Document text
Ideal Binary Pulse Transmission
by AM and FM
By E. D. SUNDE
(Manuscript received May 18, 1959)
In binary pulse transmission by carrier amplitude or frequency modula-
tion it is ordinarily desirable, both for efficient bandwidth utilization and for
improved performance under adverse noise conditions, to use bandpass chan-
nels of the minimum practicable bandwidth, as determined by considerations
of intersymbol interference and filter design. It is shown that intersymbol
interference can be avoided in binary pulse transmission by FM without the
need for a wider channel band than in double-sideband AM, for equal pidse
transmission rates. Explicit general expressions are derived for the appropri-
ate shaping of the bandpass channel and for the shapes of received pulses, for
cases in which rectangular binary pulses are transmitted by FM, unthout
premodulation or postdetection pulse shaping by low-pass filters. Illustra-
tive comparisons are made of binary pulse transmission by AM and FM
for two special cases of general interest in communication theory and pulse-
system design. The more general case of partial pidse shaping by premodula-
tion and postdetection low-pass filters is also considered.
The performance of FM and AM systems in the presence of noise de-
pends on the division of channel shaping between transmitting and receiving
filters. The optimum division with FM and AM is determined for random
noise, and comparisons are made of signal-to-noise ratios for optimized FM
and AM systems. It is shown that there is a single universal relation between
error probability and signal-to-noise ratio, applying to an infinite universe
of optimized baseband systems and optimized AM systems with ideal syn-
chronous detection, and that this relation is the same as for baseband trans-
mission over an idealized fiat channel of minimum bandwidth. The analysis
indicates that, with binary FM and appropriate postdetection low-pass fil-
ters, it is possible in principle to realize an improvement in signal-to-noise
ratio over bipolar double-sideband AM with synchronous detection {phase
reversal), for equal channel bandwidths, average signal power and pulse
transmission rates, although this may not be feasible with practicable filters.
1357
1358 the bell system technical journal, novembeh 1959
Table of Contents
I. Introduction 1358
II. Frequency and Amplitude Modulation by Binary Pulses 1360
III. Transmitted Frequency Shift-Wave _. 1363
IV. Frequency-Shift t*ulse Transmission Characteristic 1365
V. Ideal Frequency-Shift Tran.?misaion Characteristics 1367
VI. Special Case of Flat Spectrum at Detector Input 1370
VII. Special Case of Raised Cosine Spectrum at Detector Input 1372
VIII. Premodulation Pulse Shaping 1376
IX. Raised Cosine Modulating Pulses 1379
X. Postdetection Pulse Spectra and Filtering 1382
XI. Pulse Trains 1386
XII. Average Signal Power 1389
XIII. Random Noise in AM and FM Systems 1391
XIV. Oplimum AM Systems 1394
XV. Optimum FM Systems 1398
XVI. Optimum FM Systems of Minimum Bandwidth 1400
XVII. Optimum FM Systems with Raised Cosine Spectrum 1403
XVIII. Signal-to-Noiso Ratios and Error Prol>abilities 1406
XIX. Summary 1411
XX. Acknowledgments 1413
Appendix A. Pulse Train Envelopes and Average Signal Power 1413
Appendix B. Interference in AM and FM Systems 1416
Appendix C. Optimum Receiving Filter Characteristics 1424
References 1426
I. INTRODUCTION
Transmission of digital or analog information by binary rather than
by multilevel pulses offers significant advantages in systems design.
For one thing, it simplifies the implementation of regenerative repeaters
and various kinds of terminal equipment, such as carrier modulators and
demodulators, and devices for timing-wave provision, coding and stor-
ing of messages and automatic error-checking or correction. For another
thing, binary pulse transmission imposes less severe requirements on the
transmission medium with respect to signal-to-noise ratio, amplitude and
phase deviations over the channel band, and tolerable transmission-
level variations. Because of these advantages, binary rather than multi-
level pulse transmission is ordinarily the more practical and economical
method, even in existing channel facilities designed primarily for voice
or other analog transmission, where consideration of the rather high
signal-to-noise ratio alone would permit a much greater number of pulse
amplitudes and, thus, a substantially greater channel capacity than
could be economically realized.
The three principal methods of binary pulse transmission by carrier
modulation now in use are double- and vestigial-sideband AM, in the
form of "on-off" keying with envelope detection, and FM in the form
of "frequency-shift" keying. With synchronous or homodyne detection
in AM, other methods are feasible that afford a bandwidth saving or
improved signal-to-noise ratio or, like FM, have the advantage over "on-
IDEAL BINARY PULSE TRAXSMISSION BY AM AM) FM 1359
off" AM that they facilitate rupid automatic compensation of trans-
mission-level variations. Among these binary methods are bipolar double-
sideband AM, also referred to as phase reversal or two-phase modulation,
and bipolar vesti^ial-sideliand AM. Another method Is bipolar double-
sideband AM on ea(^h of two carriers at quadrature, also referred to as
quadrature double-sideband AM or four-phase modulation.
For optimum performance in binary pulse transmission by AM or FM
it is essential that the transmission-frequency characteristics of the chan-
nels be appropriately shaped with respect to amplitude and phase, so
that intersymbol interference is avoided or at least reduced to a practi-
cable minimum. A second re<|uiremeut for optimum performance in the
presenre of noi.se is an appropriate division of channel shaping between
transmitting and l■ecei^■ing filters. In addition, there are various other
requirement-s not pertaining to the channel, such as exact timing in the
transmission and reception of pulses and ideal AM and FM modulators
and demodulators.
The purpose of this presentation is a determination of these optimum
channel cha.racteristi(^s and the optimum signal-to-noise ratios for vari-
ous error probal)ihties in binary pulse transmission by AM and FM, with
particular emphasis on FM.
The analysis of both analog and digital pulse transmission for FM is
more complex than it is for AM, since FM is a nonlinear modulation
method. For this reason, the sideband spectrum of a given signal is wider
than it is in double-sideband AM, and a wider bandpass chamiel is re-
quired for analog signal transmission without distortion.
In analog transmission it is possible to realize improve<l performance
inthepresenceofnoi.se in exchange for the inerea.sed bandwidth. In binary
pulse transmission, however, it is ordinarily desirable, both for optimum
performance under ad\'('r.se noise conditions and for efficient bandwidth
utilization, to use the minimum channel bandwidth practicable from the
standpoint of intersymbol interference and filter design.
While a wider l)antlpass channel than in AM is required for distortion-
less analog transmi-ssiun, this does not preclude the possibility that, un-
der appropi'iate conditions, pulses can be tran.smitted by FM with no in-
tersymbol interference, without the need for a greater channel band
than is required in double-sideband AM. This depends on the po.ssibility
of controlling pulse distortion resulting from bandwidth limitation so
that zero points in the received pulses occur at uniform intervals from
the peak pulse amplitude. When pulses are transmitted by AM, this
can be accomplished l)y appropriate shaping of the transmission-fre-
quency characteristic of the channel, as shown elsewhere.'-- The analysis
is extended hercui to ideal l>inary pulse transmission by FM.
1360 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
A basic criterion for performance in digital transmission through noise
is the error probability as related to the signal-to-noise ratio, which has
been dealt with elsewhere for baseband transmission,^ "on-off" double-
sideband AM with envelope detection,'*'^ bipolar AM or carrier phase
modulation with synchronous detection^'^ and frequency modulation. ^-^
In the above analyses random noise is assumed and the signal-to-noise
ratio is stated in terms of signal power during a steady "mark" or
"space," which, in bandlimited channels, is usually not equal to average
signal power, even in binary phase or frequency modulation systems.
Moreover, ideal flat baseband or bandpass channels of the minimum
bandwidth required to avoid intersymbol interference are assumed or
implied in AM, although they are not practicable in actual systems. In
the case of FM, no consideration is given to bandwidth requirements
and channel shaping for optimum performance.
As an aid in systems design, specific consideration is given in Sections
II through XI of this presentation to appropriate bandwidths and chan-
nel shaping for AM and FM systems. The remainder of the analysis is
concerned with signal-to-noise ratios as related to channel shaping and
to appropriate filter shaping for optimum performance in the presence
of random noise.
II. FREQUENCY AND AMPLITUDE MODULATION BY BINARY PULSES
The original signal ordinarily would consist of rectangular baseband
pulses of duration T, with a negative polarity to indicate "space" and a
positive polarity to indicate "mark," or conversely. These rectangular
pulses may be applied du-ectly to the frequency modulator, or they may
be applied to a premodulation low-pass filter for preshaping.
The modulator output is applied to a bandpass channel with a certain
transmission-frequency characteristic, which, in the ideal case, would be
symmetrical about the midband frequency and have a linear phase char-
acteristic. The envelope of the received pulses at the channel output and
the frequency modulation of the carrier within the envelope depend on
the shape of the modulating pulses and on the transmission-frequency
characteristic of the bandpass channel. With an ideal detector at the re-
ceiving end, the demodulated signal is proportional to the time deriva-
tive of the phase of the carrier within the envelope, i.e., to the "instan-
taneous frequency deviation."
Detection of the phase derivative is facilitated by conventional fre-
quency discriminators or zero-crosshig detectors when the channel band-
width is narrow in relation to the carrier frequency, but this is not a basic
theoretical requirement if appropriate detectors are postulated. Nor,
IDEAL BINARY PLTLSE THAXSMISSION" BY AM AND FM 13fil
with appropriate ideal balanced FM detectors as assumed herein, is a
Uniitcr necessary for elimination of amplitude modulation effects, al-
though it is highly desirable with unbalanced detectors and for preven-
tion of undesirable effects of sudden level changes.
At the detector output, a postdetection low-pass filter may be used for
final pulse shaping, but it is not essential for this purpose. Such a
filter may be required to eliminate unwanted demodulation products
(carrier ripple), but the bandwidth reciuired for this can be much greater
than that of the bandpass channel, particularly when the channel band-
width is small in relation to the midband frequency. This condition can
always be realized by frequency translation before demodulation, which
may also be required for optimum performance with conventional fre-
quency discriminators or zero-crossing detectors.
A more important function of the postdetection filter in conventional
analog signal transmission is elimination of higher frequency noise com-
ponents, in order to realize the inherent FM noise advantage. In binary
pulse transmission with a bandpass channel of no greater bandwidth
than is required to avoid intersymbol interference, as considered here,
the noise advantage that can be derived from the use of a low-pass filter
may be rather limited. To realize a significant noise advantage the filter
must, in this case, have the appropriate shape, depending on its band-
width, to avoid excessive intersymbol interference, as will be shown
later.
In frequency modulation, the signal applied to the input of a band-
pass communication channel is of the general form
E<{t) = sinM + ^ + ^.(OL (1)
where m is the radian frequency of the unmodulated carrier, tp is the
carrier phase and ^,(0 is related to the modulating voltage T^■(0 by
V-.XO = ^1 [ V,{t)dt, (2)
with oil the frequency deviation in radians per second per volt.
In the case of bipolar binary pulse transmission, the original signal,
W{t), is ordinarily in the form of rectangular pulses of amphtude Vq and
duration T, of either positive or negative polarity. In this case, the origi-
nal signal is constant during a signal interval of duration T and is
given by
Fo(0 - ±ro. (3)
In general, with a premodulation low-pass filter, the carrier modulating
1362 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
voltage, Vi{t), differs from Vo(t). Without such a filter, as assumed here,
F,(0 = y^it) = ±1^0 and
Ml) - dzwi [ Vo dt
Jo
(4)
where the frequency deviation w is
Ci = (iiFo ■ (5)
The voltage applied to the input of the bandpass channel in this case
is, in accordance with (1),
Ei{t) - ain [(wo ± ^)t + ^]. (6)
Equivalent performance could accordingly be obtained if the outputs
of two oscillators of frequency wo + w and m — i^ were gated by the
voltage V(,{t), so that carrier step pulses of duration 3' and one or the
other of the above two frequencies would be applied directly to the band-
pass chamiel. If the latter method is actually used, the two oscillators
must be interlocked to avoid excessive phase discontinuities and resultant
transmission impairments that would otherwise be likely to occur in
switching from one oscillator to the other.
With phase modulation, rather than frequency modulation as con-
sidered above, \f/,it) in (1) is related to the modulating voltage by
^^{t) = ^iF,(0, (7)
where \pi is the phase modulation in radians per volt.
In the case of bipolar binary pulse transmission, (1) becomes
(8)
Ei(t) = sin [mt + ^ ± ^iViit)]
= sin (woi + >p) cos [<piVi{i)] ± cos {wo^ + <p) sin U'lViit)],
where the negative sign is used for a space and the positive sign for a
mark, or conversely.
The fii-st component in (8) is independent of the pulse polarity. In
an optimized system this component must be minimized and the second
component, which depends on the pulse polarity, be maximized. The
optimum condition is obtained when ^i is so chosen that i^-it^i = ir/2,
where f , is the peak amplitude of F,(i). In the particular case of rec-
tangular modulating pulses, V,(t) = f. = Voand (8) becomes
Ei(t) - ±cos (o,d + <p), (9)
which represents a sudden phase reversal from space to mark.
IDEAL BINARY PUI.SE TRANSMISSION BY AM AND FM 1363
In amplitude modulation, the signal applied to the bandpass channel is
E,(t) = [a, + a^V,it)/V,] cos (wo« + ^), (10)
where at, and a^ are constants that determine the degree of modulation,
as discussed below for two special cases.
In unipolar or "on-off" binary pulse transmission, ai = a^ and, in the
particular case of rectangular modulating pulses,
Ei(t) = ao(l ± 1) cos (woi + <p)
= for space (11)
= 2ao(cos oiot -\- <p) for mark.
In bipolar AM, Oo ^ and, for rectangular modulating pulses, (10)
becomes
Eiit) - iaicos (woi + ^), (12)
which is identicul to (9) with a^ = 1.
With phase reversal or bipolar AM, the signal can be recovered with
the aid of a product demodulator, i.e., by homodyne or synchronous de-
tection. To this end, a synchronous demodulating carrier, cos (woi -1- ^),
must be derived from or controlled by the signal, which may entail more
complicated instrumentation at the receiving end than is required with
frequency modulation. Unipolar AM permits the use of simple envelope
detection in exchange for a sacrifice in signal-to-noise ratio compared to
the other methods. A further disadvantage of imipolar AM is that it is
more susceptible to errors during sudden level changes than is bipolar
AM or FM.
