Ionospheric Clutter Model for High Frequency Surface Wave Radar Maryam Ravan and Raviraj S. Adve Department of Electrical and Computer Engineering University of Toronto Toronto, Ontario, Canada Email:
[email protected] and
[email protected]
Abstract—The detection performance of a high frequency surface wave radar (HFSWR) system is primarily limited by clutter, especially ionospheric clutter. Therefore, in order to analyze and/or simulate the capabilities of an HFSWR system a model for the clutter is required. This paper develops and tests a new radio wave propagation theory to model ionospheric clutter. The model is based on path integrals of ray tracing equations which predict the phase power spectrum of clutter due to irregularities in the plasma. This power spectrum is then used to simulate three-dimensional space-time-range radar data cubes. The accuracy of the model is tested by comparing the simulated data to measured data cubes. As an application, the data is then used to evaluate the performance of the newly developed fast fully adaptive processing scheme to mitigate clutter.
I. I NTRODUCTION Ionospheric irregularities have a considerable effect upon the radio waves that propagate through them. Consequently high frequency surface wave radars (HFSWRs) can also be affected by the presence of strong irregularities. For successful detection, an HFSWR needs its targets to be well separated from clutter (unwanted returns) in Doppler domain. However, the clutter tends to spread through Doppler space and, hence, has the potential to mask targets. As a consequence there is a need for a quantitative model of the spatial properties of ionospheric clutter. This would help quantify the impact of the ionosphere and the performance of clutter mitigation techniques such as space time adaptive processing (STAP) techniques applied to radar systems. For example, one approach to evaluating clutter mitigation schemes is to compare probability of detection versus signal-to-noise ratio (SNR) curves based on a common false alarm rate. However, it is rare to have at hand adequate measured data to accurately estimate these parameters. Therefore, a quantitative space-time model of the clutter is needed to generate enough data for this purpose. Furthermore, the clutter returns themselves provide a valuable tool for investigating the ionosphere, and a theory of clutter spreading is essential in unraveling the data that is produced by the radar. In the HFSWR configuration, a radar waveform is transmitted along the surface of the earth, typically over the ocean, and the signal diffracts around the curvature of the earth to The authors would like to acknowledge the use of measured data provided by Defence Research and Development Canada.
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illuminate ships and low-flying aircraft targets beyond the line-of-sight horizon. However, during transmission, a certain amount of radiation, inevitably, propagates in the vertical direction and reflects off the ionosphere. This reflection from the ionosphere is observed as intense radar clutter. In turn, this clutter imposes detection range limitations in practical long range HFSWR systems. These limitations are largely due to irregularities in the ionospheric plasma. Based on the published literature, there are two different approaches to model ionospheric clutter, stochastic signal processing and radio wave propagation. For example, the work in [1] for the Jindalee radar operated by the Defense Science and Technology Organization (DSTO) in Australia is based on a Gaussian assumption. In [2], [3] Sevgi et al. also use a Gaussian assumption to develop a simulation package called HFSIM. The theory is based on the work in [4]. Another stochastic model is that of [5]. In this study, we develop and implement a model based on the theory of HF radio wave propagation in the earth’s ionosphere that accounts for the effects of ionospheric plasma density irregularities. The closest scheme available in the literature is that of Coleman [6], [7] for OTHR systems. This model determines the Doppler variance of downrange ground clutter after propagating along a sky wave path. In our theory, a radio signal is transmitted from the ground toward the ionosphere, undergoes reflection in the ionosphere, and returns to the ground receiver. The theory determines the spatial-temporal phase spectrum of the ionospheric echoes. To test the theory, we use the model to simulate radar data. This data is compared to measurements of ionospheric clutter. Finally, as an application of the model, we use simulated data to test the efficacy of a newly developed clutter mitigation STAP algorithm; named Fast Fully Adaptive (FFA) algorithm [8]. II. M ODELING I ONOSPHERIC C LUTTER In this section, we implement and examine a radio wave propagation approach to model ionospheric clutter in the context of HFSWR systems. This method, first reported in [9], [10] uses a ray tracing model and treats irregularities as perturbations of a quiescent path solution without irregularities.
