters make a good compromise forlow time crosstalk and frequency crosstalk. In effect, in order to mini-mize the tails of the output pulses and consequently the time crosstalk,it would be best to have both filters Gaussian. On the other hand, inorder to minimize frequency crosstalk, it would be best to have bothfilters maximally flat (Butterworth) of high order. A reasonable com-promise is achieved by making one of them Gaussian and the othermaximally flat of high order. A transfer characteristic relatively easy toachieve without too much midband attenuation is that of a maximallyflat filter of third order, which we adopt.

If the transmitter and receiver are centered at the same frequency,the transmission through them measures the insertion loss through achannel. If they are centered at different frequencies, the transmissionthrough them measures the frequency crosstalk.

Each transmitter includes three filters and a nonlinear amplifier. Thefilter la preceding the amplifier and the combination of the two fol-lowing it, 1{3, are Gaussian. Each receiver consists of a linear amplifierbetween two filters 2a and 2-y whose combined transfer characteristic isapproximately maximally flat of third order. This scheme allows one toevaluate the effects of the amount of filtering preceding and followingeach amplifier.

~

TA

BL

EI-

DE

FIN

ITIO

NS

OF

SY

MB

OL

S

Filte

rsin

Fig.

1Tr

ansf

erC

hara

cter

istic

s3

dbB

and-

Cen

ter

Not

esw

idth

Freq

uenc

y---

IT

rans

mit

ter,

excl

udin

gG

auss

ian

Y1=

exp

l-a[

2,--......--....--,

2 .....----r---....,...--...,

If. -f.1 1f.-f.1ABSCiSSAS: P=_'_3_ =-'-"-3Fz 2FzORDINATES' 20 LOG E, (0) Eo(o)

Fig. 2 - Insertion loss due to first-order neighboring channels.

choices of signs correspond to frequencies 13 and 14 of the neighboringchannels on the same or different sides of it . The first case introducesless insertion loss, but for the ranges of values selected in Figs. 2 and 3,and for small abscissas, the difference in ordinates for the two cases issmaller than 0.5 db. As a compromise, Figs. 2 and 3 have been calculatedby averaging the results.


32

2 .................,oy---.---...,2.----,-r--.,----,

_ SECOND-ORDER _"> FILTERS (q=r=2) - ......

, I ,',/ I \'

,/ / \',~/ I \,

0 0 0, 4 I 2 3 4 , 2 3 4F17 !!z. F17 F27 !!Z. F27-=,. F -1.5 F;""" = 1.5; F;"" =1.1I =2; --=1.5F1 ' 2 F. F2

2 2 2

0 0 0I 3 4 , 4 I 4

~ F' 7 .2z. ..sz =2; F27--=11I' =2 --=2F2

=2 F, ., F2 F, F22 2 2

O!-.--~--==-~-~4 O!-,---=-........;""""'!:--~4 O!-,---!:--...=.:~~~

If. - f I 1f.-f.1ABSCISSAS: P =_'_3_ =_'_4_2Fz 2Fz

ORDINATES' 20 LOG E.{o) Eo{o)

F,--F;=1.3

F,----=1F2

Fig. 3 - Loss due to second-order neighboring channels.

Let us concentrate on Fig. 2, (q = r = 1 j that is, channel-droppingfilters of first order). What parameters yield a system with low insertionloss due to neighboring channels? For a given abscissa (normalized chan-nel spacing p), the smallest ordinate occurs in the lowest line in the upperleft set of curves. This line is characterized by


As could be expected, the system should have(a) long input pulses,(b) wide and equal bandwidths in transmitters and receivers,(c) narrow and equal channel-dropping filters at transmitting and

receiving ends.In order to compare Figs. 2 and 3 we compute an example:

For

F1 = 1 FlY = F2y = 15 TF = 1 and = (Jl - fa) = 2F 'F1

F2 ., 7f 1 , P 2F2 '

the insertion loss due to neighboring channels is 1.25 db if q = r = 1 inFig. 2 (channel-dropping filter of first order) and 0.25 db if q = r = 2 inFig. 3 (channel-dropping filter of second order). Second-order filtersmake the insertion loss due to neighboring channels 1 db better thanthat of first-order filters. But this does not necessarily mean that thetotal insertion loss will be 1 db better in a practical case. There are tworeasons for this statement:

i. A second-order maximally flat filter, has more cavities and conse-quently more heat loss than does one of first order. We don't know howmuch the difference is, but assuming it is 0.5 db, the increase of heat lossin the two channel-dropping filters is 1 db, and consequently the ad-vantage cited previously is only apparent. This point becomes more im-portant for smaller tolerable insertion loss due to neighboring channels(large p).

n, Consider the transmitter's total Gaussian transfer characteristic-a[(I/I-!Jl/FJ!2YI = (! (19)

and the transfer characteristic of its qth order channel-dropping filter1

(20)

The transfer characteristic of the rest of the filter should be YI!Yly . Thereare values of the frequency f for which this function is larger than unity,and therefore it is impossible to realize a passive filter with the desiredtransfer characteristic. The difficulty is solved by allowing the transfercharacteristic of the transmitting filter to be not (9) but

(21)K

e-aliit = ---;-;:---

where K is a constant greater than unity and equal to the maximum


value that YtlYly can achieve; K expressed in db is an added insertionloss.

