MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LINCOLN LABORATORY
DOPPLER MEAN VELOCITY ESTIMATION:
SMALL SAMPLE ANALYSIS AND A NEW ESTIMATOR
E. S. CHORNOBOY
TECHNICAL REPORT 942
25 MARCH 1992
LEXINGTON
MASSACHUSETTS
TABLE
OF CONTENTS
. ..
111
Abstrwt
List of ~ustrations
List of Tables
vii
xi
1. INTRODUCTION
1
2.
3
BACKGROUND
2.1
2.2
.
3.
ESTIMATOR
3.1
3.2
3.3
4.
5.
6.
ML and BAYES Formulations
Time-DomainProcessing:
Pulse Ptir(PP)
Frequency Domain Processing: Wind Profiling (WP)
13
15
20
WITH ARBITRARY oANDq
23
23
32
37
Sensitivity tOlncOrrectrf
Sensitivity to Incorrecto
Summary
WITH ADAPTIVE oANDq
39
Constrairred Inverse Filter(r)
Selection
Suboptimd Estimation ofrrandq
Decision Rules anda Partition for W
Adaptive Estimator Performance
Summary
39
46
47
49
51
PERFORMANCE
6.1
6.2
6.3
6.4
6.5
7
11
11
13
WITH KNOWN oANDq
Zero Mean Velocity
Nonzero Mean Velocity
Summary
PERFORMANCE
5.1
5.2
5.3
7
EQUATIONS
PERFORMANCE
4.1
4.2
4.3
3
4
Statistical Model
Experimental Design
59
7. CONCLUSIONS
v
LIST
OF ILLUSTRATIONS
F~ure
No.
Page
10
1
ML matched filter windows: u and q dependence.
2
Optimal velocity estimation standard error for zer-mean
Doppler weather.
14
3
Optimal performance: standard error and him results for nmrzero velocities.
(a) v = 5 m/s, UW= 0.038 VN,q.
16
Optimal performance: standard error and bim results for nonzero velocities.
(b) V = 13 m/s, UW= 0.038 UN,,.
17
Optimal performance: standard error and bias results for nonzem velocities.
(c) v = 5 m/s, Uw = 0.192 VN,~.
18
Optimal performance: standard error and him results for nonzero velocities.
(d) V = 13 m/s, UW= 0.lg2 UNVg.
19
ML algorithm performance sensitivity: effect of sigrrd-t-noice
(a) T = 5 m/s, UW= 0.038 UNYq.
parameter q.
24
ML algorithm performance sensitivity: effect of signal-t-noise
(b) U = 13 m/s, UW= 0.038 VNVQ.
parameter q.
ML algorithm performance sensitivity: effect of signd-t~noise
(c) v = 5 m/s, aw = 0.192 UN,,.
parameter q.
ML algorithm performance sensitivity: effect of signd-t~noise
(d) V = 13 m/s, UW= 0.lg2 VN,,.
parameter q.
3
3
3
4
4
4
4
5
5
5
5
6
6
25
26
27
Bayes algorithm performance sensitivity: effect of signal-to-noise parameter
V. (a) V = 5 m/s, UW= 0.038 VNVQ.
28
Bayes algorithm performance sensitivity: effect of signal-t-noise
q. (b) V = 13 m/s, UW= 0.038 VNvg.
parameter
29
Bayes algorithm performance sensitivity: effect of signd-t~nrrise
q. (c) v = 5 m/s, Ow = 0.192 VNv~.
parameter
Bayes algorithm performance sensitivity: effect of signal-t-noise
q. (d) V = 13 m/S, ow = 0.lg2 VN~g.
parameter
30
31
ML algorithm performance sensitivity: efiect of spectrum width parameter
u. (a) 5 = 5 m/s, uw = 0.038 VNUq.(h) 3 = 13 m/s, o~ = 0.038 VNvg.
33
ML algorithm performance sensitivity: effect of spectrum width parameter
C. (c) ~ = 5 m/S, OW= 0.lg2 UNv~.(d) T = 13 m/S, Cw = 0.lg2 VNVg.
34
vii
LET
OF ILLUSTRATIONS
(Continued)
Figure
Page
No,
Bayes algorithm performance sensitivity: effect of spectrum width parameter
u. (a) 3 = 5 m/s, UW= 0.038 VNvi. (b) V = 13 m/s, u~ = 0.038 WNvq.
35
Bayes algorithm performance sensitivity: effect of spectrum width parameter
u. (c) 6 = 5 m/S, UW= 0.lg2 UNvq. (d) ~ = 13 m/S, OW= 0.lg2 UNvq.
36
Bayes algorithm performance: suboptimrd u and v using best sin~e.weighting
mfficient set. (a) U = 5 m/s, OW= 0.038 WNvg.
41
Bay= algorithm performance suboptimd o and q using best sin~e.weighting
coefficient set. (b) V = 13 m/s, u“ = 0.038 VNV$.
42
Bayes algorithm performance: suboptimd u and q using best sin~e.weighting
coefficient set. (c) U = 5 m/s, uw = 0.192 u~vq.
43
Bayes algorithm performance: suboptimd u and q using best sin~e-weighting
coefficient set, (d) U = 13 m/s, UW= 0.lg2 VNvg.
44
9
Divergence minimization for thrw simple partitions.
45
10
A partition design using simple decision rules.
47
11
K-Compartment
49
12
Optimum putition
13
Set point convex hull with incre=ing
14
Bayes algorithm with adaptive u and T 15 compartment
cient set (I). (a) V = 5 m/s, UW= 0.038 VNV$.
weighting coeffi-
Bayes ~gorithm with adaptive o and T 15 compartment
cient set (I). (b) V = 13 m/s, u~ = 0.038 UNYQ.
weighting coeffi-
Bayes algorithm with adaptive u and v 15 compartment
cient set (1). (c) V = 5 m/s, aw = 0.192 ONvg.
weighting coeffi.
Bayes algorithm with adaptive u and q: 15 compartment
cient set (I). (d) T = 13 m/S, ow = 0.lg2 VNvg.
weighting coeffi-
Bayes algorithm with adaptive u and T 15 compartment
cient set (II). (a) SNR = -5 dB. (b) SNR = O dB.
weighting cwffi-
7
7
8
8
8
8
14
14
14
15
summed divergence.
51
ad’ set point locations.
52
K.
53
54
55
56
57
“
LIST
OF ILLUSTRATIONS
(Continued)
F~ure
No.
15
Page
Bayes algorithm with daptive u md T 15 compartment
tient set (II). (c) SNR = 10 dB. (d) SNR = 20 dB.
ix
weighting cwffi58
LIST
OF TABLES
~ble
No.
Page
1
Model Sample Correlation vs Spectral Width
2
Optimal Thresholds ad
Set Point Values (K = 15)
.
xi
5
50
1.
~TRODUCTION
Mvdocity(i.e., firat spectral moment) estimation ismntrd
topulwd-Doppler
weatherradar pro=sing.
However, some radars, such as the Termirrd Doppler Weather Radar (TDWR)
and Airport Survdance
Wdar (ASR), operate under conditions that test the fimits of current
upablhties.
For example, high scan rate and fine (azimuth) samptirrg requirements yield a smd
number of prdse samples per gatel (i.e., small-sample size) while clear-air conditions result in low
signd-t~ noise levds. Hence, coupled with general interest for optimal velocity estimation, specific
needs motivate an investigation of improved performance under the constraints of small-sample size
and low signal strength.
As is often the case, the issue of practicability Umits the degree to which optimal performana can be attained. hrther comphcating matters in this report is an assumed smd-sample
constraint that makes the definition of optimdity itself problematic. Indeed, the natural appeal
to the (asymptotic) optimfllty of maximum likehhood (ML) could fail; verification is clearly required. The Cram&r-Rao (CR) performance bound could be too optimistic, hkely unattainable by
any wtimator and therefore useless in gauging what room there is for improvement. Because few
thenreticrd results exist to serve M guides, numerical methods typically must be re~ed upon for the
relevant anrdysis.
This report presents a study of the velocity estimation problem and provides arguments for
frequenc~domtin
smoothing (autocorrelation-lag weighting) as a means of achieving improved velocity estimation performance. Although the emphaais is on smdl-sarnple anrdysis, the results are
not restricted to the small-sample caze. Note that the Bayes (ccmditiond mean) estimator (defined
by the usual formulation) minimizes the mean-squared estimation error over W estimators. This
property holds regardlas of sample size and hence ratimrtizes the use of Bayes performance w a
benchmark for optimcdity in this smrdl.sample case. Unfortunately, even assuming the conventional
Gaussian signal model, the conditional mean is computatimrdly impractical. It nevertheless prvidw useful insight to the development of improved (and practicable) frequency-domain velocity
atimators.
At the heart of this study is a Monte Carlo analysis examining smaH-sample performance for
selected mean-frequency estimators. In addition to the pulse pair (PP) estimator and a typical
Fast Fourier Trasform (FFT) estimator fi.e., a periodogram based estimator derived from windprofiler (WP) literature], maximum Ukefihood (ML) and Bayes estimators (these two baaed cm a
Gaussian sigrrd model) are examined and compared. Sample size is fixed to a small (M = 20)
value, and performance at low signal-t-noise ratios (SNR) is dso a point of focus. The results of
]For TDWR, on the order of 40 pulse samples are coherently processed per output velocity estimate;
for ASR-9, cm the order of 18 (dthcmgh data sharing has been used to extend this value to 27).
these andym are used to recommend an optimal smoothing (weighting) strategy for periodogram
(autowrrdation-lag)
based estimation.
As stated, the motivating
and the content of ttis report
radts of Section 6 should be
dedng with the fundamental
Utited sample size.
iuterest is improved velocity estimation for Doppler weather radars,
is clearly oriented toward weather-radar processing. However, the
of a more general interest, presenting a new and novel method for
problem of nuisance parameter removal under the constraint of a
2
2.
2.1
Statistical
BACKGROUND
Model
For this report, it is sssumed that a frequently adopted Gaussian model (for sin~e-source
weather returns in additive noise) is appropriate (see, for example, Doviab and Zrnit [1]). This
modd is tiarwterized
by the sample covariance relation
~m
=
~e-S(fi..mr/A)2 e-~4”cm’1~ + N6m,
(1)
where the modcd parameters w and o. represent mean Doppler velocity and spectrum width, A
is the radar wavelength, and S and N respectively represent signal and noise power magnitudes.
