Convolutional codes and recursive signal processing for digital communication

Pickholtz, R. L.

English

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Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19700009900 2020-03-12T01:00:55+00:00Z
tINTERIM REPORT
September 15, 1969 - February 15, 1970
DIGITAL CW21RINICATIONS TECHNIQUES
for
National Aeronautics and Space Administration
under
NASA GRANT NCR-33-006-020
Prepared by
Raymond L. Pickholtz, Associate Piuiessor
PIBE 70-001
1970
DEPARTMENT OF ELECTRICAL ENGINEERING
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Introduction	 1
The work reported in this interim report represents a brief summary
of three of the activities being carried out at the Polytechnic Institute
of Brooklyn under NASA Grant NGR-33-006-020. The three areas are:
I. Study of Convolutional Codes
II. Coding for a Dispersive Channel
III. Recursive Signal Processing
We have obtained some new results for convolutional codes by extending
some error bounding techniques due to Forney. In particular, we have developed
a unified treatment of erasure and variable list decoding, which, we hope,
will be useful in analyzing the performance.of proposed practical algorithms
for decoding using a list. The main objective of the continued research in
this area is to develop a simple and practical algorithm for decoding con-
volutional codes which does not require as much memory as the viterbi algorithm
for maximum likelihood decoding nor as much computation as the various sequen-
tial decoding schemes.
The second area attempts to salve an old problem via a different
approach then usual - i.e. reducing the effects of intersymbol interference
in digital communications by using error correcting codes rather than by
linear operators using equalizers. The main idea is to use the dispersion
in the channel to perform part of the coding operator. Standard decoding
methods are used. Under certain conditions, several orders of magnitude
improvement is possible.
The third area, on recursive signal processing, is a continuation of
work initiated over a year ago. The main emphasis is on discrete, quantized
-1-
processing of data. Several new algorithms were developed and preliminary
simulation results indicate significant improvement over approaches derived
from classical techniques when quantization effects are important.
The common thread of thest.. three areas (and others not reported here)
is the emphasis of discrete digital processing for digital communications.
Details will be produced in the final report and in expected published
papers and theses.
The following faculty and their students are being supported in this
program.
Raymond L. Pickholtz, Associate Professor
Jack K. Wolf, Professor
Richard Haddad, Associate Professor
-2-
I. Study of Convolutional Codes
We have extended Forney's method for analyzing random tree
codes, to allow investigation of a broader class of decoding algorithms.
Specifically, consideriu maximum-likelihood decoding of a terminated
tree code, Forney established the equivalence of a decoding error and
the existence of a single unmerged span over which some incorrect word
has greater likelihood than the correct one. We, then, actually loosen
this bound by calculating the probability that some incorrect word has
greater likelihood over one or more unmerged spans; the resulting bound
is found to be asymptotically equal to Forney's. The advantage we gain
is ti-.at we are now able to analyze algorithms where the event of an
error cannot be equated quite so neatly with the occurrence of a single
unmerged span with greater likelihood (viz., list decoding).
As an initial application of the method, we obtain, for tree
codes, a unified treatment of Erasure and Variable List Size decoding,
analogous to that developed, by Forney, for block codes. Referring to
Forney's papers (and using his notation), where he obtains complementary
exponents E i (R,T), i = 1,2 for block codes, we find analogous exponents
e i (r,T), i = 1,2 for tree codes, given by:
e i (r,T) = inf
µe(0,1)
E i (µn, (1-µ)T
1- µ ; i = 1,2