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In this paper we briefly describe this method and compare simulation results based on this method to measured data. The model of [9] develops a theory of High-Frequency (HF) radio wave propagation in the earth’s ionosphere that accounts for the effects of ionospheric plasma density irregularities. The model assumes an anisotropic ionospheric plasma that is inhomogeneous in the vertical direction. The effects of the irregularities are modeled as small perturbations to the quiescent solution. The perturbations are evaluated for the case of random plasma density irregularities with a power-law wave number spectrum, which leads to the predicted phase power spectrum of the signal properties as a function of wave number and frequency. A. Path integral formulation The model is developed in some detail in [9], [10] and we describe it briefly here. The plasma dispersion relation is written as an implicit function 𝐺(r, 𝑡, k, 𝜔) = 0 (where r is the ray path, 𝑡 is the time, k is the wave number, and 𝜔 is the angular frequency) that accounts for slow spatialtemporal variations in 𝜔𝑝𝑠 = 𝑞𝑠2 𝑁𝑠 /𝜀0 𝑚𝑠 (plasma frequency of species 𝑠 (oxygen ions or electrons) ) and 𝜔𝑐𝑠 = ∣𝑞𝑠 ∣ 𝐵0 /𝑚𝑠 (cyclotron frequency of species 𝑠), where 𝑚𝑠 is the mass of the species, 𝑁𝑠 is the species density, 𝑞𝑠 is the species charge, 𝜀0 is the permittivity of free space, and 𝐵0 = 𝐵0 𝑧ˆ is the magnetic flux density which is assumed to be in the 𝑧ˆ direction. As the wave propagates through the plasma, it must always satisfy the plasma dispersion relation such that 𝐺 = 0 along the entire wave trajectory in (r, 𝑡, k, 𝜔)-space. If 𝜏 is a variable parameterizing this trajectory, and 𝐺 is always identically zero along this trajectory, then 𝑑𝐺/𝑑𝜏 = 0: ∂𝐺 𝑑r ∂𝐺 𝑑𝑡 ∂𝐺 𝑑k ∂𝐺 𝑑𝜔 𝑑 𝐺(r, 𝑡, k, 𝜔) = + + + = 0. 𝑑𝜏 ∂r 𝑑𝜏 ∂𝑡 𝑑𝜏 ∂k 𝑑𝜏 ∂𝜔 𝑑𝜏 (1) The radar pulse is considered as a wave packet in the form ∫ ∫∫∫ 1 𝐸(k, 𝜔)𝑒𝑖(k.r−𝜔𝑡) 𝑑k𝑑𝜔. (2) 𝐸(r, 𝑡) = (2𝜋)2 If the spatial variation of 𝐸(k, 𝜔) is slow compared to 2𝜋/ ∣k∣, and the temporal variation is slow compared to 2𝜋/𝜔, then constructive interference occurs when the integrand phase is a constant. Differentiating this phase with respect to t and k, and equating to zero, yields:
∂𝐺 𝑑𝜔 =− . (7) 𝑑𝜏 ∂𝑡 A more explicit description can be achieved by assuming that the medium is plane-stratified, such that 𝜔𝑝𝑠 and 𝜔𝑐𝑠 vary only with altitude. The wave packet properties can be written as path integrals of (4)–(7) [10]. For instance, the direction of arrival (DOA) is ∫ ∂𝑘𝑧 𝑑𝑧. (8) Δk = ∂r The background plasma density is denoted as 𝑁0 where the irregularity density is denoted as 𝑁1 , such that the total plasma density is 𝑁 = 𝑁0 +𝑁1 , and 𝑁1 has zero mean. The perturbed wave number is given by: ∂𝑘𝑧 ∣𝑁 ≡ 𝑘𝑧0 + 𝑘𝑧1 . (9) ∂𝑁1 0 Using (9), the first-order density perturbation to (8) can therefore be approximated by [9], [10] ∫ ∂𝑁1 ∂𝑘𝑧 .𝑑𝑧, (10) Δk1 = 𝑁1 ∂r ∂𝑁 𝑘𝑧 (𝑁 ) = 𝑘𝑧 (𝑁0 ) + 𝑁1