Complex-valued rtiar returns, ZT = [zOzl . .. ZM-1].2 cOrrespOnding tO a singe range cell me
assumed to represent M equally-spaced samples (separated in time by pulse separation r) of a
mmplex Gaussian process with covariance matrix R = E[ZZt]. In view of Equation (1), R can be
given the parametric representation
R = D [SG tN~D*
= SDGDD t NI,
(2)
where
D = D(u) = diag[l e-JW . . . e-~(M-l)W],
G=
G(u)=
:I ;
[ P(o)
p(1)
...
p(M-
p(1)
p(o)
...
p(M-2)
.. .
.
P(M - 1)
p(M -2)
. . . p(o)
~
1) 1
1’
I = diag[l 1.. .1],
p(m)
= e-*(*m”)2,
2Notationdly, a superscript “T” will be used to indicate a matrix tmnspose, a superscript “*” win
be used to indicate complex conjugate, and a superscript “t” will be used to indicate conjugate
transpose.
3
It is ~sumed that N cm be reliably estimated and treated a3 known, and for convenience
S ad N me mmbined into an unknown SNR q = S/N. The complex data vector Z is therefore
governed by the probabfity density
p(Zl@)
(3)
= z-MIR]-’e-ztR-lz,
and wtimation of the Doppler velocity u must be done in the context of the unknown parameter
vector @T = [u 0 q].
2.2
Experimental
2.2.1
Design
W=ther-Radar
Simulation
Weatherhke Doppler signals were simulated m described by Zrnit [2]. For rdl simulations,
digitd sampUng of Gaumian shaped power spectra was emulated, including Nyquist folding, until
d tail values with power greater than 0.01 were accounted for. Given an M point input power spectrum, Zrnid’s method generates a random vector of M consecutive process samples. A brief study
of sample estimator performance wa3 made to examine (and thereby ensure bmits for) uncertainties
due to power spectral sampfing and Monte Carlo sample size. As a result of this investigation, a
1024 point power spectrum representation w= used and, from the corresponding output process
vector, a block of 20 contiguous samples wa3 extracted for use w a data sample (the remaining
1004 points were discarded). Also, a total of 10,000 (independent) realizations were used for each
assessment of sample mean and standard error.
Error results are presented in both normalized and unnormdized form, In view of the definitions in ~uatimr (2), spectrum width and standard error values we normrdized to the magnitude
of the Nyquist velocity UNvg. Biw errors are normrdized to the magnitude Of the true underlying
signal velocity. Unnormdized vrdues are presented in the context of weather-radar processing using a 10.4 cm radar operating with a pulse repetition frequency of 1000 s-l. (The unambiguous
Nyquist velocity therefore equrds 26 m/s.)
Provided also, for reference, are curves corresponding to the CR bounds that were computed
using the well-known result3
fi,j
=
tr
R-’ ~
R-’ ~
{
,
(4)
}
3A derivation is provided as an appendix.
4
where F = [~iti] represents the Fisher information matrix
The diagond entries of F provide the desired hrmnds:
(5)
E[(di - Oi)z] z 7i,i,
where F-l = ~iti] md ii is arry unblmed estimator of component 0; of e.
2.2.2
Test-Signal
Parameters
Performance evaluations/comparisons
are based on estimation error that clearly carr v~y as
a function of the true parameter value. Error quantification for dl possible values of the (true)
parameter is not feasible, Evacuations are bwed, therefore, on comparisons for selected parameter
values.
Target velocities of O, 5 (0.192 v~vq), and 13 m/s (0.500 o~vq) are used in the present evduatirm. Preliminary studies obtsined similar error profiles for velocities in the range O to 0.5 UNYg.
Velocities greater than 0.5 UNv~are not examined to avoid the added complexity of spectral folding.
Values for weather spectrum width can rarrge from O to UNVq.For coherent Processing, Only
a small range is of interest. A further simphficatimr is made by considering only extreme points of
this range providing two classes of input signal: narrow ad wide spectrum width. Table 1 conttins
TABLE 1
Model Sample Correlation vs Spectral Wdth
the correlation
values for the first three lag indices in the model [Equation (2)] corresponding
5
to thrm vdu= of normrdized spectrum width. For a normalized width of 0.2, there is significant
decorrelation betw~n samplw separated by two or more places. (Improved estimation using hlgherorder lagproduct statistics wotid seem unfikely beyond this point.) The normtized value 0.2 is
Given the setting of the simulation, this sign~
used to repr=nt
wide spectrum width si~ds.
mrresponds to a weather signal with a spectrum width of 5 m/s. At the other end, the value 0.038
(mrresponding to a 1 m/s spectrum width) is representative of narrow spectrum width siqds.
Mltidy, performmce mmparisons are made for discrete SNR values in the r~ge -10 to 10 dB.
Thwe (initial) rmults are the basis for a later shortening of the examined range to -4 to 10 dB.
6
3.
3.1
ML and BAYES
ESTIMATOR
Formulations
The ML solution corresponding
mization of the “Ukehhond” function
L(e)
EQUATIONS
to Equation (3) is obtained, in principle, from joint maxi.
= - h IRI - Zt R-l Z,
(6)
but this has tirnited prmticd apphcation because the resulting system of equations is neither
exphcit nor aepmable with respect to the components of ~. The Bayes estimator that minimizes
the mean-squared emor E[(d - w)*] is the mean of the condition
(posterior) density p(@lZ) (see,
for example, Van Tres [3]) where the posterior density is derived from application of the Bayes
fommla
P(@lz)
p(z[e)p(e)
= Jp(zl@)p(@)d@.
(7)
This too has hmited pruticd
use U, notationdly, “do * = “~ da dqm; the resulting estimatOr
requires substantial numerical integration to cover the parameter space volume. Approximations
of some form ace clearly required in order to proced further. The approach considered here first
steps b~k and examines a simpler model for which pr~ticd (ML and Bayes) solution is possible
and, later, examines adaptation of the resulting solutions to the model of Equation (3).
Examirdng the (somewhat) degenerate situation wherein one aasumes the spectrum width
(u) and signaf-tenoise (q) parameters known is, of itself, very useful. Under these conditions the
aforementioned computational blocks are largely avoided, and one ~h]eves optimal estimators that
fom a foundation for further mdysis.
In the caae of ML estimation, knowing u and q reduces Ukelihood function, Equation (6), to
the term ZtR-iZ
defining the ML solution
(8)
In the above, the coefficients 7i,~ are the elements of the matrix r resulting from the convenient
redefinition
r = r(O, q) = [SG t NI]-].
(9)
~uation (8) is not expUcit for w; however, a change of variable and remrmgement
interwting frequency domtin form:
results in an
(lOa)
(lOb)
where fir) is the r-weighted autocorrelation
f(r)
.
=
M-m-1
~
estimate defined by
(11)
Z: ~i,i+~ Zi+~
i=O
From inspection of Equation (10) it is clear that a solution to this ML equation can be obttined
by implementing a standard FFT transform of the weighted autocorrelation estimates f:). The
transform smnples Equation (10) with a resolution of 2T/NFFT (NFFT is the FFT size) and thus,
by zero filfing, computes bML to a desired resolution.
A similar computational reduction occurs for the Bayes formulation.
prior4 for U, the (one-dimensiond) Bayes estimator cm be written w
If one aasumes a uniform
(12)
The above FFT computation, which yields the ML solution, dso provides the exponent in Equ&
timr (12) and therefore a starting point for computing the posterior mean.
Comment
1. If, in Equation (11),~i,i+m z 1, then f$) is equivalent tO the (biwed) sample autocorrelatimr estimate, and an FFT implementation of Equation (10) is equivalent to the
(windowed) power spectral density estimate obtained using the Bartlett window. Hence, there is
4For this report only noninformative
priors are considered, which, for w, can be stated w p(w) =
lf2rr, -% s w < T.
8
&an
eqrdvdence between the FFT computation for Equation (10) and the periodo~am
wtimate
spectral
(13)
The rescdting ML estimator would therefore be equivalent to the Periodogram Maximization estimator considered by Mahapatra and Zrnit [4]. (The apparent contr,?diction in which a minimization
of the power spectrum in Equation (10) defines the ML estimate is accounted for by the sign of the
weights ~i,i+~. )
a cl~sic smoothed periodoaam
Comment 2. If ~i,i+m s ~m, then Equation (10) implements
atimate. However, r (the inverse of a TnepHtz matrix) is only ~ymptoticaOy Twplitz (though r
is persymmetric: ~i,j = ~M_jM_i). This nevertheless does suggest an alternative frequency domain
solution and approximation based on implementation of a smoothed periodogram estimate.
S.1.1
Smoothed
Periodogram
Approximation
For this report, the weighted autocorrelation, Equation (11), is viewed ~ posing no computaticmd difficulty (the results presented here assume such an implementation). However, a Toeptitz
aPPrOfimatiOn tO r prOvides a useful heuristic within which to view the relevance of the paracrceters u and q. This, in turn, may suggest approaches to the more general case where u and q are
unknown.
b treating r aa ToepUtz, one can compute smoothing weights by tM1ng averages rdong the
diagonrds Of r:
(The minus sign is introduced here so that the ML solution of Equation (10) can be associated with
finding the maximum of the power spectral density estimate.) This maximum hkehhnod window
function, indexed by v and u, can be used to define rm approximate ML scdntion
(14)
where fm represents the rrormd (unweighed)
autocorrelation
estimate.
The rdaticmship between Tm and the parameters u md o is documented by Figure 1 where
frequency-domain smoothing windows (Fourier transforms of the coefficients Tm; interpolated for
9
1
0
I
1
1
o
4
1
SNR = -10 dB
n
0
w
MLma[chedfilier
windows: uandq
dependence.
windows obtained by Foum’er tmnsfoming the weighting
ihme values of weother SNR and a mnge o]weatherspecimm
Figuwf.
10
Frequency domoin smoothing
coeficienis Tm am plottedjor
width values.
clarity) are presented for three SNR values: .10, 0, and 10 dB, and a range of norcntized spectrum
widths. At 10 dB SNR and for low to moderate spectrum widths the smoothing windows appear
to be, przctidy
speaking, square-window averagers. This suggests the genertized view that the
spectrum width parameter u primarily affects window width. As SNR decreasw, the dominant
f-ture change is a scdng of the windows to smder magnitudw (although shape rounding and
bandwidth broadening, as measured by 3 dB width, are clearly evident). It is helpful to add
to the generdzation by stating that the windows are scaled according to the SNR parameter q.
