Estimation of the Doppler frequency and direction of arrival of the ionospherically propagated HF signals

Authors


Abstract

[1] High-frequency (HF) signals reflected from different points within each ionospheric layer may have slightly different Doppler frequencies and angles of arrival. The superposition of these signals leads to time varying and nonplanar wavefronts. Investigation of temporal and spatial characteristics of the ionospherically propagated HF signals plays an important role in designing the signal processing algorithms for the HF over-the-horizon radar (OTHR). A cost-efficient superresolution algorithm for simultaneously estimating the Doppler frequencies and angles of arrival of the ionospherically propagated HF signals is proposed in this paper. The effectiveness of the proposed algorithm is verified by the experimental data from a trial HF OTHR. Furthermore, the superposition model with the HF signal reflected by a smooth ionospheric layer consisting of a number of submode signals is also confirmed by the experimental data processing results.

1. Introduction

[2] In the high-frequency (HF) band (3–30 MHz) the electromagnetic waves can propagate beyond the line of sight by sky waves refracted by the ionosphere. Since the HF sky wave over-the-horizon radar (OTHR) can provide over the horizon detection of targets in a large surveillance area, it is widely used in both military surveillance and civilian applications. The wavelength of the OTHR is generally of 10–100 m which approximates the physical size of ships and large aircrafts, and the radar cross section (RCS) of these targets is more dependent on their gross dimension than on their shape details. Accordingly, OTHR has the potential to detect stealth targets for which the RCS reduction technique has been used by proper shape design. Besides, the long-wavelengths characteristic of the OTHR also provides a means for sea-state monitoring [Headrick and Skolnik, 1974; Headrick and Thomason, 1998; Anderson, 1992; Wyatt, 1990].

[3] Although the HF OTHR has many advantages, the nonstationary ionosphere creates many challenges to the HF OTHR. The ionosphere can be divided into several regions according to the electron density: D layer (50–90 km), E layer (90–140 km), F1 layer (140–210 km), and F2 layer (above 210 km) [Davies, 1990]. It is well known that the regular motion of the ionosphere close to the reflection points will impose the Doppler shift on the reflected signals. The ionosphere imposed Doppler shift is usually less than 1 Hz [Davies, 1990]. The temporal fluctuations of ionization density may impose a phase disturbance on the echo signal. The phase disturbance imparted by the nonstationary ionosphere to the clutter echoes will broaden the clutter Doppler spectrum which may severely degrade the low-velocity target detection ability of the OTHR [Bourdillon and Gauthier, 1987; Anderson and Abramovich, 1998]. On the other hand, the stratified nature of the ionosphere often results in an HF signal being reflected by a number of different ionospheric layers. One phenomenon occurring in the ionospheric propagation is the magnetoionic splitting of a signal into an ordinary ray (O mode) and an extraordinary ray (X mode). Moreover, each of these modes can propagate via two different paths known as high-angle and low-angle paths. Owing to the dynamic and spatially inhomogeneous nature of the ionosphere, the received signal may well depart from the ideal plane wave [Warrington et al., 1990; Davies, 1990]. With the fast development of array signal processing techniques in the past two decades, many efficient superresolution algorithms have been proposed [Krim and Viberg, 1996]. Various superresolution algorithms have been applied to estimate the direction of arrival (DOA) of the ionospherically propagated HF signals [Warrington, 1995; Fabrizio et al., 1998; Zatman et al., 1993]. But the experiments are mainly focused on the spatial characteristics of the ionospherically propagated HF signals. Additionally, the HF signals in these experiments are from long-distance radio broadcasts. Since the signals from the radio broadcasts are uncorrelated with the locally generated waveform of the HF OTHR, different propagating modes cannot be resolved in ranges (time of flight or time delay), which prevents HF signals via different propagating modes from being analyzed separately.

[4] As mentioned above, HF signals reflected by a smooth ionospheric layer over the one way oblique circuit consist theoretically of two pairs of magnetoionic components known as high-angle and low-angle rays. However, the four theoretically expected rays are not usually resolvable in ranges. When two or more of the theoretically expected rays remain unresolved, the received signal is a superposition of different rays. A popular model for the ionospherically propagated HF signals at the receiver consists of a number of discrete modes with each having a planar wavefront [Warrington et al., 1990]. Accurately estimating the parameters of these plane waves plays an important role not only in designing the signal processing algorithm to improve the system performance but also in helping the system designer to well understand the effects of the ionosphere on the reflected signals. Two-dimensional parameter estimation algorithms can automatically pair the parameters when the parameters are estimated. However, conventional two-dimensional algorithms are computationally expensive. In this paper, we propose a computationally efficient superresolution algorithm to jointly estimate the Doppler frequencies and angles of arrival of the ionospherically propagated HF signals.

