G .A. Tsi h ri ntzis U. Tu re I i C.L.Nikias Indexing terms: Ambiguity functions, Delay Doppler estimation, Impulsive interference, Radar, Sonar Abstract: The problem of estimation of differential-delay Doppler in an environment in which the signal of interest is embedded in highly impulsive interference is addressed. A signal model is formulated with wide applications in radar, sonar, communications and biomedicine and the construction of a new ambiguity function is proposed which is based on the recently- developed concepts of fractional, lower-order statistics and is, therefore, resistant to the presence of severe outliers in the observed time series. The performance of the proposed differential-delay Doppler estimators is theoretically analysed, verified via Monte Carlo simulation, and compared to the performance of traditional, second-order statistics-based estimators. 1 introduction The estimation of the location and the velocity of a tar- get is an important problem in radar [l] sonar [l], com- munications [l], and biomedical [2] applications. In all of these applications two signals are measured: a refer- ence signal and a version of it that is both delayed and Doppler shifted. This scenario is satisfied by the two modalities used in practice: the active modality in which a reference signal is radiated and information is extracted from its return (echo), and the passive one in which the information is extracted by measuring the relative (differential) delay and Doppler shifts between signals emitted from a common source and received across multiple sensors. Clearly, the active modality is a special case of the passive one, on which we concen- trate. The signal model used is summarised as follows: A source emits the narrowband signal x(n) (as measured at a reference sensor) while it moves at a constant 0 IEE, 1996 ZEE Proceedings online no. 19960607 Paper frst received 30th November 1995 and in revised form 7th May 1996 G.A. Tsihrintzis and U. Tureli are with the Communications Systems Laboratory, Department of Electrical Engineering, University of Virginia, Charlottesville, VA 22903-2442, USA C.L. Nikias is with the Signal and Image Processing Institute, Department of Electrical Engineering - Systems, University of Southern California, Los Angeles, CA 90089-2564, USA 358 velocity v. We use K sensors to measure the signals Yl(4 = .(n) + w(n) y2(n) = s(n - d2)eZwzn. + w2 (n) yK(n) = ~(n - dK)eZWKn + wK(n) where dk and mk, k = 2, 3, ..., K, are differential delays and Doppler shifts, respectively. The goal is to use the measurements {yk(n); n = 0, ..., N-I; k = 1, 2, ..., K} to estimate the differential delays and Doppler shifts. Conventional approaches [l] to this problem assume that the noises wk(yl) are Gaussian of known correla- tions across sensors. For the signal x(n) the assumption is that it is either deterministic or Gaussian of known spectral characteristics. These assumptions allow the application of maximum-likelihood estimation proce- dures for this problem. For cases where the signals and/or the noises have unknown distributions, second- (SO) [l] and higher-order (HO) [3-5] ambiguity func- tions have been proposed to address this problem. In particular, the rationale hehind the use of HO ambigu- ity functions is that they tend to suppress Gaussian noise of arbitrary correlation. However, this fact holds only asymptotically, in the limit of very long data records. Recently, it was shown that signal processing algorithms designed on the basis of higher-order statis- tics may in fact perform worse than their second-order statistics based counterparts in narrowband applica- tions, even in the limit of very long data [6]. This result is derived from the fact that estimates of higher-order statistics contain high error variances when compared with the corresponding estimates of second-order statis- tics. In this paper, we develop ambiguity functions which are based on the recently proposed fractional lower- order statistics (FLOS) [7, 81. FLOS are statistical sig- nal-processing tools that tend to resist outliers in observed time series and are therefore more appropri- ate for processing highly impulsive signals and noises. A paradigm for designing and testing FLOS-based sig- nal processing algorithms is the case of time series modelled as symmetric, alpha-stable (SaS) random processes. These processes have been shown [8] to arise as canonical, statistical-physical models for virtually all processes observed and parameterise various degrees of ‘impulsiveness’ with only one parameter, their charac- teristic exponent [Note 11. The performance of optimum and suboptimum receivers in the presence of SaS impulsive interference IEE Proc -Radau, Sonar Nuvig , Vol 143, No 6, December 1996
Abstract: The problem of estimation of differential-delay Doppler in an environment in which the signal of interest is embedded in highly impulsive interference is addressed. A signal model is formulated with wide applications in radar, sonar, communications and biomedicine and the construction of a new ambiguity function is proposed which is based on the recently- developed concepts of fractional, lower-order statistics and is, therefore, resistant to the presence of severe outliers in the observed time series. The performance of the proposed differential-delay Doppler estimators is theoretically analysed, verified via Monte Carlo simulation, and compared to the performance of traditional, second-order statistics-based estimators.
