Corresponding author: M. Ravan, Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Rd., Toronto, ON M5S 3G4, Canada. (firstname.lastname@example.org)
 The detection performance of high frequency surface wave radar (HFSWR) and high frequency over-the-horizon radar (OTHR) systems is heavily influenced by the presence of radar clutter. In HFSWR systems, the clutter has its origins in vertical-incidence ionospheric reflections, whereas in OTHR systems, the origin is Bragg backscatter from plasma structures in the auroral zone. This paper models the spreading of the radar clutter signal in the Doppler and angle-of-arrival domains that arises from forward-scattering effects as the radar pulse propagates through regions of ionospheric plasma irregularities. The models use a geometric optics approach to determine the power spectrum of the radar signal phase. This power spectrum is then used to simulate three-dimensional space-time-range radar data cubes. The accuracy of the models is tested by comparing the simulated data to measured data cubes. As an application, the data are then used to evaluate the performance of the newly developed fast fully adaptive (FFA) space-time adaptive processing (STAP) scheme to improve the extraction of target echoes from a clutter background.
 In a high frequency surface wave radar (HFSWR) system, the transmitted high frequency (HF) radar waveform propagates primarily along the surface of the Earth, typically over the ocean. The signal diffracts around the curvature of the Earth to illuminate ship and low-flying aircraft targets beyond the line-of-sight horizon [Sevgi et al., 2001a, 2001b]. However, during transmission, a small amount of radiation is unavoidably radiated vertically and the direct reflection from the ionosphere is observed as intense radar clutter [Chan, 2003]. This clutter imposes detection range limitations in long-range HFSWR systems. In this paper we refer to this type of clutter as “ionospheric” clutter, following HFSWR literature conventions.
 In an over-the-horizon radar (OTHR) system, the ionosphere is used as a reflecting surface to permit illumination of targets at distances beyond the horizon [Headrick, 1990]. The transmitter beams a radar signal at an oblique angle to the ionosphere, the signal reflects from the ionosphere, and the target, such as an airplane or a ship, is illuminated by the signal. The target scatters the signal back to the transmitting location along a similar propagation path. At high latitudes, the target signals are often masked by radar clutter comprising echoes from the Earth's aurora zone [Elkins, 1980], which we denote as “auroral clutter”.
 For successful target detection, HFSWR and OTHR systems require target signals to be well separated from clutter in range, Doppler, or angle-of-arrival. In quiet ionospheric conditions, this separation can often be achieved. However, in disturbed ionospheric conditions, plasma density irregularities in the ionosphere can lead to the spreading of the clutter signal in the Doppler and angle-of-arrival domains [Vallieres et al., 2004]. The ionospheric plasma density irregularities represent refractive index fluctuations, which serve to scatter the radar signal wave packet. The spread clutter signal can occupy a large number of radar range-Doppler-angle resolution cells, leading to the masking of targets.
 Given the size and cost of these radar systems, target detection analysis based purely on measured data is difficult. 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 and a corresponding detection threshold. However, it is unusual to have at hand adequate measured data to accurately estimate these parameters. Quantifying the impact of clutter on target detection schemes therefore requires simulations. In turn, these simulations must be based on a quantitative model of the space-time radar clutter properties, such as the Doppler and angle-of-arrival spectra of the clutter.
 It is important to distinguish the source mechanism of the clutter versus the mechanism by which the clutter signal is spread in Doppler and angle-of-arrival. In the case of HFSWR, the clutter source mechanism is the reflection from the bottomside of the Earth's ionosphere. In the case of OTHR, the source mechanism is the Bragg backscatter from the aurora. However, the mechanism of spreading in Doppler or angle-of-arrival is of much greater consequence for target detection. It is the existence of forward scattering during the propagation of radar signals through volumes of plasma irregularities that leads to the spreading phenomenon. More precisely, in phased-array radar systems, the Doppler spread is due to the randomization of the signal pulse-to-pulse temporal phase relationships and the angle-of-arrival spread arises from a similar randomization of the signal sensor-to-sensor spatial phase relationships. Thus the central feature of the propagation models is the phase scintillation experienced by the radar signal upon transiting the ionospheric medium. It should also be noted that some radars suffer from range ambiguity such that range-folded ground clutter can mask non-range-folded targets. While this is a waveform issue and is beyond the scope of this paper, it should be noted that ground clutter spreading could be modeled with the same tools that are provided in this paper.