Further, analogous to the Feedback and List exponents, E f (R), EI(R),
respectively, that Forney obtains as limiting cases of E i (R,T), i = 1,2,
we find exponents e f (r), e,(r) given by:
Ef(^)(µr)	
-
e f( ^ ) (r) µe(n^fl)	 1-µ
as Feedback and List exponents for tree codes.
(Ex-very noisy channel)
E  (r, T)Rcomp 0
-2-
In the case of the Very-Noisy channel, we know e f (r) in closed
1
form; e l (r) is known in closed form for r < c /2, but must be calculated
by computer for higher rates.
Along the way, we have had to establish certain properties of
the exponents E i (R,T), i = 1,2 (viz., convexity in (R,T)). We have not
yet characterized the properties of e i (r,T), i = 1,2, but we can lower
bound them by relatively meaningful quantities as:
1	 E1(u'r' 
T)
e (r,T) > e (r,T) = mf
1	 — 1	
µe(0.1)	
1-µ
e (r,T) > e2(r,T) = e 1 (r,T) + T
2
For a fixed value of T. we can show that the behavior of e i (r,T) will
have the form:
I R 
comp ( T )	 C(T)
where 
RcomP 
(T), C(T) may be calculated for the given value of T (the above
result can also be seen quite clearly from Forney's sketch of the
e i (r,T) surface).
II. Coding For A Dispersive Channel 	 ►
The problem considered is that of reducing intersymbol interference caused
by time dispersive channels. This is not a new problem, and much recent work has
centered on the use of the Tapped Delay Line (TDL) equalizer. The conventional
approach is to choose the no dispersion channel as the desired channel, and then
to minimize some measure of the intersymbol interference.
The approach taken in this report recognizes the encoding properties of
time dispersive channels. These channels process the transmitted data in much
the same way as the generator of a cyclic algebraic code. Two methods of attack
are taken. In the first method the code generator coefficients are used as the
desired response for an otherwise conventional TDL equalizer. This method is
termed the Coded Equalizer method. Transmission of k q-ary symbols through the
channel in cascade with the coded equalizer results in a code word. Thus, error
correction can be obtained with an ordinary algebraic decoder, and without trans-
mission of parity symbols. In the second method the channel encoding is accepted
without further processing by a TDL equalizer, and is subsequently decoded by a
channel decoding matrix. This matrix is designed to minimize the additive noise
variance subject to the constraint that intersymbol interference be eliminated.
A modification of this method allows trading computation time for a limited amount
of intersymbol interference.
Upper founds on the probability of error are derived for both methods.
These bounds are in the form of easily calculated error functions. The parameters
of code block length, number of information symbols per block, and alphabet size
appear explicitly.
Computer simulations confirm the derived bounds and show that under certain
conditions orders of magnitude improvement in error rate can be obtained.
III. Recursive Signal Processing
Two different areas in optimal filtering of discrete-time data are under
investigation. The first study deals with optimal state estimation based on noisy,
quantized data, while the second considers bias effects in mismatched minimum
variance polynomial filters.
(I) Nonlinear Recursive Estimation with Quantized Data.
The problem considered here is the optimal estimation of the state of a
dynamic system based on qua,itized, noisy measurements. The results obtained are
particularly applicable when the granularity is coarse and the usual quantization
model (additive, independent, uniformly distributed noise) is inadequate.
The state of the system is generated by the vector difference equation
x(k) = f[x(k - 1), k] + G(k - 1) u(k - 1), k > 1 	 (1)
and the noisy measurements before quantization are described by
y ( j ) = h[ x ( j ), j ] + v(j)
	 (2)
Upon quantization, this measurement is degraded, and the information at the quantizer
output (upon which the state estimate is to be conditioned) is that
a (j), < y(j) !:-1. b(j);i = 	 1, 2, . . . , k
	