1 introduction
The estimation of the location and the velocity of a tar- get is an important problem in radar [l] sonar [l], com- munications [l], and biomedical [2] applications. In all of these applications two signals are measured: a refer- ence signal and a version of it that is both delayed and Doppler shifted. This scenario is satisfied by the two modalities used in practice: the active modality in which a reference signal is radiated and information is extracted from its return (echo), and the passive one in which the information is extracted by measuring the relative (differential) delay and Doppler shifts between signals emitted from a common source and received across multiple sensors. Clearly, the active modality is a special case of the passive one, on which we concen- trate.
The signal model used is summarised as follows: A source emits the narrowband signal x(n) (as measured at a reference sensor) while it moves at a constant 0 IEE, 1996 ZEE Proceedings online no. 19960607 Paper frst received 30th November 1995 and in revised form 7th May 1996 G.A. Tsihrintzis and U. Tureli are with the Communications Systems Laboratory, Department of Electrical Engineering, University of Virginia, Charlottesville, VA 22903-2442, USA C.L. Nikias is with the Signal and Image Processing Institute, Department of Electrical Engineering - Systems, University of Southern California, Los Angeles, CA 90089-2564, USA
358
velocity v. We use K sensors to measure the signals Yl(4 = .(n) + w ( n ) y2(n) = s(n - d2)eZwzn. + w2 (n)
yK(n) = ~ ( n - d K ) e Z W K n + w K ( n ) where dk and mk, k = 2, 3 , ..., K, are differential delays and Doppler shifts, respectively. The goal is to use the measurements {yk(n); n = 0, ..., N-I; k = 1, 2, ..., K} to estimate the differential delays and Doppler shifts. Conventional approaches [l] to this problem assume that the noises wk(yl) are Gaussian of known correla- tions across sensors. For the signal x(n) the assumption is that it is either deterministic or Gaussian of known spectral characteristics. These assumptions allow the application of maximum-likelihood estimation proce- dures for this problem. For cases where the signals and/or the noises have unknown distributions, second- (SO) [l] and higher-order (HO) [3-5] ambiguity func- tions have been proposed to address this problem. In particular, the rationale hehind the use of HO ambigu- ity functions is that they tend to suppress Gaussian noise of arbitrary correlation. However, this fact holds only asymptotically, in the limit of very long data records. Recently, it was shown that signal processing algorithms designed on the basis of higher-order statis- tics may in fact perform worse than their second-order statistics based counterparts in narrowband applica- tions, even in the limit of very long data [6]. This result is derived from the fact that estimates of higher-order statistics contain high error variances when compared with the corresponding estimates of second-order statis- tics.
In this paper, we develop ambiguity functions which are based on the recently proposed fractional lower- order statistics (FLOS) [7, 81. FLOS are statistical sig- nal-processing tools that tend to resist outliers in observed time series and are therefore more appropri- ate for processing highly impulsive signals and noises. A paradigm for designing and testing FLOS-based sig- nal processing algorithms is the case of time series modelled as symmetric, alpha-stable ( S a S ) random processes. These processes have been shown [8] to arise as canonical, statistical-physical models for virtually all processes observed and parameterise various degrees of ‘impulsiveness’ with only one parameter, their charac- teristic exponent [Note 11.
The performance of optimum and suboptimum receivers in the presence of S a S impulsive interference
IEE Proc -Radau, Sonar Nuvig , Vol 143, No 6, December 1996
was examined in [9], both theoretically and via Monte Carlo simulation, and a method was presented for the real-time implementation of the optimum nonlineari- ties. From this study it was found that the correspond- ing optimum receivers perform in the presence of S a S impulsive interference quite well, while the performance of Gaussian and other suboptimum receivers is unac- ceptably low. It was also shown that a receiver designed on a Cauchy assumption for the first-order distribution of the impulsive interference performed only slightly below the corresponding optimum receiver, provided that a reasonable estimate of the noise dispersion was available, which for real-time sig- nal processing purposes could be obtained via the fast algorithms in [lo].