 The modeling of wave propagation through random media is a well-studied problem. As already mentioned, we use the term “propagation” in a broad context to include both the movement of a radar wave packet in a smooth medium and the forward-scattering effects that occur when the wave packet moves through a medium containing irregularities. A recent monograph on the subject [Wheelon, 2001] promulgates a three-level hierarchy for describing the methods that have been applied to the problem, which we briefly discuss here. The first and most basic method in the hierarchy is geometric optics, whereby the scattered wavefield can be expressed in terms of a line integral of the refractive index fluctuations [Tatarskii, 1971; Rufenach, 1975; Coleman, 1996]. The line integral is taken along the zero-order signal ray trajectory, namely the ray that would occur in the absence of first-order refractive index fluctuations. Geometric optics provides good modeling of the resulting fluctuations of the signal phase and angle-of-arrival, but does not inherently contain diffractive effects. It is thus unable to accurately model fluctuations in signal amplitude or intensity. Geometric optics can be extended in an ad hoc manner to include diffractive effects by modeling the ionosphere as a series of phase screens interpolated by free space [Dana and Wittwer, 1991; Nickisch, 1992; Nickisch et al., 2011].
 As a result of this paper's focus on pulse-to-pulse and sensor-to-sensor signal correlation, our main interest is in the signal phase. A geometric optics approach is thus suitable for our application. As a supporting example from the literature, if we take the plane wave limit ofKnepp [1983, equation 30] (by setting the transmitter-phase screen distancezt to infinity), it is clear that the spatial and temporal correlation lengths are controlled dominantly by the phase variance σϕ2. However, if there was interest in amplitude or intensity fluctuations, for the purpose of modeling signal fading for instance, one could consider the second method in the hierarchy of [Wheelon, 2001], referred to as the method of smooth perturbations [Tatarskii, 1971] or the Rytov method [Gherm et al., 2005]. In this method, the scattered wavefield is expressed as an integral over a volume of refractive index fluctuations, which allows for the accounting of diffractive effects. A key feature of this method is the expression of the scattered wavefield as an exponential of a function to be determined. By allowing the function to be complex-valued, both amplitude and phase effects are captured in a straightforward manner.
 For completeness, we mention the third method in the hierarchy, which is referred to as the method of strong fluctuations, where the scattered field is expressed as a functional integral with respect to the refractive index fluctuations [Rytov et al., 1989]. As the name suggests, this method handles large-amplitude refractive index fluctuations where the other methods do not provide convergent solutions.
 In this paper, we report on the use of geometric optics models developed to simulate radar data. These data are tested using measurements of ionospheric and auroral clutter. In addition, as an application of the models, we use simulated data to test the efficacy of a newly developed clutter mitigation space-time adaptive processing (STAP) algorithm named the fast fully adaptive (FFA) algorithm [Saleh et al., 2009]. The aim of this algorithm is to reduce the radar clutter sample support requirements of the adaptive processor. Sample support is required to estimate radar clutter sensor covariances matrices, which are the essential components for most STAP schemes. In particular, HF radar STAP schemes are vulnerable to sample support shortages, since the relatively coarse resolution severely restricts the number of independent samples available for clutter covariance estimation. The FFA alleviates the sample support problem by casting the STAP algorithm as a multistage process, where each stage operates with a greatly reduced number of degrees of freedom with similarly reduced requirement for sample support.
