A table look-up algorithm for estimating underlying signal and noise parameters from quantized observables is examined. A general mathematical model is developed, and a look-up table designed specifically for estimating parameters from four-bit quantized data is described. Estimator performance is evaluated both analytically and by means of numerical simulation, and an example is provided to illustrate the use of the look-up table for estimating signal-to-noise ratios commonly encountered in Voyager-type data

Rodemich, E. R.

Vilnrotter, V. A.

English

NASA Technical Reports Server

N87- 15341 
TDA Progress Report 42-87 July-September 1986 
Table Look-Up Estimation of Signal and Noise Parameters 
From Quantized Observables 
V. A. Vilnrotter and E. R. Rodemich 
Communications Systems Research Section 
In this article we examine a table look-up algorithm for estimating underlying signal 
and noise parameters from quantized observables. A general mathematical model is 
developed, and a look-up table designed specifically for estimating parameters from 
four-bit quantized data is described. Estimator performance is evaluated both analytically 
and by means of numerical simulation, and an example is provided to illustrate the use of 
the look-up table for estimating signal-to-noise ratios commonly encountered in Voyager- 
type data. 
1. Introduction 
In this article, we consider the problem of estimating 
signal and noise parameters from quantized samples of an 
observed waveform, by means of a table look-up algorithm. 
The waveform consists of binary antipodal signal plus addi- 
tive Gaussian noise. The justification for the look-up table 
approach hinges on the observation that if the parameters of 
interest are single-valued functions of some computable 
quantities associated with the samples, then it should be 
possible to construct a table whose entries at the appropriate 
coordinates are the desired parameter estimates. Indeed, we 
shall show that for the class of problems under consideration, 
the first two absolute moments of the quantized samples can 
serve as entry coordinates to a two-dimensional look-up table. 
Thus, the problem reduces to  that of finding accurate esti- 
mates for the first two absolute moments. Since accurate 
estimates of the required absolute moments can usually be 
found (as long as a large number of independent samples are 
available), the table look-up algorithm may often be employed 
to obtain quick and accurate parameter estimates. 
The above approach was originally suggested to the authors 
by W. J. Hurd of the Communications Systems Research 
Section and was used in non-real-time symbol-stream com- 
bining of Voyager telemetry data (Ref. 1). In the following 
sections, suitable mathematical models are developed and 
employed to evaluate estimator performance. An example is 
provided to illustrate the use of the look-up table for estimat- 
ing signal-to-noise ratios in data streams typical of actual 
Voyager data. 
II. Estimator Model 
The table look-up algorithm may be used to estimate signal 
amplitude and noise standard deviation from a sequence of 
random variables obtained by synchronously integrating the 
received noise-corrupted waveform over the duration of each 
binary symbol. Thus,if the received waveform is expressed as 
r ( t )  = D ( t )  (A/T)  -+ n( t )  (1) 
104 
https://ntrs.nasa.gov/search.jsp?R=19870005908 2020-03-20T12:23:26+00:00Z
(where A is the integrated symbol amplitude and n(r)  is 
additive Gaussian noise) then synchronous T-second integra- 
tion over the ith symbol interval yields 
ri =si= r(r) dt = D/ + ni ( 2 )  
(i-l)T 
where Di is the integral of the antipodal modulation D(r) 
(hence Di takes on the value *1 with probability Pl and 
P-l, respectively) and the noise samples ni are assumed to be 
independent, zero-mean Gaussian random variables with 
variance u2. The probability density of each sample can be 
found by averaging the conditional densities (conditioned 
on the data) over the a priori statistics of Di. Assuming 
stationary statistics we can suppress the subscript "i" and 
write 
(3) 
Consider the case where the time samples are subjected to 
L-bit quantization prior to processing. In particular, let the 
quantized samples 4 take on integer values in the range 
(4) Lp-0  Q q < 2 ( W  - 1 
and let the probability that Q takes on the integer value j be 
denoted pj for any i: 
In typical applications the a priori probabilities of the data 
symbols are equal (P, = P - l ) ,  in which case the quantization 
probabilities may be expressed as 
where 
and 
It is desired to obtain estimates of the parameters A and 
u (denoted A" and ;> from the sample absolute mean 8, and 
sample mean-square e2 which are defined as 
N 
1 
82 = c4; 
where N is the number of samples observed. Since the quan- 
tized observables are random variables, the absolute moment 
vector 8 4 (el ,  d 2 )  is a random vector whose statistics depend 