The study in [9] was extended in [ll], in which the optimum demodulation algorithm for reception of sig- nals with random phase in impulsive intereference and its corresponding performance was derived and tested against the traditional incoherent Gaussian receiver [12], Chap. 4. Finally, the performance of asymptoti- cally optimum multichannel structures for incoherent detection of amplitude-fluctuating bandpass signals in impulsive noise modelled as a bivariate, isotropic sym- metric, alpha-stable (BISaS) process was evaluated in [13]. In particular, we showed in [13] that exact knowl- edge of the noise distribution was not required for almost optimum performance. We also showed that receiver diversity did not improve the performance of the Gaussian receiver when operating in non-Gaussian impulsive noise and therefore a non-Gaussian detection algorithm could substitute for receiver diversity.
2 Symmetric alpha-stable distributions
2. I Llnivariate SaS distributions A univariate symmetric, a-stable ( S a S ) probability den- sity function (PDF) fa(y,4.) is best defined via the inverse Fourier transform integral [7, 141
f c y ( Y 1 4 .) = - exp(i6w - ylwla)e-wzdw (1)
and is completely characterised by the three parameters a (characteristic exponent 0 < a 5 2), y (dispersion y > 0), and 6 (location parameter AX < 6 < w). The charac- teristic exponent a relates directly to the heaviness of the tails of the S a S PDF: the smaller its value, the heavier the tails of the distribution. The value a = 2 corresponds to a Gaussian PDF, while the value a = 1 corresponds to a Cauchy PDF. For these two PDFs, closed-form expressions exist, namely
25r r pm
(3)
For other values of the characteristic exponent, no closed-form expressions are known. All the S a S PDFs can be computed, however, at arbitrary argument with the real time method developed in [9]. The dispersion y is a measure of the spread of the S a S PDF, in many ways similar to the variance of a Gaussian PDF and equal to half the variance of the PDF in the Gaussian Note 1: An alpha-stable model fitted to an observed time series wiU generally consist of four parameters [8], but the heaviness of the tails of the modal is governed by only one of the four parameters, namely the characteristic exponent.
case ( a = 2). Finally, the location parameter 6 is the point of symmetry of the S a S PDF.
The non-Gaussian ( a # 2) SaS distributions maintain many similarities to the Gaussian distribution, but at the same time differ from it in some significant ways. For example, a non-Gaussian S a S PDF maintains the usual bell shape and, more importantly, non-Gaussian S a S random variables satisfy the linear stability prop- erty [14]. However, non-Gaussian SaS PDFs have much sharper peaks and much heavier tails than the Gaussian PDF. As a result, only their moments of order p < a are finite, in contrast with the Gaussian PDF which has finite moments of arbitrary order. These and other similarities and differences between Gaussian and non-Gaussian SaS PDFs and their impli- cations on the design of signal processing algorithms are presented in detail in the tutorial paper [7] and in greater depth in the monograph [8] to which the inter- ested reader is referred. For illustration, we show in Fig. 1 plots of the S a S PDFs for location parameter 6 = 0, dispersion y = 1 and for characteristic exponents a = 0.5, 1, 1.5, 1.99 and 2, as they have been produced by calculation of the inverse Fourier transform integral in eqn. 1.
0.6 I
0 2
0 -15 -10 -5 0 5 10 15
PDF argument Fig. 1 dispersion, and various characteristic components
Symmetric, alpha-stable PDFs of zero-location parameter, unit
a = 2 a = 1.99 a = 1.5 a = 1 a = 0.5
_ _ _
_ _ ~ ~ _ . . . . . . . . . . . . . .
2.2 Bivariate, isotropic SaS distributions The class of bivariate isotropic symmetric alpha-stable (BISaS) probability density functions fa,y,61,62(x1, x2) are defined as the subclass of multivariate stable distribu- tions specified by the inverse Fourier transform
f c u , Y , s l , 6 2 ( m > 5 2 )
e - - 2 ( 2 1 u l + 5 2 W Z ) d WldW2
(4) where the parameters a and y are termed the character- istic exponent and the dispersion, respectively, and d1 and 6, are location parameters. The characteristic exponent generally ranges in the interval 0 < a s 2 and relates to the heaviness of the tails with a smaller expo- nent indicating heavier tails. The dispersion y is a posi- tive constant relating to the spread of the PDF. The two marginal distributions obtained from the bivariate
distribution in eqn. 4 are univariate S a S with charac- teristic exponent a, dispersion y, and location parame- ters 6, and a,, respectively [7, 81. In the rest of the paper, we assume 6, = 8, = 0 without loss of generality and drop the corresponding subscripts from all our expressions.