Predictions that result from these simplifications are as follows. Clearly [from tiuation
(10)],
sdng
the window does not affect ML estimation, and hence the value of the SNR parameter
v wotid not be expected to have a strong effect on the performance of an ML implementation.
However, in the case of the Bay= implementation, scaling coupled with exponentiation implicitly
introduces a form of signal isolation. That is, by its nmdinear nature, the exponentiation in
~uation (12) enhmces (isolates) large spectral fines in the smoothed periodograrn, and scd,ng of
the window (by q) modulates this isolation by adjusting the power of expmrentiatirm. Hence, one
would anticipate that, at least, the biw of the Bayes algorithm would be affected by the v value
used.K
s.2
Time-Domain
Processing:
Pulse
Pair (PP)
The PP estimator is stadard
in weather radar apphcations, and its formulation can be
obttined from many sources. Originally proposed in the context of the independent samphng of
PP measurements (for example, see Miller et d. [5], time-domain formul~ for PP estimators have
been routinely apptied to vector me~urements such as those represented by the sample Z. For the
purpose of this report, the PP estimator is defined by the equation
&pp
M-2
= arg ~
Z; Zi+].
i=O
(15)
When M = 2, this dso defines the ML estimate, u can be seen by comparison with Equation (8).
Clearly, as u increases and the correlation between samples decreases one would dso predict Opp
to approach the performance of OML.
S.S
Prequency-Domain
Processing:
Wind
Profiling
(WP)
Among the various pubtished frequenc~domain methods is one that has been developed for
use in WP networks—a t-k with inherently low SNR vdrres (see May and Strauch [6]). The WP
‘The prewnce of a uniform noise floor in the spectral density estimate induces a b]as toward the
value zero when the mean of the density is computed.
11
wtimator of May and Strauch [6] is periodogram derived and has led to general claims of improved
Doppler mean atimation in the case of low SNR; hence, it is of interest for the present report.
The ~timator, WP, was derived with one major departure from the description given in May
and Strauch [6] (their F]rst Moment or FM algorithm). Computaticmd aspects of this algorithm
are pr-nted
bdow hy way of a comparison with the proposed (rme-dlmensiond) ML and Bayes
~orithms.
Note, however, that there is one important difference between the WP algorithm defined
here and the algorithm described in May and Strauch [6]. The WP network makes generous use of
periad~mm averaging as a means of stabihzing the periodogram estimates (Bartlett’s procednre;
see the next section). Because the emphaais of this report is on estimation using a fixed smdlsample size, no such averaging is possible. (This report does not consider the possibility of data
averaging across range gates or neighboring radids.)
Estimate
Stabilization.
At the heart of the WP algorithm is an implementation of Bartlett’s
procedure for estimating the power spectral density. That is, the data segment for anrdysis (length
M data segment corresponding to a single range ceU observation) is divided into g equrd length subsegments, each of which is used to obtain an M/q-length periodogram estimate. The g power spectral =timates are averaged to improve the stability of the power spectral estimate. The Bayes implementation [Equation (12)] dso can be viewed as having a power spectral estimator—a smoothed
periodogram-at
the heart of its procedure. Because Bartlett’s method per se is not appropriate for
the smaH sin~e.sample case (the primary interest here) it may be argned that smoothing windows
therefore are necessary to stabifize the estimates. (It should be remarked that, generdy speaking,
the gods of mean Doppler velocity estimation and power spectral density estimation are not one
and the same. Hence, general arguments for improving power spectral estimation do not necessarily
carry over to improved velocity estimation.)
Signal Isolation. After obtaining an estimate of the power spectral density, the WP method
proceeds with an ad hoc attempt to isolate signal from noise. A noise floor for the spectral estimate
is determined, and censoring (zeroing) of d spectral coefRcients beyond the first crossing of the
noise floor, when proceeding from that frequency index with maximum power, is performed. The
noise power level is dso subtracted from the remaining interval of ncmzem values, and a mean
frequency value is computed by constructing a density from the remaining spectral coefficients and
computing its mean. As previously mentioned for the Bayes implementation, the combination of
window sc~ng md expcmentiaticm can be given the heuristic interpretation of signd-from.noise
isolation.
12
4,
PERFORMANCE
WITH
KNOWN
a AND
q
This -tion
presents a bastilne analysis corresponding to a one-dimensiond parameter space
(i.e., u and q assumed known). These Monte Carlo results provide optimal performance measures
for each method. In the c~e of the Bayes implementation, results for known u and q dso provide
a ~~twt
lower bound for the standard error of estimating mean Doppler velocity.e In later
sections, the Bay- curves determined here are employed with the label “Bayes Bound” for velocity
cztimation.
4.1
Zero Mean
Velocity
Figure 2 praents the simplwt comparison: estimation of a zer-mean Doppler weather target
(&minating, for the moment, the contribution of bias7 in standard error comparisons). The figure
plots standard error vs input SNR for two ewes: a narrow input spectrum width (o = 0.038 UNuq)
and a wide input spectrum width (u = 0.192 VNv~). The corresponding CR lower bounds are
included in each panel.
4.1.1
Narrow
Spectrum
Widths
For narrow spectrum widths ML, PP, and WP estimates d exhibit
similar functional relationships with rezpect to SNR. A uniform ranking of these estimators (across
SNR) cannot be deduced from the data ML is clearly best at high SNR (7 to 10 dB) values but
the WP method appears relatively better at low SNR values (less than O dB).
ML,
PP,
and
WP.
CR Bound. For narrow spectrum widths, the CR bound is well below the error curve of
any of the above three estimators—lower by nearly a factor of two at the high SNR end. This
discrepancy between CR bound and observed performance, which is even more substantial for low
SNR values, could erroneously lead to speculation that much improved performance is possible.
Bayes Bound.
The curve for the Bayes standard error shows the CR bound to be overly
optimistic. However, the Bayes estimator clewly exhibits a performance gain for SNR values in
tbe range O to 10 dB (at 10 dB, the Bayes standard error is respectively 0.72, 0.66, and 0.52%
8Comparisons involving the ML, Bayes, and WP performance statistics must concede an error (hiss)
that results from the finite length FFT implementation. For these frequency domain estimators, the
UNyq is exhibited = a bim in the range+ l/NFFT
UNvq (and therefore
samphng resolution of 2/NF~
maps an interd about the error values reported here). All frequency domain computations in this
report were computed using an FFT length of 64.
‘For this zero mean Doppler signal, each of the three frequency domain methods was found to
exhibit an estimated bias below the resolution defined by the 64 point FFT implementation (i.e.,
< 1/32), regardless of the q value.
13
7,
I
I
I
KNOWN a ANO q
[
I
1
,-?s2
I
I
I
~ 0.269
0.154
i
0.115
0.077
0,036
I
n
~?
I
I
I
I
o
0.269
I
0
z
OW= 5 mls (0.192 v~m)
7= Omls
~6
m
CR BOUND
– 0.231
(10,OW Realizations)
5
-
0.192
4
-
0.154
-
0.115
-
0.077
3
L
2
A BAYES
0 PP
❑ ML
0.036
1
o~o
-10
+4+-2
4
02
6810
SNR (dS)
Figure 2. Optimolvelociiy estimation standard emorforzero.mean
Doppler weather.
Both panels plot esiimated standard emrforjour
methods oJDoppter velocity estimation:
ML, Bayes, WPdetived,
and PP. Theupperpanel
comsponds
ioweaiherhauing
anawow
apecimm tidth ond ihe Iowercomsponds
10 weather having a wide spectmm m’dth.
14
that of the ML, PP, and WP wtimators; at O dB, the Bayes standard error is r=pectively 0.58,
0.60, and 0.54% that of the others). For SNR values less than OdB, adequate separation of signal
from noise becomes more diffimdt and the Bayes standard error drops as its estimate bemmes
increaain~y biased toward zero. Comparisons among the estimators for SNR values below O dB
shordd therefore include hiss error, and this is included in the sections to fo~ow.
4.1.2
Wide
Spectrum
Widths
For wide spectrum widths rdso, the general observations of tbe previous section apply, but of
particular note for these input signals is the following. ML and PP performances are predictably
dew, dthougb ML is better at higher SNR vafues and appears to approach the CR bound (as SNR
increas-), whereas the PP curve appears to plateau at a level distinctly above the CR bound. From
inspection of Table 1, one may surmise that at a normtized spectrum width of 0.2, the decorrelation
between sampl~ is such that very httle improvement can be realized from using higher lag terms
in the estimation process. In confirmation, there is less difference between performance of PP and
ML and the CR and Bayes bounds; however, note that WP stands done as a clearly suboptimd
estimate. This exceptionally marked degradation in WP performance persists to high SNR values
and confirms a wide-spectrum-width weakness identified by other investigators. The ML estimator
aPPears to achieve the lower Bayes bound for SNR values in the 5 to 10 dB range, and both ML
mrd Bayes improve upon PP performance over this range (at 10 dB, the Bayes standard error is
respectively 0.98, 0.81, and 0.37% that of the ML, PP, and WP estimators). For SNR vdrres O to
5 dB, ML and PP performances are essentirdly equivalent, and both depart appreciably from the
Bayes bound M SNR approaches
dB (at OdB, the Bayes standard error is respectively 0.83,0.83,
and 0.64% that of ML, PP, and WP).
As a secondary note, one should observe that there is no conflict in the fact that the CR
bound and Bayes error curves cross (there is an impfied crossing of these two curves for the narrow
spectrum width case as we~). This only serves as a reminder that the CR bound appfies to the
performance of unbi~ed estimators, which the Bayes estimator, generally spetilng, is not.
4.2
Nonzero
Mean
Velocity
Standard error and bias results for nonzero mean velocities (V= 5 m/s and v = 13 m/s) and
narrow and wide spectrum width c~es, as above, are presented in F)gnre 3. Standard error results
are summarized in the upper hdf of emh panel, and hiss results are summarized in the lower.
4.2.1
Standard
Error
For SNR vafues greater than O dB, there is general agreement (within the resolution of the
Monte Carlo parameterizatirm) with the results of Figure 2. Below OdB, there is a notable departure
(most evident for the Bayes results) due to inclusion of bias error.
15
KNOWN o ANO q
7,
\
6 -
‘m’”l
I
I
I
I
1
I
V = 5 m/s (0.192 VNY9)
(1O,OWReabzalions)
0.269
0.231
=
g
0.192
&
o
K
A BAYES
4
0.154
❑ ML
;
~
x WP
0.115
3 -
g
~
i
0.077
2 –
;
a
g
1
CR BOUNO
1
I
0.033
1.2
I
I
I
I
I
0
0
A
BAYES
+.2
❑ ML
-1
x WP
Ow= 1 Ws (0.ow vNyq)
F = 5 tis (0.192 vNyq)
(to,~
Reahzations)
/
I
-2
-2
1
0
I
I
4
2
I
I
6
6
10
SNR (dB)
Figure 9.
