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On the Stability of a Double Integration Delta Modulator
Nielsen, Palle Tolstrup
Published in:
I E E E Transactions on Communications
Link to article, DOI:
10.1109/TCOM.1971.1090641
Publication date:
1971
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Nielsen, P. T. (1971). On the Stability of a Double Integration Delta Modulator. I E E E Transactions on
Communications, 19(3), 364-366. https://doi.org/10.1109/TCOM.1971.1090641
364
IEEE
TRANSACTIONS
ON COMMUNICATION
TECHNOLOGY,
JUNE
1971
Fig.
3
is
a
simplified block diagram. The input signal voltage
is commutated to the storage condensers by successive closures
of the input switches
at
a
rate set by oscillator
I,
and commutated
thence to the output terminal by successive closures of the output
switches at
a
slower rate under control of oscillator
2.
The latter
runs at
a
fixed rate of
16
000
Hz,
a
frequency .great enough to
avoid generation of audible components due to commutation.
Oscillator
1
has
a
higher frequency which is adjustable. The fre-
quency demultiplication ratio
is
equal to the ratio
of
the oscillator
frequencies; for example, it
is
50
percent with oscillator
1
set
at
32
000
Hz.
Both switch control circuits begin each reset cycle simultaneously
with the first input and output switches closed. Then the switch
closures progress to the right in Fig.
3,
the input rate being greater
than the output. With no glottal synchronizing signal present,
the input circuit reaches its last position and waits there until the
output circuit reaches its last switch. Then the next pulse from
oscillator
2
resets both circuits and the cycle repeats.
If
a
glottal
synchronizing signal is received, the circuits are reset immediately
regardless of how far the current storage-readout cycle
has
pro-
gressed.
Evaluatiork
of
this unscrambler was performed with taped helium
speech stimuli,
as
well
as
in on-line operation during dives. Highly
successful performance was demonstrated with divers at pressure
depths
as
great
as
1000
ft.
Objective word intelligibility test results
were
88
percent at
500
ft
and
78
percent
at
800
ft,
as
compared
with previously reported intelligibility of
5
percent for uncorrected
helium speech at
a
depth of
600
ft
.
ACKNOWLEDGMENT
Dive test facilities were pravided by Ocean Systems, Inc. The
equation relating frequency and depth was derived by Dr.
G.
R..
Gamertsfelder, and the unscrambler was designed by
J.
W. Gray,
both of Singer-General Precision.
ItEFERENCES
111
L.
J.
Gerstman,
G.
R.
Gamertsfelder.
and
A.
Goldberger,
J.
Acoust.
[21
S. S.
Stevens,
Handbook
of
Experimental Psychology.
New York:
[31
Lieberman
ef
al., Psychol.
hfonogr.,
vol.
68,
no.
8,
1954.
pp.
1-13.
Soc.
Amer.,
vol.
40,
1966.
p.
1283(A).
Wiley,
1951.
On
the Stability
of
a Double Integration Delta Modulator
P.
TOLSTRUP
NIELSEN
Abstract-A
delta modulator with a second-order network in
the feedback path is considered.
For
zero input signal the possible
modes of oscillation are determined as a function of the
zero
of
the linear
network.
The method could be employed with other
types
of
feedback networks.
A
comparison
is
made with experi-
mental results.
I. INTRODUCTION
It
is
well known that the performance
of
a
delta modulation
system in many cases can be improved by choosing
a
more com-
plicated network than
a
single integrator in the feedback path of
the modulator. With the configuration in Fig.
1
an often suggested
transfer function for the linear network is
Communication Technology
Group
for
publication
without
oral
presen-
Paper approved
by
the
Wire Communication Committee
of
the
IEEE
tntion.
Manuscript
received
February
4
1970
The
author
is
with
the
Laboratoyy
for' Communication Theory,
Technical University
of
Denmark, Lyngby, Denmark.
-
Fig.
1.
Delta modulator under considerat,ion.
The choice of the parameters
71
and
72
chiefly depends on the
spectrum of the transmitted waveform,
s(t),
whereas the zero is
needed because of the demand for stability. From a quantizing
noise, viewpoint it is desirable to keep
70
as small
a.~
possible. In
fact it can be shown that with all other parameters fixed, the ratio
of rms signal to rms noise is approximately inversely proportional
to
70.
However, if
70
is decreased below
a
certain limit, the system
becomes unstable, and the follow-up characteristics of the modu-
lator are destroyed. Hence there is an optimum value of
70
which
yields the best overall system performance. The purpose of this
paper is to show how the degree of unstability depends
on
the
choice
of
70
when all other parameters remain fixed.