2. Signal Processing Model

[5] In this section, we first describe the signal model and the signal processing of the HF OTHR, and then derive a signal interference model, which is a superposition of multiple plane waves with slightly different Doppler frequencies and angles of arrival, for the ionospherically propagated HF signal.

2.1. OTHR Signal Model and Signal Processing

[6] In order to achieve long-range target detection a high peak transmitting power is often needed. The linear frequency modulated continuous waveform (LFMCW) is a compromised choice between range resolution and high peak transmitting power. The transmitted LFMCW has the form of

equation image

where s0 is the amplitude of the transmitted waveform, Tr is the waveform repetition interval (or pulse repetition interval, PRI), B is the bandwidth of the transmitted waveform, f0 is the carrier frequency, and M is the number of waveforms in one coherent processing interval (CPI). The received signals are demodulated, low-pass filtered, and sampled. The digital samples received by the receiving antenna array in one CPI are range processed, beamformed and Doppler processed to detect and estimate the target radial velocity and DOA [Khan et al., 1994; Wang et al., 2003].

[7] Consider the receiving antenna array which has N linear equispaced omnidirectional antennas with spacing d. With some reasonable approximation, the demodulated and range transformed signal acquired by the nth antenna, in the mth PRI and at the kth range bin can be approximated as

equation image

where sm,k is the signal observed by the antenna in the mth PRI at the kth range bin, λ is the wavelength, fd is the Doppler frequency, and θ is the DOA of the target with respect to the receiving antenna array boresight. When the HF signals propagating through different ionospheric layers can be isolated in ranges, the HF signal reflected by each ionospheric layer can be processed separately. In the following discussion, we focus mainly on the signal at one range bin, and we drop the range index k in (2) for notational brevity. Hence the N-dimensional received data vector in the mth PRI and at the kth range bin is rm = [rm,0, rm,1, equation image, rm,N−1]T, where T denotes the transpose operator. And the array steering vector is defined as a(θ) = [1 ej2πdsinθ/λequation image ej2π(N−1)dsinθ/λ]T.

2.2. Ionospherically Propagated HF Signal Model

[8] The typical coherent integration time (CIT) of the HF OTHR is on the order of a few seconds. Although the electron density distribution of the ionosphere changes in a random way and the ionosphere exhibits temporal and spatial nonstationarity, in such a short time interval, under the circumstance of quiet ionosphere, each of the four theoretically expected rays can be regarded as a specular reflected plane wave. As a result, the received signal is frequently composed of multiple plane waves or submodes which correspond to the four theoretically expected rays or a subset of them. The pth submode signal acquired by the receiving antenna array is

equation image

where θp is the DOA of the pth submode signal, fd,p denotes the Doppler frequency of the pth submode signal which is composed of the Doppler frequency of the original signal and the Doppler shift imposed by the ionospheric movement close to the reflection point on the signal. The submode signal after specular reflection by the ionosphere is incident upon the receiving antenna array from an azimuth angle α and an elevation angle β. Unfortunately, a uniform linear array (ULA) does not have two-dimensional resolutions. So the measured DOA of the submode signal by a ULA is actually the cone angle (subtended by the submode signal wave vector and the boresight of the ULA) of that signal. The relationship among the cone angle, true azimuth angle and true elevation angle of the submode signal is sin θ = cos β sin α. Different reflection points within a single ionospheric layer may have different heights. As a result, different submode signals originating from a single source via different propagation paths may have the same azimuth angle but different elevation angles. Accordingly, different submode signals can be resolved in cone angles with a ULA.