 The rest of the paper is organized as follows. The clutter models for the HFSWR and OTHR scenarios are presented in sections 2 and 3, respectively, and a brief comparison with measured data is provided. In section 4, the suite of available FFA algorithms is summarized. In section 5, results of simulations based on the data models are presented and the performance of FFA approaches in terms of the probability of target detection versus SNR is evaluated. In section 6, the paper concludes.
2. Modeling Ionospheric Clutter for an HFSWR System
 In this section, we adopt a geometric optics approach to develop a simple expression for the spatial-temporal phase spectrum of signals reflected from the bottomside ionosphere in the presence of ionospheric irregularities. We consider an unmagnetized model of the ionosphere where the dispersion relation for the HFSWR pulse as it propagates in the ionosphere is given by [Chen, 1984]
where N is the index of refraction, c is the speed of light, k = |k| is the wave number, ω is the radar frequency, and ωp is the electron plasma frequency, defined as
where e is the charge on an electron, n is the density of plasma in units of ionized electrons per unit volume, ϵ0 is the permittivity of free space, and m is the mass of an electron.
 We assume a plane-stratified plasma model wheren is linearly related to altitude z, slowly varying compared to the radar wavelength, and independent of the horizontal coordinate:
which implies (for z > 0)
We define z= 0 as the bottom of the F-region ionosphere, which is approximately 200 km above the ground, andz = z0 as the height where reflection occurs. Consider an HFSWR radar signal vertically incident on the ionosphere. Since the index of refraction, given by (1), is only a function of z, the horizontal components of the wave number k remain exactly zero as the radar pulse propagates through the ionosphere, or [Budden, 1985]. Furthermore, since the plasma model is unmagnetized in this treatment, the direction of propagation is equal to the local wave number k. Thus, the wave packet travels vertically up, reflects from the ionosphere, and travels vertically back to the ground along the same path. The total phase accrued during propagation is equal to
where k(z) is the local radar wave number, as determined from (1). Let us consider small-amplitude irregularities in the ionospheric plasma, such that the total ionospheric plasma density isn = n0(z) + n1(r), where n0(z) is the zero-order quiescent ionospheric plasma density profile,n1(r) is the first-order irregularity, andr = (x, y, z) is a three-dimensional coordinate on the ray trajectory, withx being the east coordinate and y being the north coordinate. By Fermat's principle, the ray follows a trajectory of minimum phase, which means that perturbations to the ray trajectory are second order in density perturbation [Coleman, 1996]. Thus, a first-order Taylor series perturbation toϕwould be evaluated along the zero-order ray trajectory and would be given by
Note that the integral is along the zero-order raypath and the fluctuations do not change the value ofz0. From (1) and (2) we can compute the derivative of k with respect to plasma density:
where re = e2/(4πϵ0mc2) is the classical electron radius (2.8 × 10−15 m) and λ is the free space radar wavelength. Unfortunately, when we insert (7) into the integral of (6), the integral diverges for finite n1 near the turning point, where z = z0. In practical terms, the radar wavelength goes to infinity at the turning point and the concept of phase is no longer meaningful. A more rigorous full-wave analysis of fluctuations near the turning point using Airy functions was given inKiang and Liu  and Wagen and Yeh [1986, 1989], which shows the phase contribution near the turning point to be minor. Following the suggestion of a reviewer for this paper, we ignore the phase contribution near the turning point by cropping the phase integration a short distance (on the order of a wavelength) below the turning point. This new integration limit will be denoted as z = z0′. The first-order phase can then be written as a convergent integral:
where 0 < z0′ < z0. To proceed, we can characterize ϕ1in terms of its statistics. For ground-based radar applications, where sensors lie strictly in thex-y plane, we are interested in the autocorrelation of the phase in the x (east) and y (north) directions, as well as the t (time) coordinate:
The lag in the z direction is assumed to be zero. Therefore, ϕ1(x, y, t) refers to the phase accrued on a vertical trajectory above a location (x, y) on the ground at time t. Combining (8) and (9), we have
