IEEE TRANSACTIONS
ON
COMMUNICATIONS,
VOL.
COM-28,
NO.
1,
JANUARY
1980
131
CHEEYSHEV
PULSE, ~=0.2T
(FAST FADING)
CHEEYSHEV
PULSE,
(SLOW FADING)
y=om
0
5
10 15
20
25
30
35
MEAN SIGNAL-TO-NOISE RATIO,
P
IN
dB.
Figu e
3.
Pe o mance o Rayleigh Fas -Fading Channel wi h Syn-
ch oniza ion E o s.
CONCLUSION
I is
o
be expec ed ha he e ec s
o
in e symbol in e -
e ence on as - ading.channels would be mo e signi ican han
on
slowly ading channels. This hypo hesis is clea ly suppo ed
by he da a shown in Figu es
2
and 3, pa icula ly in he la e
case whe e he in e symbol in e e ence is enhanced by a
la ge synch oniza ion e o . When he signal- o-noise a io is
low, he in e symbol in e e ence has li le e ec on he
channel pe o mance, bu as he mean signal- o-noise exceeds
10 dB he a io
Pe,/Pe
begins o inc ease apidly. Fo mean
signal- o-noise a ios in excess
o
25
dB,
P,,/P,
app oaches
uni y, i.e., he o al bi e o p obabili y is almos en i ely due
o in e symbol in e e ence. Thus he e ec o he in e sym-
bo1 in e e ence on he as - ading channel is o in oduce an
i educible e o p obabili y ha limi s pe o mance a high
signal- o-noise a ios. This i educible e o p obabili y is
7.38
X
o he Gaussian pulse wi h no synch oniza ion
e o . Fo he Chebyshe pulse i is 3.39
X
wi h no
synch oniza ion e o .
While eliable communica ion o e slowly ading channels
can
be ob ained wi h la ge mean signal- o-noise a ios o
di e si y echniques, he da a p esen ed he ein clea ly indi-
ca e ha in e symbol in e e ence p esen s a e y se ious
p oblem in Rayleigh as - ading channels. Applica ion
o
equaliza ion echniques should also be conside ed o comba
he pe o mance deg ada ion due o in e symbol in e e ence
in hese cases.
REFERENCES
[I]
E. Y.
Ho
and Y.
S.
Yeh. "A new app oach o e alua ing he e o
p obabili y
in
he p esence o in e symbol in e e ences and addi i e
Gaussian noise,"
BeNSys . Tech.
J.,
Vol. 49, pp. 2249-2266, 1970.
[2]
0.
Shimbo and M.
1.
Celebile , "The p obabili y o e o due o
in e symbol in e e ence and Gaussian noise
in
digi al communica ion
sys ems,"
IEEE T uns.
Commun.
Technol..
Vol. COM-19, pp. 113-
119, Ap il, 1971.
13)
S.
Benede o.
G.
De Vincen is and A. Lu ison, "E o p obabili y
in
he
p esence o in e symbol in e e ence and addi i e noise o mul ile el
digi al signals,"
IEEE T ans.
Commun.,
Vol. COM-21. pp. 181-
190.
Ma ch 1973.
[4] James C. Vanelli and Nazmi M. Shehadeh, "Compu a ion o bi e o
p obabili y using he apezoidal in eg a ion ule,"
IEEE T ans.
Commun..
Vol.
COM-22, pp. 331-334, Ma ch 1974.
[SI
I.
S.
G adsh eyn and
I.
W.
Ryzhik,
Table o ln eg uls. Se ies
and
P oduc s,
New Yo k: Academic P ess, 1965.
[6] Howa d H. Ma, "The pe o mance o Rayleigh as - ading channels
wi h
in e symbol in e e ence and addi i e Gaussian noise," Mas e Thesis,
Dep .
o
Elec. Eng., Uni e si y o Hous on, May 1978.
[7]
S.
D.
Poisson, "Memoi e su le Calcul nume igue des in eg ales
de ines."
Mem. Acad.
Sci.
Ins?.
F ..
Vol. 6, pp. 57 1-602. 1823.
Commen s
on
and Ex ensions
o
Wol 's
Signal- o-Channel
Noise Fo mulas
o
Del a-Modula ed Sys ems
ANIBAL R. FIGUEIRAS-VIDAL, MEMBER, IEEE,
JOSE
B.
