IEEE TRANSACTIONS ON COMPUTERS, VOL. c-24, NO. 11, NOVEMBER 1975
1074
[6] W. D. Strecker, "An analysis of the instruction execution rate
in certain computer structures," Ph.D. dissertation, Dep.
Elec. Eng., Carnegie-Mellon Univ., Pittsburgh, Pa. (Rep.
AD-711408) June 1970.
[7] M. Pirtle, '"Intercommunication of processors and memory,"
in 1967 Fall Joint Computer Conf., AFIPS Conf. Proc., vol. 31.
Washington, D. C.: Thompson, pp. 621-633.
[8] E. G. Coffman, Jr., "A simple probability model yielding performance bounds for modular memory systems," IEEE Trans.
Comput. (Short Notes), vol. C-17, pp. 86-89, Jan. 1968.
[9] G. J. Burnett and E. G. Coffman, "A study of interleaved
memory systems," in 1970 Spring Joint Comput. Conf., AFIPS
Conf. Proc., vol. 36. Montvale, N. J.: AFIPS Press, pp. 467474.
[10] A. Parzen Stochuastic Processes. San Francisco, Calif.: HoldenDay, 1962.
[11] W. R. Franta and P. A. Houle, "Comments on 'models of
multi-processor multi-memory bank computer systems,'" in
Proc. Winter Simulation Conf., Jan. 14-16, 1974, pp. 87-97.
_
Kuchibhotla V. Sastry (S'71-M'72) was
born in Vijayawada, India, on October 8,
1945. He received the B.E. degree from
j Andhra University, India, in 1966, the M.E.
from the Indian Institute of Science,
degree
Bangalore, India, in 1968, and the M.S.
7 and Ph.D. degrees from the University of
Minnesota, Minneapolis, Minn., in 1970
and 1973, respectively, all in electrical engiFrom 1968 to 1973 he was first a Teaching
Assistant and later a Reasearch Assistant in the Department of
Electrical Engineering, University of Minnesota while working on
his doctoral dissertation. He joined Sperry Univac, Roseville, Minn.,
in 1973 where he has been working on multiprocessor system organization and virtual memory management techniques. His current
interests include computer architecture, operating system design,
and modeling techniques.
Dr. Sastry is a member of the IEEE Computer Society.
Richard Y. Kain (S'56-M'58) was born in
Chicago, Ill., on January 20, 1936. He received the B.S., M.S., and Sc.D. degrees in
electrical engineering from the Massachusetts
Institute of Technology, Cambridge, in 1957,
1959, and 1962, respectively.
He has been an Associate Professor of
Electrical Engineering at the University
of Minnesota, Minneapolis, since 1966.
Prior to that he was a Teaching Assistant,
Instructor, and Assistant Professor at M.I.T.
in the Department of Electrical Engineering. He has worked on
time-shared computing, automata theory, computer architecture, and
computer operating systems. He has published one book and numerous technical papers.
Dr. Kain is a member of the American Association for the Advancement of Science, the Association for Computing Machinery,
Sigma Xi, and Eta Kappa Nu.
A/D Conversion Using Geometric Feedback AGC
DENNIS R. MORGAN,
Abstract-A digital signal processing technique is described which
utilizes a built-in automatic gain control (AGC) function. A particularly attractive algorithm called "geometric feedback" is developed which has certain desirable properties. A simple analytic
solution of the response is derived for the special case of linear geometric feedback.
Index Terms-AID conversion, automatic gain control (AGC).
I. INTRODUCTION
TN many signal processing applications, the long-term
dynamic range of the signal is too large to be accommodated using practical A/D converters. The use of automatic gain control (AGC) prior to A/D conversion is
commonly used in this situation to reduce the long-term
dynamic range. Short-term signal variations must still, of
course, be within the range of the A/D converter.
Manuscript received May 15, 1974; revised April 15, 1975.
The author is with the Electronies Laboratory, General Electric
Company, Syracuse, N. Y. 13201.
MEMBER, IEEE
Minicomputers are now commonly used in many realtime signal processing systems. In this case, it is desirable
to use the vast amount of computational power that is
available in order to minimize the external hardware required. The present correspondence is concerned with this
aspect of design as it applies to AGC.
A readily available component that is particularly useful for digital AGC is the multiplying D/A converter
(MDAC). A block diagram of a signal processing system
using this component in an AGC loop is shown in Fig. 1.
Here a digitally controlled attenuator is formed by using
the MDAC in the feedback loop of an op-amp. The attenuation sequence that controls the MDAC is computed
from the input samples using some algorithm.
A particularly attractive algorithm called "geometric
feedback" will be developed in this correspondence which
has certain desirable properties. In particular, the feedback function can be tailored to give the desired largesignal transient response while the small-signal bandwidth
remains independent of input level.
MORGAN:
A/D CONVERSION
1075
USING FEEDBACK AGC
GAIN
CONTROL
OUTPUT
MINICOMPUTER
ANALOG
SIGNAL
DATA
INPUT
DIGITALLY-CONTROLLED
ATTENUATOR
Fig. 1. Block diagram of signal processing system with digital AGC.