(.j
Optimal performance: standard emr and
T = 5 m/i,
aw = 0.038 VNV,
16
bias msulfs for nonzem uelocifies.
,m7M
KNOWN O AND q
7 ~
““g
\k \
.6t
&
K
o
.
g4
–
u
o
a3
-
A
BAYES
=
g
❑ ML
0.192
x WP
(lo,m
a
o
z
a
+2
m
0.231
o PP
K
n
a
~
Realizations)
w
~
CR BOUNO
1
2’6~o’2
g
L
m
a
d
z
-2”
t/
,
:W = 1 Ws (0.033 VW)
v = 13 Ws (0.500 vNn)
(10,000 Realizations)
w
Optimal perjomance:
standard
3.
(b) T = 13 m/s, UW = 0.038 VNy#.
Figun
emr
17
i
and bias results for nonzem
velocities.
KNOWN o AND q
,-7.$
0.269
7
A
OW= 5 mls (0.192 VNyq)
BAYES
0.231
6
~
o
K
(10,WO Realizations)
0.192 K
n
K
e
O.lM g
a
+
fs~
K
o
=4
K
w
o
U3
<
n
z
q
+2
03
0,115 :
u
~
A
0.077 ;
CR BOUND
a
g
1
0.038
I
0
I
I
I
I
1
)
10
o%
0 -
z
z
0
u
1
$
N
i
i
q
g
Q PP
z
❑ ML
-1
x
–
+.2
g
WP
[
A
-2 ~
4
(10,000 Realizations)
44
-2
0
10
2
SNR (dB)
Figum$. Op!imalperfomance:
standordemrandbios
(c) E= 5 m/s, Ow= 0.192 VN,,.
18
msulisjor nonzeroue/ocifies.
KNOWN o AND q
?
I
h
,-7s.6
1
I
I
0,269
1
6
0.231
i
:W = 5 ds
t
\\\
K
K
u
o
a3
a
n
z
<
+2
m
i\
Vu,,.)
~“”’54
o
K4
.
(0,192 vN”q)
v= 13m/s(0.500
\\
4092
1
I
x
WP
I
0
2.6
I
A
BAYES
o
PP
I
I
I
I
I
❑ ML
x
WP
h
o –
&
A
*
A
I
g
m
~
m
-2.6<
:W = 5 m/s (0.192 VNyq)
v= 13tis(o.500vNyq)
(l0,000RealizatiOns)
-5.2
I
4
I
-2
I
I
I
I
I
0
2
4
6
8
-
1 4.4
10
SNR (dB)
Opiimalperjomance:
standordemor
3.
(d) T = 13 m/8, Uu = 0.lg2 UN,,.
Figure
19
and bias msullsfornonzew
velocities.
4.2.2
Bias
For both narrow and wide spectrum widths there is a breakdow of dl four methods w SNR
decreases bdow O dB. For SNR values greater thm O dB, the hi= of each estimator appears to
be wittin acceptable fimits and generaRy near zero. A large proportion of tbe improved (standard
error) performance of the Bayes method, for SNR <0, is at tbe cost of increaaed bias (toward zero).
GenerWy, the PP estimator appears to have the best (i.e., smdest) bi= performance. Departure
from zer~bias performance in the narrow spectrum width caae occurs eartier (i.e., at higher SNR
vdrrw) in comparison to the wide spectrum width c~e.
PP and ML resdts do not appear to change appreciably when the weather velocity is changed
from 5 to 13 m/s. Although the theoretical CR bound does not depend on weather velocity, this
is not true for the bound provided by the Bayes estimate. Performance for the Bayes and WP
mtimators, which botb compute a spectral mean, deteriorate when weather velocity is incremed to
13 m/s—a performmce loss due to spectral folding. (Interestingly, moving weather velocity from
5 to 13 m/s has its most significant bias effect in increming that of the PP estimator.)
4.9
Summary
4.3.1
Narrow
Spectrum
Widths
For narrow spectrum widths ML, PP, and WP methods d have similar performance characteristics (over the range of SNR values examined). Nevertheless, it can be argued that ML provides
better performance at higher SNR values. Clearly, the indication is that higher SNR values are
required to bring out a decisive advantage here from the ML implementation; a more extensive
Monte Carlo anrdysis (including higher SNR values) would be needed to measure the extent of
these improvements. As SNR decreases away from O dB, aU methods begin to fail although the
blm performance of PP is uniformly best. The CR bound is much lower than the performance of
d, but the Bayes results show the information bound to be overly optimistic for this small-sample
(M = 20) case. The asymptotic optimtity of ML estimation w= dso demonstrated to be of no
consequence for this smfl-sample case. The Bayes estimator demonstrates a markedly improved
performance, but for SNR values below zero, this is at lemt in part, at the expense of increaaed
bias. Bias does not appear to contribute appreciably to estimation error for SNR values above
O dB. Ml four methods, however, do exhibit notable biw for SNR values in the range -5 to O dB.
Bi= comparisons appear to always favor PP estimation.
4.3.2
Wide
Spectrum
Widths
At wider spectrum widths ML and PP are for the most part similar, but the ML results
show a slight improvement at higher SNR vafues (greater than 5 dB). Although Bayes performance
represents the optimum, ML and PP are close to its bound (compare to the narrow spectrum case);
d three estimators perform near the CR bound (again, in comparison to the narrow spectrum
20
-e).
However, the WP method clewly has an undesirable performance at wide input spectrum
widths. An aplanation for this is offered in Section 5.2.
5.
PERFORMANCE
WITH
ARBITRARY
u AND
q
The previous section indicated that a small performance improvement (as measured by etandard error) might be obttined using an ML formulation (relative to PP and at high SNR values
ordy), and that a much improved (and optimal) performance might result from a Bayes implementation. The task at hand, now, is to preserve these performance gtirrs while addressing the issue
that u and q are never known exactly.
h the remainder, the Bayes and ML algorithms will process data using approximate values
for the parmueters u and q aud, in that sense, represent suboptimd algorithms. (However, for
convenience, the labels Bay= and ML will stiU be used.) Before evaluating practical approaches to
=timating o and q it is useful to examine the effect of arbitrary u and q values cm Bayes and ML
performance. h other words, for velocity estimation, first consider whether it is important that
either u or ~ be known at dl. This examination is made by keeping one parameter fixed at its
known value while varying the other among appropriate candidate values,
6.1
Sensitivity
to Incorrect
q
Figures 4 and 5 continue tbe narrow/wide spectrum width anrdysis of before by exarrcining
perform~ce when data are processed assuming either one of two fixed q vrdues: Oor 10 dB. These
cboicw represent logical test cases in the sense of asking whether reasonable performance cm be
obtained by categoricdy treating tbe data as either low or high SNR data. Fjgure 4 considers
ML estimation for the narrow and wide spectrum width case; Figure 5 repeats the arrdysis for
the Bayw estimator (comparisons for Doppler weather targets of 5 m/s (0.192 VNV*
) and 13 m/s
(0.500 VNvg)are presented). In this, and dl following figures, the Bayes performance curve for tbe
case of known u ad rf (Section 4) is repeated as the “Bayes Bound.” The results for PP estimation
are dso reproduced for continued comparison.
5.1.1
ML Algorithm
In tbe case of narrow spectrum width weather [Figures 4(a) rural 4(b)], tbe parameter q as
predicted (Section 3.1.1 ) h= no apparent effect cm ML performance. This is not quite the cue,
however, for wide spectrum width weather [Figures 4(c) arrd 4(d)] where mismatch between assumed
and actual SNR results in increased estimation error. As will be seen dso in the case of the Bayes
dgoritbm, using a 10 dB vrdue for tbe SNR parameter results in a performance loss (relative to
PP) for input SNR values below 6 dB. Hence, tbe view of a smoothing window shape independent
of q is, in places, an oversimplification.
5.1.2
Bayes
Algorithm
From Figures 5(wd), it is clear that the Bayes implementation, using an arbitrary fixed q, has
a substantidy
altered performance. In neither case (O or 10 dB window) did performance match
23
~ SENSITIVITY:
7
1
1
ML
!*
6
0.269
I
I
I
o PP
0.231
❑ ML (0 dB)
■ ML (10 dB)
0.192
g5
OW= 1 mls (0.038 VNyq)
K
o
K4
K
u
o
53
G = 5 mls (0.192 VNyq)
0,154
(10,000 ReaNzations)
AYES BOUNO
0.115
0
z
a
~2
0.077
CR BOUND
1
0.038
1~
0
I
-2
(10,000 Realizations)
~.
I
‘4
4
SNR (dB)
Figure 4. ML algotiihm perjomonce
(a) T=
5 m/s,
sensitivity:
ou = 0.038 .~vf.
24
effeci
oj signal-to-noise
parameter
q.
q SENSITIVITY:
ML
IW,s.s
7 ~
““g
0 PP
0.231
6
D ML (0 dB)
~
= ML (10 dB)
g5
a
o
u4
K
w
o
-
0.192 t
o
m
BAYES BOUND
0.154 ;
a
~
(10,000 Real,zat)ons)
a3
a
o
z
a
+2
m
0.115
0
w
N
3
g
a
0.077
1
2,6
o
0.038 z
CR BOUND
I
1
I
1
I
0.2
I
,0
0
~
E
n
w
&
2
a
z
g
~
o PP
5
D ML (0 dB)
-2,6
+.2
■ ML (lOdB)
g
(10,000 Realizations)
-5.2
4
I
-2
I
I
0
2
I
I
I
6
8
+.4
10
SNR (dB)4
Figure 4. ML olgotiihm performance
(b) V = 13 m/$, UU= 0.038 WNvf.
settsifiuiiy:
25
eflect of signal-to-noise
pammetcr
q.
u SENSITIVITY:
7
v
1
I
I
ML
,*
I
o
0.269
I
I
PP
❑ ML (O dB)
■ ML (lOdB)
6
BAYES
BOUNO
$5
a
o
K4
K
u
n
K3
a
o
z
a
+2
m
SW= 5 m/s (0,192 VNyq)
v = 5 m/s (0.1,92VNYQ)
(10,000 Realizations)
\ \
CR BOUND
1
0
.