1 We take the quantities 𝑁1 and ∂𝑁 ∂r to be zero-mean random variables, thus the mean of Δk1 is also zero.
B. Second-order statistics We can write (10) in the form ∫𝑧 ℎ(𝑥, 𝑦, 𝑧, 𝑡) = 𝑔(𝑥, 𝑦, 𝑧 ′ , 𝑡) 𝑓 (𝑧 ′ ) 𝑑𝑧 ′ ,
(11)
0
where f is deterministic and g is random. Using (11) we can write the power spectrum of h as: 𝑆ℎ (𝜅𝑥 , 𝜅𝑦 , 𝜅𝑧 , Ω, 𝑧) = ∫𝑧 2𝜋𝛿(𝜅𝑧 ) 𝑓 2 (𝑧 ′ )𝑆𝑔 (𝜅𝑥 , 𝜅𝑦 , 0, Ω, 𝑧 ′ ) 𝑑𝑧 ′ .
1 where 𝑔 denotes the quantity ∂𝑁 ∂r . The spectrum of this quantity is related to the spectrum of 𝑁1 by the identities
𝑆 ∂𝑁1 = 𝜅2𝑥 𝑆𝑁1
(13)
𝜅2𝑦 𝑆𝑁1
(14)
𝑆 ∂𝑁1 = 𝜅2𝑧 𝑆𝑁1
(15)
∂𝑥
𝑆 ∂𝑁1 = ∂𝑦
∂𝑧
∂𝜔 ∂𝐺/∂k 𝑑r/𝑑𝜏 𝑑r = =− = . 𝑑𝑡 ∂k ∂𝐺/∂𝜔 𝑑𝑡/𝑑𝜏
(3)
So we have
𝑑r ∂𝐺 = . 𝑑𝜏 ∂k If the arbitrary parameter 𝜏 is defined such that
(4)
∂𝐺 𝑑𝑡 =− , 𝑑𝜏 ∂𝜔
(5)
∂𝐺 𝑑k =− , 𝑑𝜏 ∂r
(6)
then from (3) we have
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(12)
0
As for 𝑆𝑁1 itself, we use a spectrum model for plasma irregularities that follows a 4th-order power law [11], and a dispersion relation for approximately perpendicular propagating drift wave turbulence [12]. The form is given by √ 4 2𝛼𝜋 2 𝐸[𝑁12 (𝑧)]𝑘0−3 𝛿(∣Ω∣ − 𝜅⊥ 𝑣𝑑 ) (16) 𝑆𝑁1 (𝜅, Ω, 𝑧) = 1 + 𝑘0−4 (𝜅2⊥ + 𝛼𝜅2∥ )2 Here we use the notation (𝜅, Ω) instead of (𝑘, 𝜔), to distinguish the wavenumber and frequency of the spectrum of plasma irregularities from the wavenumber and frequency of the propagating wave. 𝑣𝑑 is the plasma diamagnetic diamagnetic drift velocity, 𝑘0 is the outer scale length parameter, 𝛼 is an
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anisotropy parameter, 𝜅⊥ is the magnitude of the component of the density irregularity wave number 𝜅, that is perpendicular to the earth’s magnetic field, 𝜅∥ is the magnitude of the component of 𝜅 along the field, and 𝐸[𝑁12 (𝑧)] is the variance of the density fluctuations which is assumed to be a function of altitude 𝑧. We suppose that the magnetic field of the earth follows a unit vector ˆl = (𝑙𝑥 , 𝑙𝑦 , 𝑙𝑧 ). Therefore, the quantities 𝜅∥ and 𝜅⊥ are given by 𝜅 ∥ = 𝜅 𝑥 𝑙𝑥 + 𝜅 𝑦 𝑙𝑦 + 𝜅 𝑧 𝑙𝑧 𝜅⊥ = ∣𝜅 − 𝜅∥ˆl∣
(17) (18)
By assuming that the magnetic field lies in the y-z plane (i.e. 𝑙𝑥 = 0), we can evaluate 𝑆𝑁1 for 𝜅𝑧 = 0 as: √ 4 2𝛼𝜋 2 𝐸[𝑁12 (𝑧)]𝑘0−3 𝛿[∣Ω∣−(𝜅2𝑥 +𝑙𝑧2 𝜅2𝑦 )1/2 𝑣𝑑 ] 𝑆𝑁 1 = 1 + 𝑘0−4 [𝜅2𝑥 + (𝑙𝑧2 + 𝛼𝑙𝑦2 )𝜅2𝑦 ]2 (19) We consider only the first-order phase perturbation due to irregularities in the ionosphere, which is related to the wave number perturbations by ∂𝜙1 , (20) Δ𝑘𝑥1 = ∂𝑥 ∂𝜙1 . (21) ∂𝑦 Hence, the wave number spectrum are related to the phase spectrum by (22) 𝑆Δ𝑘𝑥1 = 𝜅2𝑥 𝑆𝜙1 , Δ𝑘𝑦1 =
𝑆Δ𝑘𝑦1 = 𝜅2𝑦 𝑆𝜙1 .