II. ANALYSIS
Fig. 2 shows an envelope equivalent block diagram of a
discrete-time AGC amplifier where x(n) is the input and
y(n) the output. For convenience, the inputs and outputs
are assumed to have been scaled to a reference of unity
with no other gain factors appearing in the loop. An error
sequence e(n) is derived by subtracting the reference from
the output and is filtered by an accumulator. The accumulator output b (n 1) controls a nonlinear attenuator
which is represented by the function f(b) and a divider.
An alternative configuration of Fig. 2 could, of course, be
arranged by replacing the divider with a multiplier and
replacing f with l/f. The function f is assumed to be a
continuous, strictly monotonically increasing function.'
The system of equations describing the model is given by
-
y(n)
=
x(n)/a(n
a (n)
=
f[b (n) ]
(ib)
b(n)
=
b(n
(lc)
e(n)
=
y(n)-1
-
-
1)
(la)
1) + e(n)
and
(1d)
The time index n for these variables implies discrete samples taken every T seconds.
It can be shown by linearization that for a nominal
trajectory x== = x and yn 1, the normalized smallsignal transfer function of Fig. 2 is given by [1]
small-signal time constant is defined as
for O < < 1.
T = -T/ln
(5)
As can be seen from the transfer function, the AGC amplifier is stable in a small-signal sense provided that a < 1,
i.e., the incremental loop gain satisfies the condition
O < K < 2.
As in the continuous case, it is noted that for exponential feedback f(b) = exp (yb), the loop gain K = is
constant and hence the small-signal response is independent of input level [1]-[3]. This is a desirable characteristic for many applications. Besides maintaining a levelindependent frequency response, exponential feedback
also affords an extra measure of protection against conditional instability since the loop gain is constant.
For exponential feedback, (lb) and (lc) can be combined as
a(n) = a(n - 1) exp [Eye(n) ].
(6)
This suggests the alternative configuration shown in Fig. 3
which is equivalent to Fig. 2 for g(e) = f(e) = exp (,ye).
Since a constant error would give rise to a geometric attenuation sequence, this configuration is deemed "geometric
feedback." The function g is required to be strictly positive,
monotonically increasing, and in addition g(O) = 1.
For steady-state conditions, it is easy to see that the
incremental loop gain is given by
a,
a
y
K
g'[g-(1)]
(7)
z1
H(z) = Z aE *(2) and the small-signal relationships (2), (3), and (5) all
hold with (4) replaced by (7). It is significant that (7)
is
independent of signal level, a desirable property as prewhere
viously mentioned. If g(e) = f(e) = exp (eye), then (4)
a1-K
(3) and (7) are of course equivalent.
The significance of the geometric feedback configuraand
tion is that the function g can be tailored to give the
K = f'[fE () ]/x > 0.
(4) desired large-signal transient response while the smallThe "loop gain" K and hence the transfer function are signal response remains independent of input level.
An alternative configuration of Fig. 3 could be arranged
seen to vary according to the average input level x. A
by replacing the divider with a multiplier and replacing
'Alternatively, a monotonically decreasing function can be used g with i/g. This configuration may be more desirable for
if the polarity of the error signal is reversed, i.e., e(t) 1 y(t). implementation in some cases.
-
=
-
=
1076
IEEE TRANSACTIONS ON COMPUTERS, NOVEMBER
1975
Fig. 2. Envelope equivalent circuit of discrete AGC amplifier with
accumulator filter.
(u)X~~~ a
E (T U)
Fig. 3. Envelope equivalent circuit of discrete geometric feedback
AGC amplifier.
y(n)
1.0
0.1 L
0
1
2
3
4
5
6
7
nT/T
Fig. 4. Linear geometric feedback AGC amplifier response to
step input.
The general system equations are
y(n)
=
x(n)/a(n - 1)
(8a)
linear difference equation
a(n + 1) - a(n) + E[x(n + 1) - a(n)] (10)
which is the discrete version of the continuous case with
exponential feedback [1].
The z transform of the solution of (10) is
(8b)
a(n) = a(n - l)gre(n)]
(8c)
e(n) = y(n),- 1
and the difference equation for the attenuation is
z
(11)
A (z) =
X(z)
=
z- a
(9)
a(n + 1) a(n)g[x(n + 1)/a(n) 1].
If g is a linear function of the form g(e) = 1 + -ye (the where, as before, a = 1 - and the same conditions apply
first-order expansion of exp (,ye)), then (9) becomes the regarding the effect of y on stability.
MORGAN:
A/D
1077
CONVERSION USING FEEDBACK AGC
0
0 0~~~~ _
0 o.1
o
N
-
:
o
0
0.001.
nT/T
response to step input for linear geometric
Figg. 5. Normalized error
feedback AGC amplifier.
For a step input
x(n) =
In practice, it is necessary to limit 1 + ye so that g >
gqin > 0, for all e. A plot of normalized error is shown in
Fig. 5 and demonstrates the asymptotic exponential behavior predicted from the small-signal theory.
