Frequency Diverse Array Auto-Scanning Beam Characteristics and Potential Radar Applications

Different from the phased-array with the same carrier frequency for all elements providing only angle-dependent beampattern, frequency diverse array (FDA) using a small frequency offset across its array elements offers new application capabilities due to its both angle and range dependent beampattern, which allows auto beam-scanning capability even without the employment of phase shifters, which are required in phased-array. This paper studied FDA auto-scanning beam characteristics and several potential radar applications. The FDA auto-scanning beam characteristics including range and angle coupling, beam auto-scanning rate and integral array again are analyzed. To fully utilize FDA specific auto-scanning beam characteristics, several potential applications such as radio frequency (RF) stealth or RF localization deception, range-ambiguous clutter suppression and mainlobe interference suppression are discussed. Moreover, several discussions about space-time adaptive processing and possible future investigations are provided. This paper is formulated from a top-level system description, which aims to introduce this promising FDA technique to radar community so that more investigations can be facilitated.


I. INTRODUCTION
Beam-scanning antenna is a key requirement in many wireless systems [1]. Since mechanically beam-scanning reflector antenna has slow speed of beam scanning, typical solution is to adopt phased-array antenna [2]. Phased-array is well known for its robust capability to electronically steer its beam with high effectiveness [3], [4], the beam steering of phased-arrays in an angle is fixed for all range cells [5], [6]. That is, a limitation of the phased-array antenna is that the beam steering is independent of the range parameter. However, the ability to allow range-dependent beampattern control is an increasingly desirable capability in many applications [7], such as range ambiguous clutter or interference suppression. On the other hand, low-cost beamscanning antennas are very promising for mobile terminals [8], [9], [10].
The associate editor coordinating the review of this manuscript and approving it for publication was Debdeep Sarkar .
In this paper, frequency diverse array (FDA) [11] is adopted to study for low-cost wide-angle beam auto-scanning antennas. As illustrated in Figure 1, different from the phased-array with the same carrier frequency for all elements, which provides only angle-dependent beampattern, FDA using small frequency offsets across its array elements enables the beampattern focusing direction to change as a function of the angle, range and even time variables [12]. FDA can be seen as a combination of spatial diversity and frequency diversity. Nevertheless, note that FDA is different from traditional multiple-input multiple-output (MIMO), orthogonal frequency division multiplexing (OFDM) and frequency scanning techniques. The comparisons between FDA and other similar array techniques such as phasedarray, MIMO, frequency scanning and time-modulated arrays are listed in Table 1. It is shown that, while compared with traditional phased-array offers only angle-dependent beampattern, FDA provides not only angle-dependent but also range-dependent beampattern which is an advantage for  range-dependent target localization and interference suppression, but FDA also has the disadvantage of range-angle coupling beampattern response, which may result in ambiguous results for joint range and angle estimation of targets.
Motivated by its range-dependent/time-variant beampattern characteristics and promising application potentials [13], a special issue about FDA was published in IEEE Journal of Selected Topics in Signal Processing [14]. In recent years, FDA has received much attention, such as beampattern synthesis, range-dependent interference suppression, radar target detection and localization, deceptive jamming [15], [16], [17], secure communications [18], and high-resolution imaging.
The FDA range-dependent transmit beamapattern characteristics for a monochromatic continuous-wave waveform were analyzed in multiple papers [19], [20]. The angle and time periodicity of FDA transmit beampattern was analyzed in [21] and electromagnetically simulated in [22]. Furthermore, the multipath propagation characteristics of FDA transmitted signals were analyzed in [23]. Since the FDA with uniformly spaced linear elements and linearly progressive frequency offsets will generate range-angle coupling S-shaped beampattern, multiple methods including nonlinearly increasing [24], random [25], [26] and timedependent [27], [28], [26], [29] frequency offsets were studied to decouple for focusing beampattern synthesis. However, it should be noted that the essence of time-variance in FDA transmit beampattern is ignored in the aforementioned papers by only considering an instantaneous time. That is to say, time-invariant beampattern cannot be implemented for practical FDA antenna [30]. Otherwise, it just simplifies into a traditional phased-array antenna. Nevertheless, we believe that FDA beampattern time-variance should be utilized for enhanced application performance, rather than suppressing the time-variance. Therefore, unlike the FDA literature that mainly suppress transmit beampattern timevariance, this paper analyzes and exploits FDA time-variance to provide wide-angle auto-scanning beam for potential radio frequency (RF) stealth, range-ambiguous clutter suppression and mainlobe interference suppression applications.
The main contributions of this paper are summarized as follows: • The auto-scanning beam characteristics of FDA antennas together with the time-variance, range-dependence, range and angle coupling, and integral array gain characteristics are systematically analyzed and compared with traditional phased-array antenna.
• FDA potential radar applications including RF stealth or RF localization deception, range-ambiguous clutter suppression and mainlobe interference suppression are presented, which shows outperformed performance over classic phased-array techniques.
• Several discussions about technical challenges in FDA radar space-time adaptive processing and possible future investigations are provided from a top-level system description to call for more investigations on such promising FDA techniques. The remaining sections are organized as follows: Section II analyzes the auto-scanning characteristics of FDA transmit beam. Section III discusses FDA potential applications including RF localization deception and mainlobe interference suppression. Furthermore, several discussions and possible future investigations are provided in Section IV. Finally, conclusions are drawn in Section V.