(23)
Combining (10), (19), (22), and (23), we have that the phase spectrum is 𝑆𝜙1 (𝜅𝑥 , 𝜅𝑦 , 𝜅𝑧 , Ω, 𝑧) = 𝑏 (𝑧) 𝛿(𝜅𝑧 )𝛿([∣Ω∣ − (𝜅2𝑥 + 𝑙𝑧2 𝜅2𝑦 )1/2 𝑣𝑑 ] 1 + 𝑘0−4 [𝜅2𝑥 + (𝑙𝑧2 + 𝛼𝑙𝑦2 )𝜅2𝑦 ]2
where ∫𝑧 𝑏(𝑧) = 𝑎
𝐸[𝑁12 (𝑧 ′ )]
0
=
𝐸(𝜙21 )[16𝜋 2
(
√
∂𝑘𝑧 ∂𝑁
)2
𝑑𝑧 ′
(𝑙𝑧2 + 𝛼𝑙𝑦2 )]
𝑘02
,
(24)
(25)
,
C. Modeling the Space-Time-Range Data Cube Here we use the spectrum modeled by (26) to simulate a space-time-range data cube akin to a data cube measured at such a radar. The data cube represents the composite of 𝑁 -dimensional antenna array snapshot X(𝑟, 𝑚, 𝑛) (𝑛 = 0, 1, ..., 𝑁 − 1) recorded at the 𝑟𝑡ℎ range bin in the 𝑚𝑡ℎ Pulse Repetition Interval (PRI). To generate the data cube with phase power spectrum of (26), we consider a range bound in the 𝑧 direction which begins from the distance to the first ionospheric range bin 𝑧0 and calculate the value of 𝑏(𝑧) for each range as 𝑏(𝑧) = 𝑏(𝑧0 ) + 𝛽(𝑧 − 𝑧0 ).
(27)
where 𝛽 is a constant coefficient which must be chosen properly to have an accurate result. For each range bin, we create a two-dimensional (array) (element-pulse) signal by filtering white noise through a two-dimensional LTI filter; the impulse response of this filter is obtained by taking the square root of the phase spectrum described in (26), followed by an inverse Fourier transform. Also, because the value of plasma diamagnetic drift velocity, 𝑣𝑑 , changes slightly in different ranges, we consider 𝑣𝑑 as 𝑣𝑑 = 𝑣𝑑0 ±𝑑𝑣 where 𝑑𝑣 is a uniform random variable. The measured data sets against which our data model is compared used a eight-segment Frank code that minimizes the effect of high range sidelobes and multiple-time round clutter. To mimic the measured data sets, we consider the same Frank code with eight segments as the transmitted signal waveform [13]. The actual code phase sequences used are: 1. {0, 0, 0, 0, 0, 0, 0, 0} 2. {−𝜋/4, −𝜋/2, −3𝜋/4, 𝜋, 3𝜋/4, 𝜋/2, 𝜋/4, 0} 3. {−𝜋/2, 𝜋, 𝜋/2, 0, −𝜋/2, 𝜋, 𝜋/2, 0} 4. {−3𝜋/4, 𝜋/2, −𝜋/4, 𝜋, 𝜋/4, −𝜋/2, 3𝜋/4, 0} 5. {𝜋, 0, 𝜋, 0, 𝜋, 0, 𝜋, 0} 6. {3𝜋/4, −𝜋/2, 𝜋/4, 𝜋, −𝜋/4, 𝜋/2, −3𝜋/4, 0} 7. {𝜋/2, 𝜋, −𝜋/2, 0, 𝜋/2, 𝜋, −𝜋/2, 0} 8. {𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, −3𝜋/4, −𝜋/2, −𝜋/4, 0} and the radar transmitted signal at complex baseband for the 𝑖𝑡ℎ code segment is represented by: 𝑃𝑖 (𝜏 ) = 𝐴(𝜏 )[cos(𝜙𝑖 (𝜏 ))+𝑗 sin(𝜙𝑖 (𝜏 ))] 𝑖 = 1, 2, ..., 8. (28)