With any of the above modulation methods the shape of the received
pulses depends on that of the modulating pulses and on the transmission-
fi-equency characteristic or ''shaping" of the bandpass channel. The ap-
propriate shaping for avoiding intersymbol interference is well known^''
for baseband transmission and amplitude and phase modulation systems,
and is determined in the following sections for binary FM.
The particular case of rectangular modulating pulses will be considered
in detail, and explicit expressions will be derived for appropriate channel
shaping to avoid intersymbol interference. The more comphcated cases
of premodulation and postdetection pulse shaping will be discussed later.
III. TRANSMITTED FREQUENCY-SHIFT W^AVE
Let a continuing "space" be represented by a steady-state transmitted
wave
E:{t) - sin [{ojo - C3)t + if>\ (13)
1364 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
and a continuing "mark" by
EJ(t) = sin [(m -{- <^)t -\- <p], (14)
where t is the time from the beginning of a signal element of duration T
and is related to the time with respect to the midpoint of a signal ele-
ment by
t = to-^ T/2. (15)
With (15) in (13) and (14):
ir(io) = sin [(o-o - w)(io + T/2) + ^1
(16)
= sin [(wo — w)io + vo — w7'/2],
EJ'(U) = sin [(wo + ^)^o + ^0 + ^T/2i (17)
where
v^o = ^ + c^oT/2. (18)
It will be assumed that
Jir = T, (19)
in which case the frequency diifercnce between mark and space in cycles
per second is 2a)/27r = l/T, or equal to the bit-rate. This assumption
need not be made at this point, but it turns out later to be a condition
for avoiding intersymbol interference and simplifies the analysis. With
the above assumption,
EfiU) = -cos [(ojo - ^)U + vo], (20)
Ej(io) = +C0S K'^o + ^)ia + vol- (21)
Assume that a single mark of duration T is preceded and followed by a
continuing space. The resultant transmitted wave can be regarded as
made up of two components. One is a steady-state component given by
the following expression applying for — oj < (o < ^ '■
E^ito) - -cos [(coo - w)fo + ^o]. (22)
The other is a transient KJ = --E/ + E^" given by the following ex-
pression applying for — T/2 < k < T/2 :
E,J(to) = cos [(wq — w)/o + <Po] + cos [(ojo + oj)to + ^o]
(23)
= cos {oioto -\- tpo)2 cos Oitii .
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1305
The spectrum of Eim° is given by
/r/2
COS (ojo'o + Vo) COS tj^o e"'"'" dio
-Tl2
COS uto COS uto dto (24)
-r/2
/r/2
COS (2wo + w)'o COS wfo dto ,
-Til
where it = w — coo is the frequency from midband.
When the bandwidth of the channel is small in relation to the midband
frequency, the spectrum need only be considered for u « wo . The second
integral in (2-4) can then be disregarded in comparison with the first, and
the amplitude of the spectrum becomes independent of ipo , i.e., inde-
pendent of the phase of the carrier with respect to the modulating pulse.
On this assumption, the amplitude of the spectrum of the carrier en-
velope at the frequency u from the midband frequency ojq is given by
the first integral in (24) , and becomes
With (IiT = TT in accordance with (19), (25) becomes
ev. ^ crv . ^ ^ ^ *^os (tw/2w) , ,
2x1 — («/^)
For u = ±w, the value of (26) is*
S'i^w) = |. (27)
IV. FREQUENCY-SHIFT PUISE TRANSMISSION CHARACTERISTIC
Let the phase characteristic of the channel be assumed to be linear,
and let A (w) be the amplitude characteristic of the channel as a function
of the frequency u = uy — o}o ■ At the channel output, the spectrum of
the received wave resulting from Esm''it<,) given by (23) is then
Siu) = A{ii)S"{u). (28)
If the amplitude characteristic is symmetrical, i.e., li A{—u) = A (it),
the spectrum is also symmetrical; i.e.,
S{-u) = Siu). (29)
This result ia obtained by determining the value of the limit 0/0 as u —* u.
1366 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
When the above conditions are satisfied, the shape of a received pulse
in response to E„^{U) is given by^
E,„M = cosMo + sPo]^.„(io), (30)
where Ej„ is the envelope of the received pulse. The shape of the en-
velope is the same as that of a demodulated pulse in AM, when the pulse
spectrum is S{u), and is given by^
p(/o) - E,,Mo) =- [ S{u) cos utodu, (31)
where the lower limit can, for practical purposes, be replaced by — so ,
since S{n) ^ for u = —uo when tj « wo ■ The received wave in re-
sponse to the steady-state component ^/(fo) given by (22) is
Esita) = —A{—o}) cos [(wo — w)to + <Pfi\
(32)
= — A(— aj)[cos (wo(o + ^'o) cosw^o + sin (oiak + i^o) sin ai(ol,
where A{—o)) = A (to) is the amplitude of the transmission-frequency
characteristic A(u) at m = ^w, i.e., at the frequencies wo =F ib.
The resultant received wave when a mark of duration T is preceded
and followed by a continuing space is
E{to) = E,(to) + EUto) (33)
= —cos (woio + ^o)[^( — w) cos Olio — p{to)]
—A{ — oj) sin (coo^o + fpo) sin wto .
This can be written in the form
Eito) - E(to) cos (a.o(a -K s^o + h), (35)
where the envelope is given by
E(lo) = ^(-w)i[cos w/o - fipf + sin^ w/ol"'
= A(-w) ll -i- mV - 2mP cos u/o!"^
and the phase modulation ^o is given by
tan^o(fo) = ^i^^^^, (37)
cos oiJo — A*p
where p = p(io) and
y. - l/A(-ti) = l/A{^). (38)
(34)
IDEAL BIISfARy PULSE TRAXSMISSTON BY AM AND FM 13G7
With an ideal detector, the received signal is proportional to i/'o' —
d^o/dt, which becomes
, > . 1 — MP cos w/o + (mp'/w) sin oik /^n^
Wo = -co ^^— — — — y, , l-^yJ
sni-w/o + [cos wto — i^pr
where p' ^ dp(i(,)/dlo .
Relation (39) gives the frequency deviation from the midband fre-
quency Wo , in which case a continuing space is represented by a fre-
quency deviation — w and a continuing mark by to. If the frequency
uo — 0} is instead used us reference, the frequency deviation at the re-
ceiving end is
yf^'W = ^ + ^p.'ito). (40)
The ratio ^'(;o)/2a) represents the pulse transmission characteristic of
the channel in response to a sudden frequency shift 2w of duration T,
from coo — CO to coo + 03, and is given by
„,, V li HP — cos o}t(i — (p /w) sin Oil
2 sm- wio + (cos coto — up)-
(41)
From (30) or (41.) the conditions for binary pulse transmission with-
out intersymbol interference can be established^ as is discussed in the
next section.
v. IDEAL FREQUENCY-SHIFT TRANSMISSION CHARACTERISTICS
In order to transmit binary pulse trains without intersymbol inter-
ference, it is necessary that the transmission characteristic P{ta) for a
single mark or pulse as considered in the preceding section be zero at
sampling instants U = ±viT, m = 1, 2, 3, etc., and that, at h = 0,
P - 1.
In viewof (iy),coH(±77tor) - ( -1)'" and sin (i^nwT) - 0. Hence
at sampling points (41) becomes, for ?». 9^ 0,
^^^"^^ ^-2MP(mr) + (-l)"' ^^'^
and
P(0) = \ -jE^^ . (43)
2 fip(O) ~ 1
Thus P(mT) = provided
p{7nT) - 0, (44)
1368 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
ampl
rjg. 1 — Properties of spectra at detector input such that, in double-sideband
jlitude modulation, p(0) - 1, p{mT) = 0,T = tt/w, m = 1, 2, 3, ■ ■ ■ .
and P(0) = 1 provided
MP(0) -^ 2.
(45)
The above conditions (44) and (45) can be satisfied provided that
S{u), in addition to being symmetrical as required by (31), has the
further property illustrated in Fig. 1. This property is the same as that
required for double-sideband transmission of pulses at intervals T with-
out intersymbol interference,^'' and can be satisfied by an infinite variety
of spectra. Among these it is convenient from the standpoint of theoreti-
cal evaluation of p(to) to assume spectra of the form shown in Fig. 2
given by the following expressions.
In the range < ti < w — cOi :
S(u) = S(-u)
T
2
(46)
IDKAL BINARY PULSE TRANSMISSION BY AM AND FM
1369
Fig. 2 — Special case of spectra with properties inriicaled in Fig. 1.
In the range w + toj. > 7/ > w — oj^: :
vr ^ t'/ \ '^' 2 /t w — w + oiA
k{u) = S{-u) - - cos 1 -I.
At u = w, (47) becomes
*S'{^) = 5(-w) = ~ .
(47)
(48)
The required amplitude characteristic of the channel obtained from
(28) is
A(w) =
S(u)
(49)
1370 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
In view of (27) and (48), .4(-w) - ^, so that (38) gives
M - 2. (50)
When the spectrum is given by (46) and (47), evaluation of (31)
gives
, , > TTi /, % Tw sin wtf, cos (jixU) ,--. ■.
where Toi/tt = 1, in view of (19), so that
p(/o) =
sin CiU\ cos (jiiU
w/q 1 - {2oiJ^/-K)-
= 1 for to =
= for U - mT, m = 1,2,3,
(52)
Hence, /xp(0) = 2 and p{mT) = so that conditions (44) and (45)
are satisfied, and single liinary pulses can be transmitted without inter-
symbol interference. This is also the case for pulse trains, as is shown
in Section XI.
To find the shape of the received pulses, it is necessary to employ (41)
for other values of (o than ta ^ and m-T, as is illustrated in the next
sections for two limiting cases of general interest.
VI. SPECIAL CASE OF FLAT SPECTRUM AT DETECTOR INPUT
The amplitude characteristic .4((() of the channel and the frequency-
.shift pulse transmission characteristic P(/o) will be determined here for a
channel of minimum bandAvidth, in which case the spectrum S(u) will
be flat for < u < w, and will be zero for ii > «.
With sharp cutoffs at u = ±a), cox will be zero in (46) and (47), so that
S(m) = Si~u) = ^ g w ^ w (53)
= it > w. (54)
The required amplitude characteristic of the channel, as obtained
from (49), is
A( ) = - 1 ~ {u/i^Y
^'^^^ ^ 4 cos (xM/2ti)
= ^ for u = (55)
= 1 for u = itdi
= for I w j > Wo.
IDEAL BINARY PtT.SE TRAXSMISSION BY AM AND FM
1371
In the case of double-sideband AM, the spectrum of a pulse of dura-
tion T with respect to the midband frequency is
„, , T sin (wr/2)
(50)
where, as before, it has been assumed that the channel bandwidth is
small in relation to the midband frequency.
To obtain a flat spectrum of amplitude T/2 at the detector input, the
required amplitude characteristic of the channel is
a{u) = ■-.
uT/2
Tru/2ui
sin {uT/2) sin (7ri//2w)
= 1 for u -
(57)
^ - for u = CO.
The amplitude characteristics A(m) and a(it) are shown in Fig. 3.
1.4 -
1.2 -
Q 1,0 -
< 0.6
0.4 -
0.2 -
-
Nv \ am/ >/
5(u) ^*^.^ ^^.^^
AMiFM ^-SPECTRUM
SPACE
RADIAN FREQUENCY
MARK
Fig. 3 — Ideal transmission -frequency eharacteriatics of bandpass channels in
AM and FM for rectangular modulating pulses and flat spectrum of minimum
bandwidth at detector input.
1372 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
For purposes of direct comparison, the value of ^(w) as obtained from
(55) has been normalized to A{u) = 1 for u = 0, by multiplication by
4/7r.
With a channel characteristic as given by (55), p{U), as obtained from
(52) with o)^ = 0, is
sin uiU
p(U) =
uito
ccto \
V 1^ = -— { COS wio
(Sitn
sin (aU
oita
y
(58)
(59)
With (58) and (59) m (41), the frequency-shift pulse transmission
characteristic of the channel becomes
Pito) =
sin Olio
1 +
sin ul-o rt -t
6 — I — — 2 cos w/o
Lcto
. sin uto /sin w^o -. \ '
4 ~^r ( r— - — cos Oito 1
OJto \ Oito /
(60)
The function P(to) is given in Table I and shown in Fig. 4,
The term sin utn/utQ in (60) represents the pulse-transmission charac-
teristic for double-sideband transmission over a channel with an ampli-
tude characteristic a(u). In Fig. 4 the double-sideband AM and the
frequency-shift transmission characteristics are compared. It will be
noted that they differ appreciably in shape, but have the common prop-
erties of unit amplitude at (o = and zero amplitude at intervals such
that uto = niir, m ^ 1, 2, 3, ■ • • .
VII. SPECIAL CASE OF RAISED COSINE SPECTRUM AT DETECTOR INPUT
In actual communication systems, channels with sharp cutoffs as as-
sumed in the previous example are impracticable for various reasons,
such as excessive phase distortion near the band edges and relatively
Table I — Frequency-Shift Transmission Characteristic P(to) for
Flat Spectrum of Minimum Bandwidth
Sta/ir
0.20
0.40
0.60
o.so
1.0
1.0
0.7860
0.5468
0.4182
0.2746
1
-0.3026
0.0072
0.1629
0.1454
2
-0.1540
-0.0330
0.0940
0.0985
3
-0.1023
-0.0339
0.0646
0.0743
4
-0.0766
-0.0303
0.0488
0.0597
5
-0.0611
-0.0267
0.0391
0.0499
10
-0.0304
-0.0160
0.0194
0.0274
19
-0.0160
-0.0091
0.0101
0.0151
IDEAL BINAllY PlILRE TRANSMISSION BY AM AND FM
1373
4.5
Fig, 4 — ShapeK of demodulated pulses in FM and AM for rectangular modu-
lating pulses and for flat spectrum of minimum bandwidth at detector input, as
in Fig. 3.
large oscillEitions in the received pulses that entail precise synchronized
sampling at fixed intervals to avoid iutersymbol interfei-ence. A prefer-
able type of channel characteristic is one that results in a "raised cosine"
spectrum of the received wave in response to the transmission of a single
rectangular pulse of duration T, as considered below. With this type of
channel characteristic, the bandwidth is twice the minimum possible con-
sidered in the previous limiting case.
With w^ = win (-17),
S{u) = S{-u) - ^cos' (W4^)
u ^ 2c
(61)
= u > 2w.
The required amplitude characteristic of the channel as obtained from
(49) is
TT 1 — cos (xw/2(I))
^^"^ 4 2 cos (7rw/2£L)
[1 - iu/ibr]
for u = Q
(62)
=
for w = w (/x = 2)
for u ^ 2u.