*
*
o
7
-1
A
&
+
0
PP
D
ML (0 dB)
■ ML (10 dB)
GW= 5 mls (0.192 VNyq)
i = 5 tis
(0.192 vNyq)
(10,000 Realizations)
I
4
-2
I
0
I
I
I
I
10
6
2
SNR (dB)4
Figure ~. ML algotiihm performance
(C)F= 5 m/S, Uw = 0.lg2 UNVq.
sensitivity:
26
effect
OJsignal-to-noise
pammeier
q.
7
I
Q
I
R
q SENS~lVIW:
I
I
ML
I
I
0
6
1
I
PP
❑ ML (0 dB)
■ ML (10 dB)
-\
OW= 5 ~s
(ol 92 vNyq)
i = 13 mls (0.500 VNyq)
(10,000 Reafizitions)
RAVF$ BOUND
\\\
c
0 PP
❑ ML (O dB)
■ ML (10 dB)
:W = 5 ~s (ol 92 vNye)
v = 13 tis (0.500 v~yq)
(1 O,OW Realizations)
-5.2 ~
4
Figure
(d)~=
-2
0
2
4
SNR (dB)
ML o(gotithm performance
13 m/8, ow = 0.lg2 .Nyq.
~.
6
8
10
sensitivity: eflecf of signal-to-noise pammeter
27
q.
~ SENSITIVITY:
7,
1
I
BAYES
,-7,.
I
I
I
I
!!
1
0.269
0,231
6
0
PP
0.192
5
:W = 1 ~s (o.038 vNyq)
V = 5 @S (o.lg2 vNyq)
4
0.154
BAYES BOUNO
3
0.115
2
0.077
1
0.038
CR BOUNO
01
I
I
I
I
I
I
1
I
I
[
I
I
I
10
0.2
70
0 G
g
E
o
w
~
2
a
=
\
$
5
A BAYES (O dB)
V BAYES (10 dB)
o PP
1
g
4.2
g
i = 5 fiS (o.lg2 vNyq)
(10,000 Realizations)
-2
4
I
-2
I
0
I
2
I
4
I
6
I
8
+,4
10
SNR(dB)
Figure 5.
q. (a)T=
Bayes algorithm performance
5 m/s, UW = 0.038 v~vq.
sensitioiiy:
28
e5eci of
signal-lo-noise
pammeier
7
q SENSITIVITY:
I
I
I
,m,s,,
BAYES
I
4
0
6
0.269
I
I
L
PP
BAYES (O dB)
V
BAYES (lOdB)
0.231
K
o
K
L~ ~
0.192 %
o
a
gw = 1 m/s (0.038 VNyq)
v = 13 tis (0.500 vNyq)
(10,000 Realizations)
BAYES BOUND
O,lw
:
a
k
0.115 0
w
~
0.077 g
a
o
1
CR BOUND
0.038 z
1
t
-5.2
r
4
OW= 1 tiS (0.036 VNYq)
? = 13 MS (0.500 vNyq)
(10,000 Realizations)
I
0
I
-2
I
I
2
I
6
)
6
1
I +.4
10
SNR (dB;
Figuw
5.
Bayes
algotiihm
perjomance
sensitivity:
q. (b) ~ = 13 m/s, OU = 0.038 vNVq.
29
effect of signal-to-noise
pammeier
q SENSITIVITY:
I
I
I
,*7!
BAYES
I
I
I
A BAYES (0 dB)
v BAYES (lOdB)
o PP
\\\
Ow= 5 tiS (o.1g2 vNyq)
5 mls (0.192 vNyq)
Walizations)
(10,000
v =
\
IAYES BOUNO
CR BOUND
1
0
1
I
I
I
I
I
I
I
I
I
I
I
1
0
A BAYES (O dB)
v BAYES (lOdB)
f
-1
Q PP
OW= 5 m/s (0.192 VNyq)
? = 5 m/S (o.1g2 vNyq)
(10,000 Realizations)
—:
I
I
I
a
-2
0
2
I
I
1
4
6
8
10
SNR (dB)
Figure
5.
q. (c)V=
Bayes algorithm perjomance
5 m/6, Uu = 0.192 v~vg.
sensitivity:
30
effect OJ signal-to-noise
pammeter
q SENSITIVITY:
SAYES
7
0,269
A BAYES (O dB)
v
6
BAYES (lOdB)
0.231 a
J
o
o PP
K
0.192 5
0
a
0,154
;
0,115
k
~
u
~
0.077
:
a
a
g
1
0.038
01
I
I
I
I
I
I
1 o
0.2
“~
0
x
x
*
x
.
o~
m
z
g
n
u
m
N
i
a
z
A BAYES (O dB)
~
m
v
BAYES (lOdB)
o PP
-2.6 (
1
+.2
:
OW= 5 mls (0.192 v~yq)
v = 13 mls (0.500 VNfl)
(10,000 Reahzalions)
-5.2
4
I
I
-2
0
I
4
I
2
I
6
1
8
4.4
10
SNR (dB)
Figw~ 5. Bayes algotiihm
perfomonce
sensitivity:
~. (d) T = 13 m/n, uw = o.lg2 v~vq.
31
effect of signal-to-noise
pammeter
the optimum Bayes Bound for d input SNR values; nor did it, at least, unifody
PP performance.
improve upon
For the narrow spectrum width case of Figure 5(a), there is au (apparently) anomdmrs
crossing of the Bayes Bound curve by the performance curve assuming q = OdB. TMISis not a
contradiction because the Bayes Bound optimrdity appties only in the sense of average performance
against the tottity of d possible weather target velocities. Hence, it is possible to obtain lower
standard errors for this purticufnr weather velocity of 5 m/s. (Clearly, tbe estimator O = 5 m/s,
an extremdy degenerate case, has zero standard error and him at this test point.) With the O dB
window, note the severe compromise in estimator bias. This bias is reflected in the (standard error)
performance loss, which is most notable at higher velocities [see Figure 5(b)]. Hence, for narrow
spectrum width weather, processing the data with M assumed SNR of O dB incurs the penalty of
increased bias restiting from inadequate signal isolation.
Assuming an q of O dB does make sense for wide spectrum width weather [Figures 5(c) and
(d)]. For low input SNR values, the optimal Bayes Bound is matched, and at higher input SNR
vduw, performance appears to be no worse than that of PP. The bias of this implementation is
dso very sitilar to that of PP.
At the other extreme, processing tbe data assuming q = 10 dB appears to better PP performance in the cwe of narrow spectrum width signals but at the cost of a performance loss (VS PP)
at low input SNR values and wider spectrum widths [Figures 5(c) and 5(d)]. The bias performance
with a 10 dB window, if anything, does improve upon that of the ided (q known) case.
6.2
Sensitivity
to Incorrect
o
For this series, the known values for q were used in the algorithm and a set of values for the
parameter u, ranging above and below the true value, were tested. In contrast to varying V, the
range of u values tested did not appreciably change tbe bim results. Therefore, bias curves for this
set are not presented.
Figures 6 and 7 summarize
the standard
error results for ML and Bayes
dgoritbms respectively.
For the narrow input spectrum [Figures 6(a), 6(b), 7(a), and 7(b)], u values ranging from
Utrue/4
to 5Utru.8 Were tested;
fOr the wide input spectrum [Figures 6(c), 6(d), 7(c)) and 7(d)l~
values ranging
from 0,,., I 5 to 9/5uir.eg were tested. In plotting the results, the performmce
region spanned by underestimating o is marked in white; the performance region spanned by
overestimating u is indicated with dark shtiing. Performance curves for euh of the tested values
are included as thin fines; the lowest curve in each panel always represents the performance when
‘Values 0.25, 0.5, 1.0, 2, 3, 4, and 5 m/s (u~,., = 1 m/s) were examined.
‘Values 1, 2, 3, 4, 5, 6, 7, 8, ad
9 m/s (u,,.,
= 5 m/s) were examined.
32
0 SENSITIVITY:
I
I
I
ML
1
,x,,.!,
I
I
I
0.269
0.231
0.192
0,154
0.115
I
I
I
I
I
I
0.077
a
g
0.036
E
~
1 o
(a)
0,269
a
a
n
z
a
~
R
y
d
0.231
~
$
5
0,192
4
O.IM
3
0.115
2
0.077
1
0.036
0
4
-2
0
2
6
6
10
SNR (dB;
(b)
Figuw 6.
ML algorithm perfomaltce
(4) T = 5 m/s,
UW = 0.038 UN,,.
sensitivity: effect of spectmm
(b) T = 13 m/s, UW = 0.038 UN,,.
33
width pammeler
o.
o SENSITIVITY:
,m,,. !e
ML
7
0,269
6
0.231
5
0.192
-
4
0.154
3
0.115
2
0.077
0.036
PP
I
I
I
I
I
K
o
K
%
I
(c)
g?
a
o
z
~6
m
v.
0.231
~
=
K
0,192
9
13 mls (0.500 VNYQ)
(10,000 Realizations)
5
4
0,154
PP
3
0.115
2
0.077
1
0.038
01
I
I
I
4
-2
0
2
1
I
I
6
6
I o
10
SNR (dS;
(d)
Figaw
(c) ~ =
6.
5
ML algotifhm perjomance
m/8,
Ow = 0.192 UNvq.
sensifiviiy:
efleci of
specfmm widfh pammefer u.
(d) T = 13 m/S, Ow = o.lg2 UN,,.
34
o SENSITIVITY:
7
BAYES
,m,,
,,
~
“269
6
0.231
t\
Os = 1 ~s (oo38 VNyq)
V -5 m/s (0.192 VNyq)
/
(10,000 Realizations)
j 0.192
0.231
Ow = I MIS (0.038 vNfq)
v .13
I
5
‘\
\
m/s (0.500 vNyq)
i
=
s
a
(10,000 Roatizations)
Y
4
3
4
2
0.077
1
01
I
I
I
4
-2
0
2
SNR (dB;
I
I
1
6
8
10
10
(b)
Figure 7. Bayes algon’thm perJomnance sensifiuify:
a. (a) v = 5 m/8, Ow = 0.038 u~v~. (b) V = 13 m/s,
35
efiect OJspectmm
UW = 0.038 UN”*.
width pammeier
?