11.
POSSIBLE
MODES
OF
OSCILLATION
Consider the delta modulator in Fig.
1.
For the purpose of this
evaluation it is assumed that the input signal
s(t)
is zero and that
the dc balance of t.he modulator is perfect, i.e., the number of
ones equals the number of zeros in the output bit stream. Under
these conditions the signal
r(t)
will be a symmetrical square wave,
wit,h a half-period equal to
nT,
where
T
is the reciprocal of the
clock frequency and
n
is some integer;
n
2
1.
We denote the
os-
cillation frequency by
.fo
=
-
-
1
f*
2nT 2n
where
f,
is
the
clock frequency. The
case
n
=
1
corresponds
to
the
highest possible degree of stability.
If
n
is
high, the oscillation will
be of relatively low frequency and will therefore appear at the
output of the demodulator with high amplitude.
Our
object
is to
determine the possible values
of
n
for
a
given value of
T.O.
All the waveforms
of
the system are shown in Fig.
2
for the case
of
a
stationary oscillation.
It
is readily
seen
that
y(t)
is a true
replica of
~(t),
except for
a
delay
td
due to the linear network
H(s)
and the phase shift of
180"
due to t,he change of sign at the input
of the comparator.
If
the oscillation is
to
continue, the following inequality must
be satisfied:
(n
-
1)T
<
td
<
n'T.
(3)
Our
next objective is
to
calculate the zero-crossing delay
td
for the
transfer function
(1).
It is shown in the Appendix that
td
can be
found by solving the equat.ion
where
c1
and
c2
are given by
c1
a-
70
-
71
71
-
72
cza-.
70
-
72
72
-
71
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CONCISE PAPERS
365
11111111111111111
T
Clock
I
X(t)
.
nT
.
td
~
r(t1
1
I
-I
y(t)
Fig.
2.
Waveforms in oscillating system.
If
we define
a
p-
nT
-
td
T
the inequality
(3)
can
also
be expressed
as
ojO1<1.
The quantity
a
was calculated by numerical methods from
(4)
for several values of
n
and
TO.
In this calculation the other param-
eters were chosen as follows:
71
=
1200
pS,
72
=
165
pS,
T
=
18
/AS.
(!)I
These are typical values
for
a
delta modulation system designed
for telephone quality speech transmission.
The result is shown in Fig.
3,
where
(Y
is plotted versus
TO
wibh
TZ
as
a
parameter. Due to the choice of finite values for
rl
at~d
71,
a lower limit forfo is seen to exist even for
70
=
0.
n
=
2
is
:L
possible
mode for high values of
TO.
Consequently, no improvemenl of
system performance could be expected when choosing
TO
>
1.52'.
111.
EXPERIMENTAL
RESULTS
Tests were carried out in the laboratory in ah attempt
lo
es-
tablish the modes of oscillation predicted in Fig.
3.
It
tranetl out
that for
TO
=
T,
the only possible output pattern was
-
101010..
.
whereas
for
TO
=
0.25T
three patterns could be stable:
-
101010..
.
__
11001100...
111000111000~.
..
The number
of
observed modes is considerably less thall (,he theo-
retically expected number which, from Fig.
3,
in the tjwo cases
should be
4
and
8.
Obviously oscillations corresponding to high
values of
ai
are hard to establish because
of
noise and jitter present
in the system.
However, if
a
random signal
is
applied to the input
of
the delta
modulator, short bursts
of
these high-order oscillations may occur
from time
to
time, and unless the frequency of oscillation is well
out of t,he voice band, these bursts will have enough inband energy
to
cause a noticeable decrease in the output
SNR.
Esperimenfal work by de Jager
[l],
Tomozawa and Kaneko
[2],
and ot.her aut,hors indicates that the optimum choice of
TO
(maximizing the output
SNR)
is very close to
2'.
From Fig.
3
this
choice is seen
to
eliminate oscillations
at
frequencies lower than
nbout
i
kHz which is in good agreement with the previous discussion.
nsl
0.5
I
.o
1:s
Fig.
3.
Stability factor
01
=
(nT
-
td)
/T.
TI
=
1200
PS
TZ
=
165
ps.
T
=
18.p~.
Oscillations
of
frequency
fo
=
fp/2n
cad
take place
if
corresponding
a
satisfies
0
<
a
<
1.
IV.
CONCLUSION
For
a specific second-order delta modulator the possible modes
of oscillation for zero input signal were determined as a function
of
TO,
t.he zero
of
t'he feedback transfer function. The approach is
general in nature and could be employed with other types of trans-
fer functions. Since the analysis presented is valid only under the
idle condition, it cannot lead to a rigorous optimization of
70.