refers to the autocorrelation of the density fluctuations. We will not attempt to model variations in the mean square density fluctuation with altitude. Following an approximation suggested in Tatarskii , we note that the half-power width is small compared to the scale length z0, and the integrand of (10) is significant only along a narrow strip where z ≈ z′:
Typically, is of a complicated form which precludes direct evaluation of the above integral. An alternate way to calculate the integral uses Fourier transforms. The Fourier transform of the left side can be written
We use the notation (κ, Ω) instead of (k, ω) to distinguish the wave number and frequency of the phase spectrum from the wave number and frequency of the propagating wave packet, as given in (1). The Fourier transform of the right side can be written as
The transforms along the X, Y, and T axes are straightforward. We can also interpret the integral over Z as a Fourier transform [Rufenach, 1975] if we evaluate the resulting transform at κz = 0:
where is the Fourier transform of For we use a very simple third-order spatial power spectrum for infinitely extended field-aligned plasma irregularities [Woodman and Basu, 1978] and assume that temporal variations are entirely due to the drifting of the spatial irregularities, which is referred to as the Taylor hypothesis [Coleman, 1996]:
where κ0 ≈ 10−4 m−1 is the “outer” scale length 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, 〈n12〉 is the variance of the electron density fluctuations at the reflection height, is the plasma drift velocity, and the spectrum is normalized such that We suppose that the magnetic field of the Earth follows a unit vector . However, the wave propagation is limited to the z direction. Without loss of generality, we can rotate x and y azimuthally such that the magnetic field vector lies in the y-z plane and lx = 0. The quantity κ∥ is given by the dot product of κ and , which under the assumption of lx = 0 is given by
whereas κ⊥ takes the following form:
where we have used the relation ly2 + lz2 = 1. In light of these definitions, and the delta function δ(κ∥), the plasma density fluctuation spectrum in cartesian coordinates is given by
To complete the evaluation of the phase spectrum, we need to insert the density spectrum into (16). Acknowledging the effect of the delta function δ(κy), we find that
The essential result is that the spectrum rolls off with the inverse third power of κx, and is impulsive in κy. The frequency dependence forms a ridge in the joint κx − Ω direction, where the direction of the ridge depends on vdx. Finally, the area under the spectrum depends only weakly on the exact choice of the truncation height z = z′0 because of the influence of the logarithm function.
2.1. Modeling the Space-Time-Range Data Cube for HFSWR
 The previous section developed the phase power spectrum for the signal received in the horizontal plane. Here we use this spectrum to simulate a space-time-range data cube appropriate for an HFSWR system.
 The data cube represents the composite of an N-dimensional antenna array snapshot (n = 0, 1,., N − 1) recorded at the rth range bin in the mth pulse repetition interval (PRI). To generate the data cube with the phase power spectrum of (21), we consider a range bound in the zdirection which begins at the distance to the first ionospheric range bin. For each range bin, we create a two-dimensional array (sensor-pulse) signal by filtering white noise through a two-dimensional linear time-invariant (LTI) filter; the impulse response of this filter is obtained by taking the square root of the phase spectrum described in(21), followed by an inverse Fourier transform. Also, because the value of plasma drift velocity, vd, changes slightly in different ranges, we consider vd as where dv is a random variable.
 The measured data sets against which our data model is compared used an eight-segment Frank code that minimizes the effect of high range sidelobes and range-folded clutter [Chan, 2003]. To mimic the measured data sets, we consider the same Frank code with eight segments as the transmitted signal waveform. The actual code phase sequences used are:
 The radar transmitted signal at complex baseband for the ith code segment is represented by:
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 into 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 correlating the returns of the eight waveforms with their replicas and coherently summing the results.