1374 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
The amplitude characteristic required for double-sideband transmission
without intersymbol interference is, in this case, given by
, ^ COS^ (7rw/4co)7rw/2w
a{u) =
(63)
sin {wu/2(^]
■m/4w
tan (7rM/4a))
- 1
for
u =
IT
" 4
for
u ^
w
-
for
u >
2w.
The amplitude characteristics A(u) and a{u) are shown in Fig. 5.
For convenient direct comparison, the value of A{u) obtained from (62)
has been normalized to A(^i) ^ 1 at w ^ 0.
With a., = win (52),
pita) -
sin 2coto
2^^011 - (2tI.V^)'l '
where the relation cos ibto sin oi/o ^ ^ sin 2cbto has been used.
(64)
1.2
1.0 -
2
< 0.6
0.2
^/^
^"^X
///
'X\
///
'\^
/ //
'A\
6(U1/ / /
\\ Y"
/a(ul/ /
\ \fm\
/ / /
/ //sCu)
\\\
SPECTRUM- N \ \
1 "Oi
RADIAN FREQUENCY
MARK
Wn + 2(3
Fig. 5 — Ideal transmission-frequency characteristics of bandpass channels in
FM and AM for rectangular modulating pulses and raised cosine spectrum at de-
tector input.
IDEAL BIN.\RY PULSE TRANSMISSION BY AM AND FM
1375
Table II — Frequency-Shift Transmission Characteristic P(/o)
FOR Raised Cosine Spectrum at Detector Input
Zil«/v
0.2
0.4
0,5
0.6
0.8
1
2
3
4
1.0
t)
0.8053
-0.0175
0.0035
-0.0012
0.0005
0.5769
0.0301
^0.0048
0.0016
-0.0007
0.4887
0.0265
^0.0053
0.0019
-0.0009
0.4030
0.0109
-0.0027
0.0011
-0.0005
0.2035
-0.0108
0.0025
-0.0009
0.0004
Differentiation of (G-l) yields
^ = 2
cos 2w/o
_ o
2w/oll - (2c^/o/7r)-]
sin 2wh[\ - '^{2C^k/Trf\
(65)
For i^o = 7r/2, p'/^ = -3/(2ir).
With (G4) and ((i;j) in (41) the frequency-shift transmission chanic-
teristic P(io) given in Talile II and shown in Fig. 6 is obtained, for a
channel with an amplitude characteristic A{u) a.s shown in Fig. 5.
For comparison with P{h), Fig. G also shows the pulse-transmission
characteristic p(io) olitained from (64) for double-sideband transmission
over a channel with the amplitude characteristic a{ii) shown in Fig. 5.
In both cases, the oscillations in the received pulses are quite small, and
for this reason the transmission-frequency characteristics shown in Fig. 5
are preferable to those in Fig. 3 in practicable AM and FM systems.
-0.02 J
Fig. 6 — Shapes of demodulated pulse.s in FM and AM for rectangular modu-
lating pulses and raised cosine spectrum at detector input, as in Fig. 5.
137G THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
VIII. PREMODULATION PULSE SHAPING
In the preceding analysis the modulating pulses were assumed to be
rectangular in shape and of duration T, m which case the modulator
input in response to a change from space to mark was given l)y (2S)
and the corresponding spectrum by (24), In the more general case of
premodulation pulse shaping, these equations are replaced by
E.Jitii) = 2 cos (mto + ^o) cos f; (l„
(66)
S^u) = 2 \ cosi^.- (/o) cosidadto, (07)
Jo
where, as before, the second component in (24) has been neglected, and
the phase modulation yj/i is related to the modulating voltage Vi{to) by
(2), or
dtn
(68)
The above relationis apply provided a continumg space is represented by
a constant frequency deviation -co and a continuing mark by w. To
this end, it is neccsHary that the individual modulating pulses V,'((o)
overlap and l)e of such form that
E F,</o + «r) - F,(o).
(69)
For example, the latter relation is satisfied when impulses are applied to
a flat low-pass filter of bandwidth w and linear phase, resulting in a
modulating voltage F,(0 ^ I^.(O) (siii (^Q/i^k . Tlie simpler case of
overlaps between adjacent pulse intervals only is illustrated in Fig. 7.
-MODULATING PULSE
-T/2 -to* i a- to T/2
Fig. 7 — Modulating pulses with overlap between adjacent pulse intervals.
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1377
The amplitude characteristic of the channel required for a spectrum
S{u) at the detector input is, as before, given by
AW^fg. (70)
To avoid intersymbol interference, it is necessary to satisfy (45) ;
i.e., /ip(0) = 2. To this end, it is necessary, in accordance with the dis-
cussion in Section V, that S(zto}) = 7'/4 and that ^(w) = ^. Hence, a
requirement imposed on S°{u) as given by (67) is that
SXci) -|. (71)
There is an infinity of pulse shapes, Vi{ia), that satisfy (69), and the
corresponding ^,(i()) can be determined formally from (68) and, in turn,
S^u) can be determined from (67). The principal problem is to deter-
mine pulse shapes, other than the rectangular shape considered pre-
viously, which also have a spectrum that satisfies (71). The solution of
this problem will not be attempted here, but two pulse shapes of general
interest in pulse transmission theory wiU be considered.
A familiar example of modulating pulses that overlap into an infinite
number of pulse intervals is represented by the ideaUzed pulse shape
obtained by applying impulses to an ideal flat low-pass filter with linear
phase characteristic. It will be assumed that the filter bandwidth is w,
in which case the modulating voltage is
VM = ViiO)'^. (72)
With (72) in (68), and with ^,^.-(0) - w,
>l^i{U) - Si («(o) (73)
where Si is the sine integral function.
In Fig. 8 the phase modulation, ^v , obtained from (73) is compared
with that for rectangular modulating pulses, for which V'i = ^to for
uto ^ t/2 and ^i = 7r/2 for C^to > ■7r/2.
With (73) in (67) and w^o = t, w3' = it, a = u/u:
S\u) =T.-! cos Si (t) cos ar dr. (74)
2 TT Jo
¥oY T > 17 the following approximation applies:
Si (r) ^ ■ . (75)
1424 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
The peak amplitudes, p(0), required to produce an error when an
impulse occurs midways between samphng points is greater than when
they occur at a sampling point by a factor of 2"^ in FM and a factor of
2 in AM. With a Gaussian amplitude distribution of the pulses, the
probability of an error from a pulse midway between two sampling
points is in the order of 1 per cent of the probability of an error from a
pulse at a sampling point in the case of FM, and is substantially smaller
for AM. Hence, virtually all the errors will be caused by pulses that
occur near samphng points. The AM advantage over FM for equal error
probability is 3 db for impulses that occur at samphng points, and would
be expected to be only slightly greater, about 4 db when impulses oc-
curring at all instances with respect to a sampling point are considered.
The above comparisons apply without a postdetection low-pass filter
in FM. With an optimum bandpass receiving filter characteristic in FM,
the reduction in peak impulse noise afforded by low-pass filter would be
expected to be about the same as the reduction in average random noise.
APPENDIX c
Optimum Receiving Filter Characteristic
The optimum receiving filter characteristic in AM and in FM with-
out a postdetection low-pass filter can be determined from the solution
of the more general case considered here, of FM with a postdetection
filter.
In the latter case, the optimum R{'u) is obtained when the product
of the two integrals in (125) is a minimum, or for the minimum value
of the product :
J = J1J2 , (256)
where Ji and J2 are functions of R(u) given by
J, = r L\u)R\ii){\ + u/^Y du, (257)
/,= r^d„ = 2r^*.. (258)
In (257), L(—u) 7^ L{ii), so that it is convenient to resolve the
integrand into one component with even symmetry with respect to u
and one with odd symmetry. The integral of the latter component van-
ishes and that of the component with even symmetry becomes
Ji= f H\u)R\u)du, (259)
IDEAL BINARY PT^LSE TRAXSMISSIOX BY AM AND FM
1379
raised cosine pulse, which overlaps into adjacent pulse intervals, and
is considered in the next section. With the latter type of pulse it turns
out that A'^w) = 0.99-t 772. It can thus be conjectured that there is
some intermediate shape of overlapping modulating pulses for which (71)
is satisfied. For practical purposes, this is the case with modulating
pulses given by (64) or for the raised cosine pulses considered in the
next section.
IX. RAISED COSINE MODULATING PI7LSES
Raised cosine modulating pulses have the shape indicated in Fig. 9,
and can be derived conveniently by appropriate gating of a biased
steady-state sine wave, with the total pulse duration, 27', equal to one
cycle of the sine wave, which may have advantages from the standpoint
of instrumentation.
SPACE
Wn + 2W
RADIAN FREQUENCY
Fig. 9 — Comparison of transmission-frequency characteristics of bandpass
channel in FM f»ir raised cosine modulating pulses (solid curve) and rectangular
modulating pulses (da.shed curve), with raised cosine spectrum at detector input
(dotted curve) in both cases. Pulse shapes are same as those for FM in Fig. 6.
1380 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
With a raised cosine modulating voltage,
Vi(k) =^[l + cos (Tto/T)] -1 < to/T < 1
^ K'8)
= -1 > to/T > I.
In accordance with (2), the resultant phase modulation is, for oi^ ^ tt,
^ ■>o (79)
= |(aifo + sin uto).
For w(o ^ TT, ^,(io) = MT) = x/2.
The spectrum of cos ^,((0) is obtained from (67) with the upper limit
equal to T, since cos ^,{fo) = for to > T. The following relation is
thus obtained:
S''iu) = 2 / cos ^{ibto + sin uto) cos lUa dU . (80)
Jo
With (iito ^ X, uiT = IT and u/ib = a, (80) becomes
»S''(m) — ^ / cos (.'c/2 + ^ sin .i;) cos ax dx
w Jo
= — / COS (j/i-t; + I sin a;) dx- (81)
X Jo
+ — / cos {v2X + i sin x) dx,
IT Jq
where
vi = I - a, V2 = ^ + a.
The above relation can be written
.nti.) = T{J.,{~\) +J.S-l)\ (82)
where Jy{z) is a so-called Anger function, which is associated with Bessel
functions and is defined by^
1 /"'
JXz) = - / cos {vx — z sin x) dx, (83)
jr Jq
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1381
, . . Sin j/]r . 3 .
JAz) = 1 - :^ -. +
+
sin fir r
22 _ „2 (2= - i;=)(4= - y"-)
6
z
~ (22 - (.2) (4= - 1/2) (6= - i*^)
3
+
(84)
_ p2 (p _ ^2) (-32 _ ^2)
.5
+
(!=_ ^2)(3^_ ,2) (5
=^- ■]■
The spectrum as a function of a obtained from (82) and (84) is given
in Table III, together with that for a rectangular modulating wave, as
obtained from (26).
Table III — ^ Values of {2/T)S''{u) for Raised Cosine and
Rectangular Modulating Pulses
a = u/Zi
0.5
1
1.5
2
Raised cosine
Rectangular
1.58
1.28
1.39
1.20
0.994
l.OOO
0.544
0.720
0.282
0.425
To obtain a raised cosine spectrum at the detector input, as given by
(61), the required amplitude chai'acteristic of the bandpass channel ob-
tained from (70) is
A{~u) = A{^i) = '°'' ^""^^"^ , (85)
2[J.,(-|)+JJ-|)]'
which gives for various values of a = w/w
a = u/^: _0_ ^ 1 1.5 2
A (u) : 0.63 0.61 0.503 0.275 '
Since .4 (w) - 0.503, rather than 0.50, the factor m = 2 in (43) is re-
placed by ^ = 2.012. The peak pulse amplitude at ^o = as obtained
from (43) is, in this case,
P(0) =^
1 2.012
2 2.012 - 1
= 0.994 .
Thus, intersymbol interference in this case results in a slight reduction
in the peak ampUtude of a pulse.
1382 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
In Fig. 9 the above channel characteristic is compared with that for
rectangular modulating pulses. In both cases the shape of the demodu-
lated pulses would be as shown in Fig. 6 for FM, aside from the shghtly
smaller peak amplitude, 0.994 rather than 1.00. It can be shown that,
with raised cosine modulating pulses, the error involved in neglecting the
second integral in (24) is not appreciable, even when the total channel
band width is equal to the midband frequency. Hence, the channel shap-
ing shown in Fig. 9 for raised cosine modulating pulses applies without
restriction on channel bandwidth relative to midband frequency.
X. P08TDETECTI0N PULSE SPECTRA AND FILTERING
At the detector output a low-pass filter may be desirable for final
pulse .shaping, elimination of unwanted demodulation products or higher-
frequency noise components, as noted in Section 11. The appropriate
transmission-freciueucy characteristic of such filters depends on the spec-
tra of the demodulated pulses, as discussed here.
Let P(to) be the shape of the pulses at the detector output in FM as
given for rectangular modulating pulses by (41) and illustrated in Figs.
4 and 6 for two special cases. These pulses have a baseband spectrum
given by
S,(^) ^ £° P{tn)e-"-'"' dto (86)
or, since P(-to) = P(ta), by
Soiw) - 2 f Pita) cos aodto. (87)
Jo
In accordance with the definition in Section IV, P(t(,) = ^'((o)/2ti.
Hence, for w - 0, (87) becomes
^^o(0) ^ ~ [ ^p'(k) dU
(88)
-^[^(«) - ^(-«=)1-
Zdi
In view of (19), the phase change caused by transmission of a mark
preceded and followed by a continuing space, is i/'( a? ) — ^( — oo ) =
2u)T = 2ir. Hence (88) becomes
5o(0) = T. (89)
IDEAL BIN.AHY PULSE TH.\NSMI5SION BY AM AND FM
1383
In the case of AM, the spectrum of the baseband pulses is equal to
the spectrum at the detector input above the midband frequency multi-
plied by a factor of 2, because of the direct addition of the two sideband
spectra. From Fig. "2 it follows that, in this case, iS'o{0) = T. Relation
(89) thus shows that the dc component of the demodulated pulses is
the same in FM as in AM. That is, the areas under the FM and AM
pulses shown in Figs. 4 and 5 are equal.
For equal spectra at the detector input in AM and F^I, the pulse shape
P{to) has a nonlinear relation (41) to the pulse shape p(to) in AM. For
this reason, the bandwidth of the demodulated pulses will be greater in
FM than in AM, and, in view of the appearance of p(to) in the denomi-
nator of (41), the bandwidth is theoretically infinite. For this reason,
part of the spectrum will be eliminated by any postdetection low-pass
filter, and intersymbol interference is thereby introduced unless the filter
has an appropriate amplitude characteristic.