The previous reasoning applies also to the receiver, but, since we are fi-nally going to be interested in signal-to-noise ratios, the value of K ap-plied to the receiver drops out.

In order to have an idea of the values that K may achieve in Gaussiantransmitters we have calculated some examples:

First-order maximally flat channel-droppingfilter (q = r = 1);

Second-order maximally flat channel-drop-ping filter (q = r = 2):

20 log K

0.3 db

< 0.1 db

In general, the broader the channel dropping filter and the flatter thetransfer characteristic of the transmitter or receiver, the smaller theadded insertion loss 20 log K. This added insertion loss can be neglectedexcept when Fly = F I and q = r = 1.

Now we turn to the increase of time and frequency crosstalk that isintroduced by the presence of neighboring channel dropping filters. It isshown in Appendix B that, for systems in which the increase of insertionloss due to neighboring channel-dropping filters is not very large, (around1 db), the increase of both time and frequency crosstalk is negligible.

IV. INFLUENCE OF THE NONLINEARITY OF THE TRANSMI'lTING AMPLIFIER

ON SIGNAL-TO-NOISE RATIO AND CROSSTALK

Let us assume that, in Fig. 1, (a) the transmitting amplifier is driven sohard that the input signal is not linearly amplified, and (b) the presenceof neighboring channel dropping filters can be neglected; this means that.fa = .f4 = 00 and, from (7) and (8),

tfli = tfl2 = 00. (22)Call E2(t) the output envelope (1) when (22) is satisfied and normalize

this output to the value Eo(O) that measures the peak of the output(t = 0) that would occur if there were no compression (AN = 0 for N ~


0), and with the transmitter and receiver centered at the same frequency,il = h . Then,

(23)

. cosh [2np(I - blA2BN) + inpABN~]I.As explained in Appendix A, this expression has a simple physical

. interpretation. The Gaussian pulse of unit amplitude and width

2T 11'1 + (7r;Fa)2entering the nonlinear amplifier Fig. 1 produces an output that is a sum-mation of Gaussian pulses each characterized by the integer N. Thelarger N is, the smaller are the amplitude and width of the corre-sponding pulse. Because of the linearity of the circuit following theamplifier, the envelope of the normalized output of the system (23) isthe envelope of the sum of the transients produced by the Gaussianpulses.

For

t = p = 0, (24)

(26)

. (23) measures the normalized maximum output intensity through achannel (p = 0) which, expressed in db, we call output compression:

20 log E 3(O) = 20 logto[1 + i: AN TN IBN e(n IJd/2)2(BN- BOI] . (25)Eo(O) N-l A o To 'V e,

Naturally, if there were no compression (linear amplifier), AN would bezero for all N and the output compression would be zero db.

The amplifier compression characterizes the nonlinear amplifier. Itmeasures in db the ratio between the amplitude of a sine wave input andthe amplitude of the output without harmonics. It is derived from (25),(5), (6), and (13), making T = 00:

01 E.(O) 0 I ( ~ AN)2 og Eo(O) = 2 og 1 +~ A o Assuming the summations in (25) and (26) to be small compared tounity, it is possible to expand the logarithms in series and to retain thefirst term; then the ratio between output compression and amplifiercompression is


tAN1 A o

(27)

00

1.0

2

0.8 0.90,4 0.5 0.8 0.7(F1/FI/3) 2

05

0.2 0.30.1

p.= ~: = 1.41TTF,I t.O L

0.75 }N=0.50 t-- -- ----- '--- ~----- __---!L}1--------------- ~-----I-__~.7S N=2--___ 1-0 .50 I":":""":"

I

9F

Jot= F: =1.08

"- 1TTF, i.o"-I---1--- -- O~}N"~-- 0.501--- '--~- i----- - ----]1--- --1-_.-- t.O 1--- --~-- t--_~,.... N=2---'---I-- 1-. 0.75

- 0.501-

0.0

0.40.8

0.7

0.5

0.8

0.4o

0.5

~-~alZco~- z:112WillII: IIIo.w:::Ii II:00.I) :::IioWI)~II:::;)wo.iL1-:3::;) 0.o.:::IiI- -

compression, filtering, and signal-to-noise ratio in a pulse-modulated system

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