 In the measurements, each element of eight phase code segments in the transmitted signal pulse is 55 μs long, and thus, each segment is 440 μs long. The pulses are emitted every 8 ms; thus, the complete code is emitted every 64 ms and the pulse repetition frequency (PRF) of the data set is 1/(64 ms) = 15.625 Hz. There are 4096 of these pulses, making the data set 262.144 s long. The antenna array contains 16 antenna elements that are separated by 33.33 m from each other.
 To consider the effect of the transmitted signal, we first calculate the ambiguity function of the transmitted signal as:
We then convolve the resulting signals of different range bins with this ambiguity function to create a data cube. To simulate the random bias, Δf(r), in Doppler frequency seen in the measured data cube at range bin r, we multiply the 2D pulse-range signal for all antenna elements byej2πΔf(r)t in the time domain where f(r) represents the random Doppler bias in range bin r and the time variable t covers the time of the appropriate pulse.
2.2. Simulation Results
 This section presents the results of simulations testing the theory developed in the previous section. Table 1 shows the values considered for the parameters in (21) to simulate the data cube. Figure 1a plots the 2D view of when vdx = 50 m/s. The variation of the diagonal entries versus Ω is shown in Figure 1b. We calculate for −1 Hz < Ω < 1 Hz because, as Figure 1b shows, the value of is much smaller than the peak value outside this range. Therefore, the boundary of κx is (−1/vdx, 1/vdx) = (−0.02 mm−1, 0.02 m−1). The parameter Δf, which shows the random bias in Doppler frequency, is considered as a uniform random variable between −0.6 and 0.6. Also, the amplitude of the simulated signal is multiplied by 200 to have the same scale as the measured signal.
Figures 2a and 2bshow the 2D angle-Doppler plots for range bin 230 in the measured data and range bin 20 in the simulated one. By comparing these results, one can see that the simulated signals are similar to the measured ones. InFigure 2b, the measured signals also show an interfering signal at an angle of about 45°; however, this is due to an external interference, not ionospheric clutter. The corresponding Doppler-range plots for antenna element 12 are shown inFigure 3. The ionospheric clutter started from range bin 210 in the measured data and extended to the last range bin (range bin 270). As the figures show the range of power and the Doppler spread of the measured and simulated data are comparable.
3. Modeling Auroral Clutter for OTHR
 In this section we use the geometric optics approach to determine the phase spectrum of reflections from Bragg scatterers in the auroral region, which is applicable to OTHR systems operating at high latitudes. We again consider an unmagnetized model of the ionosphere. Therefore, the dispersion relation for the OTHR pulse as it propagates in the ionosphere is the same as (1). Furthermore, we continue to assume a plane-stratified plasma model of(3).
 Consider an OTHR signal obliquely incident on the ionosphere at z = 0. We consider a cartesian system of coordinates, with x east, y north, and z vertical. The cartesian components of the radar wave number can be written in terms of standard spherical coordinates:
where θ is the beam elevation angle with respect to zenith and φ is the beam azimuth angle counterclockwise with respect to east. However, without loss of generality, the azimuthal coordinate can be taken with respect to the Earth's magnetic field, such that x is magnetic east and yis magnetic north. This re-orientation is useful later when the spectral model of ionospheric irregularities is introduced. Since the index of refraction, given by(1), is only a function of z, the horizontal components of the wave number (kx, ky) remain constant as the radar pulse propagates through the ionosphere. Therefore, these components maintain their free space (z = 0) values:
The value of kz in the ionosphere is given by
Let us define ρ as a radial coordinate in the x-y plane, such that ρ2 = x2 + y2. The slope of the pulse trajectory above the x-y plane is given by
By direct substitution into the right side of (30), it can be shown that within the ionosphere, the ray trajectory has the form of an inverted parabola:
It should be noted that the point (x, y) = (0, 0) occurs at the peak of the parabolic trajectory. It should also be emphasized that this parabolic form is for altitudes z > 0, or equivalently, heights at least 200 km above the ground, where the wave packet travels in the ionosphere. In the free space below the ionosphere, the wave packet travels in a straight line.