for 0 < 𝜏 ≤ 𝑇 , and 𝑃𝑖 (𝜏 ) = 0 for 𝜏 > 𝑇 , where 𝑇 is the segment length. To implement the phase codes, the segment length, 𝑇 , is divided in to eight subintervals. In each subinterval, the phase, 𝜙𝑖 (𝜏 ), is set to one of the elements of the 𝑖𝑡ℎ segment. These eight codes are transmitted sequentially. The matched filter (MF) response is obtained by processing the returns of the eight waveforms with their replicas and coherently summing the results. 2𝑏 (𝑧) ∣Ω∣ √ (26) 𝑆𝜙1 (𝜅𝑥 , Ω, 𝑧) = In the measurements, each of the eight phase code segments (𝑙𝑧 𝑣𝑑2 ) (Ω2 /𝑣𝑑2 − 𝜅2𝑥 ) in the transmitted pulse is 55 𝜇s long, and thus each segment 1 . × is 440 𝜇s long. The pulses are emitted every 8 ms, thus the (1 + 𝑘0−4 [𝜅2𝑥 + (𝑙𝑧2 + 𝛼𝑙𝑦2 )(Ω2 /𝑣𝑑2 − 𝜅2𝑥 )/𝑙𝑧2 ]2 ) complete code is emitted every 64 ms and the pulse repetition This spectrum is nonzero in the region ∣Ω∣ > 𝜅𝑥 𝑣𝑑 . frequency (PRF) of the data set is 1/64 ms=15.625 Hz. There
√ 𝑎 = 8 2𝛼𝜋 3 𝑘0−3 , and 𝐸(𝜙21 ) is the variance of the phase. This phase spectrum is four-dimensional (𝜅𝑥 , 𝜅𝑦 , 𝜅𝑧 , Ω). However, an HFSWR with a linear array can only resolve two of these dimensions, which are the 𝜅𝑥 and Ω dimensions for the case of an East-West receive array. By integrating out the 𝜅𝑦 and 𝜅𝑧 dimensions in the East-West receive array case, we have
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TABLE I L IST OF MODELING PARAMETERS IN (26) AND (27)
100 50
Values 102 − 103 √1/2 3/2 (50 ± 𝑑𝑣) 10−4 m−1 [−0.25, 0.25]/m 50 km 3000 −103 90 m
95 40
Range number
Parameters 𝐸(𝜙21 ) 𝑙𝑦 𝑙𝑧 𝑣𝑑 𝜅0 𝑘𝑥 𝑧0 𝛼 𝛽 𝜆
90 30 85 20 80 10 75
0 −0.8
−0.6
−0.4
−0.2 0 0.2 Doppler (Hz)
0.4
0.6
0.8
(a)
∫ 𝑆(𝜁) =
110 260 105 100
250
Range number
are 4096 of these pulses, so the data set is 262.144 seconds long. The antenna array contains 16 antenna elements that are separated by 33.33 m from each other. To consider the effect of transmitted signal, we first calculate the ambiguity function of transmitted signal as:
95 240 90 230
∞ −∞
𝑄(𝜏 )𝑄∗ (𝜁 − 𝜏 )𝑑𝜏, 𝑄 =
8 ∑
𝑀 𝐹 (𝑃𝑖 ). (29)
𝑖=1
and then convolve the resulting signals of different range bins with this ambiguity function to create a data cube. To simulate the random bias, △𝑓 , in Doppler frequency seen in the measured data cube, we then multiply the 2D pulse-range signal for all antenna elements by 𝑒𝑗2𝜋△𝑓 𝑡 in the time domain with the time variable 𝑡 covering the appropriate range.