MARIRO-ACEBAL,
AND
MIGUEL A. LAGUNAS-HERNANDEZ,
MEMBER, IEEE
Abs mcGThe channel noise e ec s on linea del a modula ion
(LDM)
sys ems ha e no ye been adequa ely analyzed. This pape p esen s
a
new
and gene al o mula ion o hese
e ec s,
based
on
he heo e ical wo k by
Wol
[I].
A
compa a i e discussion o ou o mulas wi h p e ious esul s is
also included. Finally, he applica ion o ou me hods and he alidi y o
ou commen s a e illus a ed by some nume ical examples.
I.
INTRODUCTION
This pape gi es a me hod o calcula ing he signal- o-chan-
ne1 noise powe a io in a linea del a modula ion (LDM) sys-
em. Al hough LDM sys ems do no o e a alid al e na i e o
PCM o applica ions equi ing
a
wide dynamic ange
[
21,
he
echnique desc ibed can be applied o o he obus DM me h-
ods, especially digi ally syllabic-companded del a modula ion
(DSCDM) sys ems.
Pape app o ed by he Edi o o Da a Communica ion Sys ems o
he IEEE Communica ions Socie y o publica ion wi hou o al p esen-
a ion. Manusc ip ecei ed Ap il 21, 1978; e ised June 21, 1979.
A. R. Figuei as-Vidal is wi h ETSI Telecomunicacion, Ciudad
Uni e si a ia, Mad id, Spain.
J.
B. Ma iilo-Acebal and
M.
A. Lagunas-He nandez a e wi h ETSI
Telecomunicacion, Jo ge Gi ona Salgado, Ba celona, Spain.
0090-6778/80/0100-0131$00.75
0
1980
IEEE
132
IEEE
TRANSACTIONS ON COMMUNICATIONS,
VOL.
COM-28, NO.
1,
JANUARY 1980
In many cases, he channel noise e ec is negligible wi h e-
s( )
+
n( )
spec
o
he quan iza ion noise (g anula and excess slope e -
o s), bu ,
in
some applica ions, whe e he quan iza ion noise
is made ela i ely negligible
in
an o he wise noisy channel, his
e ec can be a c i ical quali y pa ame e . E en in cases whe e
he wo noise componen s a e o equal impo ance, one can
calcula e he channel noise e ec s by he echniques desc ibed
he ein.
The quali y c i e ion employed in his pape is he classical
mean-squa ed e o measu e. Al hough in mos applica ions o
oice o image ansmission subjec i e c i e ia may be mo e ap-
p op ia e, he mean-squa ed exp essions a e a i s -o de indi-
ca o . Fu he mo e, hey can be modi ied easily in o a weigh ed
mean-squa ed e o c i e ion, which is in close ag eemen wi h
subjec i e e alua ions.
Finally, i should be poin ed ou ha ex ensions
o
he
gi en o mula ion a e easily ob ained o hose cases in which
he independen channel e o model canno be used (e.g., in
comme cial elephony) in he same way ha Wol indica ed
[
11
o he Gilbe bu s -noise model. The only modi ica ion
necessa y would be in he digi al au oco ela ion unc ion
E(ni
ni+m)
(see Sec ion
11).
"
LINEAR DELTA DEMODULATOR
Fig.
1:
Linea
del a-modula ion ecei e .
and he powe spec al densi ies:
He e
Rpp( )
and
SDp( )
a e he au oco ela ion unc ion
and he ene gy spec al densi y o
p( ),
espec i ely. Then, he
signal and noise powe s a he ou pu can be calcula ed wi h
he help o he ollowing o mulas:
11. SIGNAL AND CHANNEL NOISE POWERS 1
&O)
=
5
E(bibi+m)[Rpp( )
*
Rhh( )l
=mT
Le
bi
indica e he i h ansmi ed symbol, whe e
bj
=
+I.
mZ-00
and
ni
he channel noise e ec on his s mbol. whe e
n..
=
0
The ecei e de ec s and egene a es he incoming channel
signal; i
p( )
is he basic shape
o
he egene a ed pulses, we exp
(j2nmT )
d ;
will ha e, a he inpu
o
he LD demodula o ,
00
and a noise e m:
n( )
=
nip(
-
iT)
(2)
j=-m
whe e
T1
is he symbol a e. The LD demodula o can be
conside ed a linea sys em, ha ing an equi alen impulse e-
sponse
h( )
and a ans e unc ion
H( ).
Figu e
1
shows he
gene al si ua ion jus desc ibed.