II. FDA AUTO-SCANNING BEAM CHARACTERISTICS A. GENERAL FDA TRANSMIT BEAMPATTERN
We consider a general uniform linear array (ULA) FDA, which modulates the same baseband waveform to a distinct center frequency for each array element. As shown in Figure 2, the mth array element transmits the signal at the center frequency: where f 0 is the carrier frequency and f m is the frequency offset for the mth element. In amplitude sense, the propagation attenuation for the distance differences between individual elements to the given position point can be ignored. In phase VOLUME 10, 2022 sense, the range r m from the observation point to the mth element can be approximated by r m ≈ r − md sin θ, where r is the reference range to the first element, d is the element spacing, which is assumed to be d = c 0 /2f 0 with c 0 being the speed of light, and θ is the array direction angle. The typical f m in an FDA radar system with the carrier frequency of f 0 = 10 GHz is about f m = 10 kHz. That is, f < f 0 /10 6 is generally adopted. Therefore, due to the fact that f 0 f m , the equal normalized electric field function e(θ) is assumed for all elements.
Generally, consider a pulsed FDA system and define the transmitted baseband waveform φ(t), 0 ≤ t ≤ T p with pulse duration T p satisfies the unit energy, i.e., T p φ * (t)φ(t)dt = 1, where * denotes the complex conjugate operator. For a given far-field position with angle θ and range r, the arrived signals can be expressed as where w m is the complex weight for the mth element. Note that, here the time variable t should be limited by r c 0 ≤ t ≤ T p + r c 0 . For notation convenience, we define Under narrow-band assumption, we have φ t − r m c 0 ≈ φ t − r c 0 . In this case, the FDA transmit beampattern can be defined as Since γ m depends on both the angle θ and range r, the FDA transmit beampattern F(θ, r, t) will change as a function of the angle θ and/or range r. For instance, the beampattern P(θ, r, t) for given range r 0 and time t 0 can be expressed, respectively, as [6] As an example, consider an FDA with M = 16 elements spaced by half wavelength and T p = 10µs. For simplicity without loss of generality, f m = m · f with f being a constant is assumed. Figure 3 shows that FDA beam will change as a function of the time (range) for a fixed range (time), which implies that FDA transmit beam will auto-scan in the range-angle dimension even for fixed range and/or time. This motivates us to adopt FDA for wide-angle autoscanning beam applications.

B. FDA RANGE AND ANGLE COUPLING BEAMPATTERN
To quantitatively analyze the specified FDA beampattern characteristics and for simplicity without loss of generality, f m = m· f with f being a constant is assumed thereafter. Considering (3), due to the fact that f m c 0 and m = 0, 1, 2, . . . , M −1 is generally smaller than 100, we then have the following approximation: which is true because both left-side and right-side are nearly 0 and thus they depend very weakly on m. The (3) can then be further approximated as In this case, substituting (7) into (4) yields the FDA transmit beampattern: The FDA beampattern maximum arrives at Even for a fixed time t = 0, the beampattern peaks will be periodic, which arrive at with λ 0 = c 0 /f 0 being the wavelength, which is generally assumed to be d = λ 0 /2. As shown in Figure 4, due to the range-angle coupling effects, the ''S'' peaks will be generated in the FDA range-angle beampattern.