With the aid of a po-stdetection low-pass filter having an amplitude
characteristic Aa{(ji) and a linear phase characteristic, it is possible to
modify the spectrum So into a desired spectrum S„, with such proper-
ties that intersymbol interference is absent. To this end, the amplitude
characteristic would be so chosen that
(90)
A o(u>) So(lx)) = S,n{(i}).
For example, by appropriate choice of ^4o(w) the pulse shape shown in
Fig. 4 for l''M could be modified into that shown for AM, or into other
shapes. The principal difficulty resides in the determination of the spec-
trum .S'(i(w) from (87), which entails numerical integration, in view of
the fairly complicated expres.sions for P(/o).
The spectrum obtained by numerical integration of (87) is given in
Table IV for the special case of a flat spectrum of minimum bandwidth
at the detector input, as considered in Section VI. In the numerical inte-
gration, contributions to the integral were neglected for wta = t > 20.
The above spectrmn is shown in Fig. 10, together with the baseband
Table IV — *S'o(to)/7' for Flat Spectrum at Dectector Input
u/Zt
0.25
0.5
0.75
0.9
1
2
3
4
1*
0.4755
-0.1321
0.0396
0.1105
0.8779
0.2932
0.2871
-0.0233
0.0650
0.7561
0.0626
0.2031
-0.0529
0.0270
0.6254
-0.2370
0.4186
-0.0220
0.0003
0.5385
-0.4609
-0.0696
0.0418
* Based on (89); compares witli computed value of 0.99999G54.
1384 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
1
1
1
s
1
\ 1
f\
N
k
\
^
J
\
\i
^
V
V
O 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
uj/u;
Fig. 10 — spectra, SqCw), of demodulated FM (solid cui've) and AM (dashed
curve) pulses shown in Fig. 4 for flat spectrum at detector input.
spectrum of the AM pulses. It will be noted that the spectrum is nega-
tive in certain ranges for w > 1.6a). This places a restriction on the choice
of the modified spectrum H^ and on the filter bandwidth, if ylo(w) is to
be positive and finite for all values of u in the filter band. To this end,
it is necessary that the filter bandwidth be less than 1.6w.
In Fig. 11 is shown the amplitude characteristic -4o{co) of the post-
detection low-pass filter obtained from (90) when *S„(a)) is assumed to
be equal to the AM baseband spectrum. With the amplitude characteris-
tic .4o{a)) shown in Fig. 11, the FM pulses shown in Fig. 4 would be con-
verted into pulses of the same shape as shown for AM.
With a raised cosine spectrum at the detector input, as considered in
Section VII, the spectrum of the demodulated pulses obtained by nu-
merical integration of (87) is given in Table V. In the numerical integra-
tion, contributions to the integral for wfo = t > 5 were disregarded.
The above spectrum is shown in Fig. 12, together with that of the
AM pulses.
The circumstance that 5o(w) is negative in the approximate range
1.9w < to < 2.1a! in this case Umits the filter bandwidth to loss than LOoj,
if .Ao(co) is to be positive and finite for all values of to in the filter band.
IDEAL BIN.\RY PULSE TRANSMISSION BY AM AND FM 1385
2.0
Fig. 11 — Amplitude characteristic, Ao(w), of postdetection low-pass filter re-
quired to convert spectrum iSo(«) into modified spectrum Sm(u)-
Table V — S(,(o})/T for Raised Cosine Spectrum at Detector Input
u/oi
0.25
0.5
0.7S
1,0
1
2
3
4
1*
0.4159
-0.0069
0.0742
0.0052
0.9492
0.2438
0.0131
0.0302
0.0096
0.8092
0.1063
0.0582
0.0093
0.0121
0.6147
0.0208
0.0926
0.0032
0.0114
0.4159
-0.0069
0.0742
0.0052
0.0084
* From (89); actual computed value = 0.99999951.
1386 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
Fig. 12 — Spectra SoM of demodulated FM (solid curve) iind AM (dashed
curve) pulses shown iu Fig. 6 for raised cosine spectrum at detector input,
By way of illustration, an appropriate type of modified spectrum of
bandwidth 1.75w is given by the following expressions.
ForO < co/w < I:
M") - T. (91)
For I < co/w < i;-.
5.(co) = T cos^ ;
(TT 4a)/w — 1 \
4^^ /■
(92)
The above spectrum S„, is shown in Fig. 13, together with the spec-
trum *So(w) and the ampUtude characteristic -4o(a)) of the low-pass filter
obtained from (90). The shape of the pulses at the output of the filter
for the above spectrum 5„,(w) can be obtained fnmi (52) with w^ = fai,
but it does not differ significantly from that shown in h'ig. 6 for AM.
With a low-pass filter having a linear phase and an amplitude charac-
teristic .4o(a)) as shown in Fig. 13, intersymbo! interference is avoided,
and some improvement in signal-to-noise ratio is realized by elimination
of higher-frequency noise components in the detector output, as will be
shown later.
XI. PULSE TRAINS
In the preceding analysis, transmission of a single mark of duration
T was assumed. When a pulse train consisting of a sequence of marks
and spaces is transmitted, (33) is modified into the following expression
IDEAL BINARY PUI.SE TRANSMISSION BY AM AND FM 1387
1.2
1.0
0.8
0.6
^
-^
r^
AoM
^^
^
Y^^f^^J 1
\
V
^
^
\ 1
\
\\!
\
J^SoC«)S^[y
\
n
\^
\
N
S;^^
\
'^
:::A-
0.2 0.4 0.6 0.6 1,0
1.2
1.4
1.6
1.8
Fig. 13 ^ Amplitude characteristic, Aaiui), of postdetection low-pass filter re-
quired to convert spectrum So(ti)) of Fig. 12 into a modified spectrum S„,(w),
for the received wave at the detector input:
W^.-('o) ^ —cos (tou/u + Vo) U(— w) COS <l!(/o — mT)
CO
- Y^a,nV{k - "if)] - A{-u)) sin (wo(o + vo) sin w(/o - mT),
(93)
where a„, ^ for a space and 1 for a mark, and m is an integer. The
alcove expression can be written in the form
Wi{k) = Tf ,-(/o)(cos wo/o + ^0 + *o), (94)
where the pulse train envelope is given by the following expression in
place of (36):
lf,(M = .■l(-w)[l + m' E a„y(fo - mT)
- 2^ E «™p('o - wi^") cos w(/o - mT)!"-,
where 2 indicates summation between the limits m ^ — a> and m = «,
as iu (93). Expression (37) for the phase ^o is replaced by-
sin w{/o — ■'"?')
(95)
tan 'I'o =
cos w(^o - mT) - /i E amp(io - mfl')
(9G)
T ' ^
1388 THE BELL SYSTEM TECHNICAL JOURNAL, NOYEMBER 1959
Expression (39) for the time derivative of ^o is replaced by
1 — M S «™p{^ — mT) cos w((o — niT)
(97)
+ ^ Z (UP (to - mT) sin J)(/o - mT) ,
0} J
where
D = sin' w{(o - mT) + [cos ^(^0 - mT) - ^ Z«m?>(io - mr)f. (98)
Expression (41) is replaced by the following expression for the demodu-
lated pulse train :
W(t,) = ^ L[E aMtf> - niT)J - S aMU - mT)
(99)
■cosii(io — niT) Z ampito — mT) sin uiU — mT) }.
CO J
At the sampling points to = 0, sin ibmT = and cos uimT = ( — 1)™,
so that (99) becomes
WiO) = ^- ^ i:<UV{mT) ^ ^
^l^i:<iMmT) -{-!)-
Since p{mT) = except for m = 0, (100) becomes
W(0) = % 7„W (101)
2 M«op(0) — 1
where y. = 2 and p(0) = 1, so that
W{Q) = ""
2ao - 1
^ 1 for ao = 1 (mark)
= for oo = (space) .
(102)
There is thus no intersymbol interference when a pulse train is trans-
mitted.
In (99) the denominator D depends on the composition of the pulse
train. For this reason the shape of the demodulated pulse train between
sampling points cannot be obtained by direct superposition of individual
demodulated pulses, such as those shown in Fig. 4 and Fig. 6.
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1389
XII. AVERAGE SIGNAL POWER
The preceding analysis was concerned with the ideal shaping of the
over-all transmission-frequency characteristic of the bandpass channel
required to avoid intersymbol interference in FM and AM systems.
This over-all shapini^ will ordinarily be divided between bandpass filters
at the transmitting and of the receiving ends. While the division between
the two ends is immaterial from the standpoint of intersymbol inter-
ference, it does affect signal power at the transmitting end and inter-
ference of various kinds at the receiving end. In order to determine the
optimum proportioning and shaping between transmitting and receiving
ends, it is thus necessary to consider both signal power and interference.
In analog systems for voice transmission and other purposes, the peak
signal power is ordinarily substantially greater than the average signal
power, by 10 db or more, and is usually a hmitation in systems design.
In binary pulse systems with representative transmission-frequency
characteristics,* however, peak signal po\ver is not much greater than
average signal power, and the latter is ordinarily a limitation, either
from the standpoint of repeater design or interference with other sys-
tems. For this reason, average signal power will be considered here in
comparing FM and AM binary pulse systems.
Let the arapHtude characteristic of the transmitting filter be T{u)
and that of the receiving filter be /?{«), in which case T{u)R(u) =^ A{u),
where A{u) is the over-all amplitude characteristic and u the frequency
from midband.
Let the peak amplitude of the carrier for a continuous mark or space
at the output of the transmitting filter be E, in which case the signal
power for a continuous mark is
P. - f . (103)
In Appendix A it is shown that the average signal power for random
pulse trains with FM and with bipolar AM are given by
^ FM — J
IT
P.M^P„J,iJ'(0)£[|gJd«. (105)
where T is the pulse interval or duration, and S{u) is the spectrum at
* This excludes idealized flat channels of minimum bandwidth.
1390 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
the channel output, or detector input, in response to the transmission of
a single pulse, i.e., in response to the transmission of a mark preceded
and followed by a continuing space.
By way of illustration, in the case of a raised cosine spectrum as given
by (61)
r. ^ r,2/-\ r" cos* (TrW/4w) , . ,,„„s
TT •'-2U K^{U)
p.,=p„.^^«^(o)£:»^^^;gf^*.. (107)
If the receiving filter has a half-cosine shape as given by
R(u) = cos xm/4co for —2C}<u<2o}
^ ' (108)
= for I w I > 2w
expressions (106) and (107) become, with R{ih) = |''^ and R(0) - 1,
-Pfm = Pam = Pr.^ \ coi Tru/^&>) du
^ir J-25 (109)
= P^.
In this particular case the average signal power of a random pulse train
in both FM and bipolar AM is equal to the signal power for a continu-
ous mark or space.
Consider next a flat receiving filter, in which case
R(u) = 1 for -2u < w < 2w
(110)
= for j w I > 2w.
In this case (106) and (107) yield
Pfm = Pm — I COS* 7rM/4ai du
TT J-25 (111)
_ 3 p
— If ^ m )
^AM = 2 ^FM — 4 Pm- \112J
Finally, let the receiving filter have a raised cosine shape as given by
R(u) = cos^{7rw/4to) for — 2w < u < 2(J
(113)
= for M > 2d).
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1391
In this case /?(co) ^ | and
4irJ-2^ """ (114)
Pam = 2PrM = 2/»„. (115)
The above relations show that, for eciual signal power in FM and bi-
polar AM during transmission of a continuing mark (or space), there
may be apprecialjle difference in average signal power for a random
pulse train, depending on the shape of the transmitting filter. For this
reason, signal power during a continuing mark, which is often used in
specifying signal-to-noise ratio, may not be an appropriate reference
signal power.
To determine the optimum division of channel shaping between trans-
mitting and receiving ends, it is necessary to consider the effect of ran-
dom noise or other interference, such as impulse noise. The effect of any
particular type of interference depends on the shape of the receiving filter,
as discussed in the following sections for random noise. Impulse noise is
discussed briefly in Appendix B.
XIII. RANDOM NOISE IN FM AND AM SYSTEMS
Certain basic equations relating to noise and interference on FM and
AM are given in Appendix B and applied to the particular ease of single-
frequency interference. In the case of a sinusoidal interfering voltage at
a frequency u from midband and of amplitude e(u) at the input of the
receiving filter, rms interference in FM and bipolar AM taken in rela-
tion to the peak-to-peak signal amplitude at the detector output is given
by
- ^ e(u)R(u) . ,., , .
T?AM —
e(-u)R{u)
(117)
2ER{0)
where e(u) = e(u)/'2^'', R{u) is the ampUtude characteristic of the re-
ceiving filter and E is the peak ampUtude of the carrier for a contiiuiing
mark or space. The above relation for tjfsi is a first-order approximation
applying if e(») is small in relation to E, as is required for transmis.sion
without excessive error rates.
The equations gi\'e the rms amplitude of the interfering voltage at the
1392 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
detector output, taken in relation to the peak-to-peak difference in pulse
amplitudes for mark and space, considering all possible phases of the
interfermg voltage equally probable. The shcing or threshold level is
ordinarily half this difference, and rms interference voltage taken in
relation to the slicing level would be twice as great.
Random noise can be regarded as the sum of a very large number of
sinusoidal waves of different frequencies, with both the amplitude and
phase of each component wave varying with time. For a single sinusoidal
wave at a frequency u from midband and peak amplitude e, the rms
amplitude is e = e/2^''. When the sinusoidal wave varies in amplitude
with time there will be a certain rms amplitude e over a long interval,
and a corresponding average noise power e^ for each sinusoidal compo-
nent. The corresponding average noise power per unit of bandwidth at
the receiving filter input will be designated n{u). In the case of white
thermal noise as assumed in the following, n{u) = n is independent
of u.
The ratio of noise power to signal power at the output of the detector
can be obtained from (116) and (117) for single-frequency waves, by
integration of the noise power density over the channel band. Thus, in
the case of FM, the ratio of average output noise power Na to the output
peak-to-peak signal power ^o between mark and space, as obtained by
integration of (116), becomes
(iVo/*§o)FM = p„ Z -^ f ■^'(«)(1 + ^^/^y du (118)
where the last expression follows since R{—ii) = R{u), so that the in-
tegral of R^2t(/(ii vanishes.