[9] Suppose a signal model with P submodes, and the received ionospherically propagated HF signal is

equation image

where A = [a1), a2), ⋯, aP)]T is an N × P Vandermonde matrix, ap), p = 1, 2, equation image, P is the array steering vector of the pth submode signal or ray, sm = diag[sm,1 eequation image, sm,2 eequation image, ⋯, sm,P eequation image]T is a P × 1 vector, and nm is an N-dimensional uncorrelated additive internal receiver noise vector. Without loss of generality, we assume the additive noise is temporal and spatial white. Furthermore, the Doppler frequency and the cone angle of each submode signal can be viewed as deterministic and constant in each CPI, but varying from CPI to CPI. The submode signals with slightly different Doppler frequencies and cone angles will result in temporal varying and spatial nonplanar wavefronts.

3. Parameter Estimation Algorithm

[10] Even though the rays can be resolved separately in the temporal and spatial domains, correctly pairing the parameters of each ray is still a difficult problem. This drawback of separate temporal and spatial parameters estimation algorithms motivates the use of two-dimensional algorithms which can jointly estimate the Doppler frequency and cone angle of each ray. Superresolution techniques which can simultaneously estimate the temporal and spatial parameters of the signals have been proposed [see, e.g., Wax et al., 1984; Zatman and Strangeways, 1994; Su et al., 2006]. These two-dimensional methods are computationally expensive. Yin et al. [1991] proposed a computationally efficient high-resolution two-dimensional DOA estimation algorithm. In this section, the algorithm proposed by Yin et al. [1991] is modified to simultaneously estimate the temporal and spatial parameters of the signal interference model developed in section 2.2.

3.1. Joint Parameters Estimation Algorithm

[11] We denote the N-dimensional received data vector in the mth and (m + 1)th PRI as xm and ym, respectively.

equation image
equation image

where ψ = diag[eequation image, eequation image, equation image, eequation image]. Matrix A in (5) and (6) relates to the cone angle of each ray, and its columns span the signal subspace. Matrix ψ relates to the Doppler frequency of each ray.

[12] The spatial covariance matrix of xm is

equation image

where E[·] denotes the expectation operator, S = E[smsmH] is the spatial covariance matrix of the signals, I is an N × N identity matrix, and σ2 is the variance of the additive noise.

[13] The cross spatial covariance matrix between ym and xm is

equation image

Define Rxx0 as

equation image

When the submode signals are noncoherent, Rank(Rxx0) = P. Wax et al. [1984] verified that the eigenvalues and eigenvectors of matrix Rxx0, denoted by {μ1μ2equation imageμN} and {v1, v2, equation image, vN}, respectively, have the following properties:

[14] 1. The minimal eigenvalue of matrix Rxx0 is zero, with multiplicity NP, i.e.,

equation image

[15] 2. The subspace spanned by the eigenvectors corresponding to the minimal eigenvalues is orthogonal to that spanned by the columns of matrix A, i.e.,

equation image

[16] From (10) and (11), it is easy to verify that

equation image

Denote Rxx0# as the pseudoinverse of Rxx0, and Rxx0# is defined as

equation image

[17] Following the verification of Yin et al. [1991], and from (9) we have

equation image

[18] Substituting (14) into (8), we have

equation image

Multiplying both sides of (15) by Rxx0#A

equation image

Define R as

equation image

[19] Substituting (13) and (17) into (16), we have

equation image

[20] Substituting (12) into (18), we have

equation image

As long as both A and S are full rank, and the diagonal elements of ψ are unequal, (19) implies that Doppler frequencies and cone angles of arrive are contained in the nonzero eigenvalues and corresponding eigenvectors of R.

[21] The Doppler frequencies and cone angles of arrival of the submode signals are estimated as follows

equation image
equation image

where μp and νp are nonzero eigenvalues and corresponding eigenvectors of R, νp(i) is the ith element of νp, and arg(·) denotes the argument of a complex number.

[22] Two parallel linear equispaced receiving antenna arrays were used by Yin et al. [1991] to estimate the azimuth angles and elevation angles of the sources. By exploiting the phase relationship of the received data vector between two successive pulses, only one linear equispaced receiving antenna array is used by the proposed algorithm. In other words, the main distinction between the proposed algorithm and that of Yin et al. [1991] is that ψ relates to Doppler frequencies of the submode signals in the proposed algorithm while ψ relates to elevation angles of the sources in that of Yin et al. [1991].