C. FDA BEAM AUTO-SCANNING RATE
It should be noted that, for a given pair (θ, r), the above (10) does not guarantee the existence of beampattern maximum, because the suitable integer n may not exist. The FDA beam auto-scanning rate can be evaluated by differentiating (9) with respect to t: It states the beam scanning rate depends on both angle θ and frequency offset f . Suppose the corresponding angles at t 1 and t 2 are θ 1 and θ 2 , respectively. The FDA beam scanning during one pulse duration T p will be This implies that FDA can complete the entire scanning from −90 • to 90 • for given f and T p . Generally, d = c 0 /(2f 0 ) is adopted. In this case, the above (12) can be simplified into If f = 1/T p , the beam can scan from −90 • to 90 • in one pulse. However, the beam direction change in one pulse will be ignorable for 1/ f T p . Figure 5 shows the FDA beam scanning in one pulse duration and across the pulses. That is, the actual beam scanning characteristics can be controlled by tuning the frequency offset, even for a fixed element spacing in the antenna without changes of the pulse duration T p and carrier frequency f 0 . However, it should be noted that the radiation pattern may be not an ideal ''S'' shape in the pulsed FDA systems.

D. FDA INTEGRAL ARRAY GAIN
Since FDA beampattern has time-variance and auto-scanning characteristics, to effectively evaluate the time-variant array gain, we define the integral transmit beampattern in the angle dimension as [31] where H is the conjugate transpose operator, a θ (θ) is the FDA transmit steering vector in the angle dimension: with T being the transpose operator, and R T ∈ C M ×M is the covariance matrix of the FDA transmitted signal with the m, m th element being  where w m and w m denote the weights for the mth and m th elements, respectively. As shown in Figure 6 T being the transmit weighting vector. In this case, the FDA will have equal array gain to conventional phased array. In contrast, if f ≥ B s , then [R T ] m,m ≈ 0 and IP(θ ) ≈ w 2 , where w is the 2-norm operator. This case implies that orthogonal baseband waveforms are employed by the FDA and omnidirectional array radiation will generate for the array antenna.

III. POTENTIAL APPLICATIONS
FDA has not only all the advantages of phased-array, but also the range-dependence, range-angle coupling and spatial variance characteristics in the transmit beampattern. It is thus expected to have a subversive impact on the existing active radar target detection, passive detection and electronic countermeasure techniques. In this section, we discuss several typical potential applications.

A. RF LOCALIZATION DECEPTION
According to (8), FDA will generate an apparent angle θ a [19]: which is different from its real scanning angle θ, as shown in Figure 7 that FDA apparent direction angle is not equal to its physical direction angle. That is, the FDA apparent direction depends on not only its physical direction but also the employed frequency offset f . This implies that FDA transmitted signals may have RF localization deception or RF stealth for low probability of identification (LPI) applications [32]. To evaluate the RF localization deception performance of the FDA transmitted signals, we consider one four-antenna reconnaissance receiver, as shown in Figure 8. The 85282 VOLUME 10, 2022 Suppose the slant range from the reconnaissance receiver to the antenna 1 is r. According to (2), the four antennas received signals can be expressed, respectively, as where φ 0 is the reference initial phase. Taking the signals received by the antenna 1 as the reference and assuming the carrier phases of y 1 (θ, r, t), y 2 (θ, r, t), y 3 (θ, r, t) and y 4 (θ, r, t) are φ 1 , φ 2 , φ 3 and φ 4 , the phase differences between the four received signals to the antenna 1 can be calculated as The direction of arrival (DOA) of the FDA transmitted signal can then be estimated from the interferometry phase aŝ As an example, we consider the following parameters: D x = 0.015m, D y = 0.0075m, f 0 = 10 GHz, f = 20 kHz and M = 16 elements spaced by half wavelength, Figure 9 compares the four-antenna receiver's interferometry reconnaissance performance for FDA and phased-array transmitted signals under SNR = 15 dB. The results verify that, while compared to phased-array, FDA will seriously degrade the interferometry reconnaissance performance, which validates that RF localization deception capability indeed can be achieved for the FDA transmitted signals.