For other choices-of g, the response will generally involve a nonlinear difference equation. Methods for analyzing this situation using the discrete Volterra series
are described in [1].
x(O),
x(O) /yo,
n <0
the solution of (11) in terms of y is
I(
-
(=
-
1/yO) an]-
n >0
(12)
III. CONCLUSIONS
A new digital AGC algorithm has been developed which
where yo = y(0) specifies the initial output level and is is called "geometric feedback." This algorithm has the
equal to the ratio of the input step transition. Equation desirable property that the small-signal response is inde(12) is plotted in Fig. 4 for various input levels. As can pendent of input level. This property holds independent
be seen convergence is faster for yo < 1 but slower for of the particular nonlinearity used. Therefore, the nonyo > 1 as would be expected since exp (ye) > 1 + ye.
linearity can be tailored to give the desired large-signal
1,1
n <0
1078
IEEE TRANSACTIONS ON COMPUTERS, VOL.
transient response while preserving level invariant smallsignal response.
In the simplest form, a linear feedback function results
in a linear difference equation which can be solved in
terms of an arbitrary input. Level-invariant small-signal
response was demonstrated by illustrating the asymptotic
convergence of the error to an exponential decay.
Other types of nonlinearities in the geometric feedback
algorithm are a subject for future investigations.
NO.
11,
NOVEMBER
1975
[3] A. G. Morris, "A constant volume amplifier covering a wide
dynamic range," Electron: Eng., vol. 37, pp. 502-507, Aug. 1965.
Dennis R. Morgan (S'63-S'68-M'69) was
born in Cincinnati, Ohio, on February 19,
1942. He received the B.S. degree in 1965
from the University of Cincinnati, Cincinnati,
Ohio, and the M.S. and Ph.D. degrees from
Syracuse University, Syracuse, N.Y., in
1968 and 1970, respectively, al in electrical
REFERENCES
[1] D. R. Morgan, "On discrete-time AGC amplifiers," IEEE
Trans. Circuits Systems, vol. CAS-22, pp. 135-146, Feb. 1975.
[2] W. K. Victor and M. H. Brockman, "The application of linear
servo theory to the design of AGC loops," Proc. IRE, vol. 48,
pp. 234-238, Feb. 1960.
c-24,
engineering.
He is a Senior Engineer in the General
Electrical Company, Electronics Laboratory,
Syracuse, N.Y., where he is involved with the
analysis and design of signal processing systems.
System Fault Diagnosis: Closure and
Diagnosability with Repair
JEFFREY D. RUSSELL,
MEMBER, IEEE, AND
Abstract-Determination of the detectability and diagnosability
of a digital system containing at most t faulty system components is
considered. The model employed is to an extent independent of the
means used to implement diagnostic procedures, i.e., whether the
tests are accomplished via hardware, software, or combinations
thereof. A parameter, called the closure index, is defined which
characterizes the capability for executing valid tests in the presence
of faults. The closure index can be thought of as the size of the
smallest potentially undetectable multiple-fault in the system as
modeled. On the basis of this parameter, results are presented which
permit the determination of t-fault detectability and t-fault diagnosabiity with repair for the system. Examples are presented to illustrate
the application of the model for systems close to those encountered
in actual practice.
Index Terms-Closure index, diagnostic modeling, diagnosable
digital systems, fault detection and diagnosis, fault-tolerant computing.
Manuscript received April 4, 1974; revised April 29, 1975. This
work was supported in part by the National Science Foundation
under Grants GK-3844 and GK-35037.
J. D. Russell was with the Department of Electrical and Computer
Engineering, University of Wisconsin-Madison, Madison, Wis.
53706. He is now with the Collins Avionics Division, Rockwell
International, Cedar Rapids, Iowa 52406.
C. R. Kime is with the Department of Electrical and Computer
Engineering, University of Wisconsin-Madison, Madison, Wis.
53706. A portion of this research was performed while he was with
the Department of Electrical Engineering and Computer Sciences,
University of California, Berkeley, Calif.
CHARLES R. KIME,
MEMBER, IEEE
INTRODUCTION
SEVERAL studies [1]-[9] have been concerned with
system-level fault diagnosis and have had as an
objective the determination of conditions for diagnosability. Interest in this topic is motivated by the need for
highly available digital systems that can continue operation, perhaps at reduced capacity, when multiple hardware failures occur. The approach to such systems via
reconfiguration or standby sparing requires that the
presence of malfunctioning elements be detected and their
location determined to within a system module, i.e., the
system must be diagnosable. Moreover, initial diagnosis to
the level of large replaceable modules can reduce system
downtime in those cases in which manual repair is performed.
Most of the previous models for system-level diagnostic analysis have considered the system to be partitioned into a number of disjoint subsystems under the
assumption that each subsystem can be completely tested
by some combination of the other subsystems. Each test so
defined involves the controlled application of stimuli to
the subsystem under test and the analysis of the ensuing
responses, resulting in the evaluation of the tested sub-