3.2. Implementation Considerations

3.2.1. Determination of the Number of Submodes

[23] An important step in the proposed algorithm described in section 3.1 is the determination of P, namely, the number of submodes. There are a few methods to estimate the number of sources [see, e.g., Krim and Viberg, 1996, and references therein]. The Gerschgorin disk estimator (GDE) developed by Wu et al. [1995] overcomes the problem in the case of a small sample number, and an unknown noise model.

[24] In the GDE method, the unitary transform matrix U is defined as

equation image

where VN−1 is an (N − 1) × (N − 1) unitary matrix formed by the eigenvectors of Rxx, and Rxx is an (N − 1) × (N − 1) leading principal submatrix of Rxx.

[25] The unitary transformed covariance matrix becomes

equation image

[26] The decision rule is [Wu et al., 1995]

equation image

where j is an integer in the closed interval [1, N − 2], ri = ∣ρi∣, i = 1, 2, ⋯, N − 1 is the Gershgorin radii, and D(M) is the adjustable factor, which could be a nonincreasing function (between 0 and 1) when the sample number increases. If GDE(j) is evaluated from j = 1, the number of submodes is j − 1 (i.e., P = j − 1) when the first nonpositive value of GDE(j) is reached. The GDE is employed in this paper to estimate the number of submodes.

3.2.2. Determination of the Number of Antennas

[27] The typical receiving antenna array of an HF OTHR consists of more than one hundred antennas, so full array processing is unrealistic. Computational complexity and available samples are two main limitations to the determination of the number of antennas. However, there are no definite guidelines for the selection of the number of antennas, especially when the signal environment is not known apriori.

[28] The computational complexity of the proposed algorithm is about O(N2M) + O(10N3) + O(P3), which is almost the same as that required by the one-dimensional ESPRIT technique [Roy and Kailath, 1989]. As indicated by Roy and Kailath [1989], the computational complexity of the one-dimensional MUSIC algorithm is about O(N2M) + O(10N3) plus the computational load required by a computationally expensive searching procedure. The computational complexity of the space-time MUSIC proposed by Su et al. [2006] is more than O(N2N12M) + O(10N3N13), where N1 denotes the number of temporal lags used. Table 1 summarizes the computational complexity of different algorithms. The number of complex multiplications required by each algorithm is shown in the last column. Moreover, for convenience of comparison, we assume the number of temporal lags in the space-time MUSIC algorithm and the temporal MUSIC algorithm equals the number of antennas in the proposed algorithm and the spatial smoothing MUSIC algorithm, that is, N1 = N.

Table 1. Estimates and Computational Complexity of Different Algorithms
AlgorithmsSubmode 1Submode 2Complex Multiplications
Cone Angle (deg)Doppler (Hz)Cone Angle (deg)Doppler (Hz)
Space-time MUSIC−29.221.01−28.120.96O(N4M) + O(10N6)
Spatial smoothing MUSIC−29.2X−28.2XO(N2M) + O(10N3)
Temporal MUSICX21.15X20.96O(N2M) + O(10N3)
Proposed algorithm−29.2321.013−28.17720.945O(N2M) + O(10N3) + O(P3)

[29] In practical applications, the true covariance matrix should be substituted by the sample covariance matrix which is estimated from a number of samples or snapshots. Reed et al. [1974] showed that at least M = 2N statistically independent sample data are needed if one wishes to maintain an average loss in the output signal to noise plus interference ratio (SINR) less than 3 dB relative to the true optimum as a result of the estimation error.

[30] Furthermore, Stoica and Nehorai [1989] proved that the Cramer-Rao bound (CRB) of any unbiased estimator of θ decreases monotonically with the increase in the number of antennas (N) or the number of samples (M). Accordingly, in order to accurately estimate the θ a large N or M is preferred.

[31] In the current application, the total number of available samples is fixed. Hence, too large an antenna array will not only degrade the estimation accuracy of the sample covariance matrix which increases the variance of the estimates but also increase computational complexity. On the other hand, too small an antenna array will also degrade the estimation accuracy of the proposed algorithm. Consequently, the number of antennas utilized in the proposed algorithm represents a compromised choice between the computational complexity and the variance of the estimates.