In case of bipolar AM, the corresponding ratio obtained by integration
of (117) is
(^»/^»)- = IpJho) /-^^'"^ '"• ^'''^
Relations (119) and (120) can be expressed in terms of average signal
power in FM and AM with the aid of relations (104) and (105). The
following expressions are thus obtained:
47ri /^FM ]_•'—<« -i
■u:mA
(121)
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM
1393
(No/ So) AM =
4wTP,
R (u) du
S'ju)
_ L-'-« RHu)
du
(122)
The above expressions for FM apply without a postdetection low-
pass filter. As discussed in Section X, it is possible to modify the pulse
shape in FM without causing intersymbol interference, with the aid of a
low-pass filter of appropriate amplitude characteristic ^o(w) that de-
pends on the bandwidth. Such a filter reduces the noise power in a nar-
row band at w by the factor Ao'io)). On a carrier basis, this is equivalent
to multiplying the noise power in a narrow band at a frequency u from
SPACE
Fig. 14 — ■Frequency modulation with postdetection low-pass filter having
transmission characteristic Aq(o:). Noise power in the narrow band at frequency
u from midband is reduced by the factor L'^{u) = Ao^(w + u) during transmission
of space with carrier at wo — ij, as assumed above.
midband by An (oi -|- u) during transinissi(m of a space, as indicated in
Fig. 1-i, or by Ao^(— w -j- u) during transmission of a mark. This equiva-
lent repre.sentation on a carrier basis is legitimate, provided the carrier
power at the detector input is substantially greater than the noise power,
in which case the ratio Nn/So obtained from (121) is much smaller than
one.
The following notation indicated in Fig. 14 will be used:
L(u) = Ao(w -I- u).
With such a filter, (118) is modified into
(123)
(No/So)vM - ^p p" -, f L\ii)R\u){l +u/C^fdu, (124)
1394 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
and (121) is modified into
47ri I FM L"*-™ -I
■KH'
(125)
where L, R and S arc functions of u.
Comparison of (125) with (121) shows that, for a given R{u), a post-
detection low-pass filter reduces the ratio (iVo/A'o)FM by the factor
r L^R\\ -^u/^fdu
p = -^ , (126)
f R\l+ n/5?) du
where L and R are functions of if.
XIV. OPTIMUM AM SYSTEMS
The minimum value of (No/So) for a given average power P = Pam
is obtained when the product of the two integrals in ( 122 ) is a minimum.
As shown in Appendix C, this is the case when R-iu) is such that the two
integrals are equal, in which case the optimum R(u) is given by
R-iu) = cS"\u), (127)
where c is an arbitrary constant independent of u that can be chosen to
give R{u) the appropriate dimension.
With (127) in (122), the optimum ratio No/ So becomes
(No/S,r - ^ [/_"^ SUl) duj. (128)
When S(u) has the properties previously discussed and illustrated in
Fig. 1 and Fig. 2, the hitegral in (128) is always equal to the area under
the rectangle in the upper part of these figures, or 2coT/2 = o>T ^ ir.
Furthermore, with (127) in (105), it follows that P„, = Pam = P-
Hence, for ah spectra S{u) of the form previously assumed, (128) be-
comes
IDEAL BIN.\IIY PULSE TRANSMISSION BY AM AND FM 1395
where
A^ ^ nib = average noise power in a flat band w at input of receiv-
ing filter,
P = average signal power at receiving filter input,
P,n = signal power for continuous mark (or space),
jVo = average noise power at detector output.
So = peak-to-peak signal power at detector output.
In (129) A'' is the average noise power in a flat band equal to that of
the minimum possible bandwidth w over which pulses can be transmitted
at intervals T = tt/w without intersymbol interference, i.e., the noise
power in a band 1/27' cps. With the above definition of A'^ and for the
above pulse transmission rate, (129) apphes for aU spectra at the detector
input with the properties illustrated in Figs. 1 and 2. There is thus no
noise penalty, but only a bandwidth penalty, in modifying the spectrum
that was indicated in Fig. 2 and discussed previously to obtain pulses
whose shape is more appropriate shape than that of systems with the
minimum possible bandwidth. Moreover, (129) also applies for optimum
bipolar baseband systems with base band spectra equal to those shown
on the right-hand side of coo in Figs. 1 and 2. Thus there is no noise pen-
alty in l)ipolar double sideband AM, but the bandwidth of the carrier
channel is twice that of the baseband channel.
This two-fold increase in bandwidth for a given transmission rate can
be overcome by providing two independent channels on two carriers at
quadrature, a method sometimes referred to as four-phase transmission.
With 3-db reduction in noise power, because of the two-fold reduction in
bandwidth, and with 3 db less signal power per channel, so that the
average signal power P is the same as for a single channel in bipolar AM,
(129) applies with A'^ defined as above An alternate means of avoiding
the two-fold increase in bandwidth is to use bipolar vestigial sideband
AM^'" with homodyne detection, in which case (129) also applies. The
last two methods are thus equivalent to bipolar baseband transmission,
both as regards bandwidth and signal-to-noise ratio.
At the detector output the pulses may be bipolar or may be biased
into unipolar (on-off) pulses, and (129), in terms of the peak-to-peak
signal power §o at sampUng instants, applies regardless of any bias.
In the particular case of a raised cosine spectrum as given by (71),
the over-all amplitude characteristic of the channel is given by (63) or
'■(") = . T^"^M-r (130)
tan (x!v4aj)
The optimum receiving filter characteristic as obtained from (127) is
R'iu) = cos(tw/4w). (131)
1396 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
The corresponding optimum transmitting filter characteristic is T° =
a(u)/R'', or
xm/4w
r(u) = -.
for u =
sin (jrM/4a!)
= 1
= - 2"^ ^ 1.12 for u = ib
4
(132)
^ - for u ^ 2a).
2
These characteristics are shown in Fig. 15.
In the case of vestigial-sideband AM, the carrier would be at wo + "
or too — w rather than at wo . Pulses can then be transmitted at twice the
double-sideband rate, provided homodyne detection is used so that the
effect of the quadrature component is eliminated.^ The optimum shape
of the receiving filter is again given by (131), but the shape of the asso-
ciated optimum transmitting filter is modified, since the spectrum at the
UJo+iO
Fig. 15 — Double -sideband AM with raised cosine spectrum at detector input:
" = optimum shape of receiving filter; T" = optimum shape of transmitting filter;
A = R''T'' = combined transmission characteristic.
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1397
channel input is now caused by a rectangular carrier pulse of frequency
0)0 ± w rather than wo . The optimum transmitting filter characteristic
is given by
mo/ \ ! n-\ T(M=b w)/4a) /ioo\
T (u) = cos {Tu/4a;) -^-^-p , -\;.-T (133)
sin [ir{u ± o})/4:Oj\
The positive signs apply when the carrier is at wq — w and the negative
signs apply when it is at wo + w; the expression apphes for — 2w ^ m ^
2w.
If the carrier is assumed at ojq + w,
rr^r..\ „„. / ..iA-\ t(" - w)/4w
V ^^/
sin [7r(w — aj)/4a!]
=
for
M g -2J)
"4^
for
U = —0}
Tr„i/2
for
u =
1 „l/2
"2^
for
U = U
=
for
u ^ 2u.
(134)
In actual systems, it may be expedient from the standpoint of design
to employ transmitting and receiving filter characteristics that differ
from the optimum characteristics shown in Fig. 15. This results in some
penalty in signal-to-noise ratio, as shown below.
By way of illustration, it will be assumed that the transmitting filter
has the shape shown in Fig. 5 for the channel transmission-frequency
characteristic, in which case the receiving filter could be flat and given
by
R(u) = 1 for -2^ ^ w ^ 2w
(135)
- for -2w > w > 2w.
In this case, (122) for AM becomes
r /.2u n r (.2u m2 -]
N./S, - ^ m^ du^ \^^ - cos^ (xV4^) du^
^2 4P ^'^^^
1398 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
where N, P and P^ are defined as in (129) and the last relation follows
from (112) for the above case; i.e., P = |Fm ■
Comparison of (186) with (129) shows that, for equal average signal
power P, the ratio No/^% is greater here than for optimum division of
filter shaping by a factor off (or 1.8 db), while comparison of (137)
with (129) shows that, for equal continuous mark power P™ , it is
greater by a factor of 2 (or 3 db). This illustrates that the penalty in-
curred in departing from the optimum division of channel shaping be-
tween transmitting and receiving filters can be significant.
XV. OPTIMUM FM SYSTEMS
The minimum ratio A'"o/*S'u is obtained \vhen the product of the two
integrals in (121) is a minimum. As shown in Appendix C, this is the
case when /?(«) is such that the two integrals are equal, resulting in the
following expression for the optimum R{u) without a postdetection low-
pass filter :
R-iu) ^ cS"\u){l -\- u'/i;^Y'^\ (138)
where c is an arbitrary constant. The corresponding optimum No/Bo ob-
tained with (138) in (125) is
2n
(No/SoT =
AtTP
p {l-^-u'/S^ySdOdul (139)
= §^^ (140)
X" = 2
(141)
where P = Pfm , ^ is defined as in (129) and
[ j" S(u) dvA
Comparison of (140) with (129) shows that, for equal average signal
power, the optimum ratio No/ So is greater in FM than in bipolar AM by
the factor X". Inspection of (141) shows that X" > 2, without a postde-
tection low -pass filter.
With (138) in (126), the factor p, by which the noise power is reduced
by a postdetection low-pass filter, becomes
f
•I—!
-2/ %o^ ^ (1 + W^y
J—oa
p =
f (l -\- u'/6:yS(u) du
J— CO
where L{u) is defined by (123).
(1 + «'/»=)'" (142)
IDEAL BINARY PULSE TKANSMTSSION BY AM AXD FM
1390
With a postdetection low-pass filter, (140) is replaced by
where
(iVo/Sor-^7,
y = \ p
(143)
(144)
is the fact.or by which the optimum ratio yVo/A'o differs in FM from bipolar
AM, for equal average signal power at the receiving filter input.
The above factor 7 is not the minimum (optimum) factor with a post-
detection low-pass filter, but rather the factor applying when a low-pass
filter is applied to a system in which R{u) is optimized without such a
filter. If a postdetection low-pass filter of specified amplitude character-
istic AoiiJ}) is assumed, the optimum R(u) is related to Ao(w), as dis-
cussed below.
With a postdetection low^-pass filter, the minimum ratio No/So in
terms of average .signal power is again obtained when R(u) is such that
the two integrals in (125) are equal. As shown in Appendix C, R{u) is
in this case given by
where
H(u) = [L\u){l -^ u/C^Y -\- L'{
- \2il/2
»)(i -u/^y]
(145)
(14G)
= blo'(Jj -\- u)(l + u/<bf + Ai{w - u)(l - u/C:)']'". (147)
In the last relation, .4(i(w ± u) designates the amplitude characteristic
of the low-pass filter at w ^ oi ± m.
The optimum ratio obtained with (145) in (125) can be written
where
7 ^
{No/S,r ^^y\
J II{h)SUi) diA
j S{u) du
-12
H{u)S(m) du
I Sill) dvA
(148)
(149)
(150)
1400 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
where (150) follows from (149) since H{~u) = H{u) and >S(-w) =
Siu).
For the reasons discussed in Section X, the bandwidth of the low-pass
filter must be less than 2w. Hence, for u ^ oi, Aoiib + m) = 0, so that
the first term in (147) vanishes and
H{u) ^A,{0)il - ii/^) for u/ih^l
(151)
= for w = w.
Thus, for u ^ w, (145) becomes
L^o(0)(l - w/")J
for ?i/ii = 1
(152)
for w = oi.
The corresponding transmitting filter characteristic is T^u) =
A{u)/R"(u) = Oforw = w.
With i2''(±w) — w, the noise power at the detector input would be-
come infinite and the signal-to-noise ratio at the detector input would
be zero. Hence, the basic premise of adequately high signal-to-noise
ratios underlying the representation in Fig. 14 and expression (125)
would be violated. In this case, No/So , without a postdetection low-pass
filter as given by (121), would become infinite. To limit the noise power
at the detector input, so that (125) is a legitimate approximation, it is
necessary to modify J^''(w) near u ^ ±win sucha way that K''(±w) ^ oj
and Nq/Sq , as obtained from (121) without a postdetection filter, be-
comes appropriately small. The value of No/So obtained from (125) after
such modification of R{u) will be greater than that obtained from (150),
but may be smaller than that given by (144). The factor 7" is thus to be
regarded as a lower bound that cannot be fully realized but may be
closely approached, at least for small ratios N/P in (148), by appropriate
modification of R°{u) near u ^ ±w. (An example of such modification
is indicated by the dotted curves in Fig. 17, to be discussed later.)
XVI. OPTIMUM FM SYSTEMS OF MINIMUM BANDWIDTH
In the limiting case of a channel of the minimum possible bandwidth,
as considered in Section YI, the channel characteristic is given by (55).
When normahzed to unit ampUtude for u = 0, this characteristic is
A(u) = ^ -/"/;?' (153)
cos (7^/20))
IDEAL BIN.\IIY PULSE TRANSMISSION BY AM AND FM 1401
With a spectrum as given by (53) and (54), the optimum receiving filter
characteristic obtained from (138) becomes, with c chosen as (2/2')^'^
K{u) = (1 -^uy^)-'''. (154)
The corresponding optimimi receiving filter characteristic,
r(u) = A{u)/R''{u),
is
^ [1 - iuM + u%r\ (155)
COS {ttu/Zco)
The above expressions apply for —1 < u/oi < 1.
The factor X" defined by (141) becomes
X" = 2
du^ (156-
= |[2^^^+log.(l + 2^'^)r
^ 2.65.
This corresponds to about 4.2 db disadvantage in signal-to-noise ratio
for an optimum FM system without a postdetection low-pass filter, as
compared to an optimum bipolar AM system, for equal average signal
power.
In the above case the spectrum So{o}) of the demodulated pulses is as
shown in Fig. 10. With a low-pass filter having the amplitude character-
tic shown in Fig. 11, this spectrum is converted into a fiat spectrum of
the minimum permissible bandwidth. With the above type of filter,
L(u) = A(,(w -\- u) ^ for u > 0. Since S(u) ^ f or w < w < —oi,
(142) becomes
i;
. 2/- ^ (1 + u/<^)^ j„
[_ (1 + u^^y
^!,-.?\^i^ ^.. (157)
du
^ 0.38 (by numerical integration).