 For the purposes of modeling auroral clutter, the phase accumulated along a path to and from a Bragg scatter event is given by
where k(r) is the radar wave number, is an element of arc length, r = (x, y, z) is a point on the ray trajectory, s0 is the location on the arc where the pulse enters the ionosphere, and s1is the location on the arc of the Bragg scatter. Using the first-order Taylor-series perturbation tok, the perturbation to ϕ is given by
From (1)–(3) we can compute the derivative of k(r) with respect to density:
From (30) the element of arc length ds can be expressed as
where ρ0 and ρ1 are the starting and ending points of the integration in the ρ coordinate. Since OTHR systems are based on the surface of the Earth, our interest is again in the autocorrelation of the phase in the x (east) and y (north) directions, as well as the t (time) coordinate, as given by (9). By combining (36) and (9), we have
where the unprimed and primed terms refer to the coordinates of the radar pulse trajectory for the integrations over ρ and ρ′, respectively. As in the HFSWR case, we make no attempt to account for any possible variations of the mean square density fluctuation 〈n12〉 with respect to z. However, the main difficulty with the integral is that x, y, and z are not linearly related to each other due to the parabolic trajectory. Some simplifying assumptions need to be made about the form of First, since we are considering auroral Bragg scatter, we assume that the high-latitude magnetic field of the Earth is approximately vertical, meaning that field-aligned plasma irregularities are stretched along the vertical direction, leading to very good vertical correlations. This means that we can ignore the dependence on zcompared to the dependence on the two cross-field coordinates (x, y). Second, since we are assuming a vertical magnetic field, we take to be rotationally symmetric around z. This means that we can set φ = π/2 by assuming north-south propagation, without any loss of generality in the results. With these two simplifying assumptions the integral reduces to
We now take the Fourier transform of both sides with respect to X, Y, and T. The Fourier transform of the left side is
Once again, the notation (κ, Ω) is used for the wave number and frequency of the phase spectrum. The Fourier transform of the right side is
The integrals without limits specified are assumed to be over the entire axis. The transforms along the X and T axes are straightforward. Along the Yaxis, we use the delay-modulation property of Fourier transforms to write
where L = |y1 − y0| is the horizontal distance traveled in the ionosphere. We have assumed that L ≫ 1/κy for values of κy that are of interest. As for the power spectral density we use a very simple third-order spatial power spectrum for infinitely extended field-aligned plasma irregularities and assume that temporal variations are entirely due to the drifting of the spatial irregularities, which was given earlier as(17). When the magnetic field is vertical, we can integrate out κz in (17) so that we can write:
where again κ0 ≈ 10−4 m−1 and the spectrum is normalized such that Inserting (42) into (41), and acknowledging the effect of the delta function, we find that
Thus, we have a third-power roll-off inκx (which is perpendicular to the beam azimuthal direction), and an impulsive dependence in κy (along the beam) as one might expect for a propagating wavefront. As we found in the HFSWR case, the frequency dependence forms a ridge in the joint κx − Ω direction, where the direction of the ridge depends on vdx. The parameter vdx in this case represents the bulk drift velocity of the plasma medium in the x direction (across the radar beam). It should be noted that the Bragg scatterers also drift with the plasma. Any motion of the scatterers in the x direction would not impart any additional Doppler shift. However, if the motion of the plasma along the radar beam vdy is nonzero, the motion of the Bragg scatterers translates the phase spectrum by a Doppler shift of ΔΩ = − 2kvdy, where k is the radar wave number. This additional Doppler shift has been disregarded in this analysis, since the emphasis has been on the mechanism of Doppler spreading.
