Level Crossing Rate of Sea Clutter: Theory
and Measured Data
M. Farshchian, A. Abdi and F. Posner
The contribution of this letter is to propose, and compute the level crossing
rate (LCR), average calm duration (ACD) and average surge duration
(ASD) in the context of radar clutter and false alarm analysis. The derived
formulas for the LCR, ACD and ASD are then compared to their measured
values calculated from experimental radar sea clutter and are shown to be
in good agreement.
Introduction: Sea clutter displays spiky behaviour at the low grazing
angles [1] [2] that can seriously interfere with the radar’s primary objective
with regard to the intended target. Posner and Gerlach empirically studied
sea spikes by defining a threshold amplitude for sea clutter spikes and
looking at the expected number of spikes per second, the mean spike
duration and the mean interval between spikes [2]. In this letter, we obtain a
theoretical formulation by deriving these three parameters through the level
crossing rate (LCR) [3], average surge duration (ASD) and average calm
duration (ACD) functions. As explained, these three functions can also be
used to study the duration of false alarms. Watts [4] studied the duration
of false alarms in noise based upon the approximate excursion shape while
the method used in this letter is based directly on the level crossing rate
[3]. As far as we are aware, the results presented here for sea clutter have
not been previously reported. It should be noted that the ACD is called the
“average fade duration” in wireless communications. Similarly, the ASD
has also been called the “average false alarm duration” [4], however, the
ASD can also have meaning in the context of radar targets which is not
discussed here.
Since the amplitude of our data fits the Weibull distribution, the LCR,
ACD and ASD of the radar clutter are studied for a Weibull stochastic
process, although the ideas in this paper are equally valid for other popular
radar stochastic models that are used to model radar noise and clutter. The
Weibull distribution itself has been used extensively for describing land
and sea clutter [5]. However, the focus of this letter is not the amplitude
distribution and the formulations for LCR, ACD and ASD are equally
applicable to other amplitude distributions [6].
measures the average duration of sea spikes and χ̌λ {ξ(t)} measures the
average duration between sea spikes.
A correlated Weibull process Z(t) with an autocorrelation function
(ACF) κ(τ ), shape factor β and scale factor Ω may be obtained from
two independent and identically distributed Gaussian processes Xi (t)
(i = 1, 2) with zero mean, variance σ 2 = Ωβ /2 and ACF ρ(τ ) [8]:
Z(t) = {X12 (t) + X22 (t)}1/β
The LCR Nλ {Z(t)} of the Weibull process can be obtained in closed form
as:
" #
r β/2
β
ρ¨0 λ
λ
Nλ {Z(t)} =
exp −
(5)
π
Ω
Ω
where ρ¨0 is given by:
ρ¨0 = −
∂ 2 ρ(τ )
1
|τ =0 =
∂τ 2
2πσ 2
Z∞
ω 2 S(ω) dω
(6)
−∞
and S(ω) is the Fourier transform of ρ(τ ).
The derivation of Equation (5) is similar to the method of [7], although
it contains an arbitrary ACF rather than the Jake’s model used in [7]. ACF
mapping between κ(τ ) and ρ(τ ) was not considered in [7]. The value of
κ(τ ) as a function of ρ(τ ) is complicated and is given in [8]. In [9], an
affine relationship between κ(τ ) and ρ(τ ) for a specific β is proposed:
ρ(τ ) ' k0 + k1 κ(τ )
(7)
where ki is a function β . This relationship is accurate for values of β
between 0.6 to 2 that are typical of sea clutter.
The analysis presented so far is for stationary clutter. One way to model
non-stationary clutter is to allow random parameters for the ACF [10]. The
LCR for non-stationary clutter can be calculated by integrating (5) with
respect to the PDFs of the random parameters of the ACF and the Weibull
distribution itself, e.g., random shape factor. For example, assuming a
Gaussian power spectral density (PSD) that has random variance γ to
model the short-term temporal variations of the sea clutter Doppler spectra
[10], Nλ {Z(t)} can be written as:
β/2
Z∞ r
2
2k1 γ
π
Nλ {Z(t)} =
0
Level Crossing Rate: For a threshold λ, the LCR Nλ {ξ(t)} is by definition
the average number of times per second that the signal ξ(t) crosses a
specific level λ with positive slope [3] and is calculated by:
Z∞
Nλ {ξ(t)} =
(1)
ξ̇pξξ̇ (λ, ξ̇) dξ̇
(4)
" exp −
λ
Ω
λ
Ω
β #
pγ (γ)dγ
(8)
Consequently, the LCR can be extended to non-stationary clutter.