Since he ou pu o he LD demodula o is a con inuous
p ocess whose powe le el does no depend
on
he ime
o igin, we can andomize he e e ence ime o he pulses
131,
ob aining he signal and noise e ms:
exp
(j2nmT )
d ;
(10)
whe e
*
indica es con olu ion, and
Rhh( )
is he au oco e-
la ion unc ion o he impulse esponse
o
he LD demodu-
la o .
We ha e hus ound he signal- o-channel noise o mula:
m
m
n( )=
2
nip( +
8
-iT)
whe e
0
is a andom a iable, uni o mly dis ibu ed o e
[0,
m
TI
.
These signal and noise e ms a e s a iona y p ocesses, wi h
I(mT)
&
sp,( )
I
H( )
l2
exp
(i2nmT )
d .
(12)
espec i e au oco ela ion unc ions
[
31
:
1-
This signal- o-channel noise o mula
is
absolu ely gene al
R,s( )
=
-
x
E(bibi+m)Rpp(
+
mT)
T
m=-m
1-
T
m=--
e ec s, bu we will es ic ou discussion o di ec ansmission.
(5)
o LDM sys ems. I can also be ex ended o adap i e ( a iable
s ep) DM sys ems, using an app op ia e ede ini ion o he sig-
nal and noise e ms, al hough he digi al sou ce model will
Rnn( )
=
-
2
E(nini+m)Rpp(
+
mT)
(6)
ha e
o
be changed. I
is
also possible o conside line coding
IEEE
TRANSACTIONS ON COMMUNICATIONS,
VOL.
COM-28, NO.
1,
JANUARY 1980
133
I
I
1
1
I
,
I
~ -.
Fig.
2:
Ma ko
sys em
model.
111.
WOLF'S MODEL
FOR
INDEPENDENT CHANNEL
ERRORS
The only p oblem emaining
o
be sol ed is he e alua ion
o
E(bibi+,)
and
E(nini+,).
Wol 's model
[
1
]
is app op ia e
o e alua ing hese ma hema ical expec a ions assuming in-
dependen channel e o s. This model consis o wo syn-
ch onized Ma ko chains o he digi ized sou ce and he chan-
nel e o s; Figu e
2
shows he model. The ansi ion p oba-
bili ies
PT
a e equal, since we assume symme y in his ega d.
Only
PT
is equi ed o he e alua ion o he digi al au oco e-
la ion unc ions, and we can ob ain i om mo e s uc u ed
(and ealis ic) Ma ko models o he quan ized signal.
We de i e he signal- o-channel noise o mula in Appendix
I.
The ob ained esul is:
he analog message; we will deno e he co esponding ansi ion
p obabili y by
PT'.
Wol 's o mula is exac when
PT
=
0.5,
i.e., when P(so)/
P(no)
=
1/4Pe, bu i will be mo e and mo e inaccu a e as
PT
app oaches ze o o uni y. This appea s o be he main eason
o
he p og essi e sepa a ion
o
Wol 's heo e ical cu e om
B aun's expe imen al alues
[
1
]
:
he i ing o low
Pe
implied
de ia ion o high
Pe
alues.
I
PT
app oaches uni y, P(so) will inc ease app eciably in
p ac ice wi h espec
o
he alue ob ained assuming sin2
nT
=
0,
due
o
he e m (1
-
2PT)
sin2 nT in he denomina o
o
(El), conside ing ha , in p ac ice, he e ec i e bandwid h
co esponding o
H( )
is less han 1/T.
P(no)
has a small a i-
a ion because o he p esence o he ac o
P,
in he second
e m
o
(1.4). Consequen ly, P(so)/P(no) exceeds Wol 's alue.
Jus he con a y occu s when
PT
app oaches ze o. The a i-
a ions a e la ge when
P,
is small, i.e., in "good" channel cases.
I is easy o see ha ou o mula ion gi es P(so)/P(no)
=
0
when
PT
=
0
o
PT
=
1 (signal absen ), bu i also implies ha
P(so)/P(no)
#
0
when
P,
=
1/2. This inco ec esul is due o
he de ini ion selec ed o signal- o-channel noise a io, as Wol
indica ed. Al e na i e de ini ions chosen in o de o sol e his
p oblem p esen o he di icul ies
[
1
]
.
Ne e heless, he cases
in
which
P,
%
1/2 co espond o ansmissions o e unusable
channels (ha ing a capaci y nea ze o), and he p e ious
P(so)/P(no) o mulas a e dec easing unc ions o
P,
which in-
dica e he sys em pe o mance in all p ac ical si ua ions.