85 80
220
75 210
−0.8 −0.6 −0.4 −0.2 0 0.2 Doppler (Hz)
0.4
0.6
0.8
(b) Fig. 1. Doppler-range plot of the (a) simulated and (b) measured data for the antenna element 12.
III. DATA P ROCESSING D. Simulation Results This section presents the results of simulations testing the theory developed in the previous sections. Table I shows the values considered for the parameters in (26) and (27) to simulate the data cube. The parameter △𝑓 , which shows the random bias in Doppler frequency, is considered as a uniform random variable between −0.4 and 0.4 and 𝑑𝑣 in the third row of Table I is considered as a uniform random variable between -10 and 10. Also, the amplitude of the simulated signal is multiplied by 200 to have the same scale as the measured signal. Figures 1(a) and (b) plot the 2D range-Doppler plot of the simulated and measured data for 12𝑡ℎ antenna element, respectively. The ionospheric clutter, that is of interest here, span range bins 210 - 270 (the last range bin) in the measured data cube. As the figures show the range of power and the Doppler spread of the measured and simulated data are the same with a random bias in Doppler frequency and a random change in plasma diamagnetic drift velocity in different range bins. The corresponding angle-Doppler plots for range bin 230 in the measured data and range bin 20 in the simulated one are shown in Fig. 2. By comparing these results, one can see that the simulated signals are similar to the measured ones. In Fig. 2(b), the measured signals also show an interfering signal at an angle of about 45𝑜 ; however, this is due to an external interference, not ionospheric clutter.
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In this section, we present an application for the data created by the simulations: analyzing two FFA algorithms using the simulated data. Consider a linear array of 𝑁 receivers, each of which samples the received signal 𝑅 times with each sample corresponding to a range bin. This process repeats 𝑀 times within a Coherent Processing Interval(CPI), forming a 𝑁 × 𝑀 × 𝑅 data cube. For each range bin, the received data can be stored in a length-𝑁 𝑀 vector, which is a sum of the contributions from interference sources, thermal noise, and possibly a target. This vector can be written as x = 𝜉v(𝜙𝑡 , 𝑓𝑡 ) + n,
(30)
where n is the vector of all interference and noise sources, 𝜉 is the target amplitude, v is the space-time steering vector corresponding to a target at look angle 𝜙𝑡 and look Doppler frequency 𝑓𝑡 [14]. The optimal non-adaptive space-time processor matches the weights to the normalized steering vector, w = v/vH v, thereby maximizing the signal-to-noise ratio (SNR) in the output statistic, 𝑦 = w𝐻 x at look azimuth 𝜙𝑡 , look Doppler 𝑓𝑡 , and the range corresponding to x. A target is declared present, at the corresponding detection statistic, if ∣𝑦∣2 is above a chosen threshold. The threshold can be chosen to control the probability of false alarm. The target detection performance of non-adaptive techniques, especially in the presence of ionospheric clutter at the far ranges, is limited. In this regard, adaptive processing appears to be a
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where 𝑁 ′ ≪ 𝑁 and 𝑀 ′ ≪ 𝑀 , and then uses the AMF within each such sub-matrix to compute an intermediate statistic. Reducing the degrees of freedom within each AMF problem allows for huge reductions in required sample support. The outputs from each successive stage form the data matrix of a subsequent stage which is again subdivided and processed. This process of repartitioning the newly formed data matrix, followed by adaptively processing each resulting partition, is repeated until the original 𝑁 × 𝑀 data matrix is reduced to a single final statistic. In the FFA process, the optimal weight vector within a single sub-division is again given by ˆ −1 v where, for each sub-division, the estimated cow = R ˆ is calculated variance matrix of the interference-plus-noise, R, from (31) using data from 𝐾 secondary range cells. However, the FFA scheme requires very few training samples, on order of 𝐾 ≃ 2𝑁 ′ 𝑀 ′ . The algorithm also has the distinct advantage that, at every stage, the entire data matrix is adaptively processed. The key to the functioning of this algorithm is the tracking of the impact that successive stages of processing have on both the data and the steering vector.