0
where the prime denotes time derivative and pξξ̇ (λ, ξ̇) is the joint
probability density function (PDF) of ξ(t) and ξ̇(t). From the LCR, two
other functions can be derived. The ASD of ξ(t), denoted by χ̂λ {ξ(t)},
signifies the average duration in seconds that ξ(t) stays above the level λ.
The ACD of ξ(t), denoted by χ̌λ {ξ(t)}, signifies the average duration in
seconds in which the signal stays below a certain level λ. The formula to
calculate the ASD is given by
χ̂λ {ξ(t)} =
1 − Fξ (λ)
Nλ {ξ(t)}
Experimental Results: Since the sea clutter is not stationary, in order to
compare the theoretical equations with experimental results, the PSD is
estimated over J blocksizes of integer samples. Subsequently, for the
experimental results, the LCR is calculated by using Equations (5) and
(7) and allowing for the variation of the shape and scale parameters over
the samples of the PSD estimate:
v
R∞
J u
k1i −∞ ω 2 ŜZi (ω) dω
1 Xu
t
Nλ {Z(t)} '
R∞
J
(2)
The formula for ACD is given by:
Fξ (λ)
χ̌λ {ξ(t)} =
Nλ {ξ(t)}
(3)
where Fξ (y) is the cumulative distribution function (CDF) of ξ(t).
For this letter, we have confined our analysis to individual fixed radar
range-cells evolving in time. However, the concept of LCR, ASD and
ACD can be extended to the multi-dimensional case which is not covered
in this letter. In the context of false alarm analysis, Nλ {ξ(t)} signifies
the average number of excursions in the false alarm region per second,
χ̂λ {ξ(t)} signifies the mean duration in seconds that the signal stays in
the false alarm region and χ̌λ {ξ(t)} denotes the average time that the
signal stays below the false alarm threshold. The inverse of the LCR is the
same as the false alarm time [4]. In a region with high sea clutter power
return, assuming that exceedence above a λ is considered a sea spike [2],
Nλ {ξ(t)} measures the average number of spikes per second, χ̂λ {ξ(t)}
ELECTRONICS LETTERS
12th January 2012
λ
Ωi
i=1
βi /2
π
−∞
" exp −
ŜZi (ω) dω
λ
Ωi
βi #
(9)
where ŜZi is an estimate of the PSD at the ith block index. It should be
noted that the PSD as well as the Weibull parameters depend heavily on
the environmental (e.g. sea state) and radar parameters (e.g. resolution).
Consequently, we have numerically estimated these while noting that their
PDFs can be substituted as was done in Equation (8). The dataset is
CFC17-001 provided by CSIR and was part of the study in [10]. The pulse
repition frequency (PRF) of the data is 5000 kHz, its center frequency
9 GHz, its range resolution is 15 meters, its grazing angle 1 degrees
and its sea state between three to four. The block lengths for our PSD
estimates were set to 29.5, 3.28, 0.82, 0.20, and 0.051 seconds. The PSDs
were estimated through the periodogram with a Hamming weight.
In Figure 1, the actual LCR, ACD and ASD are plotted relative to
those predicted by Equation (9). The theoretical ASD and ACD were
calculated by using Equation (9) in Equations (2) and (3). The measured
LCR, ACD and ASD values are shown for a 0.1 threshold stepsize. In
Figure 2, the Weibull CDF was fit to the envelope of the data using the
Vol. 00
No. 00
3
10
0.999999
0.99999
0.9999
0.999
actual
29.5 sec
3.28 sec
0.82 sec
0.20 sec
0.051 sec
2
10
1
10
0.96
0.90
0.75
0
CDF
10
LCR
Actual CDF
Weibull Fit
0.99
−1
0.50
10
0.25
−2
10
−3
10
0.10
−4
10
0
0.5
1
1.5
2
λ
2.5
3
3.5
0.05
4
ASD
−3
10
−4
10
−5
10
−2
−1
10
0
10
λ
4
10
actual
29.5 sec
3.28 sec
0.82 sec
0.20 sec
0.051 sec
2
10
0
ACD
10
−2
10
−4
10
−6
10
−4
10
−3
10
−2
10
−1
λ
10
0
10
2
λ
3
4
5
1 Ward, K.D., Tough, R.J.A., and Watts. S: ‘Sea Clutter: Scattering,the K
Distribution and Radar Performance’ (IET, 2006)
2 Posner, F., and Gerlach, K.: ‘Sea spike demographics at high range
resolutions and very low grazing angles’, Proceedings of the 2003 IEEE
Radar Conference, pp. 38-45, 2003.