3.1. Modeling the Space-Time-Range Data Cube for OTHR in the Auroral Zone
 The algorithm for modeling the 3D space-time-range data cube for OTHR is the same as the HFSWR case and only the spectrum formula and parameters were changed to match the OTHR setup for the measured data. The OTHR setup has two transmit antennas each emitting distinct waveforms, consisting of pulses at carrier frequencies of 4.94150 MHz and 4.94152 MHz. This setup provides 20 Hz of separation between the signals in Doppler to prevent Doppler-aliasing of the clutter signals. On the receive side, four antennas are summed together in hardware to maximize gain in the boresight direction. Therefore, we have a slow-time multiple-input multiple-output (MIMO) radar system [Mecca et al., 2008] comprising two transmit channels and one receive channel. A pulse generator creates a 5-ms duration pulse waveform every 25 ms. The relatively long pulse duration is chosen to maximize average transmitted power within the power limits of the existing HF amplifier power supplies. The simple pulse waveform provides a range resolution of 750 km and the pulse repetition frequency offR = 40 Hz provides an unambiguous range of Ru = c/(2fR) = 3750 km. Thus, the waveform provides 3750/750 = 5 uncorrelated range bins, the first of which is eclipsed by the transmitted pulse. Among these range bins, only the fourth range bin contains clutter. The coherent processing interval (CPI) is 18 s. Therefore, each receive element receives 18 × fR = 720 pulses in the time domain or Doppler bins in the frequency domain.
 In a slow-time MIMO radar, all elements in the transmit array radiate the same basic waveform, though Doppler-shifted to allow for orthogonality. In the case of havingL transmit elements, this leads to L orthogonal channels after Doppler processing, i.e., the entire unambiguous Doppler region of width fR is divided into Lnon-overlapping segments of lengthfR/L.The length-M received vector is, therefore, the stacking of L vectors of length M/L. This forms a M/L = 360 (Doppler bins) by L = 2 (channels) by R(ranges) experimental data cube in the Doppler domain. In order to generate a slow-time MIMO radar data cube, we again create a two-dimensional array (element-pulse) signal for each range bin by filtering white noise through a two-dimensional LTI filter, where the impulse response of the filter in this case is obtained by taking the square root of the phase spectrum in(43), followed by an inverse Fourier transform. We then modulate the two-dimensional range-pulse signals of the two transmit channels to two different carrier frequencies each separated byfR/2 = 20 Hz.
3.2. Simulation Results
 To simulate the data cube, we set the values shown in Table 2 for the parameters in (43). Figure 4 plots the 2D view of and the variation of the diagonal entries versus Ω. As is shown in the figure, the value of is very small in comparison to its peak for |Ω| > 0.5 Hz. The boundary of κx is (−0.5/vdx, 0.5/vdx) where vdx = 100 m/s is considered.
 In order to mimic the experimental set up described in section 3.1, we separate the two-dimensional range-pulse data of each transmit channel by 20 Hz in Doppler frequency.Figure 5ashows the Doppler-range plot for these two channels when the center frequencies for the channels are −10 Hz and 10 Hz. The Doppler spectrum of these two channels at range bin 25 are shown inFigures 5b and 5c. To separate these two channels, we first shift the carrier frequency of first channel to baseband (0 Hz) and filter this channel by a low pass filter with a passband from −10 Hz to 10 Hz. We apply the same procedure on the other channel. Figure 6shows the data for the first and second transmit channels of the measurement setup for range bin 4 (the only range bin with auroral clutter). Again, the simulated signals are similar in appearance to the measured ones in terms of overall shape and Doppler spread. However, it must be noted that the apparent noise floor in the measured data is somewhat elevated due to finite clutter-to-interference-plus-noise ratio produced by the experimental system, which is not contained in the auroral clutter model. This feature in the experimental data could be remedied by an increase in radar transmit power.
4. Data Processing
 Consider a linear array of N receive elements, each of which samples the received signal R times with each sample corresponding to a range bin. This process repeats M times within the CPI, forming a N × M × Rdata 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
where n is the vector of all interference and noise sources, ξ is the target amplitude (assuming a constant radar cross section with time), vis the space-time steering vector corresponding to a target at look angleϕt and look Doppler frequency ft [Ward, 1994]:
where ⊗ represents the Kronecker product of two vectors, T the transpose operator, fs = (d/λ)sin ϕt the normalized spatial frequency, λ the wavelength of operation, and fR the PRF.