With the above low-pass filter, (144) gives
7 = X-'p^l.O, (158)
1402 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
SO that the signal-to-noise ratio is virtually the same as for an optimum
bipolar binary AM system, for equal average signal power.
The factor 7" given by (150) becomes
7 ^
/ Aoiic — w)(l — w/w) du
I
= 0.60 (by numerical integration).
(159)
For reasons mentioned in Section XV, the minimum factor 7 that can
be realized will be less than that given by (158) but greater than that
given by (159), which is to be regarded as a lower bound. Since this
minimum factor 7 is less than one, some advantage in signal-to-noise
ratio can be realized with FM, as compared to an optimum bipolar base-
band or AM system with synchronous detection, for equal average signal
power at the input of the receiving filter. This advantage would be
small, and is principally of theoretical interest as an indication that a
noise advantage can be derived from the unavoidable two-fold increase
in channel bandwidth with FM as compared to baseband transmission
or equivalent AM methods.
1.0 -
3 o.a
0.4 -
^7"----
i
1
^^*- \
J
-'5'
' NTs ^
1
//
N^. A(u) =
\
1
\\R°(u)T°Cu)
\
/
//
] \ \
\
/
//
\ \
\
/
//
' \^
\
1
//
\ \
\
1
//R°(u)
>\
\
1
//
\\
\
1
//
\\
\
1
//
N\
\
-
\
\
/ /
\ \
1 /l
A \
1 /f
vV \
1 / f
A, \
//
SPACE
(Ui*. MARK
Wn + t*;
Oi^ + ZCO
Fig. 16 — Frequency modulation with raised cosine spectrum at detector input
and no postdetection low-paas filter: li" = optimum shape of receiving filter;
T" = optimum shape of transmitting filter; A = R"T'' = combined transmission
characteristic.
IDEAL BINARY PULSE TRANSMISSION' BY AM AND FM
1403
XVII. OPTIMUM FM SYSTEMS WITH RAISED COSINE SPECTRUM
With a raised cosine spectrum at the detector input, as considered in
Section VII, the over-all amplitude characteristic of the bandpass chan-
nel is given by (62). When normaUzed to unit amplitude for u = 0, the
characteristic is
,ioo^ \+^"^/"';/y (i-,.7^-).
160)
2 cos {tu/2u)
The o|5tinnuu characteristic of the receiving fiUer as ()l)t!iincd from
(138) is, in tliis case,
cos (x!i/4(il)
R" =
(1 + mVw^)'"'
(101)
The corresponding optimum characteristic of the transmitting filter is
r = A(u)/ir, or
cos {Tru/2o>)
= 1 for ti =
= -(1)^'"^ 1.09 for u = w
(162)
=
for u = 2w.
The above filter characteristics are shown in Fig. 16.
With a raised ctjsine spectrum in (141),
X" = 2
= 2
i[i::
(l + w /o) ) COS {iru/^bj) (ill
[i:
cos" (ir»/4ai) du
(1 4- .x-')'^'cos' (Trx/4:)d.
■J
(163)
^ 2.8 (by numerical integration).
With the aid of a postdetection low-pass filter, the noise power in the
output is reduced by the following factor p obtained from ( 142) :
.2;
P =
I Gill) cos^ (iru/4:c^) du
(1 + u/Ci'Y'' cos (xw/4w) du
■2S
(164)
1404 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
where
G(u) = L\u)
(1 + U/C:)''
(1 + uysi'^y'^'
The factor 7° given by (150) in this case becomes
U25 -|2
H{u) cos" {ru/^d) du
(165)
7 =
[/:
(166)
eos^ {iru/A.Ci) du
where, as in (147),
H{u) = [Ao\^ + w)(l + w/^)' + A,\ib - u){l - u/wff.
The optimum receiving filter characteristic is, in accordance with
(145),
/?"(„) = c2"'cos (7r«/4w)[iJ(w)r". (167)
With a raised cosine spectrum at the detector input, the spectrum of
the demodulated pulses is as shown in Fig. 12. For a low-pass filter with
a transmission-frequency characteristic as shown in Fig. 13, approximate
values of L{u) = Aa{ui + u), G(u) and H(u) are given in Table VI by
way of illustration.
Table VI — Illustrative Values of L, G and H
-
w/u)
-0.75
-0.S
0.5
0,7S
I.O
1.5
2.0
L{u)
G{u)
H{u)
0.27
0.65
0.8
1.1
1.2
1.44
1.7
1.15
0.30
1.1
1.05
0.05
0.27
1.00
1.15
0.185
0.58
1.2
0.51
1.2
For the particular low-pass filter above, the optimum receiving filter
characteristic obtained from (167) and the corresponding transmitting
filter characteristic T''(u) = A(u)/R''{u) are as shown in Fig. 17, where
they are normalized to unity for w = 0. It will be noted that, for u = zbw,
Jl" = <x> and T° = 0. Hence these optimum characteristics cannot be re-
alized physically, though they can be approached, as indicated by the
dotted lines near w = ±w.
For the above low-pass filter, numerical integration of (164) and (166)
gives p = 0.5 and 7° = 0.84.
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM
1405
The significance of the above various numerical results are, in sum-
mary, as follows.
The factor X" = 2.8 indicates about a 4.5-db disadvantage in signal-
to-noise ratio, for an optimum FM system without a postdetection filter
as compared to an optimum bipolar AM or baseband system, for equal
average signal power at the input of the receiving filter. The factor
p = 0.5 indicates a 3-db improvement in signal-to-noise ratio obtained
with the aid of the particular postdetection low-pass filter assumed
above, so that the above FM disadvantage is reduced to 1.5 db; i.e.,
7 - rp = 1.4.
In accordance with the discussion in Section XV, a somewhat lower
factor 7 could be realized when the division of channel shaping between
transmitting and receiving bandpass filters is optimized for this particu-
lar low-pass filter. The lower bound is represented by y" = 0.84, and the
minimum value that could be realized with an appropriate modification
in filter characteristics near u = u) would be greater than 0.84 but less
than 7 = ] .4. With the modification indicated in Fig. 17, it turns out
Fig. 17 — Frequency modulation with raised cosine spectrum aud post-detec-
tion low-pass filter with transmission characteristic Ao(t^) shown in Fig. IS: R" =
optimum shape of receiving fiUer; T" = optimum shape of transmitting filter;
A = R'T" = combined transmission characteristic.
1406 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
that the ratio No/So obtained from (125) is smaller than that for bipolar
AM by a factor y = 0.94. Thus, with optimum design in FM, the signal-
to-noise ratio would be very nearly the same as for an optimum bipolar
AM or baseband system.
If the spectrum ;So{ai) shown in Fig. 12 is converted into a flat spectrum
of bandwidth 2oi by an appropriate low-pass filter, it turns out that
7° = 0.64, corresponding to about a 2-db advantage in signal-to-noise
ratio over an optimum bipolar AM or baseband system.
As an illustration of the penalty incurred in departing from the opti-
mum division of channel shaping between transmitting and receiving
bandpass filters, it will be assumed that R(u) is given by (135), as pre-
viously considered for AM. In this case, the receiving filter is flat be-
tween — 2w < w < 2a), and (125) becomes
A'o/A'o = j^ f " L'(u){l + uM' du
^1^8)
■ / T" COS (7rtt/4w) dii,
J-1Z 4
which can be written as
with
7 = ^
A^o/^'o -^7, (169)
4) / L\u){\ + u/ib)' du- I cos' (7rw/4w)dw (170)
6J/ J-2ai -^0
= 1.66 (by numerical integration),
where the numerical result appUes for a postdetection low-pass filter
with the same amplitude characteristic as assumed previously.
The value y = 1.66 corresponds to a 2. 2-db disadvantage in signal-to-
noise ratio as compared to an optimum bipolar AM system, and about a
0.7-db disadvantage compared to an optimum FM system (7 = 1.4).
With the above type of flat receiving filter, about a 1.8-db penalty in
signal-to-noise ratio was incurred in AM, as shown in Section XIV.
XVIII. SIGNAL-TO-NOISE RATIOS AND ERROR PROBABILITIES
In the case of baseband transmission or AM with homodyne detection,
the probability of exceeding the rms noise amplitude by a specified factor
follows the normal law.
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1407
If the polarity of the noise voltage is specified, (i.e., positive or nega-
tive) the probability of exceeding the rms noise amplitude by a factor
k is
p = hevic(k/2"'), (171)
where erfc is the error function complement.
If the noise amphtude at a samphng instant t is [N'oi.t)^'', an error
will occur if the ratio [No(t)/Sof'^ exceeds | in the presence of a space
or negative pulse, or exceed —^ in the presence of a mark or positive
pulse. The rms ratio, (A'^o/'So)"', must be held smaller by an appropriate
factor l/k. The probability of an error in a digit is accordingly p = pe ,
as given by (171), provided
(W^o)'" = -^. (172)
In accordance with (129),
(P/Nr = KS./N,). (173)
From (172) and (173),
/,: = (P/Ny'\ (174)
Hence, for an optimum AM system as assumed above, the error prob-
ability in binary bipolar pulse transmission is, with (174) in (171),
p. ^ lcde{P/2Nf\ (175)
In Table VII the signal-to-noise ratios P/N obtained from (175) are
shown for various probabilities of an error in the digit. In accordance
with (129), under the optimmn condition the average signal power P
in bipolar AM is equal to the continuous mark power P^ ■
Ideal synchronous detection, as assumed above, caunot be fully real-
ized with symmetrical bipolar AM methods. Derivation of a demodulat-
ing carrier from the signal wave, or adequate phase control of a locally
supplied carrier, entails some increase in either signal powder, noise power
or bandwidth, depending on the particular method used, and thus en-
tails a somewhat greater ratio P/N than that given in Table VII.
In the case of unipolar or "on-off" baseband or double-sideband AM
with homodyne detection, the maximum tolerable peak noise power is
G db less than it is with bipolar transmission, but the average signal
powder is reduced by 3 db, so that the ratio P/N must be hicreased by
3 db, as indicated for unipolar AM in Table VII. The average signal
power is, in this case, 3 db less than the signal power during a continu-
ous mark.
1408 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
Table VII — Optimum Signal-to -Noise Ratios P/N in Decibels
FOR Binary AM and FM Systems*
AM with Ideal Synchronous
FM with Ideal Frequency Discriminator
Detection^
Detection
Probability of an Error
Unipolar
With I-ow-
With Low-
(Two-Phase)
{On-Off)<^.°
Pass Filter*-
Pass Filter^. ^
Pass Filter'^. *^
10-2
7.3
10.3 (11.3)
11.8
8.8
6.5
10-*
11.4
14.4
15.9
12.9
10.6
10- "5
13.6
16.6 (16.9)
18.1
15.1
12.8
10-8
15.0
18.0
19.5
16.5
14.2
10-10
16.0
19.0
20.5
17.5
15.2
10-12
17.0
20.0
21.5
18.5
16.2
P = Average signal power at input of receiving filter
N = Average noise power in flat band W = 1/2T cps at input of receiving
filter
T = Interval between pulses, in seconds
Notes:
* Signal-to-noise ratios in terms of noise power 2N in double-sideband channel
of bandwidth 2W are 3 db smaller than in table.
^ Applies for baseband transmission, double-sideband, double-sideband on two
carriers at quadrature and vestigial-sideband AM.
^ Signal-to-noise ratios in terms of steady mark power are 3 db greater than in
table.
^ Values in brackets are for double-sideband with optimum envelope detection.
^ Band-pass filter shaping as in Fig. 16.
^ Low-pass filter shaping as in Fig. 13.
o Band-pass filter shaping as in Fig. 17.
For the reasons discussed in Section XIV, the optimum signal-to-
noise ratios shown in Table VII for bipolar AM apply for bipolar base-
band transmission, for bipolar double-sideband AM (phase reversal or
two-phase transmission), for bipolar quadrature double-sideband AM
(four-phase transmission) and for bipolar vestigial-sideband AM. Sim-
ilarly, the ratios shown for unipolar AM with s3mchronous detection
apply for unipolar baseband, double-sideband, quadrature double-side-
band and vestigial-sideband transmission. Furthermore, with optimum
division of channel shaping between transmitting and receiving filters,
the above optimum signal-to-noise ratios apply for all spectra of the
pulses at the detector input with the properties illustrated in Figs. 1
and 2. Moreover, the signal-to-noise ratios apply not only for pulses of
duration T equal to the pulse interval, as assumed in the previous anal-
ysis, but also for pulses of shorter duration. When the duration of the
pulses is less than T, the receiving filter characteristic remains un-
changed, but the shape of the transmittmg filter is modified because of
the different spectrum of the modulating pulses. This also applies with
other than rectangular shapes of the modulating pulses.
IDEAL BINAHY PULSE TRANSMISSION BY AM AND FM 1409
There are thus an infinite number of optimum AM systems with a
performance from the standpoint of aignal-to-noise for a given error
probability equivalent to that of a baseband system of the minimum
possible bandwidth, provided ideal homodyue (synchronous) detection
is used, in which the pulse train is applied to a product demodulator
together with a constant demodulating carrier of proper phase.
The ratios shown in Table VII for unipolar AM also apply in a first
approximation to "on-off" double-sideband AM with envelope rather
than homodyne detection. However, in the latter case both the average
noise power and the probability distribution at the detector output differ
between mark and space/ During a mark, the rms noise ampHtude, and
the probabiUty that this noise amplitude is exceeded by a factor k, is
virtually the same as it is with homodyne detection, for large signal-to-
noise ratios. During a space, however, the rms noise amphtude is in-
creased by a factor of 2"^ and the probabihty that the rms amplitude
is exceeded by a specified factor k follows the Rayleigh law
p = e-'" (176)
rather than the Gaussian law (171).
For the above reason, the optimum slicing or threshold level is not
one-hah the peak pulse amplitude, but slightly greater, depending on
signal-to-noise ratio and error probabihty. The optimum slicing levels
with binary double-sideband AM and envelope detection and the cor-
responding optimum signal-to-noise ratios versus error probability have
been determined elsewhere,^ and are indicated in Table VII for two
cases. For an error probability of 10~^ the optimum threshold level is
about 52 per cent of the peak pulse amplitude and the signal-to-noisc
ratio is about 0.3 db greater than for unipolar AM with homodyne de-
tection. Hence, for error probabilities in the range ordinarily considered
acceptable, the difference in signal-to-noise ratio with envelope and ho-
modyne detection is insignificant.