Unfortunately, no closed-form expressions exist for the general BISaS PDF, except for the special cases of a = 1 (Cauchy) and a = 2 (Gaussian):
where d = xf + xj. For the remaining (non- Gaussian, non-Cauchy) BISaS distributions, power series exist [7, 81, but are not of interest to this paper and, therefore, are not given here. However, we do note that the tails of the non-Gaussian BISaS distributions are algebraic, i.e.
lim Pcy+2fcy,y(P, Q ) = B(a, 7) (6) P+CC
where B(a,y) is a positive constant.
2.3 SaS and BlSaS models for impulsive noise Consider wideband reception of interference arising as the superposition of elementary pulses emitted at ran- dom time instants by a random distribution of sources. It can be shown [8] that under general, realistic assumptions, the first-order distribution of the interfer- ence w(t) asymptotically follows a S a S law, i.e.
Consider now a narrowband receiver operating in S a S impulsive interference n(t) of dispersion y. Let
f ( w ) = fcy(7, 0; w) (7)
n(t) = nc(t) cos(2nfot) - ns(t) sin(2nfot)
= w k ( t ) + z n s ( t ) ) exp(z2Tfot)) ( 8 ) where nc(t) and n,(t) are the in-phase and quadrature components of the interference aud n,(t)+in,(t) is its corresponding complex amplitude. The joint PDF of the two components can be shown [8] to be the BISaS PDF of dispersion y, i.e.
From eqns. 4 and 9, one can see that the in-phase and quadrature components of narrowband S a S interference are not independent, except in the Gaussian case (a = 2) [8]. Moreover, the joint distribution of the two components is heavy tailed compared with the Gaussian case. Finally, transformation of the Fourier integral in eqn. 4 in radial coordinates (p = d[n? + E:], 8 = arctan(n,/nJ), gives the following joint PDF of the envelope p and the phase 8 of narrowband SUS interference:
f (%, n s ) = f a , y ( n c , n s ) (9)
- = ' f e p , , ( P ) ; P > 0,Q E [o, 2n) 2T
Eqn. 10 clearly shows that the phase 19 of narrowband S a S interference is uniformly distributed in the interval [O, 2 4 and independent of the corresponding interference envelope. These facts are in agreement with the corresponding results for the Gaussian case [8]; however, the envelope distribution is heavy tailed when compared with the Gaussian model, as seen from eqn. 6.
In Fig. 2 we show the amplitude probability
360
distribution (APD), i.e. Pr{A > a> of the amplitude A of BISaS random variables of unit dispersion and various characteristic exponents and compare them to the Gaussian distribution (dashed line).
APD of envelope of BISuS noise for unit dispersion and various u
2.4 Fractional, lower-order statistics of SaS and BlSaS processes Consider two real S a S or complex BISaS random vari- ables and y~ with zero location parameters and finite pth-order moments, where 0 < p < a. We define their pth-order fractional correlation [ 151 as
where (5, &J = 4 5 ( r l ) ( P - 1 9 (11)
(.)(~-1) I . I(P-l)sgn(.) (12)
(13)
for real-valued random variables and
for complex-valued random variables. In eqns. 12 and 13, sgn( ) denotes the signum function, whle the over- bar denotes complex conjugation, respectively. These definitions are clearly seen to reduce to the usual SOS and HOS in the cases where those exist and can be eas- ily extended to include random processes and their cor- responding fractional correlation sequences. For example, if {X,), k = 1, 2, 3, ..., is a discrete-time ran- dom process, one can define its fractional, pth-order correlation sequence as
(14) which, for p = 2, gives the usual autocorrelation sequence.
The FLOS of a random process have been found useful in designing algorithms that exhibit resistance to outliers and allow for robust processing of impulsive, as well as Gaussian, data. This property of FLOS- based algorithms will be again demonstrated for the differential-delay Doppler estimation problem addressed in this paper.