0 ~ - Sds (o. 192 VNYQ)
v. 5 Ws (0.192 v~~
6
\
(~o,ooo Realizaf;ons)
--
\
0,231
PP
0.192
5
0.154
4
3
0.115
2
0.077
0.036
i
1
I
I
I
I
I
u
o
a
~
I
0 ~. 5 mls (0. 192 VNYQ)
-
\i
V. 13 M[S (0.500 VNYQ)
(f 0,000 Realizations)
Pp
\
5
i
1
0.154
4
0.115
3
0.077
2
1
0.036
1
~o
0
a
-2
0
4
2
6
6
10
SNR (dB)
(d)
Figure
7.
u. (c)5=
Boye$ algotithm performance sensitivity:
eflecf oj specimm width pammeier
5 m/s, uu = O.lgz UN”q. (d)V=
13 m/st Ow = 0.lg2 uNvq.
36
there is a perfect match between the assumed u value and that of tbe weather. The heavy curve
in =h panel is PP performance for reference.
h general, signifiwt
deviation from optimal performance occurs for o parameter values
grossly in error (using values 3, 4, and 5 m/s when et,., = 1 m/s, and using values 1 and 2
m/n when at,= = 5 m/s). Because ML performance, assuming u md v known, is close to PP
performance, not knowing the correct value for o generaUy results in a significant performance loss
[Figura 6(&d)]. For narrow spectrum widths and SNR vd.es in the range O to 5 dB, a marked
performance gtin stiU exists for the Bayes estimator, regardless of the value used for the parameter
u. The potential compromise at larger SN R values, however, indicates that the algorithm does not
perform weU with a u value that is too distant from the underlying true value.
Comment.
The WP method, prior to noise subtraction md censoring, can be viewed as
a Bayes rdgorithm wherein the parameter o is assumed to have an arbitrary narrow width (no
frequency domain smoothing is done). The plots in figure 7(c) and 7(d) confirm this notion as it
can be seen that the Bayes performance curves approach those of the WP method (see Figure 3)
when a nmrow u is assumed but the weather possesses a wide spectrum width.
6.S
Summary
It is unclear whether arbitrary fixed values for u and q can guarantee an estimator performance
that is consistently better thao that of PP. Bayes performance relies heavily on knowledge of both
u and q, ML performance, more appreciably on o
In the smoothing window view of Section 3.1.1, it is not enough to smooth arbitrarily. It
is important that the data be processed with an amount of smoothing that matches correlation
strength between samples. Treating the data as being highly correlated (low u) demonstrated a
severe performance loss when weather signals in fact had wide spectrum widths. This explains
why periodogram b~ed rdgorithms, such as WP, do not perform we~ given wide spectrum width
weather (too much weight is given to higher lag products). Unfortunately, treating the data as
being largely uncorrelated (high u) resulted in a corresponding performance loss with input signals
having narrow spectrum widths and high SNR values. Nevertheless, the indications are good that
improved performance (relative to PP) is possible with a smoothing methodology requiring less
than perfect knowledge of u and q.
The ML implementation at best only matches PP performance; any performance gain at
higher SNR levels would clearly be compromised by inaccurate knowledge of u. (Hence, the ML
implementation wiU not be considered further in this report.) A successful Bayes implementation
will require an adaptive selection of smoothing (weighting) coefficients, which is the focus of the
next section.
37
6.
PE~ORMANCE
WITH
ADAPTWE
u AND
q
With an emphasis on the estimation of w, one can adopt the view whereby u and q are
treated as nuisance parameters. Here, there exists a natural Bayesian method of treatment: removal
through expectation. This lo@cd recourse unfortunately does not lead to algorithm simplification
(in the present case), requiring as much computation w that n~ded for solving the vector parameter
problem. (It appears that the integrations required for nuisance parameter removal must be done
numerically and dso require the data vector to be in band.) Coupled with this observation, the
results of the previous sections motivate an approach that s~ks to adapt the computaticmd form
of Equation (12) using (suboptimd) estimates for o and V. Simple estimation of u and q (using
method of moments estimates described below) and direct substitution into Equation (9) and (12)
(smple by sample) were found to yield a performrmce clearly worw thw that of PP. This is not
(entirely) unexpected because, Uke w, u and q are being estimated from a small sample and (uperformmce) sensitivity to large deviations from the true u and q values csn, on average, do more
harm than good. Clearly, an approach is n~ded whereby the estimated values d and fi, substituted
into Equation (12) via Equation (9), are suitably constrained to minimize penalties that result from
their inaccuracy. This section describes one such approach that was found to be successful.
The general processing strate~ considered is as follows. Given a data sample Z, suboptimd
estimates of o and q are first computed and used to select a weighting coefficient array (matched
filter) r from a small, fixed, and predetermined family of matrices; the chosen matrix is used to
process the data as per Equation (12) and provide a velocity estimate. The number of matrices
required, their coefficient specification, and the criteria used to choow from among them is the
subject, then, of this present section.
6.1
Constrained
Inverse
Filter
(r)
Select ion
This section focuses on the nuisance pair (u, V) as an element of a (parameter)
assumed to be partitioned into K disfilnt pieces, i.e.,
set Q that is
K-1
V=
UV&,
where ~in~j=O(i#j).
k=o
assigned an optimal representor (o, q)k c V, and a corresponding weighting
matrix rk is computed as per tbe definition [~uation
(9)]. A representor for a given ~k is
determined by mems of a minimization involving the directed divergence [7] (i.e., KuUback-Leibler
information)
Each
region
~k
is
I(p:q)=EP
[1
logs
(16)
.
39
The divergence 1(P : g) has a useful interpretation as a tistarrce~o me~uring ~ a~lity tO discriminate probablhty density p (and, hence, its corresponding model) from alternative g (note:
1(P : g) ~ O and 1(P : g) = O # p s g). Furthermore, the divergence [Equation (16)] is e~ily
mmputed for the Gaussian case [Equation (3)]: if rl and rz correspond, respectively, to densities
PI and M, then
l(P1:
~)
= 10g
[rll - 10glr21 t tr(rzrl-]
- 1).
A somewhat natural approach, then, is to define the optimal representor for Vk as that pair (u, V)
whose corresponding density [Equation (3)] minimizes the discrimination information (divergence)
averaged over W densities g corresponding to parameter pairs in the set Wk:
(17)
and ( is an index to pairs (u, q) in wk. In words,
where Fk is the density corresponding to ~k
as mpreaentor for the set Wk, select that density (model) which, in an average sense, is lemt
distinguishable from the feasible densities (models) in wk. Note, no restriction is made that the
point ~k
must be conttirred in wk.
where it is assumed that W = (O,0.25] x (0,20] arrd
dl data can .be processed with the weighting coefficients
corresponding to one (arNJtrary) set point (~.
FOr t~s cme it is an e~y matter tO sOlve Equation (17)?]
and one obtains the solution (u, q). = (0.165, 8.4). figure 8 shows the Performance
of the resulting estimation algorithm. Al”tbmrgh performance is near optimal for wide spectrum
widths and high SNR levels (locations new the set point), uniform improvement for the entire range
of parameter values in V is absent, and performance is severely compromised in places as weU. One
must conclude that this V is too large to be represented by one set of weighting coefficients.
Ezample.
Consider
tbe
(trivial)
case
K = 1. That K = 1 is specified ~sumes
Figure 9(a) is a plot of the divergence for the above example. Divergence is near zero in the
vicinity of the set point and cnrves upward (away from zero) as weather spectrum width and SNR
deviate from tbe set point. Note, also, that the curvature is not uniform in direction—the most
acute curvature occurs with respect to spectrum width as weather SNR becomes large. Clearly, the
gnrd is to devise an adaptive method that has a composite divergence surface as flat and as nem
]oTe&njc~lY,
the divergence
f~ls w a true
distance
because
it does DOt SatiSfY the tri~~e
jneqU~-
ity. Th]s faitirrg, however, dries not prevent its use in the present application.
]] OPtjmjzat ion w= ac~fieved bY means of an implementation of the Nelder-Mead modified POlytOPe
(direct-search)
algorithm (see, for example, Gill et rd. [8]).
40
7,
I
I
BEST SINGLE o AND q
I
I
I
I
‘w”
‘; 0.269
0.231
6
;
~5
g
K
o
K4
a
u
n
23
0.192
K
n
K
O.lw
;
(0.038 VNyq)
Ow.
1 mls
v -5
m/s (0.192 VNyq)
a
~
o
z
a
● 2
0.115
n
w
~
0.077
:
m
~
CR BOUND
1
0.036
A
+
o -
A
*
A
Y
A
v
A
w
A
4
A
A
A
v
Ak Om
~
m
~
o
w
~
g
m
~
1
m
-1.0
A
BAYES
o
PP
A
a
z
4.2
OW. 1 mls (0.038 VNyq)
v.
g
z
5 m/s (0.192 VNYQ)
(10,000 Realizations)
-2.0 ~04.4
4
-2
0
SNR (dB;
Figure 8,
coefficient
Boyes algotithm per]omance:
suboptimal
set. (o) V = 5 m/s, UW = 0.038 WNj$.
41
u and v using best single-weighiing
BEST SINGLE o AND q
I
1
I
I
,-7.
!,
0,269
I
I
Y
A BAYES
0.231
0.192
Ow= 1 mls (0.038 V.yq)
BAYES BOUND
?. 13 mls (0.192 VNYQ)
(10,000 Realizations)
0.19
0.115
0,077
CR BOUND
0.03B
0.2
A BAYES
Q PP
-2.6
~W = 1 m/s (o.036 VNYQ)
t = 13 mls (0.500 vNyq)
‘Y
1
(10,000 Realizations)
t
I
-5.2
4
-2
I
o
I
I
2
I
I
6
8
1 +.4
10
SNR (dB;
Figure
8.
cocficieni
Bayes aljorifhm
performance:
suboplimal
set. (b) T = 13 m/s, UW = 0.038 UNYq.
42
a and v using best single-weighting
71\’
I
BEST SINGLE o AND q
I
I
I
‘%75m 0.269
I
6
.
1
o –
)0
A BAYES
~
o
PP
-1
4.2
0..5
v -5
M/S (o.192
vNyq)
m/s (0.192 vNyq)
(10,000 Realizations)
-z~
4
44
-2
0
2
4
6
8
10
SNR (dB)
Figure 8.
coeficienf
Bayes algorithm performance: suboptimal u and q tising besi single-weighiing
set. (c) V = 5 m/s, OW = 0.192
VNyq.
?