 The optimal non-adaptivespace-time processor matches the weights to the normalized steering vector, , where His the conjugate transpose operator, thereby maximizing the signal-to-noise ratio (SNR) in the output statistic, at look azimuth ϕt, look Doppler ft, and the range corresponding to x. A target is declared present if |y|2is 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 promising approach to deal with such interference.
 The adaptive matched filter (AMF) provides optimum adaptive processing in terms of output signal-to-interference-plus-noise ratio (SINR). The weight vector is given by , where is the estimated covariance matrix of the overall interference using Ksecondary space-time data snapshots :
In HF radar systems, we note that the noise floor is almost always set by external interference sources, so in this paper will not distinguish interference and noise, and we refer to SINR as simply SNR.
 We use the modified sample matrix inversion (MSMI) statistic that has the very useful property of constant false alarm rate (CFAR) in Gaussian interference [Cai and Wang, 1991]:
The problem with this fully adaptive approach is that to accurately estimate the covariance matrix we need at least 2 × N × M statistically homogeneous range cells [Reed et al., 1974] which is rarely available in practice. In the following subsections we review two alternative STAP approaches, regular and randomized FFA, that exploit all available degrees of freedom while simultaneously reducing computational complexity and required sample support [Saleh et al., 2009].
4.1. Regular FFA
 The most intuitive form of the FFA approach is the “regular” FFA. The algorithm sub-divides theN × Mspace-time data matrix into rectangular sub-matrices of dimensionsN′ × M′ where N′ ≪ N and M′ ≪ M, and 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 stageform 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 × Mdata 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 where, for each sub-division, the estimated covariance matrix of the interference-plus-noise, , is calculated from (50) using data from K secondary range cells. However, the FFA scheme requires very few training samples, on the order of K ≃ 2 × N′ × 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.
4.2. Randomized FFA
 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 anysubset of the space-time data vector. The key to the randomized FFA algorithm is taking manyrandom 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 [Saleh et al., 2009]:
 1. Given the available training data and computation resources, choose NDoF, the maximum number of adaptive DoF that can be processed. 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.
5. Performance Evaluation
 In this section we evaluate the performance of the FFA algorithms for the HFSWR and OTHR setups using ideal targets. To illustrate the use of the simulated data, we focus on probability of detection for a chosen false alarm rate, something that cannot be obtained using measured data. In the case of the HFSWR data model, we generated a data cube with 10,000 ranges, 4096 pulses and 16 receive channels to mimic the experimental set up. The MSMI statistic for each of the 10,000 ranges is computed in the absence of a target. The thresholds required for the MSMI statistic that correspond to a probability of false alarm Pfa= 0.1 for this data are 10 dB, 12 dB, and 34 dB for the randomized FFA, regular FFA, and non-adaptive methods, respectively.Figure 7aplots the probability of detection versus SNR for the two methods using the pre-computed thresholds mentioned above. 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.
 We examine the performance of the algorithms in the case of the OTHR data model. In this case, we again generated a data cube with 10,000 ranges, 2 transmit channels, one receive channel, and 360 pulses. For this data, the thresholds required for the MSMI statistic that correspond to a probability of false alarm Pfa= 0.1 are 8 dB, 12 dB, and 32 dB for the randomized FFA, regular FFA, and non-adaptive methods, respectively.Figure 7bplots the probability of detection versus SNR for the three methods. As can be seen from the plots, the randomized FFA method is up to 18 dB better than the non-adaptive method while the regular FFA method is up to 12 dB better than the non-adaptive method for a probability of detection of 0.5.
 This paper has developed a theoretical model of ionospheric clutter for HFSWR and auroral clutter for OTHR. 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 that represents ionospheric and auroral 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.
 This work was supported by Defense Research and Development Canada.