on the number of quantization intervals and on the total 
number of samples as well as on the signal and noise param- 
eters embedded jn the received waveform. 
Under the assumption that a large number of independent 
samples are used, the joint distribution of 8 can be determined 
by means of the multivariate central limit theorem (Refs. 2 ,  
3). Defining the components of the auxiliary random vector 
x as 
and forming the sum 
N 1 z =- xi 
fl i= l  
(9) 
it f~!!ows that as N grows without bound, ?he distribution 
function of z approaches that of a jointly Gaussian random 
vector with covariance matrix 
Az = E [zTz]  = E [ x T x ]  = Ax (10) 
105 
where, for any i, the components of Ax are 
Here, as well as in equations (8), the overbar denotes expec- 
tation. The kth absolute moment is defined as 
i 
Since the absolute moment vector can be expressed as 
1 6 = - z t T  
The central limit theorem implies that 
N, 8 may be modeled as a Gaussian 
I mean value vector 8 and covariance matrix 
0 = [max (o., 1.3921 1 - 0.0519 e, - (</e,))] ‘1’ 
in order to improve resolution over certain regions of interest. 
A graphical representation of the resulting look-up table is 
shown in Fig. 1, illustrating the functional dependence of the 
parameter estimates on the transformed coordinates (e,, 0). 
Note that some regions in the (e,, 0) plane fall outside the 
range of the look-up table. In general, points within these 
external regions may be assigned special values (i.e. negative 
values) to distinguish them from valid parameter estimates. 
The user may then invoke a different estimator, or take other 
appropriate action, when one of these special values is encoun- 
tered. In our case, the estimates depend on whether 0 is less 
than or greater than 0.3: 
for sufficiently large 
random vector with 
Given measured values of 8, and 0 falling within the range 
of the look-up table, the table entry coordinates are the 
integers I d ,  and I O  most nearly satisfying 
Observe that as N approaches infinity, each component of 
A* approaches zero, implying that for any choice of input 
parameters and quantization levels 8 approaches the associated 
mean value vector 8. It is this property of the absolute moment 
vector that originally motivated the construction of a table 
look-up algorithm for estimating signal and noise parameters. 
A look-up table may be constructed in the following man- 
ner. Given L ,  the mean value vector 8 may be determined for 
any choice of (A,o), using equation (12), and a dense matrix 
generated for each component of over the desired range of 
A and u. Numerical interpolation then enables one to compute 
estimates of A and u corresponding to a uniformly spaced 
grid over (e,, g,). 
In the current application, a look-up table was constructed 
for obtaining the estimates (A^,;) from four-bit quantized 
samples (t = 4),over the range 0.5 Q(A,o )  G5.5. It was useful 
to introduce the predistortion transformation (e,, e,) + 
(e,, 0) where 
106 
0 = 0.005 (I0 - 1) (1 7b) 
with 1 < I O  < 95 and 1 < IO, < 81. All table entries for A 
and (I were converted to integers in the range 0 to 255. Desig- 
nating the integer table entries for A and u as IA and Io ,  
respectively, final estimates are obtained using 
Since the table entries are integers, resolution is limited to  
0.02 for both estimates. Examples of the effects of this 
“granularity” will be provided in the next section. 
111. Estimator Performance 
The performance of the table look-up estimator has been 
evaluated both analytically and by means of numerical simula- 
tion. The details of the analysis are presented in the Appendix. 
We shall consider the analytical results first. 
A. Pedormance Analysis 
It was shown previously that the absolute sample moment 
vector can be decomposed into the sum of a deterministic 
and a zero-mean Gaussian component, provided the total 
number of observed samples is sufficiently great. For conve- 
nience, represent the sample moment components as 
where C is the Jacobian nf the forwsrd transfnrmation: 
An explicit cafculation of the components of the Jacobian is 
performed in the Appendix, assuming four-bit quantization. 
With C-' the inverse of C, the estimation error vector becomes 
where the mean values are defined in Eq. (13b), and (al , a,) 
are small Gaussian random variables of mean zero and covari- 
ance matrix 12, = 4 (see Eq. [14]). The estimates (2,;) are 
obtained by inverting the formulas for (el, 6,) in terms of 
A and u. Assuming that the inverse equations can be linear- 
ized over small enough regions in the (el, 0,) plane, the 
estimates may also be approximated as the sum of deter- 
ministic and zero-mean Gaussian components 
2 A + E 1  (20a) 
Therefore, el and E, represent random estimation errors 
resulting from the effective additive noise components 
(a1, a,). Due to the Gaussian assumption, the error vector is 