Nevertheless it should be clear that some relation exists between
the degree of stability in the idle case and the follow-up char-
acteristics
when
a
signal
is
applied.
The
observation by other
authors that a value of
70
close to
T
seems
to
be an optimal choice,
is
in good agreement with the results presented here.
APPENDIX
Let
u(t)
be the function defined in Fig.
4.
Then the Laplace
transform of
u(t)
is
U(s)
=
-
1
1
-
exp
(-as)
s
1
+
exp
(-as)
'
This signal is applied to the input of
H(s).
The transform
of
the
output is
V(s)
=
__
H(s)
1
-
exp
(-as)
s
1
+
exp
(-as)
.
From the theory of Laplace transforms we know that
if
G(s)
=
d:(S(t)l
(
12)
where
d:
(
-
)
denotes Laplace transform, then
(1
3)
where
q
=
0,1,2,.
and
a
is a rea1 constant,
a
>
0.
To invoke
(13)
we set
and
Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 2, 2009 at 08:44 from IEEE Xplore. Restrictions apply.
366
with
IEEE TRANSACTIONS ON COMMUNICATION TECHNOLOQY, JUNE
1971.
and
cz=-.
io
-
72
72
-
71
From
(11)
and
(13)
we obtain
+
c2 exp
(-
)]
.
Fig.
4.
Deflnition
of
function
UU).
Only the stationary solution is of interest,
so
‘we look for an ex-
pression for
v(t)
in the interval
am
<
t
5
a(m
+
1)
where
m
is
a
positive integer. We define
t’
=
t
-
am
and obtain
for
the stationary solution in the interval
(19)
r(t’)
=
lim
v(t’)
m-m
But
Substituting (22) in (21) we get in the limit
2c2
+
1
+
exp
(-a/72)
exp
(-
f)]
.
(23)
Using the same terminology we have
z(t’)
=
lim
u(t’)
=
(-1)m.
(24)
By now it should be clear that the value
to‘,
obtained by solving
r(t,l)
=
0
(25)
with
a
=
nT,
is identical to the delay
td
of
Fig.
2.
It
is easy to verify
that
r(t’)
is increasing through the zero determined by
(25)
when
m
is even and decreasing when
m
is odd, provided only that
m
..-
70
<
max
(r1,72).
(26)
This
completes the proof of
(4).
REFERENCES
[l]
F..
de Jager, “Delta ,modulation,
a
method of
PCM
transmission
uslng
the
I-unit
code
Phillips
Res. Rep.,
vol.
7,
1952.
pp.
442-466.
[Z]
A.
Tomozawa and
H.
Kaneko,
“Companded
delta
modulation
for
tele
hone
transmission
IEEE
Trans.
Commun.
Technol., vol.
[31
P.
P.
Wing,
“An
absolite
stability
critefion for delta modulation.”
COG-16
Feb.
1968
pp’
149-157.
IEEE
Trans.
Commun. Technol.
(Concise Papers),
vol.
COM-16.
Feb.
‘1968,
pp.
186-188.
Correspondence
Intelligible Crosstalk in Multiple-Carrier FM Systems
with Amplitude Limiting and AM-PM Conversion
Abstract-Intelligible crosstalk will occur
in
a
multiple-carrier
FM
system when a gain slope versus frequency characteristic
is
followed by
AM-PM
conversion. This phenomenon can be particu-
larly serious in a communication satellite repeater that utilizes
a
traveling-wave tube for output power amplification.
A
model that
includes both amplitude saturation and
AM-PM
conversion
is
developed and the two-carrier intelligible crosstalk
is
calculated.
of
the
I8EE
Communlcation Technology
Group.
Manuscript
received
Corres
ondence apprpved
by
the
Communication
Theory
Committee
December
7,
1970.
I.
INTRODUCTION
The simultaneous transmission of
frequency-division-multi-
plexed
FM
signals through
a
nonlinear system can result
in
in-
telligible ,crosstalk. In a voice-communication system this effect
is particularly severe because it is manifested
as
intelligible whisper-
ing and can have
a
psychological impact considerably more severe
than one might expect from the quantitative crosstalk level.
In
order for intelligible crosstalk to occur it
is
only necessary
to
have
a
frequency-dependent gain slope followed by
AM-PM
conver-
sion
[l];
however, amplitude limiting will affect the crosstalk
magnitude.
The intelligible crosstalk level is determined in
this
analysis
for
a
gain slope versus frequency characteristic followed by
a
Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 2, 2009 at 08:44 from IEEE Xplore. Restrictions apply.