Apth-order random process {X,}, k = 1, 2, 3, ..., is called pth-order stationary if its corresponding pth- order correlation sequence pp(n, m) in eqn. 14 depends only on the difference 1 = m-n of its arguments. Sample averages can be used to define the FLOS of an ergodic stationary observed time series {X,}, k = 1, 2, 3, ..., similarly to ensemble averages
(.)(P-1) 3 I . l ( P - 2 ) U
pP(n,m) = (Xn ,Xm)p = & { X n ( X m ) ( p - l ) }
k=O
IEE Proc-Radar, Sonur Navig., Vol. 143, No. 6, December 1996
The basic properties of the FLOS of S a S and BISaS been used. Thus processes are summarised as (see [I51 for more details): P.l For any and c2, (%Cl +a2<2,& =a1(51,rl)p+a2(52,rl)p (16)
( C , r l ) P = 0 (17)
P.2 If 5 and 7 are independent,
while thLe converse is not true. P.3 If ql and q2 are independent, (a1771 +a2772,a1771 + a z r l z ) p = lalIp(771,771)p+ la2/P(772,772)p
(18) P.4 For a stationary pth-order random process {X,}, k = 1,2,3, ..., its pth-order correlation and the correspond- ing sample average satisfy
We return to the data collection problem in Section 1 and attempt to estimate the differential delays and Doppler shifts by using the measurements {yk(n); n = 0, 1, ..., N-1; k = 1, 2, ..., K}. Assume that the signal x(t) is a BISaS process of arbitrary and unknown depend- ence structure while the noises are BISaS processes, independent of the signal x(t) and of each other, but of otherwise arbitrary and unknown dependence struc- ture. Allso assume that both the signal and the noises have zero location parameters. If this assumption is violated, these location parameters need to be sub- tracted from the observations.
We propose ambiguity-function-based estimation methods rather than maximum-likelihood ones to over- come the otherwise required knowledge of the exact distribution of the signal and the noises. Moreover, the proposed algorithms rely on the use of FLOS and exhibit resistance to the presence of heavy-tailed noise. More specifically, for k = 2, 3 ,..., K, we define
Pyk,yl(nJ) = ( Y k ( n + ' T ) ,ydn) )p
= ( z (n + 'T - c ik)ezwb(n+T) , x (n ) ) ,
(4. + T - dk),x(n))p
+ ( W k ( 1 2 + T ) , W ( 4 ) p
- - e z w k ( n + T )
where we have invoked property P.2 (eqn. 17) for the noise terms. Similarly, for k = 2, 3, ..., K
- N-1
n=O N-1
= lim - 1 x - e z ( w L - a ) n ZwkT e (z(n + 7 - d k ) , .(.))p
(.(n i- T - & ) , z ( n ) ) , = &{z(n + T - dk)(z(TL))'"-l '}
N+cc N n=O But
- = E{Z(TL + 7 - dk)lz(n)l"-2.(n))
I E{ IZ (TL + 7- - dk)llz(n)l"-l>
- < &P{Jz(n + 7- - d,)""-"/"(lz(n)I")
5 E { 1 4 n ) l p }
where the Holder inequality (see also property P.4) has been applied and the stationarity of the signal x(n) has
, N-1
n=O is the sample fractional, lower-order cross-ambiguity function between the kth and the first sensor output.
4 Theoretical estimator justification and computer illustration
4.1 Theoretical justification The proposed estimate is justified from the following fact: Fact: If x(n) and wk(n), k = 2, 3, ..., K, are mutually independent BLSaS processes, for p < al2, the estimate &(a,z) converges to the true ambiguity function A,(a,z) in the mean-square sense, as N - 00.
Proof: From eqn. 23, 1 N-l
€{&(a, T ) } = - E { y k ( n + T)(yl(n))'"-l'e--zan
n=O ~ N-1
N
and
1 N
= --E{(z(n + T - d,)121z(n)l""-2]
where, to write the last two equations, we made use of the stationarity of the signal x(n). Thus, ~{I&a,z)1~} is the joint FLOS of several jointly SaS random variables and is therefore [15] finite if the sum of the moment orders is strictly less than the characteristic exponent, i.e. if 2 + 2p - 2 = 2p < a or p < al2. It follows from the law of large numbers [I61 that, as long as p < al2, A,(a,z) is a consistent estimator of ~{,(a,z) as N + 00
and additionally 4i'!4&(a,z)-Ak(a,z)] is asymptotically normal (Gaussian) of mean zero and variance E { ~x(n+~--d,)/~/x(yl)1~p-~}--l~~(a,z)l~. Subsequently, the estimated ( c o ~ , ~ , ) = argmax/Ak(a,z)l in eqn. 22 is a con- sistent estimator of (w,d).