completely characterized (in the statistical sense) by its 
covariance matrix Ae, with components 
I ~ I  order io evaiuate this matrix, observe that under the iinear- 
izing approximation (which is valid for large N) the transpose 
of the effective noise vector can be written as 
while its covariance matrix may be represented as 
This covariance matrix can be evaluated for any choice of A ,  
a, L and N using equations (1 l), (14) and (A-5). The error 
covariance matrix was determined at the four internal points 
designated in Fig. 1, namely at coordinates (2,2), (3,2), (4,2) 
and (3,3) in the ( A ,  u) domain. The results are displayed in 
Table 1. The expectations in Table 1 refer either to ensemble 
averages (obtained from the linearized analysis) or to  sample 
averages (obtained from the simulation described in 
Section 1II.B). 
B. Performance Simulation 
The performance of the estimator can also be evaluated by 
means of numerical simulation. This can be accomplished by 
generating random sequences with the appropriate statistics, 
performing the quantization operation, taking the magnitude 
and square of each sample, and adding up the desired number 
of terms to obtain a simulated sample absolute moment 
vector. Next, the look-up table is entered at the coordinates 
specified by the components of the simulated vector, and 
estimates of the desired parameters are obtained. The appro- 
priate sample statistics may be determined by repeating the 
above procedure a large number of times, using independent 
random sequences each time. However? if each random se- 
quence consists of a large number of terms, then the multi- 
variate central limit theorem may be invoked to generate 
samples of the absolute moment vector directly with the 
proper statistics. This latter approach was adopted in the cur- 
rent application. 
107 
The purpose of the simulation is to  characterize estimator 
performance at a given value of signal level A ,  noise standard 
deviation u, quantization level L and sample size N .  cnce  A 
and u are specified, one computes 5, = 141, g2 = 42 and 
simulates the noise vector a by means of the central limit 
theorem approximation. The simulated value of a, denoted 3, 
can be represented as a zero-mean, unit variance Gaussian 
random vector u (with independent components) scaled by a 
matrix I‘ as 
z = ur 
where 
It is readily verified that 
EGTG = A& = 
hence, in the limit of large N ,  the simulated approximation Z 
has the same statistics as the actual n2ise vector a. The simu- 
lated sample absolute moment vector 6 is obtained using 
Next, the look-up table is entered at the coordinates specified 
by g, and the estimates (2, 3 recorded. This procedure is 
repeated K times, after which sample statistics are computed 
from the entries of the resulting (2 X K )  array. The value 
K = IO4 was chosen for the simulation. The distribution of 
the parameter estimates are displayed in Fig. 2 for typical 
simulation runs, using N = lo3 in each case. These “scatter 
diagrams” serve to  reveal the structure of the look-up table 
grid, illustrate noise-induced scattering, and indicate correla- 
tion in the estimates near the input coordinates. However, it 
is not possible to form an accurate mental picture of either 
the mean or the “spread” of the resulting sample distributions, 
since the number of times each visible grid point occurred is 
not evident. Therefore, sample statistics were computed at 
the four designated coordinates to provide this additional 
information. The simulation statistics are displayed in Table 1, 
directly under the analytical results. When computing these 
sample statistics, points that were out of range were ignored, 
in order to remove dependence on the choice of the ad hoc 
estimator employed. Most simulation runs did not register 
out of range points, but even when such points occurred, 
only a few were recorded per simulation run. 
Analysis and simulation results were found to be in good 
agreement. Errors in the sample means were negligible, indi- 
cating that the estimates are unbiased. Discrepancies in covari- 
ance are attributed to  deviation from the linear model. Indeed, 
by letting N = lo4 at the point (3,3), the agreement between 
simulation and analysis improved, as shown in the last two 
rows of Table 1. 
C. An Application 
Finally, the look-up table was employed to estimate symbol 
signal-to-noise ratio (SNR) for Voyager data typical of that 
used in the symbol stream combiner. Here symbol SNR is 
defined as 
SNR = A 2 /2  u2 (30) 
while its estimate is obtained directly from A^  and g a s  
A 
SNR = 2 * / 2  z2 
The estimator bias B,  defined as 
A 
B = E(SNR) - SNR (32) 
is a measure of the difference between the mean of the esti- 
mate and the actual SNR. For an unbiased estimator, B = 0. 
Normalizing by the true SNR, one may define the ratio 
A B 
R = E(SNR)/SNR = 1 + - SNR (33) 