Comparison of binary FM and AM on the basis of signal-to-noise
ratios is legitimate provided that, for a given ratio Nq/So , the error
probability is the same in FM and AM. For high signal-to-noise ratios,
this is approximately the case, since the normal law (171) is then closely
approximated in FM.^ On this premise, comparison on the basis of sig-
nal-to-noise ratios is legitimate for small error probabilities.
In accordance with the discussion in Section XV, the optimum signal-
to-noisc ratio in binary FM without a postdetection filter is related to
the optimum ratio in bipolar binary AM by
(P/NY^M- nP/N^u, (177)
1410 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
where X" = 2.65 (4.2 db) for a flat spectrum as considered in Section
XVI and X" = 2.8 (4.5 db) for a raised cosine spectrum at the detector
input, as considered in Section XVII. In Table VTI the latter case is
assumed as being the more representative, and the signal-to-noise ratios
are taken 4.5 db greater than they are in l:)ip(>lar AM.
With the aid of an appropriate postdetection low-pass filter, the sig-
nal-to-noise ratio can be improved such that
iP/N^M- p\\P/N)1m. (178)
With a postdetection filter having an amphtude characteristic as
shown in l''ig. 13, p ~ 0.5, corresponding to 3-db improvement in signal-
to-noise ratio, and this case is assumed in Table VII.
As discussed in Sections XV, XVI and XVII, when a postdetection
low-pass filter is used, it is possible with FM to realize some improvement
in signal-to-noise ratio over bipolar AM. In this case, a lower bound is
given by
iP/NnM = y(P/N)lM, (179)
where y" = 0.84 with the type of filter shown in Fig. 13. This corresponds
to about an 0.8-db improvement in signal-to-noise ratio over bipolar
AM and entails transmitting and receiving bandpass filter character-
istics as hidicated in Fig. 17. This lower bound is given in the last column
of Table VII, but it (lannot be fully realized, for reasons discussed in
Section XV.
As noted before, the optimum signal-to-noise ratios given in Table
VII for synchronous AM are universal and apply to a variety of op-
timized systems, including the special case of ideal flat channels of min-
imum bandwidth assumed in other analyses of baseband, synchronous
AM or PM systems.^'^ With appropriate allowance for different defini-
tions of signal power (continuous mark versus average power) and of
the bandwidth used in specifying noise power (flat single-sideband
versus flat double-sideband), the results given in Table VII for syn-
chronous AM conform with those in the above references.
In the case of FM, however, the optimum signal-to-noise ratio depends
on several factors that need not be considered in AM, such as the shape
of the spectrum at the detector input and the shape of the post-detection
low-pass filter. There is thus no universal optimum signal-to-noise ratio
for a given error probability in FM, and the ratios given in Table VII
for FM apply for the particular conditions indicated. For this reason,
significant comparisons cannot be made with sigual-to-noise ratios for
FM given elsewhere^ '^ that are based on simplified mathematical models
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1411
that ignore the various factors above. One analysis* indicates about a
1-db advantage of bipolar AM (or phase reversal) over FM for an error
probability of 10 .
XIX. SUMMARY
It has been shown that binary pulses can be transmitted without
intersymbol interference by FM over a channel of the same bandwidth
as is required for double-sideband AM. To this end, a first i-equirement
is that the total frequency shift i>etween space and mark be equal to
the pulse transmission rate, for example 500 cps for 500 bits per seconds.
A second basic requirement is that the pulses at the input of the fre-
quency modulator have the appropriate shape, a condition that is met
with rectangular modulating pulses of duration equal to the interval
between pulses. A third requirement with i-ectangular modulating pulses
is that the channel bandwidth be smaU in relation to the midband fre-
quency. A fourth requirement is that the bandpa.ss channel must have
the appropriate amplitude-versus-frequency characteristic and a linear
phase characteristic.
The appropriate amplitude characteristic of the bandpass channel is
not the same as for AM, nor is the shape of the received pulses the same.
By way of illustration, a comparison is made in Fig. 3 of the amplitude
characteristic of the bandpass channels for FM and AM, for a channel
of the minimum possible bandwidth, if intersymbol interference is to be
avoided. With the channel characteristics shown in Fig. 3, the transmis-
sion of a single pulse, (i.e., transmission of a mark, preceded and fol-
lowed by a continuing space) will give rise to a flat frequency spectrum
at the detector input with both FM and AM, as indicated in Fig. 3.
Such a flat spectrum at the detector input mil give rise to a pulse at
the detector output, but the pulse shape is not the same in FM and AM,
as illustrated in Fig. 4. However, a common property of the pulse shapes
shown in Fig. 4 is that they have zero points at intervals T equal to the
duration of the rectangular modulating pulses. Thus, pulses can be trans-
mitted at these intervals without intersymobl interference by FM or
AM.
In actual pulse systems, channels of the minimum possible bandwidth
are not practicable for various reasons. In Fig. 5 comparison is made of
the appropriate amplitude characteristics with FM and AM for chan-
nels with twice the minimum bandwidth. In this case, transmission of a
single pulse by FM or AM gives rise to a "raised cosine" spectrum at the
detector input and to pulse shapes at the detector output, as shown in
Fig. 0. Because of the small oscillations in the tails of the received pulses,
1412 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBEE 1959
the channel characteristics Hhown in Fig. 6 are desirable for FM and
AM systems.
In the above illustrations, rectangular modulating pulses were as-
sumed. With modulating pulses of other shapes that overlap between
pulse intervals, certain restrictions are imposed on the shape of the pulses
if intersymbol interference is to be avoided, which renders determination
of the exact appropriate shapes difficult. For example, (sin x)/x modu-
lating pulses, which are often considered in AA-I, are inappropriate in
FM. However, with raised cosine modulating pulses, as shown in Fig. 9,
it is possible by appropriate channel shaping to virtually avoid intersym-
bol interference. With such modulating pulses and a channel characteris-
tic as shown in Fig. 9, the shape of the demodulated pulses will be
virtually the same as those shown In Fig. 6 for FM. Although intersymbol
interference cannot be avoided \vith raised cosine modulating pulses, it
is small enough to be disregarded (less than 1 per cent).
The pulses at the output of the FM detector, such as those sho-wn in
Figs. 4 and G, have baseband spectra of infinite bandwidth, as in Figs.
10 and 12. It is possible to modify the shape of the pulses in such a way
that intersymbol interference is not introduced, with the aid of a post-
detection low-pass filter having the appropriate amplitude characteris-
tic. For example, the FM pulse of Fig. 4 has a spectrum as shown in Fig.
10. This spectrum can be converted into a flat spectrum of minimum
bandwidth with the aid of a postdetection low-pass filter having the
amplitude characteristic shown in Fig. 11 and a hnear phase characteris-
tic. The pulse shown in Fig. 4 for FM would thereby be converted into
the same shape as shown for AM.
The ideal amplitude characteristics of the bandpass channels in FM,
such as those exemplified in Figs. 3 and 5, are obtained mth the aid of
an appropriate combination of transmitting and receiving bandpass
filters. From the standpoint of intersymbol interference, as considered
above, the division of channel shaping between these filters is immate-
rial, but there is an optimum division from the standpoint of perform-
ance in the presence of noise.
By way of example, for the over-all amplitude characteristics of the
bandpass channels shown in Fig. 5 for AM, the optimum division of
channel shaping between transmitting and receiving filters is shown in
Fig. 15 for the case of random noise. As discussed in Section XVIII,
with optimum di\'ision of channel shaping and ideal synchronous (homo-
dyne) detection, there is an infinity of optimum AM systems mth per-
formance equivalent to that of a baseband system of the minimiun pos-
sible bandwidth, as regards signal-to-noise ratio for a given pulse trans-
mission rate and error probabihty.
IDEAL BINAUr PULSE TRANSMISSION BY AM AND FM 1413
With an over-all channel characteristic as shown in Fig. 5 for FM, the
optinmm division of channel shaping is as shown in Fig. 16, assuming no
postdctcction low-pass filter. Such an optimum FM system has about
a 4.5-db disadvantage in signal-to-noise ratio compared to an optimum
bipolar AM or phase reversal system, for equal average signal power. By
providing a postdetection low-pass filter with a transmission characteris-
tic as shown in Fig. 13 and linear phase, the signal-to-noise ratio is im-
proved about 3 db, and the FM disadvantage of 4.5 db is reduced to
about 1.5 db. This a.ssumes the same division of liandpass channel shap-
ing as without a low-pass filter. However, the optimum division with the
above low-pass filter is different, and is approxiinately as indicated in
Fig. 17, except near ojo ± ti. With an optimum division, it appears possi-
ble, in principle, to realize an advantage m signal-to-noise ratio of at
most 0.8 db over an oplimum bipolar AM or baseband system, for equal
average signal power.
With appropriate po.ft<ietection low-pass filters of the minimum per-
missible bandwidth, it i.s possible in principle to realize an advantage in
signal-to-noise ratio of at mo.st 2 db over an optimum bipolar AM or
baseband system. However, the above FM advantages cannot be fully
attained in practice. They are principally of theoretical interest in that
they indicate that an advantage in signal-to-noise ratio can be derived
from the unavoidable two-fold increase in channel bandwidth with FM
as compared to baseband transmission, or equivalent AM methods.
XX. AOKNOW^LEDGMENTS
The writer is indebted to H. O. PoUak for a critical review that re-
sulted in a radical modification of Sections VIII and X, to Miss Marion
C. Gray for supervision of computer programming carried out by Miss
D. G. Becker to determine the pulse spectra given in Section X, to S.
0. Rice for pointing out the Anger function relations used in Section
IX and for helpful suggestions at other points in the analysis, and to G.
Wascheck for many editorial comments.
APPENDIX A
Pulse Train Envelopes and Average Signal Power
In Section IV the transmission of a single mark or pulse was con-
sidered, and the rosultant wave at the channel output (detector input)
was, in this case, given by (35) and (36), or
E{k) = cos (woio + vo + ^o)E{U), (180)
1414 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
where
E(to) = /l(-w)[l + nV(t,) - 2nv{k) coswM (181)
and
ti - 1A4(-^) = lM(w). (182)
When a train of pulses is transmitted, the resultant envelope is given
by (95) or
W(/o) = A{-ih)
1 + m' E flm p'(^o - mT)
- 2/1 E «mp('o - m-T) cos 03(^0 - mT)
where T is the interval between pulses and
flm = for space
= 1 for mark.
r,
(183)
(184)
In view of (184), the rms value of the envelope at a particular time,
^(1 , with respect to a sampUng point is for equal probability of marks
and spaces:
lf(/n) = A{-ih)
1 + !^" E Vit, - mT)
where the limits of the summations are as in ( 183).
The mean squared value of the envelope taken over a pulse interval
Tis
1 J-T/2
^ A'{-C:)\l +^" /"'''EpU-mT).
- ^ f E Vik -mT) cos (i(/o - mT) dto ,
dlo
(186)
which can be transformed into
1+^j^ p\t,)dt-'^j_ vito)Gosibt,dkj. (187:
IDEAL BINARY PULSE TRANSMISSION BY AM AXD FM 1415
In (32) for p(tu), the lower limit of integration, — wq , can be replaced
by — CO J since S( u) = lor u ^ — wo , in which case (32) can be written
p(;o) ^ -L / 2Siu) cos ido du (188)
or, by inversion,
2S{u) = f p(to) GQsuUdto. (189)
In view of (189), the last-bracket term in (187) with w = w becomes
-2MA'(w)/r. Since S(^) - T/4:, in accordance with (48), and m = 2,
in accordance with (50), the last-bracket term becomes —1, and (187)
simpUfies to
W' = A\ -J)) ^ f_^ p'(to) fit, . ( 190)
In view of (189),
f viU)(U,^ ^ ( [2Siu)fdu. (191)
Hen(!e (190) can be written
]V' - A'(-^) ^- f S"-(u.) (hi. (192)
IT J J-„
This is the mean squai'ed value of the envelope at the output of a
channel with an over-all amplitude characteristic A{u). If the trans-
mitting filter is assumed to have an amplitude characteristic T(u) and
the receiving filter to have a characteristic R(u), then
Tiu)R(u) = A{u). (193)
The spectrum at the transmitter output, or at the input of the re-
ceiving filter in a channel without transmission loss, is in this case
T(u)S{u) ^ S{u)/R(u). The mean squared value of the envelope at
the transmitter output is obtained by replacing -4(w) by T{ii}) and
S(u) by S(u)/R(u) in (192), so that in this case
where -4 (w) ^ ^.
When a continuous mark with unit amplitude of the carrier at the
input of the transmitting filter is transmitted, the envelope of the car-
141 G THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
rier at the output in 3'(w). It will be assumed that the carrier amplitude
at the input is so cliosen that the peak amplitude at the output is E
for a continuous mark or space. With !/'(w) replaced by E in (194), and
with.4{£l)) - i
W'^E'^r§p\du. (195)
The average signal power within the envelope is, in view of the factor
cos (ojo^o + ^0 + ^o) in (180), smaller by a factor of |, or
In bipolar AM (183) is replaced by
W{t,) = A(0) E a^p(to-mT), (197)
m=— CO
where a^ = —1 or 1.
Equation (190) is replaced by
(198)
1 J~00
and (194) is replaced by
When the peak amplitude of the transmitted carrier for a continuous
mark is E, the average signal power in this case is, with A{^) = 1,
APPENDIX B
Interference in AM and FM Systems
B.i. General
B.i.i Frequency Modulation
Let a carrier of frequency wo — co representing a space be transmitted,
and let its peak amplitude at the input of the receiving filter be E. If
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1417
the receiving filter Iuih an amplitude characteristic R(u) as a function
of the frequency u from miciband, the signal at the detector input is
eMo) - -ER(-C^)[tiOB (wo - C^)to + ^o]. (201)
An interfering voltage at the detector input can be written in the
general form
ei(to) = ri(to) cos (wo/o + Vo) + Qi{to) sin (oioto + <pi\). (202)
The signal at the detector input in the presence of interference is then
Cs + e,- = —cos {mto + (pD)[ER(—h}) cos w^o — I'iik)]
(203)
— sin (wo'o + (pa)[ER(—u) sin w/o — qi(io)]-
The phase at the detector input in the presence of interference is given
by
tan^M- -g^ ^"'"""^^'- , (204)
cos Mo — fliVi
whore
The demodulated signal in presence of interference is proportional to
d^l/Q.i/dU , which becomes
(206)
^0./ = — -^ 1 — i^i(ri cos will + Qi sin w/i,)
+ - (r/ sin (hta — q/ cos ulo) + ^ (f/Zr,- — r/gO ,
0} w J
where ?■/ ^ dri/dto , q/ = dqi/dh and
7) ^ (sin wio — Ui9i) + (cos w/o — ;j,rO"'
(207)
^ 1 + Mi'(n^ + qh — 2/i.(ri cos u^o + 9v sin win).