4.2 Computer illustration For illustration, we have simulated the performance of traditional SO and the proposed FLO ambiguity func-
tions in three different experiments. In all three experi- ments we ran 500 Monte Carlo simulations 'and plotted the resulting average ambiguity functions. In all cases, the signal of interest was a 16-QAM communication signal [ l l ] embedded in various alpha-stable noises of high level. We considered K = 2 sensors available, N = 32 samples measured, a differential delay d = 5 (which corresponds to z = -5 in the ambiguity functions, the minus sign indicating a true delay from the first to the second sensor), and a Doppler shift w = xl2.
. . . , . . . .
L
i
C
-2
-4 0 10 20 30
Fi .3 nag and its delayed Doppler-sh$ed version at first sensor ~ real part _ _ _ - imaginary part
Typical realisation of noise-fvee ( a = 2, 2 = 128) 16-QAMsig-
301
20 301
..
0 10 20 30 Typical realisation of noisy ( a = 2, d = 128) 16-QAM signals Fi .6
an2 its delayed Donpler-shifted version ut second sensor - - ~ real part
imaginary part _ _ _ _
40 501 1 3 0
20
10
0
0
Fig.7 a = 2, y = 64
Ambiguity function of order p = 2 for first experiment: 3 -0 plot 2oi
-20 t -301
0 10 20 30 Typical realisation of noisy ( U = 2, d = 128) 16-QAM sigmls Fi .4
an2 its delayed Doppler-shifted version at first sensor ~ real part
imaginary part
1
1
c
--L
- 4 10 20 30
Fi .5 nag and its delayed Doppler-shifted version ut second sensor ~ real part
Typical realisation ofnoise-free (a = 2, d = 128) 16-QAMsig-
imaginary part _ _ _ _
I
d 2 t U I
First we assumed complex white Gaussian noises wl(n) and w2(n), independent of each other and both of variance d = 128. Typical realisations of the noise-free and the measured (noisy) signals at the two sensors are shown in Figs. 3-6, while Figs. 7-10 show the resulting average ambiguity functions of orders p = 2 and p = 0.8. From this simulation one can conclude that the SO ambiguity function has a slightly better performance than its FLO counterpart, however the degradation is minimal.
Next we assumed white BIS(a = 1.5)s noises wl(n) and w2(n), independent of each other and both of dispersion y = 15. Typical realisations of the noise-free
IEE Proc -Radar, Sonar Navig , Vol 143, No 6, December 1996 3 62
2 . 7 2.0
0
1.5
1.0
0.5
0
0
. . . . . . . . . . .
. . . . . . . . . . -;
Fig.9 Ambiguityfunction of order p = 0.8 forf;rst experiment: 3-D plot a = 2 , y=64
. . . . . . : i . . . . . .
. . . . . . . . . . . . . . . . I
and the measured (noisy) signals at the two sensors are shown in Figs. 11-14, while Figs. 15-18 show the resulting average ambiguity functions of orders p = 2 and p = 0.8. From this simulation one can conclude that the SO ambiguity function fails to estimate the differential-delay Doppler between the two signals and contains several artifacts. The FLO ambiguity function, on the other hand, is robust to the presence of outliers in the measurements and correctly estimates the differential-delay Doppler between the two signals.