In p ac ical sys ems, he inal s ep in he LD demodula o
is a bandpass il e , wi h lowe and uppe cu o equencies
cl,
c2,
espec i ely, and 1/T
>
c2.
A
i s app oxima ion
would be
o
assume sin2 nT e
0
in he e ec i e band o in-
eg a ion. Then:
and
IV.
EXAMPLES
When he in o ma ion consis s o a oice signal and he
sys em is app oxima ely op imized wi h espec o g anula
and excess slope noises, Kikke 's simula ion esul s [4] allow
us
o conclude ha
P,
will no be e y a om
0.5.
In his
case, he applica ion o Wol 's o mula will be accep able. This
emains ue as long as
1
/Tis la ge enough o main ain sin2 nT X
0
in he in eg a ion band. Ne e heless, we will p esen mo e
gene al calcula jons o illus a e he depa u e om Wol 's
esul s.
Le
us
conside a inal ideal bandpass il e ha ing cu o
equencies
cl
=
300 Hz,
c2
=
3400 Hz, le
us
also assume
T
=
1/5600 symbols/s, and:
P(0
=
wm
(1
7)
whe e:
This is Wol 's o mula. No e ha Wol did no conside he
qx)=
1,
i
Ix
I<
1/2
inal bandpass il e . We no e ha he app oxima ion is ac-
0,
i
I
x
I
>
1/2.
cep able when 1/T is la ge wi h espec o ,2. I we accep he
app oxima ion, he esul ing o mula will apply independen
F om
he
abo e
exp ession
o
p( ):
o
he shape
o
he egene a ed pulses. Wol assumed a pe ec
in eg a o , bu his esul is also use ul o leaky and double in-
spp(n
=
pp
sinc2
~
eg a ion sys ems and o del a-sigma sys ems. In he la e
case,
PT
would co espond o a model o he digi ized in eg a- whe e:
ed message signal. The same model p e iously indica ed
would apply o he p ocess esul ing om he in eg a ion
o
~inc
X
4
sin
nx/nX.
134
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO.
1,
JANUARY
1980
Fig.
3:
Pe ec in eg a ion linea del a demodula o
(T
:
delay; he in? block is
an
ideal bandpass il e ).
Figu e
3
shows a ci cui ha can be used as a pe ec in e-
g a ion demodula o when
p( )
is a ull ec angula pulse. I s
powe ans e unc ion may be w i en:
whe e
c
=
( cl
-
cz)/2 is he cen al equency, and
I,
=
cz
-
cl
is he bandwid h
o
he inal bandpass il e . Table
I
p esen s nume ical esul s om ou o mula ion, showing he
small
di e ences wi h espec o Wol 's alues (in b acke s) i
P,
is no e y di e en om 0.5, and he inc easing di e -
ences when
PT
goes
o
ze o o
1.
I a single
RC
(leaky) in eg a o is used, we will ha e:
(22)
whe e
3
is he cu o equency
o
he
RC
il e . Table 11
shows nume ical alues co esponding o a ypical
3
equal
o
150
Hz.
I 'PT
=
0.5 and we pu sinc2
T
%
1 in
[ cl, cz],
we will
ob ain Johnson's o mula o
P(n0)
[6] [71.
A
double in eg a o is shown in Figu e
4.
Typical pa am-
e e s allow
us
o w i e:
TABLE
I1
SIGNAL-TO-CHANNEL NOISE POWER RATIOS (IN
dB),
RC
INTEGRATION
'+,
PT
la' lo3 105 107
0.05 11.57
34.78
54.02 X02
0.20
6.73
29.80
~BI
69.81
0.35
-1.69 17.90
37.97
57.97
0.60
1.51
2.30
41.30 61.30
0.65
3.98 23.96
43.90
63.98 0.50
620 26.63 4663
66.63
0.95 -8.36
n.21
3.21
51.21
1
I.
Fig.
4:
Double in eg a o .