(a)
B. Randomized FFA (b) Fig. 2. Angle-doppler plots of the (a) simulated data for the range bin 20 and (b) measured data for the range bin 230.
promising approach to deal with such interference. The optimum adaptive processing in terms of output signalto-interference-plus-noise ratio (SINR) is the fully adaptive processing or adaptive matched filter (AMF), where the weight ˆ is the estimated coˆ −1 v, where R vector is given by w = R variance matrix of the overall interference using 𝐾 secondary space-time data snapshots x𝑘 (𝑘 = 1, 2, ..., 𝐾): 𝐾 1 ∑ ˆ x𝑘 x𝐻 R= 𝑘 . 𝐾
(31)
𝑘=1
We use the modified sample matrix inversion (MSMI) statistic that has the very useful property of CFAR in Gaussian 2 interference [15] 𝜌MSMI = ∣𝑦∣ /w𝐻 v. The problem with this fully adaptive approach is that to accurately estimate the covariance matrix we need at least 2𝑁 𝑀 statistically homogeneous range cells [16] which is rarely available in practice. In the following subsections we review two alternative STAP approaches, regular and randomized FFA, that exploits all available degrees of freedom while simultaneously reducing computational complexity and required sample support. Details of the FFA algorithm are available in [8]. A. Regular FFA The most intuitive form of FFA approach is the “regular” FFA. The algorithm sub-divides the 𝑁 × 𝑀 space-time data matrix into rectangular sub-matrices of dimensions 𝑁 ′ × 𝑀 ′
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In contrast to the regular FFA, described above, the randomized FFA algorithm is not limited to any specific size, or location, of partition. In fact, there is no need to restrict choices to rectangular partitions. As long as the process keeps track of the steering vector at each stage, the AMF can be applied to any subset of the space-time data vector. The key to the randomized FFA algorithm is taking many random subsets of the data vector. The resulting statistics can be grouped into a new data vector for the next stage of processing; furthermore this process can be repeated as many times as necessary. The steps of the randomized FFA scheme are as follows [8]. 1) Given the available training data and computation resources, choose 𝑁DoF , the maximum number of adaptive DoF that can be processed. Also, vectorize the space-time data and steering matrices. 2) Randomly interleave (rearrange) the data vector and apply the same interleaver to the steering vector. 3) Choose blocks of length 𝑁DoF from within the interleaved vectors and process these blocks using the AMF. For example, in the zeroth stage, there would be approximately 𝑁 𝑀/𝑁DoF blocks. 4) The output statistic of each block forms the data and steering vectors for the following processing stage. Repeat steps 2 and 3 until a single “final” complex statistic is obtained. 5) Repeat steps 2-4 as many times as computationally feasible to form multiple “final” statistics that can be grouped to form a new data and steering vector. Repeat steps 2 and 3 until the truly final statistic is obtained. IV. P ERFORMANCE EVALUATION In this section we evaluate the performance of the FFA algorithms for the HFSWR setup using ideal targets. Note that this is done to illustrate the use of the data model for
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Fig. 3. Probability of detection versus SNR using non-adaptive, regular FFA, and randomized FFA methods when PFA=0.1.
simulation of an HFSWR radar system. In this regard, we focus on probability of detection (for a chosen false alarm rate) plots, something that cannot be obtained using measured data. In the case of the HFSWR data model, we generated a datacube with 10000 ranges, 4096 pulses and 16 receiver antenna elements to mimic the experimental set up. The MSMI statistic for each of the 10000 ranges is computed in the absence of a target. The thresholds required for the MSMI statistic that correspond to a probability of false alarm 𝑃𝑓 𝑎 = 0.1 (10% of the MSMI statistics lie above the threshold) for this data are 10 dB, 12 dB, and 34 dB for the regular FFA, randomized FFA, and non-adaptive methods, respectively. Figure 3 plots the probability of detection versus SNR for the two methods using the pre-computed thresholds mentioned above when an ideal target is injected in range bin 250, Doppler frequency of -0.2 Hz and at broadside. As can be seen from the plots, the randomized FFA method is up to 6.5 dB better than the regular FFA, and 28 dB better than non-adaptive methods for a probability of detection of 0.5.