1 0 10 20 30
Fi . I 1 na! and its delayed Doppler-shifted version at Jirst sensor
~ real part -~~~ imaginary part
Typical realisation of noiselfree ( a = 1.5, y = 15) 16-QAM sig-
Finally, we assumed white BIS(a = 1.1)s noises wl(n) and w2(n), independent of each other and both of dispersion y = 1.5. Typical realisations of the noise-free
IEE Proc.-Radar, Sonar Nuvig., Vol. 143, No. 6, December 1996
-30 0 10 20 30
Fi . I 2 an!! its delayed Doppler-shifted version at first sensor
Typical realisation of noisy ( a = 1.5, y = 15) 16-QAM signals .~
~ real part _ _ ~ _ imaginary part
0 10 20 30 Fi . I 3 naf! and its delayed Doppler-shifted version at second sensor __ real part _ _ ~ _ imaginary part
Typical realisation of noise-jiree (a = 1.5, y = IS) 16-QAM sig-
0 10 20 30 Fi . I 4 an9 its delayed Doppler-shifted version at second sensor __ real part ~ ~ _ _ imaginary part
Typical realisation oj noisy ( a = 1.5, y = 15) 16-QAM signals
and the measured (noisy) signals at the two sensors are shown in Figs. 19-22, while Figs. 23-26 show the resulting average ambiguity functions of orders p = 2 and p = 0.8. From this simulation one can conclude that again the SO ambiguity function fails to estimate the differential-delay Doppler between the two signals and contains more severe artifacts than in the previous case. The FLO ambiguity function, on the other hand, is again robust to the presence of outliers in the measurements and again correctly estimates the differential-delay Doppler between the two signals.
363
100
50
0
0
Fig. 15 3-0 plot a = 1.5, y = 15
Ambiguity function of order p = 2 for second experiment:
. . . .
., . . .
. . . . . .
V
' 0 U 0 0 0 0 0
. . . . .
-30 -20 -10 0 10 20 30 delay T
Fig.16 Ambiguity Jidnction of order p = 2 for second experiment: 2-0 plot a = 1.5, y = 15
I 4 I 2
0
0
Fig. 17 Ambiguity function of order p = 0.8 for second experiment: 3-0 plot a = 1.5, y = 15
'1
0
. . . .
-30 -20 -10 0 10 20 30 delay T
Fi . I 8
a = 1.5, y = 15
Ambiguity fwlction of order p = 0.8 for second experiment: 2-8 plot
I
0 10 20 30 Fig.19 signals and its delayed Doppler-sh fted version at first sezsor ~ real part
30
Typical realisation of noise-free ( a = 1.1, y = 1.5) 16-QAM
_ _ _ _ imaginary part
- * O r
0 10 20 30 Fig.20 and its delayed Doppler-sh f ledersiox at first sensor __ real part
Typical realisation o nolsy ( a = 1.1, y = 1.5) 16-QAMsignals
- _ _ _ imaginary part
-41 0 10 20 30
Fig.21 Typical realisation of noise-fvee ( a = 1.1, 1.5) 16-QAM signals and its delayed Doppler-shified version at SecondYSsor
~ real part imaginary part - _ _ _
2o t
-101 i
I 0 10 20 30 Typical realisation ofnoisy ( a = 1.1, y = 1.5) 16-QAMsignals Fig.22
and its delayed Doppler-shifted version at second sensor ~ real part - - - - imaginary part
Ambiguity function of orderp = 2 for third experiment: 3-D plot
delay T Fig.24 a = 1.1, y = 1.5
Ambipityfunction of order p = 2 for third experiment: 2-D plot
’O1
6 8l
.. Fig.25 Ambiguity function of order p = 0.8 for third experiment. 3-0 dot a = 1.1, y = 1.5
. . . . .
0
1 -20 -10 0 10 20 3 delay T
Fig.26 Ambiguity function of order p = 0.8 for third experiment: 2-0 plot a = 1.1, y = 1.5
We have extended the concepts of SO and HO ambiguity functions to the domain of FLOS to define statistical signal processing tools for differential-delay Doppler estimation that are resistant to severe outliers in observed time series. We obtained asymptotically consistent estimates of the differential-delay Doppler parameters by seeking the maximum of the sample FLOS-based cross ambiguity function and compared the performance of the proposed algorithm to that of its SO counterpart. We found that the proposed algorithm outperforms the SO one even in slightly impulsive environments. Future research in this area seems to be in order, especially in the direction of extension of the present results to wideband signal processing applications. Two different approaches that seem promising in carrying this extension out include: distributed narrowband processing, and combination of wavelet transforms and FLOS. Both approaches are presently being investigated and extensively compared to derive their relative merits. The results of that investigation are expected to become available shortly and will be announced elsewhere.
6 References
1
2
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