which, when expressed in decibels, becomes 
A 
R (dB) = 10 loglo [E(SNR)ISNR] (34) 
(Thus, if R (dB) = 3,  E(SNR) = 2 SNR,whereas ifR (dB) = -3, 
E(SNR) = 0.5 SNR). In addition, if the bias is small compared 
to the true SNR, then Eq. (34) reduces to 
R (dB) 2: 10 logIo(e)[B/SNR] (35) 
providing a convenient measure of the fractional bias. 
The quantity R (dB) was determined by simulation for true 
SNRs ranging from -1 to  + I  dB, and various signal levels char- 
acteristic of Voyager data. (In the simulation, expectation is 
approximated by the sample mean.) The number of samples 
used for determining each estimate w a s N =  5700 in agreement 
with the value used for the near-real-time symbol-stream com- 
biner. The results, shown-in Fig. 3, confirm that SNRestimates 
are virtually unbiased over the indicated range. The sample 
standard deviation of estimation error was also computed for 
108 
each signal level, and found to be nearly constant, approxi- 
mately equal to  0.018 over the same range. These results indi- 
cate that the table look-up estimator examined in this article 
can be used to  obtain unbiased estimates of symbol SNR over 
the range of interest for deep-space reception. 
IV. Summary and Conclusions 
A look-up table was constructed for the purpose of esti- 
mating signal and noise parameters from quantized samples 
of a noise-corrupted antipodal signal. Since the table is entered 
at coordinates derived from the first two sample absolute 
moments of the quantized observables, substantial errors in 
moment estimates lead directly to errors in the corresponding 
parameter estimates. Hence, the look-up table performs best 
when a large number of independent quantized samples are 
available to allow accurate determination of the required 
sample statistics. It should be observed, however, that perfect 
moment estimates do not necessarily yield perfect parameter 
estimates, due to  the limited resolution inherent in any finite 
element table. The parameter estimates were found to be 
unbiased over the central region of the table, with random 
components that depend both on the sample size and on the 
spectral level of the additive noise process. Thus, the covari- 
ance matrix of the estimation error is a strong function of the 
coordinates at which the table is entered. It was also demon- 
strated that unbiased estimates of symbol SNR can be obtained, 
over a range of values and sample sizes characteristics of those 
encountered in processing actual Voyager data. 
References 
1. Hurd, W. J., Rabkin, J., Russell, M. D., Siev, B., Cooper, H. W., Anderson, T. O., 
and Winter, P. U., “Antenna Arraying of Voyager Telemetry Signals by Symbol 
Stream Combining,” TDA Progress Report 42-86, Jet Propulsion Laboratory, 
Pasadena, Calif., pp. 131-142, Aug. 15,1986. 
2. Wozencraft, J .  M., and Jacobs, I. M., Rrinciples of Communication Engineering, 
Chapt. 3., John Wiley & Sons, New York, 1965. 
3. Wilks, S. S . ,  Mathematical Statistics, Chapt. 9,  John Wiley & Sons, New York, 1962. 
. 
109 
Table 1. Comparison of analysis and simulation results 
E& E(?) E ( Z @  E&-% (e-5) E(G-F)Z 
N = 103 
Analysis 2.0 2.0 1.258 X -8.370 X 1.089 X 
Simulation 1.988 2.008 1.578 X -9.995 X 1.206 X 
Analysis 3.0 2.0 5.242 X -1.390 X 4.783 X 
Simulation 2.998 2.002 5.606 x -1.428 x 5.159 X 
Analysis 4.0 2.0 4.53s x 1 0 - ~  -1.640 x 1 0 - ~  3.562 x 10-3 
Simulation 4.001 1.997 5.640X 7.2 x 10-5 3.572~ 10-3 
Analysis 3.0 3.0 2.889 x -2.379 X 3.640 X lo-* 
Simulation 2.967 3.024 4.344 x -3.429 x 4.375 X 
N = 104 
Analysis 3.0 3.0 2.889 X -2.379 X 3.640 X 
Simulation 2.996 3.002 3.651 X -2.623 x 3.509 X 
2 4 
4 
Fig. 1. Look-up table repmmtation 
111 
<b 
<b 
3.0 2,1 .0 . .. . (2.2) 
1.5 
1 .o 
1 .o 1.5 2.0 2.5 3.0 
.:. - 0  . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . ...... . . ... 
. .  
. .  
0 .  
. . ' .e*- 
** :. *. .. 
. *  
8 .  I f 
(3.3) 
2 . 0 ~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  
2.0 2.5 3.0 3.5 4.0 
A A 
A A 
Fig. 2. Scatter diagrams at coordinates (A, U) 
-0.04  
-1.0 -0.5 0 
SNR, dB 
0.5 1 
Fig. 3. The quantity R (de) as a function of the true SNR 
112 
Appendix 
In this appendix, formulas for the components of the 
Jacobian matrix C are derived, assuming four-bit quantization 
(L = 4). I t  is convenient to write the Ilth absolute moment of 
the quantized observable q as 
where 
‘ j + l  
“1 
f i=  I e-u2/2 d u l a  (A-2a) 
ai = (i+A)/a - 7 G j G 7  (A-2b) 
and a8 = + -, a4 = -00. Letting 
-a212 
, / = e l  , - 7 G j G 7  
e8 - e4 = 0 
(A-3) - 
it follows that 
Therefore, the components of the Jacobian C can be computed 
using 
(A-Sa) 
(A-Sb) 
Having determined the components of C, C-’ can be com- 
puted and used in Eq. (25) to  evaluate the covariance matrix 
of the estimation error. 
113 


Table look-up estimation of signal and noise parameters from quantized observables

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