(3)
In Eqs. (1), (2), (3) u(k) and v(j) are sequences of mutually independent gaussian,
vector random variables with zero mean and covariances
E(u(i) u'(j)}	 = 4(i) b ij ; E(v(j) v'(i)}	 = R(i) bij
We also assume that p[x(o)], the initial state probability density is known;
that each component of f(.,.) has continuous derivatives, and that p(x(o)) and
h(.,.) have continuous second partials.
A direct approach to the determination of approximate algorithms for the
recursive calculation of the mean and mode of x(k) conditioned on the quantized
data is taken. An exact equation for the conditional mode estimate is derived
which requires the solution of a nonlinear two-point boundary value problem.
The latter is linearized to give an approximate recursive state estimation
algorithm of the predictor-corrector form
x(i) = x(i) - P(i) hx [x(i)] d[x(i), R(i)] 	 (4)
where	 x(i) = f(x(i-1)) represents the predicted MLE
and
	
P(i) = (1-P(i/i-1) L[x(i)]) -1 P(i/i-1)	 (5)
P(i/i-1) = fc[x(i)] P(i-1) fx'[x(i)] + A(i-1)
In Eq. (5), hx, fx represent matrices of partials of h, and f respectively,
A(.) = G(.) Q(.) G'(.), P(o) is related to the initial state density; d is
calculated from the known sequence of gaussian density functions, and L[x(i)]
is the matrix obtained from the first partial of the product hx(.) d(.,.)
evaluated at x(i).
An approximate conditional mean estimate is derived upon making the
simplifying assumption that at each step the a priori density of x(i) given
the priori measurements is gaussian. This approximate estimator also has a
predictor-corrector form
N	 /y
x(i) = x(i/i-1) + W(i) d[x(i/i-1), Py(i)]
where	 b(i)
j[y(i)-h[x(i/i-1)]g(y(i)-h(x(i/i-I),Py(i)]Idy(i)
d[x(i/i-1),Py(i)] = Py-1(i)	 b(i)
9(y(i)-h[x(i/i-1)], Py(i))dy(i)
a(i)
and
Py(i) = hx[x(i/i-1)] W(i) hx'[	 + R(i)
W(i) = P(i/i-1)
and h jxx is the matrix formed from the second partials of h j (.) and where g(.,.)
represents a gaussian density.
Monte Carlo simulation of these algorithms demonstrated the superiority
of these over the Kalman filter in which the quantization error is approximated
by additive measurement noise, at least for the examples considered (first-order
and a second-order system) whereinthe granularity is coarse. The improvement
in the RMS error is by a factor of 10 for one example.
II Bias Effects in Mismatched Minimum Variance Polynomial Filters
A steady-state dynamic error, or bias, results whenever the model of the
signal process (and the filter designed on the basis of that model) is an inadequate
representation of the actual filter input. Suppose a finite-memory polynomial
filter is -orrectly designed to smooth polynomial inputs of degree M. Bias errors
result when this filter is excited by non-polynomial inputs, or by polynomial
inputs of degree L > M. Our study is devoted to the latter aspect, which we view
as resulting from the fact that the state vector of the input is (L+1) dimensional
whereas the filter state is only (M+1) dimensional.
Under matched conditions, the filter algorithm for a p-unit prediction
interval is
N
J(n + p(n)) = H'(p) f
where f' _ [f(n) f(n-1) ... f(n- )] is the noisy data sequence, and t(n) is the
state vector of the input, and ^(.) its estimate, and
H ( p ) = H( o ) i(p)
v'here	 is the state transition matrix, and
H(o) is the product of known matrices of the form
H(o) = P1 C-2
By partitioning the state vector, and all the associated matrices according
Fo
E
•m
Fm+l
^C(t)
^Ll(t)
F2(t)
to
LN _ I
we can determine the bias according to
b l (n + p/n) =
	 (p) E2 (n)
where N2 (t) represents the bottom part of the partitioned state vector
vm+1 x(t)/(rf+l): 1
N2 (t) _
	
	
where x(t) is the noise free input.
p x(t)/L:
: 11	 112
and B(p) _ [ill (p)	 + i12 (p) x223, where J(p) =	 is the transi-
*21	
*221tion matrix
r 22 is a submatrix of r,
a known matrix
and	
= (-1)m+111	
5 21 +	 22 E
21 '111 ' Jl, all known matrices
and
	
b l (n+p/n) = E{F l (n+p) - F n(r+p/n)} is the bias error in prediction.
These results are g-zncralizations of specific work found in texts by
Morrison and Blackman. Specific cases can be evaluated to minimize or keep
within tolerable bounds, the bias by suitable choices of p, the prediction interval,
and N the filter memory. Usually p = - N/2 is a good choice; but this represents
a smoothing estimate as distinguished from filtering (present-time esLimating)
or prediction.


Digital communications techniques Interim report, 15 Sep. 1969 - 15 Feb. 1970

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