4. Experimental Results

[32] In this section, the proposed algorithm is applied to the data from a trial HF OTHR. During the experiment, a cooperative transmitter transmitted an LFMCW signal, which was located approximately 1000 km from the receiving antenna array and offset approximately −30 degrees from the boresight direction of the receiving antenna array. A communication link between the cooperative transmitter site and the receiving antenna array site makes it possible to synchronize the transmitted LFMCW signal with the waveform generator in the receiving antenna array. In order to indentify the beacon signal, a Doppler shift of 21 Hz is imposed intentionally on the beacon signal. Furthermore, Only 60 range bins covering the echoes of beacon signal are retained for further investigation.

[33] Figure 1 shows the range profile of the beacon signal which is calculated by averaging all the range profiles of every antenna in each PRI. As illustrated by Figure 1, three distinct peaks are resolved in group ranges. Since the oblique incidence ionogram is unavailable, the relationship between ionospheric layers and the peaks cannot be determined. However, samples corresponding to each peak can meet the requirement for investigating the temporal and spatial characteristics of the submode signals. In the following investigation, samples from range bin 29 are used.

Figure 1.

Average range profile.

[34] The performance of the MUSIC algorithm is severely degraded for coherent or highly correlated signals. The spatial smoothing scheme described by Shan et al. [1985] is a technique for eliminating the rank deficiency problem caused by coherent or highly correlated signals. In practical applications, the spatial smoothing scheme is adopted to maintain the performance of the MUSIC algorithm under these circumstances. Figure 2 illustrates the estimated spatial smoothing MUSIC spectra using 32 receiving antennas with the assumption of a different number of submode signals. The number of submode signals estimated by (23) is 2 which will be utilized in the following experimental data processing. As shown in Figure 2, an underestimated number of submode signals makes it impossible for the signals to be resolved correctly, and an overestimated number of submode signals leads to the spread and deviation of the peaks.

Figure 2.

Spatial smoothing MUSIC spectra with different numbers of submode signals.

[35] Figures 3 and 4 show the estimated spatial smoothing MUSIC spectrum and the temporal MUSIC spectrum, respectively, using 32 receiving antennas. Since two submode signals are resolved separately in the temporal and spatial domains, the problem of correctly pairing them occurs. The drawbacks of two separate one-dimensional estimation algorithms motivate the usage of jointly temporal and spatial parameters estimation algorithms.

Figure 3.

Spatial smoothing MUSIC spectrum.

Figure 4.

Temporal MUSIC spectrum.

[36] Figure 5a shows the estimated 3-D space-time MUSIC spectrum using 16 receiving antennas and 16 temporal lags. Figure 5b shows the top view of Figure 5a. Two distinct peaks are resolved by the space-time MUSIC algorithm as shown in Figures 5a and 5b. As demonstrated in Figures 5a and 5b, the Doppler frequency and cone angle of each submode signal are automatically paired. However, the eigen decomposition of a 256 × 256 space-time covariance is very time consuming.

Figure 5.

(a) Space-time MUSIC spectrum. (b) Top view of Figure 5a.

[37] Figure 6 shows the estimates of the proposed algorithm using 32 receiving antennas. As illustrated by Figure 6, two submode signals are resolved and parameters of each submode signal are correctly paired. As mentioned in section 3.2.2, the computational load of the proposed algorithm is almost the same as that of the one-dimensional MUSIC algorithm and is much lower than that of the space-time MUSIC algorithm.

Figure 6.

Estimates of the proposed algorithm.

[38] Parameters of each submode signal estimated by different superresolution algorithms are listed in Table 1. As demonstrated by Table 1, estimates of different superresolution algorithms are well matched. Slight deviations among different superresolution algorithms may be due to the tolerance of algorithms to various errors such as the position, amplitude and phase error of the receiving antenna array.

5. Conclusion

[39] Although parameters of the ionospherically propagated HF signals can be estimated in the temporal and spatial domains separately, correctly pairing them is not easy. Jointly estimating the parameters is preferred because pairing is achieved automatically. Conventional subspace based two-dimensional space-time high-resolution algorithms are computationally expensive. A computationally efficient superresolution algorithm is proposed in this paper, which can simultaneously estimate the Doppler frequencies and angles of arrival of the ionospherically propagated HF signals. Experimental data processing results show the effectiveness of the proposed algorithm. Additionally, the model with the HF signal reflected by a smooth ionospheric layer consisting of a number of submode signals with slightly different Doppler frequencies and angles of arrival is also verified by the experimental data processing results.

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