[3] ——, “An integrated maritime surveillance system based on highfrequency surface wave radars, part 2: Operational status and system performance,” IEEE Ant. Prop. Mag., vol. 43, no. 4, pp. 52–63, 2001. [4] L. Sevgi, “Stochasting modelling of target detection and tracking in surface wave radars,” Int. J. of Numerical Modeling, vol. 11, no. 3, pp. 168–181, 1998. [5] R. H. Anderson and J. L. Krolik, “Track association for over-the-horizon radar with a statistical ionospheric model,” IEEE Trans. Signal Proc., vol. 50, no. 11, pp. 2632–2643, 2002. [6] C. J. Coleman, “A general purpose ionospheric ray tracing procedure,” Defence Sci. and Technol. Organ., Salisbury, S.A., Australia, Tech. Rep. SRLO131TR, 1993. [7] ——, “A propagation model for HF radiowave systems,” in Proc. IEEE Milit. Commun. Conf., vol. 3, 1994, pp. 875–879. [8] O. Saleh, R. S. Adve, and R. J. Riddolls, “Fast fully adaptive processing: A multistage STAP approach,” in Proc. IEEE Radar Conf., May 2009. [9] R. J. Riddolls, “A model of radio wave propagation in ionospheric irregularities for prediction of high-frequency radar performance,” Defense Research Development Canada, Tech. Rep. TM 2006-284, December 2006. [10] M. Ravan, O. Saleh, R. S. Adve, K. Plataniotis, and P. Missailidis, “KBSTAP implementation for HFSWR,” Final report for Defense Research and Development Canada, Tech. Rep. W7714-060999/001/SV, March 2008. [11] V. E. Gherm, N. N. Zernov, and H. J. Strangeways, “HF propagation in a wideband ionospheric fluctuating reflection channel: physically based software simulator of the channel,” Radio Sci., vol. 40, no. 1, pp. 1–15, 2005. [12] M. C. Kelley, The Earth’s ionosphere. San Diego, USA: Academic Press, 1989. [13] H. C. Chan, “Characterization of ionospheric clutter in HF surfacewave radar,” Defense Research Devlopment Canada - Ottawa, Tech. Rep. DRDC Ottawa TR 2003-114, September 2003. [14] J. Ward, “Space-time adaptive processing for airborne radar,” MIT Linclon Laboratory, Tech. Rep. F19628-95-C-0002, December 1994. [15] L. Cai and H. Wang, “Performance comparisons of modified SMI and GLR algorithms,” IEEE Trans. Aerospace Electron. Syst., vol. 27, no. 2, pp. 487–491, 1991. [16] I. S. Reed, J. Mallett, and L. Brennan, “Rapid convergence rate in adaptive arrays,” IEEE Trans. Aerospace Electron. Syst., vol. 10, no. 6, pp. 853–863, 1974.
V. C ONCLUSIONS This paper has developed and implemented a theoretical model for ionospheric clutter for HFSWR. The motivation for this work arose from the need to characterize the performance of radar systems based on simulations. In this regard, we used the theoretical models to develop simulated space-time-range data cube that represents ionospheric clutter. We then used the simulated data cube to evaluate the performance of two newly developed clutter mitigation techniques, the regular and randomized FFA algorithms. Note that the key contribution here is the ability to generate sufficient simulated data in order to obtain probability of detection curves. The data itself could be used to test any chosen scheme. R EFERENCES [1] G. Fabrizio, “Space-time characterization and adaptive processing of ionospherically propagated HF signals,” Ph.D. dissertation, The University of Adelaide, 2000. [2] L. Sevgi, A. Ponsford, and H. C. Chan, “An integrated maritime surveillance system based on high-frequency surface wave radars, part 1: Theoretical background and numerical simulations,” IEEE Ant. Prop. Mag., vol. 43, no. 4, pp. 28–42, 2001.
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