3 Rice, S. O.: ‘Mathematical analysis of random noise’, Bell Syst. Tech., vol.
23, pp. 282-332, July 1944 and vol. 24, pp. 46-156, Jan. 1945.
4 Watts, S: ‘Duration of radar false alarms in band-limited Gaussian noise’,
IEE Proceedings of Radar, Sonar and Navigation, vol. 146, no. 6, pp. 273278, Dec 1999.
5 Long, M.W.M.: ‘Radar Reflectivity of Land and Sea’ (Artech House, 3rd
edition, 2001)
6 Abdi, A., Wills, K., Barger, H.A., Alouini, M.S., and Kaveh, M.:
‘Comparison of the level crossing rate and average fade duration of
Rayleigh, Rice and Nakagami fading models with mobile channel data’,
Vehicular Technology Conference, vol. 4, pp. 1850-1857, 2000.
7 Sagias, N.C., Zogas, D.A., Karagiannidis, G.K., and Tombras, G.S.:
’Channel capacity and second-order statistics in Weibull fading’,
Communications Letters, IEEE, vol. 8, no. 6, pp. 377-379, June 2004.
8 Gang, L. and Yu, K.: ‘Modelling and simulation of coherent Weibull clutter’,
IEE Proceedings, vol. 136, no. 1, pp. 2-12, Feb 1989.
9 Farina, Russo, A., and Scannapieco, F.: ‘Radar detection in coherent Weibull
clutter’, IEEE Transactions on Acoustics, Speech and Signal Processing,
vol. 35, no. 6, pp. 893-895, Jun 1987.
10 Watts, S.: ‘A new method for the simulation of coherent sea clutter’,
Proceedings of the 2011 IEEE Radar Conference, pp. 52-57, 2011.
−2
10
10
1
Fig. 2: Actual and Estimated CDF β = 1.05 and Ω = 0.42.
actual
29.5 sec
3.28 sec
0.82 sec
0.20 sec
0.051 sec
−1
10
0
1
10
Fig. 1: The actual and estimated LCR, ASD and ACD
method of moments. From Figure 1, it can be seen that (9) calculates the
LCR and ACD accurately for the clutter region. The blocklength of 0.82
seconds provided the most accurate fit for the ASD and LCR. This suggests
that amongst the blocklengh sizes used, 0.82 provides the best trade-off
between minimizing the variance of the PSD estimate and capturing the
essential non-stationarity features of the clutter signal that is needed to
calculate the LCR. However, the results for all block length are close for
most regions of the envelope. The ASD, plotted in Figure 1, was calculated
accurately up to a threshold value of 2.2. This roughly corresponds to a
probably of false alarm value of 0.001. However, beyond this threshold, the
ASD falls of much slower than that predicted by the theoretical equation. A
possible explanation of this phenomenon is that not enough samples were
used to accurately calculate the ASD. For example, the measured ASD
value at a threshold level of 4.0 was greater than that of threshold 3.9.
Similarly, the measured ASD value at a threshold of 3.8 was greater than
that of 3.7. Despite this, the ASD is within less than half the magnitude for
the block length of 0.82 seconds for all threshold values.
Conclusion: We studied the LCR, ACD and ASD functions in the context
of radar sea clutter and false alarm analysis. Theoretical results were
derived and applied accurately to experimental sea clutter. One application
of these functions is the calculation of the false alarm duration and rate
[4] in the clutter region. For the high clutter region, this corresponds to the
spike duration and the number of spikes per second. Another application
would be measuring the fidelity of synthetic clutter generation models by
calculating these functions for the synthetic clutter models and measuring
them against real clutter data.
M. Farshchian and F. Posner (Naval Research Laboratory, USA)
A. Abdi (New Jersey Institute of Technology, USA)
References
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