BEST SINGLE o AND q
I
I
I
\
1%7’ “
I
I
0.269
f
0.231 =
g
6
~5
g
K
o
=4
K
w
o
K3
a
0
z
a
+2
m
BAYES BOUND
V = 13 m/s (0.500 VNyq)
(10,000 Realizations)
0.192 K
o
K
a
0.154 ~
a
~
0.115 0
w
~
CR BOUND
0.077 g
K
o
0.038 z
1
I
I
I
I
I
I
Ow= 5 m(s (0.192 VNyq)
V. 13 m/s (0.500 VNYQ)
(10,000 Realizations)
I
A
-2
I
0
I
2
I
I
I
4
6
6
10
SNR (dB)
Figure 8.
coeficienf
Bages algotiihm
perJomance:
set. (d) V = 13 nl/s, UW = 0.192
suboptimal
UNyf.
44
u and v using best
single-weighting
o“
0.2;
SPECTRUM
~o
‘~
o
SPECTRUM
WIDrn
D:vcrgecce mi”imiza{ion
of iti compatiment
0.25
SPECTRUM
for three simple
gence between an optimal compatimcni
densities
~o
o
0.25
Figaro g.
MD~
WID~
patiifions. Computed direcfed diver-
mpmsentor [Eqnaiion (1 7)] and the feasible model
w plotted. (The divergence scale is arbitmq.)
mm as possible. The logical extension, of course, is to swk improvement by solving the situation
for K > 1. Two simple extensions, a two and thrw compartment model, are dso tilustrated in
Figure 9(&c). Here, the divergence surface for each compartment is computed as per Equation (17);
dthcmgh not yet striking, the notion of flattening the composite surfwe is clear.
There is, however, a problem with the simple extension of Equation (17) to K > 1; Equ&
ticm (17), as written, has the imphcit assumption that one would (could) distinguish perfectly
between the opposing hypotheses w to whether data Z were more consistent with parameters from
the set ~k vs rdternatives from its complement Vi = V – wk. clearly, for }{ >1, Equation (17)
must be modified to ucmrnt for the accur~y of the decision process that matches the data to one
of the subsets V&:
~k
=
(Csv)tv
t
. ..
.
Pr(SeleCt
~k I ~kc).
~kc
~(q(
: ~~)ti}
(18)
.
In this way, the desire to optimally match (u, q)k to ~k is balanced against the probability that
the data will be incorrectly matched with Wk.
6.2
Suboptimal
Estimation
The cdculatimrs
of u and q
for Equation
(18)
require
specification
of a set of decision
rules
and
the
be used; however, once this is done dl information required
to solve ~uation
(18) is present and the selection of rk! (k = 0,.. ., I{ - 1), can be completed
prior to the processing of any data. Therefore, atthough computaticmdly formidable, the solution of
~uatimr (18) is quite achievable. This section will focus on decision rules using essily computable
suboptimd estimates for a and V. For suboptimd estimates, an appeal to the assumed correlation
structure of Equation (1) and (method of moments) estimates for & and O, derived from a weighted
least-squares fit to the data, can be used. The le~t-squares equations
correspondlcrg
statistics
(estimators)
to
(19a)
and
M-1
M-1
~ urmm21n
m=o
I?m] =
~
wmm21rr[S
t~~m]
1 * *M-l
- ~rr
u
~
wmm4
(19b)
m=o
“=0
@n be used to solve for o ad S (which provides an estimate for q); the lag estimates ?~ are
unweighed here and the (least-squares) weights w~ can be used to weight or select the lag estimates
46
to be used. M w~ = 6~-1, for example, 6 becomes equivalent to the common single-lag spectrum
width estimator [see Zrni6 [9], hls Equation (5.1)]. For the results of this report,
1 f0rm<4
~m d=~
{
was used for wtimation
O otherwise
of u and q.
ME PARAMETER
SPACE W
0,2!
0
c
0
+1
Y,
000
(
20
Figure 10.
6.S
Decision
Rules
A partition
and a Partition
design using simple
decision
wles
for W
At this point it is necessary to be more specific regarding the description of the ~k’s. For the
remainder, it is =sumed that V is the pardlelepiped of the previous example: (O,0.25] x (O,20]. To
simphfy definition of the ~k’S and to provide decisions b~ed on simple rules, consider a partition
derived from a sequence of threshold tests—first, for O and second, for ~. The general design is
illustrated in Fjgure 10. A threshold test of Oagainst (yet undetermined) SNR values divides V into
47
Ewh column is further subdivided along spectrum width
values (rdso to be detemined ) and a requirement is imposed whereby increased petitioning of a
column with respect to spectrum width requires comespondin~y high values for SNR. (The sufiace
curvature of Figure 9(a) suggests that more detailed representation is desired for high SNR values
than low.) To simphfy implementation, each new column to the right is allowed core addltiond
division with respect to spectrum width. The placement of SNR and spectrum width thresholds,
the selection of (u, q)k values, and specification of K are rdl unknowns to be determined.
columns
of (yet
undetermined)
tidth.
For K fixed, Equation (18) can be used to identify optimal values ford threshold boundaries
and rdl (u, V)k pairs. A K-compartment summed divergence error can be defined by summing the
divergence emor in Equation (18) over each set in the partition of V. The K-comp=tment
summed
divergence is monotonic (ncmincreaing) in K. Certairdy adding extra compartments in Figure 10
can only lower the total divergence error. For example, optimrd threshold placement would force
new compartments to become degenerate (collapse to nothing) if they could not improve the overaH
error vduq lower resolution compartments to the left would get squeezed by higher resolution
compartments from the right if the data and estimators for o and q could support finer levels of
partition. Hence, az K increaaes, the K-compartment summed divergence error (bounded below by
zero) must converge. A stopping criteria can be established for selecting K by arguing diminishing
returns with further incremes in K. In this way, K, optimal threshold placement, and optimal set
point values can be obtained.
Figure 11 plots the K-compartment summed divergence for the pmtition scheme illustrated
in Figure 10. Evduaticm of Equation (18) waz approximated by using a Monte Carlo simulation
to obtain a mean and standmd deviation characterization for the ~ and * of Equation ( 19), and
a Gaussian approximation was used to evaluate Pr( select Vk[~k ) and complement. Based on the
results presented in Figure 11, K = 15 waz selected for continued analysis. The corresponding
thresholds and set point values for K = 15 are summarized in Table 2 and illustrated in Figure 12.
In Fjgure 12, optimal set point locations are indicated by a labeled (squwe) dot. The convex
hu~ of the set points in W is illustrated by shading. The most striking result of the optimization is
that the best set point for a region Vk is not necessarily contained within ~k. The convex huU of the
set points represents the constraint required of 6 and Oto hrdance the effect (cm velocity estimation)
of their uncertainty. F1grrre 13 plots the zequence of convex hulls for the values of K considered
in Figure 11. As K incre~es, the convex hull expands but there is a fundamental hmitation to its
eventual extent. To increaae the extent of the limiting hull requires more precise estimators for o
and q or, equivalently, more data. The hmiting hull will rdways be strictly contained within V: for
the hu~ to be equivalent to (i e., cover) W impfies that perfect estimation of u md q is possible.
6.4
Adaptive
Estimator
Performance
The adaptive procedure defined by Table 2 (Figure 12) was used to process simulated data as
in the previous sections. The results for narrow and wide spectrum width weather are presented
in Figure 14.
48
.
.
K
Figure 11.
K-Compartment
summed
divergence.
For narrow spectrum width weather, Panels a and b, the adaptive method demonstrates a near
uniform improvement on the performance of PP. The performance is not only better than that of
PP, it is much closer to the optimum bound established in emlier sections. The slight deterioration
at the higher SNR values may be due, in part, to the decision to use the 15 compartment pwtition.
Figure 13 indicates that the 21 compartment partition added refinement specific to higher SNR
values; therefore, some further improvement at higher SNR values may be possible. Note, also,
that in parallel with approaching the standard error bound for velocity estimation, the adaptive
estimator dso approaches the bias performmce of the optimal Bayes (bound) estimator.
For wide spectrum width weather, Panels c and d, the room for improvement was not =
evident (except at low and very high SNR values). Nevertheless, the tiaptive method demonstrated performance improvement relative to PP at low and high SNR vdues—thrrse regions where
improvement relative to optimrd performance was most likely.
49
Optimal
Thresholds
2
and
Point
Set
Values
(K
Upper q
Threshold
Set Point
Compartment
I
TABLE
= 15)
Upper u
Threshold
o
(0.0983, 2.386)
1.367
-
1
3.545
0.0909
2
(0.1095, 3.892)
(0.1659, 3.794)
3
4
5
(0.1041, 6.440)
(0.1452, 7.082)
(0.1921, 6.960)
7.047
0.0549
0.1279
6
7
(0.0907, 9.554)
(0.1317, 9.723)
(0.1667
12.006
0.0380
0.0903
8
9
in
.“
11
12
13
14
9, 10.916)
(0.203$
I (n nnll 17 n07\ I
,“,.
”..,
.“....,
(0.1186, 13.822)
(0.1554, 13,927)
m
(0.1848, 15.( ,
(0.2155, 15.438)
....
E
I
0.0354
I
0.0821
0.1252
0.1844
The data of the previous figure was combined with measurements at additionrd weather
spectrum widths to produce Rgure 15, which illustrates estimator performance as a function
of weather spectrum width for fixed levels of SN R. There is an almost uniform improvement in
performarrce relative to PP and this improvement for a fixed SNR level is seen to apply across the
range of spectrum widths considered. The improvement is most striking for the two low SNR levels
(Panels a and b). Improvement does not require a corresponding performance loss at Mgher SNR
levels—the adaptive method continues to improve performance there u well.
6.5
Summary
Applying adaptive procedures to situations with small-sample sizes is a difficult t=k. This
sectimr h~ shown that by adopting suitable constraints, an adaptive methOd cOuld be develOped
for the smaB sample velocity estimation problem.
Performance for the adaptive (Bayes) estimator is close to the optimal performance bounds
established earlier. Significant improvement relative to PP wu estabhshed.
50
.
.
0.25
,.,:;
;,.
,.,,..
i,
E
‘ .,
,,.
.
b
,.,.
*?,’;’.’.
“J
10
4
0
o
1
3
20
SNR
Figure 12.
Optimum
partition
51
and sei point locations
0.25
20
o
SNR
Figure 13. Set poini convez hull with increasing K. Fundamental Iimitaiion
of set point convez hull is illustrated Jor a se9uence with increasing [{ ualue.
52
on eztent
?