TABLE
111
SIGNAL-TO-CHANNEL NOISE POWER RATIOS (IN
dB),
DOUBLE
INTEGRATION
XI
lo3
10-1
0.05
-0.37 11.20 3120 51.20 0.95
-1.70 17.97
3.97
57.97
0.00
1.50 21.30 41.30 61.30 0.65
3.98 23.98
43.98
63.98
0.50
6.30
26.65
46.65 66.65 0.35
8.80
29.90
49.4
69.91 0.20
11.83 3540
55.45
7545
whe e
l
=
1/2nR,C1(1
+
C2lCl) (24)
z
=
1/2nR&z[ 1
-RZCz(l
+
R1/R2)'/4R1C1I (25)
Table 111 p esen s nume ical esul s co esponding o a
ypical
z
equal o 1 kHz. I
PT
=
0.5 and we app oxima e
sincz
T
by uni y, ou o mula ion will gi e an al eady known
esul o
P(n0)
[21:
P(n
0)
k
8pe
z l
T{
l/ c
1
-
1
/ c
2
-
[ an-l( ,z/ z)- an-lC cl/ dl/ ~). (26)
In he case
o
del a-sigma modula ion, he esul s ob ained
will no be di ec ly compa able wi h he p e ious ones, since
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO.
1,
JANUARY
1980
135
TABLE
IV
SIGNAL-TOCHANNEL NOISE
POWER
RATIOS (IN dB),
DELTA-SIGMA MODULATION
PT
Pe
los
10"
104
o7
0.05
0.50
6.20 2653
4653
6653
0.35
8.36
29.28 4929 6929
0.20
9.88 31.54 51.56 7.56
1.53
2l.33
41.33 61.33
0.65
3.98 23.98 43.98
63.98
0.80
-1.65 18.02 38.01 58.a
/*
0.95 -8.32 1125
3125
51.25
4) By p ope ly ede ining signal and noise e ms, he
me hod can be ex ended o coded
LDM
sys ems and o o he
p ac ical
DM
sys ems (adap i e del a modula ion,
ADM,
and,
pa &ula ly, digi ally syllabic companded del a modula ion,
DSCDM).
We a e ob aining esul s along hese lines a he p e-
sen ime.
APPENDIX
I
F om he model, i is easy
o
ob ain
[
1
]
:
E(bibi+,)
=
(1
-
2PT)lm1
(1.1)
PT'
#
PT
in a gene al case. The ecei e has: whe e
P,
is he e o p obabili y. Then, ea anging he powe
l.V11'=.(b)+ l(~).
- c
+ ,
exp essions, we can w i e:
Some signal- o-channel noise a io alues a e included in
PT'
=
0.5
and sinc2
T X
1 in
[ cl,
c2]
will lead o John-
m=-m
The case
o
linea del a-sigma modula ion can be sol ed by
pT(l
-PT)
Table
IV.
son's o mula o
P(no)
[
51
[
61.
calcula ing he co esponding
Z(mT)
in he ime domain. Sine-
in eg al unc ions appea in he esul ing exp ession, which is
di icul
o
manipula e and o in e p e . Making sinc2
T
1,
we
ob ain:
m
=
x
(1
-
2PT)lm
I
exp
(j2nm T >
=
-
-
PT
'
+
(1
-
2PT)
sin'
nT
(1.3)
and he esul ing P(so)/P(no) can be exp essed wi h he help
o
(1 1). The same app oxima e me hod can be applied in o he
cases, bu he in eg alsl(mT) ha e
o
be calcula ed nume ically.
PT(
1
-
2PT)
-
(1
-
PT)
sin
n
PT2
+
(1
-
;?PT)
sin2
'
"1
(1.4)
V.
SUMMARY AND CONCLUSIONS
whe e he p ime sign in he summa ion indica es exclusion o
The gene al o mula ion o he signal- ochannel noise
powe a io in (di ec )
LDM
sys ems has been p esen ed and
compa ed wi h ea lie esul s. The main conclusions a e:
he ze o index e m. The a io o he in eg als o hese unc-
ions mul iplied by
SppO
I
H( )
1'
will be he signal- ochan-
ne1 noise a io, o mula (13)
in
he main ex .
1)
A
sligh modi ica ion o Wol 's wo k allows
us
o ex end
i
o all
LDM
sys ems;
2)
Wol 's o mula is accu a e enough in p ac ical sys ems
111
ha ing
PT
no e y di e en om
0.5,
bu i ails when
PT
goes o ze o o
PT
goes o 1, and especially when
P,
is small o
121
when
Tis
la ge.
Some nume ical esul s a e compa a i ely p esen ed.
Addi ional commen s a e:
3)
The me hod can be easily ex ended
o
co ela ed channel
151
[31
141
e o s (e.g.,
o
he case
o
bu s e o s acco dingly o he Gil-
be model
[7],
conside ed by Wol ), he only modi ica ion
being he al e a ion
o
E(nini+m)
alues (no e y p oblema ic
171
161
in he case
o
he Gilbe model);
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