The frequency deviation with respect to the frequency wo — w is i/-/ =
w + i/'o,,', and the ratio ij,- — i/'//2w becomes
^i(fo) = ^ ^i(r/ + qi') - Ti cos w^o - Qi sin w/o
in' sin (i/o — ql cos wfo) — ^ iqi'vi — r/qi) .
0) w J
(208)
1418 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
This is the amplitude of the interfering voltage taken in relation to the
peak amplitude of a demodulated pulse.
At sampling points tu = mT, sin w^o = and cos oit% = ( — 1)'". At
these points
,,(mT) = ^[Mi(n^ + ?/) - {-1)"'(^. - Ui')
- ^ (q/ri - r/qi) ,
(^ J
(209)
D{7nT) = -{-\)"'2n,n + ^iCirC + q'). (210)
If second-order terms are neglected, which is permissible for ade-
quately low amphtudcs of the interfering voltage, the ahove expression
reduces to
( ^n ^ 1 ^^("^^) - g/(mr)^ , .
1
2ER{-^)
^, [uimT) - q/imT)/^]. (212)
B.1.2 Bipolar AM or Two-Phase Modulation
In bipolar AM w^thhomndyne detection, let a carrier —E cos (wnfo -|-
^o) represent a space and a carrier E cos {wo^o + fa) fi mark. The sig-
nal plus interference at the detector input during a space in this case is
6= + Ci = -cos (mta + ip(,)[ER(0) - ri] -f sin (wu^o + pn)9> (213)
The demodulated output after eUmination of high-frequency demodula-
tion products by low-pass filtering is
7.+, - -hlERiO) - r,(io)]. (214)
If a space is represented by at the output, rather than by -~ER{0)/2,
the demodulated output is V,+i + ER(0)/2, or ri/2. The resultant
interference taken in relation to the amplitude ER(0) of a mark is
Viik)
Vi =
2ER{0) ■
(215)
B.1.3 Unipolar AM with Envelope Detection
In unipolar or "on-off" pulse transmission mth envelope detection,
the demodulated interference voltage in the presence of a space (zero
carrier) is
y/^> = (,./ + q:y\ (216)
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1419
The amplitude of the interfering voltage taken in relation to the ampli-
tude of a demodulated mark is
''' ER(0)
(s)
(217)
When a mark is transmitted, the voltage at the detector input is
e^ /"" ^ cos (coo^o + <Po)[ER{0) + n] + sin (o},U, + ^o)?^ (218)
and at the detector output is
Vj'"^ - {[ER{0) +r,f + g.V'' ^^,^.
(21 Jj
^ ER{0) + r, .
The amplitude of the interfering voltage taken in relation to the ampli-
tude of a demodulated mark in this case is
^' ER{^)
B.2 Single Frequency Interference
In the particular case of a sinusoidal interfering voltage of frequency
wn + u and amplitude e{u) at the input of the receiving filter, the inter-
fering voltage at the detector input is
SiiM) - c{u)R{u) cos [{ojo + u)U + <Pi] (221)
= e(u)R{u) cos {mtii + ^o) cos (uto -(- (p)
'-e{u)R{u) sin (wo^o + ^'o) sin {uU + <p),
(222)
where tp — tpi — <pfs .
In tliis case,
Tiih) ^ e(u)R{u) cos (h/q + '^),
Qiik) ^ — e(w.)i?(i() sin (ii^o + f).
(223)
B.2.1 Frequency Modulation
In the case of FM, (212) becomes
c(u)R(u]
'''^ 2^i2(-«)
^ +'^)cos (u/o + ^). (224)
1420 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBEH 1959
The rms interference in FM with all phases, ¥>, equally probable is
e{u)R(u) /, , iA fl f 2 r 1 I ^ j 1
^ ' ^ ^ ' ' ' ' - / cos {uk ->r<p)d(p\
TT Jo J
'''-2S72{-ci) V '^ ^
e{u)R{u) ( ^ I ti.
2ER(-w) \ ^
where
(225)
e{u) ^ '-^ . (226)
B.2.2 Bipolar AM
For bipolar AM, (215) gives
e(u)R(u) cos (uto + (p)
Vi =
2ER(0)
and the rms value with all phases <p equally probable is
e{u)R(u)
Vi =
2ER(0) '
(227)
(228)
B.2.3 Unipolar AM with Envelope Detection
For unipolar AM with envelope detection, (223) in (217) yields, for
the interference during a space,
(,) ^ e{u)R{u) ^ 112 eju)
''' ERiO) ER(0)
= 2'" ^ly , (229)
and (220) gives, for interference during a mark,
Vi = ER(0) '
with an rms value
_ („) ^ e{u)R{u)
"^^ ER(0) ■
(231)
B.3 Impulse Noise
In the case of idealized impulse noise, the interfering voltage is of
the general form
ei(ta) ^ pi{to) COS {cJo^o + <Pi)
y ZoZ )
= Pi(to) COS ^ COS (wo/o + ^o) — Pi(to) sin ^ sin (umto + ^o),
IDEAL BINARY PULRE TKAN'SMISSIOX BY AM AND FM 1421
where ip = ip-, — >pf] and p,(^o) is the envelope of the noise pulse at the
detector input. The shape of the latter will depend on the receiving
filter characteristic, R(u). The phase ip represents the difference between
the phase of the carrier within the noise pulse envelope and the phase
of the carrier within the signal pulse envelope. This phase difference
depends on the instant at which the impulse occurs and can have any
value.
B.3.1 Frequency Modulation
Comparison of (232) with (202) shows that, in this case,
ri(to) = piik) cos <p,
(233)
qi(tii) = —Pi(to) sin tp
In (208) the various quantities become:
rf + q.' ^ p.'(U), (234)
Ti cos oito + qi sill oita = p,(/o) COS {wto + <p), (235)
r/ sin u}to — q/ cos oito = p/{to) sin (ciyto + ip), (23G)
g/ri - r/qi = 0. (237)
With these values in (208), the amplitude of a noise pulse after de-
modulation becomes
ViUo) = ^ iiipi'(to) - pi(to) cos (uk + <f>)
,,,, . (238)
pi {to) ■ /-, I X
— - — ; — Sin (uto -\- tp)
D = 1 -\- niW(to) - 2tiiPi{to) COS (a/o + <p). (239)
An error will occur if, at the sampling instant, 17,(^0) ^ 21 which gives
the following relation for determining the peak amplitude of a noise
pulse at the detector input that will produce an error in the demodu-
lated signal:
htii fiipi\to) - pi(to) cos (wto + tp) - - ■ ^' - " I .
L "Sin (wio + ^)J (240)
^ |[1 + fiiPiito) - 2niPi(to) cos (o}to + (p)]
With iii = 1/ER( — w), this can be written
PiUo) cos (oj/o + ^) — - pi (to) sin (uto + <p) /o^^^
o) ^ 1 (.-^41;
2ER(-ib) '■
(242)
1422 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
B.3.2 FM VS. Bipolar AM
In the case of bipolar AM, the corresponding relation is
Pi(to) COS (gjfo + ip) ^ 1
2ERiO) " '*
If it is now assumed that the impulses occur at sampling instants
^Q = 0, then p/(/o) = and the following relations apply.
For FM:
Pi(0) cos (utg + <p) ^ 1 /243)
2Si2(-w)
For bipolar AM :
Pi(0) cos (uto -\- v) _ 1
(244)
2ER(0)
The peak amplitudes of the impulses that will cause errors are thus
smaller in FM than in AM by the factor
P(0)fm _ fi(-c^)
p(0)am ^(0)
(245)
In accordance with (104) and (105), the following relation applies
between the average signal powers in FM and AM :
Pam L fi(0) J
From (245) and (246) it follows that
p(0)r
FM
p(0),
( "fM \ il/2
Pam/
(247)
Thus, for equal probability of errors when the impulse occurs at
sampling instants and equal average signal power Pfm = Pam , the
impulses at the detector input can be greater in AM than in FM by a
factor 2^^'\ corresponding to 3 db.
As another Umiting case, assume that the impulses occur midways
between sampling points corresponding to ibta = t/2 or (o — 7r/2w. In
this case (241) and (242) become
for FM and h - 7r/2ii:
— piik) sin^ + il/o})pi (to) cos^ ^ 1 (248)
2ER(-cb) ' "
IDEAL BINARY PULSE TRANSMISSION BT AM AND FM 1423
for AM and to = 7r/2ai:
-piik) sin ^ _ 1
2-
(249)
2ER(0)
Since tp may have any value, it is permissible in (249) to substitute
^j = (^ — 7r/4) = COS 7r/4 sin <p -\- sin 7r/4 cos <p, in which case, for
AM
— pi{to){^in<p + cos <p) _ ii
12
1
., . (250)
2ER(0)
The following approximation applies for a representative pulse shape*
and (o ^ 7r/2a):
-lp/(U)^Pi(k) ^ipiiO), (251)
where ?j,(0) is the peak pulse amplitude at the detector input, which
will occur at a time ta = Tr/2w from a sampling point.
Hence (248) and (250) can be written
for FM:
for AM:
-^•("^'^^fF^^^ '''''
The peak amplitudes that \vill cause errors are thus smaller in FM
than in AM by the factor
p(0)fm _fi(-w)
(254)
In view of (24G),
p(0)^m~2'/^^(0)'
(255)
P(0)PM _ 1/PfM^'^'
?>(0)am 2VP.,
Thus, for equal probabiUty of errors when the impulses occur mid-
ways between sampling points and equal average signal power Pfm =
Pam , the peak ampUtude of the impulses at the detector input can be
greater in AM than FM by a factor of 2, corresponding to 6 db.
* In the case of a pulse shape obtained with a raised cosine spectrum, p,(/o) =
^ PiiO) and Pi'ita)<^ = 0.475 p.(0).
1424 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
The peak amplitudes, p(0), required to produce an error when an
impulse occurs midways between samphng points is greater than when
they occur at a sampling point by a factor of 2"^ in FM and a factor of
2 in AM. With a Gaussian amplitude distribution of the pulses, the
probability of an error from a pulse midway between two sampling
points is in the order of 1 per cent of the probability of an error from a
pulse at a sampling point in the case of FM, and is substantially smaller
for AM. Hence, virtually all the errors will be caused by pulses that
occur near samphng points. The AM advantage over FM for equal error
probability is 3 db for impulses that occur at samphng points, and would
be expected to be only slightly greater, about 4 db when impulses oc-
curring at all instances with respect to a sampling point are considered.
The above comparisons apply without a postdetection low-pass filter
in FM. With an optimum bandpass receiving filter characteristic in FM,
the reduction in peak impulse noise afforded by low-pass filter would be
expected to be about the same as the reduction in average random noise.
APPENDIX c
Optimum Receiving Filter Characteristic
The optimum receiving filter characteristic in AM and in FM with-
out a postdetection low-pass filter can be determined from the solution
of the more general case considered here, of FM with a postdetection
filter.
In the latter case, the optimum R{'u) is obtained when the product
of the two integrals in (125) is a minimum, or for the minimum value
of the product :
J = J1J2 , (256)
where Ji and J2 are functions of R(u) given by
J, = r L\u)R\ii){\ + u/^Y du, (257)
/,= r^d„ = 2r^*.. (258)
In (257), L(—u) 7^ L{ii), so that it is convenient to resolve the
integrand into one component with even symmetry with respect to u
and one with odd symmetry. The integral of the latter component van-
ishes and that of the component with even symmetry becomes
Ji= f H\u)R\u)du, (259)
IDEAL BINARY PULSE TRANSMISSION BY AM AND FM 1425
wliere
H-(w) = L-(m)(1 + u/i^f + L\-n){\ - u/i^^f. (260)
When a small variation, bR{u), is made in /?(«), the resultant vari-
ation in / is
= J. I 2R{u)H-(u)SRiu) du
The optimum R{u) is obtained when 8J = 0, or
I" {j,R(u)H%u) - ^^^^] 5R(u) du = 0,
which is the case when
or
J.Mn)H\u) - ^^ =
R^{u)
R{u) = RKu) =
where c = (Ji/JiY'* is a constant.
With (265) in (258) and (259),
H"'{u]
Ji = c'2"' I S(u)Hiu)du,
Jo
J2 - -. 2'
c-
S(u)H(u) du,
^ 1
~ 2
f Siii)H{u) du ,
S{u)H{u) du .
(261)
(202)
(263)
(264)
(265)
(266)
(267)
(268)
The optimum ratio, No/So , obtained with (268) in (125) becomes
(iVo/So)™ = r^^\r S(u)Hiu) dJ. (269)
^TTl JTfM L"'"" J
1426 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1959
In the case of FM without a postdetection filter, L{u) ^ 1 and, in
this ease,
H\u) ^ 2(1 -^u'/^'), (270)
in which case (265) gives (138).
In the case of AM, the term ii^/u is absent in (270) and
H\u) = 2, (271)
so that (265) gives (127).
REFERENCES
1. Nyquist, H., Certain Topics in Telegr.iph Transmission Theory, Trans.
A.I.E.E., 47, April 1928, p. 617.
2 Siinde, E. D., Theoretical Fiindiiinentals of Pulse Transmission, B.S.T.J., 33,
May 1954, p. 721 ; July 195-1, p. 987.
3. Oliver, B. M., Pierce, J. R. and Shannon, C. E., The Philosophy of PCM,
Proc. I.R.E., 36, November 1948, p. 1324.
4. Bennett, W. R., Methods of Solving Noise Problems, Proc. I.R.E., 44, May
1956, p. 609.
5. Montgomery, G. F., A Comparison of Amplitude and Angle Modulation for
Narrow-Band Communication of Binarj-^-Coded Messages in Fluctuation
Noise, Proc. I.R.E., 42, February 1954, p. 447.
6. Cahn, C. R., Performance of Digital Phase Modulation Communication Sys-
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7. Turin, G. L., Error Probabilities for Binary Symmetric Ideal Reception
Through Nonselective Slow Fading and Noise, Proc. I.R..E., 46, September
1958, p. 1603.
8. Watson, G. N., A Treatise on the Theory of Besscl Functions, Cambridge Univ.
Press, London, 1944.
9. Rice, S. O., Statistical Properties of a Sine Wave Plus Random Noise, B.S.T.J.,
27, January 1948, p. 109.