I
I
I
ADAPTIVE
I
o AND q
I
6
tm7,
I
27
0,269
1
I
0,231
A BAYES
0 PP
0.192
0. = ‘ “s ‘0:036 ‘Nyq)
i = 5 mls (0.1 92 VNyq)
a
o
=4
u
O.lM
(10,000 Reabzations)
u
n
BAYES BOUND
a3
a
0
z
a
+2
w
0.115
-
0.077
1
0.036
CR BOUND
2’6~
“2
4
4
I
-2
0
2
6
8
10
SNR (dS;
Bayes algorithm with adaptive u and V: 15 compatiment
set (I). (a) T = 5 M/S, au = 0.038 *N”q.
Figure
Il.
53
weighting coeficieni
,-7!
ADAPTIVE o AND q
?
I
A BAYES
0.231
6
OW= 1 mls (0.OW VNYQ)
25
-
BAYES BOUND
0.269
0.192
C = 13 dS (0,500 VNYQ)
(1O,OWRealizations)
;4
0.154
a
w
o
53
0
z
<
+2
m
0.115
0.077
1
0.0%
CR BOUND
01
I
I
I
~..4
-2
0
I
I
I
4
6
6
2
o
10
SNR (dB)
14. Bayes algoriihm with adoptive o and q: 15 compatiment
(I). (b) U = 13 m/s, u~ = 0.038 VNyq.
Figure
set
54
weighting coeficieni
ADAPTIVE
I
u AND q
,W,.a
I
i
I
0.269
I
71<’
0.231
6
0 PP
0,192
5
v = 5 m/s (0:192 vNyq)
.
4
0.154
BAYES BOUNO
3
0,115
2
0.077
1
0.038
A
BAYES
Q PP
f
Ow= 5 m/s (0.192 VNyq)
T = 5 m/s (0.192 VNm)
(10,000 ReaUzalions)
f
.
I
-2
-2
4
I
I
0
I
4
2
I
6
I
8
SNR (dS)
Figun
Eel (I).
14. Bayes algon’thm
(c)
T
=
5
m/S,
OW
=
with adaptive
0.192
o and q:
VNvq.
55
15 compatiment
weighting coeficieni
03
o
g
z
o
u
~
m
a
z
A
i
>
K
BAYES
Q PP
-2.6 (
Ow .5
i
=
m/s (0.192
4.2
g
vNyq)
13 mls (0.500 vNyq)
(I 0,000 Realizations)
Li
-5.2
4
I
-2
I
I
0
2
I
4
I
6
I
8
10
SNR (dB)
Figure 14.
Bayes algorithm with adaptive u and O: 15 compotimeni
sei (1). (d) T = 13 m/s,
a~ = 0.192 VNY9.
56
weighting coefficient
10 [
I
9
‘p~
I
ADAPTIVE a AND q
I
1
I
I
0,346
8
0.306
?
0.269
‘AyE’~
6
5
0.231
0.192
BAYES BOUND
0.154
4
0,115
3
~=-5dB
v = 5 m/s (0.192
2
CR BOUND
z
2
-1
a
vNyq)
0.077
(10,000 Realizations)
0.038
7/
E
u
n
u
,W7, ,!
I 0,385
(a)
10
I
I
I
I
o
%9
q=OdB
~
V -5
6
0,365
0.346
m/s (0.:92 VNYQ)
(10,000 Realizations)
0.306
?
0.269
6
0.231
5
0.192
‘p~
4
0.115
3
0.077
2
1
:::%*
’038
~o
0
1
0.154
BAYES
0.05
0.15
0.10
NORMALIZED
SPECTRUM
0.20
0.25
WIDTH
(b)
Bayes algorill,m will, adnptive u avtd q: 15 coIt]part!!Ler,t
Figure 15.
set (11). {e) SNR = -5 dU. (b) SNR = O d~.
57
weigljting
coeficieni
i;k?i
a
u
<
(c)
0,385
~
n
u
– 0.346
;
a
~
~ 10
a
:9
a
+
m8
– 0,308
g
7
-
0.269
-
0.231
5
-
0.192
4
-
O.l M
3
-
0,115
q=lOdB
v -5 Ws (0.192 vNyq)
6
(10,000 Reakzations)
‘~:
o
0.05
0.15
0.10
NORMALIZED
SPECTRUM
0.20
0.25
WIDTH
(d)
Figure 15. Bayes algotithm with odopiiue o and q: 15 compotimelit
(c) SNR = 10 dB. (d) SNR = fO dB.
set (II).
58
weighting cmficient
7.
CONCLUSIONS
Ewh of the previmrs sections included a brief summary and these statements
repeated here. Some concluding remarks are, nevertheless, in order.
will not be
This report’s primary objective h= been to examine the challenge of Doppler velocity estimation when confronted with a small-sample size. Applying a generally accepted model for the
measurement of (metmrologic~y
generated) Doppler signals, lower bound performance Imitations
were first wtabhshed. These bounds were shown to be clearly different (greater) than those prvialed by standard CR analysis, but room for improvement (relative to the standard PP estimator)
ww nevertheless indicated. Optimal velocity estimation, under the aasumed model, by definition
requires the joint estimation of a vector parameter. Previous attempts at such optimal estimation
have been hampered by the technical difficulties that arise in solving the comphcated system of
equations that result. The potential of approximate methods that seek to treat some of the vector
parameters = fixed quantities w= explored. Sensitivity to these approximating usumptions w=
studied md insight into the relative importance of these nuisance parameters ww provided. An
adaptive method w= proposed. The new method does not require excessive computations—nothing
more extensive
than previously
proposed FFT methods. The find adaptive estimation scheme w=
shown to provide near optimal performance for a span of SNR and spectrum widths likely to be
associated
with
meteorological
signals.
59
APPEND~
Proof of Equation
Preposition
A.1 kt
P(ZIG)
where R = R(O)
Parumeter
vector.
can be abtained
ZT = [zOZI . . . ZM-1] be a complez
=
A
(4).
mndom
vector with Gaussian demity
rr-MIRl-le-z+R-’z,
is the M x M covan’ante matm’z and @T = [00 0, . . . 0“-]]
Then, the Fisher Information matriz F = [ji,j] dejned by
equivalently
is a rwl. valued
jrom
(Al)
Pwfi
Define the unit vectors of order n as
o
1
0
,e~=
l:!
1
,.. ., en=
o
and the elementary matrix Eiti (of order n x n) as
Ei,j = e<e~.
.
For an arbitrary
example):
matrix A = [ai,j], the following identities are standard
aii = e~Aej,
(MC Graham [10], for
(A,2a)
61
(A.2b)
ij
ij
trA = ~
(A.2c)
e~Aei.
,
To prove ~uation
(Al),
begin by taking the logmithm of the density,
lnp(Zl@) = -Mlnz
-lnlR[
- ZtR-]Z,
and compute the derivative, term by term, with respect to @. Only the latter two terms are of
importance. For the first of these, Oin ]R1/~@,
=
‘r{R-’%}
where the l~t tine fo~ows from application of Equation (A.2):
For the second term, note that
ZtR-]Z
= tr{ZtR-’Z)
= tr{ZZtR-l)
from which it follows that
~zt
~=
R-12
Otr{ZZtR-’)
O@
.
62
(A.3)
Continuing,
&r{ ZZtR-l)
88k
=
F?tr{*}$
=
=
~~-t~{ZZtR-l
ij
Ei,jR-’}~,
where the last Ene fo~ows from the result
Additionrd reamangement
results in
8tr{ZZtR-’)
86k
= -tr
{
,
R- 10R
— R-1 zzt
~ok
}
(A.4)
Combining Equations (A.3) and (A.4) provides the intermediate result
Olnp(Zl@)
~ek
= tr
{
R-l ~
The next step is to evaluate the expectation.
and Equation (A.5),
Fid
=
E
=
E
tr
From the definition of the Fisher Information matrix
89j1
81np(Zl@)
08i
[
[{
81np(Zl@)
R-’ ~
R-l (ZZt -R)
t
Worting with the terms inside the expectation
tr
R-l ~
{
R-l (ZZt -R)
,
(A.5)
R-l (ZZt - R)}.
}{
}{
tr
R-l ~R-]
>
(ZZt -R)
and again using (A.2):
tr
R-l ~
63
J
R-l (ZZt -R)
}
}1
.
=
Now,
(ZZt - R)e~e~(ZZt
- R)
is an M x M matrix with ninth entry given by
(z”zI- r“,k)(ziz%
- Ti,m)
for which the expectation
is commonly known (see Miller [11], for example):
E {(znz~ - r~,k)(zlz~
- rl,m)} = r~,mrl,k.
Therefore,
E {(ZZt
– R)e~e~(ZZt
- R)} = Rri,~.
Flndly, using Equations (A.6) and (A.7) one obttins
wh]ch
is the desired result.
64
(A.7)
REFERENCES
1.
R.J. Doviak and D.S. ZrniL. Doppler Radar and Weather Obsemations.
Orlando, Florida, 1984.
2.
D.S. Zrnit. Simulation of weatherfike Doppler spectra and signals. Journal of Applied Meteoro~y,
14:61$620,
1975.
3.
H.L. Van Trees. Detection, Estimation,
Sons, he., New York, New York, 1968.
4.
P.R. Mahapatra ad D.S. Zrnit. Pr~ticd
algorithms for mean velocity estimation in pulse
on Geoscienm
Doppler weather radars using a smd number of samples. IEEE Tmnsactions
and Remote Sensing, GE21(4):491-501,
1983.
5.
K.S. MiUer and M.M. Rochwarger.
.
IEEE fi~actions
6.
on Injomation
P.T. May and R.G. Strauch.
Journal oj Atmospheric
and Modulation
Academic Press, he.,
Part I. John Wiley and
Theory:
A covariance approach to spectral moment estimation.
Theoy, IT-18(5):588-596, 1972.
An examination
and Oceanic Technolqg,
Theory and Statistics.
of wind profiler signrd processing algorithms.
6:731-735, 1989.
John Wiley and Sons, New York, New York,
7.
S. KuUback. Injomation
1959.
8.
P.E. GiU, W. Murray, and M .H. Wright. Practiml Optimization.
York, New York, 1981.
9.
D.S. Zrnit. Estimation of spectral moments for weather echoes.
Geoscience Electronics, GE17(4):1 13-128, 1979.
Academic Press, Inc., New
IEEE
Tmnsactions
10.
A. Graham. Kmnecker Products and Matfiz Calculus with Application.
Wiley and Sons, New York, New York, 1981.
11.
K.S. Miller. Complex gaussian processes. SIAM Review, 11(4):544-567, 1969.
t
65
on
Hdsted Press: John