B. RANGE AMBIGUOUS CLUTTER SUPPRESSION
Since FDA transmit steering vector depends on the angle, range and time variables, it is necessary to adopt space-timerange joint processing [33] for FDA radar data processing. In [34], a range-angle coupled signal model for simultaneous adaptation in the spatial and range dimensions was utilized to mitigate the joint space-range sidelobes inherent VOLUME 10, 2022 to FDA radars. A reiterative minimum mean square errorbased space-time adaptive processing (STAP) algorithm was proposed in [35] to address FDA radar range-couple adaptive processing. Additionally, Xu et al. [36] studied robust adaptive beamforming for FDA-STAP radar fast-moving target detection and Wen et al. [37], [38] investigated STAP in slowtime FDA-MIMO radars.
Particularly, STAP for range ambiguous clutter suppression in airborne FDA-MIMO radar was presented in [39] by exploiting the range dependence property and relationship between spatial and Doppler frequencies. Due to the fact that FDA steering vector consists of the angular, range and time variables, the spatial frequency of the forward-looking FDA radar with azimuth angle θ and elevation angle ϕ should be formulated as The above (24) simplifies into the spatial frequency for phased-array forward-looking radar when f = 0, which states that FDA radar produces both angle and range dependent spatial frequency, while only angle dependent spatial frequency can be generated in phased-array radar.
Consider also the Doppler frequency where f prf is the pulse repetition frequency (PRF) and v is the relative velocity between the radar platform and observed target. The spatial and Doppler frequencies relationship in the FDA radar can then be expressed as [39] λ Obviously, this elliptical relationship is different from that of the phased-array radar [40]. It was concluded in [39] that the elliptical curves in forward-looking FDA radar will be shifted in the spatial frequency domain with respect to the slant range while that of traditional phased-array radar will remain invariant. This implies that the clutters from ambiguous and unambiguous ranges cannot be separated in phased-array radar STAP processing, but they can be discriminated in the STAP spatial frequency domain for FDA radar.
To evaluate the STAP performance of FDA radar, we consider the system improvement ratio (SIR), which is defined as [41] where P in s /P out s and P in n /P out n denote the signal and noise (and clutter) input/output power, w is the space-time adaptive weighting vector, s is the assumed steering vector for a target, R is the clutter-plus-noise covariance matrix, and Tr(·) denotes the trace of a matrix. According to the SIR, the minimum detectable Doppler frequencies at two sides of the clutter notch can then be determined, respectively, as where L SIR = SIR/γ 0 with γ 0 being the coherent accumulation gain, and L 0 is the decision threshold. Defining the Doppler frequency at the clutter notch as f d = 0, the normalized minimum detectable Doppler frequency can then be defined as Considering the forward-looking FDA radar system with M = 8 transmit elements and N = 8 receive elements spaced by half wavelength and the other parameters are listed as f 0 = 8 GHz, f = 3 kHz, PRF = 15 kHz, platform altitude is 3 km, platform velocity is v = 90 m/s, signal-to-noise ratio (SNR) is 0 dB, clutter-to-noise ratio (CNR) is 20 dB, and 10 snapshots are assumed, Figure 10 compares the FDA radar SIR loss versus normalized Doppler frequency with that of the phased-array with the same parameters except f = 0 Hz, where the cases without and with range ambiguity are considered separately. Note that the decision threshold is typically considered at −2 dB in statistical detection algorithms [41]. The results show that narrower notch width can be achieved for the FDA radar, which states that smaller minimum detectable velocity can be implemented for FDA radar than traditional phased-array radar, because the detectable velocity has a linear relationship to the minimum detectable 85284 VOLUME 10, 2022 Doppler frequency in moving target indication (MTI) applications.

C. MAINLOBE JAMMING SUPPRESSION
It is well known that phased-array can effectively suppress sidelobe interferences, but it cannot efficiently deal with mainlobe interferences because of its only angular-dependent array factor. In contrast, FDA offers a solution to counteract mainlobe interferences. We initially discussed the functionality of FDA subarrays in suppressing mainlobe jamming signals in [42], and exploited the combination of FDA and MIMO in counteracting mainlobe jammers in [43] by jointly utilizing FDA in providing rang-dependent beampattern and MIMO in increasing degrees-of-freedom (DOFs).
Different from that in traditional radars the targets should be diagonally distributed in the transmit-receive spectral distribution due to the same transmit and receive spatial frequencies, the targets in FDA radar may be non-diagonally located in the transmit-receive spectra because of the frequency offset caused spatial frequency 2 f c 0 r (see also (24)). This property can then be utilized to suppress mainlobe interferences.
After multichannel matched filtering, the FDA radar received signals can be modeled as [44] Y = ξ 0 w H T a θ (θ) a r (r) ⊗ b(θ) + a r (r) ⊗ n (31) where ξ 0 is the target reflection coefficient, n denotes the noise vector, ⊗ denotes the Kronecker product operator, and a r (r) and b(θ) denote the transmit steering vector in the range dimension and receive steering vector: According to (31), the incoming mainlobe interference located at (θ j , r j ) can be expressed as where ξ j denotes the incoming interference strength.
Once if the jammer and target have distinct range positions, the interference can be canceled by designing a range-related blocking matrix, even if the jammer and target are located in the same angular. Using the projection matrix: we can get This implies that the interference is successfully suppressed and the desired target with the same angle (i.e., θ = θ j ) will be preserved in the final suppressing results. Suppose the following simulation parameters: The target and jammer are located at (10 • , 10km) and (10 • , 15km), f 0 = 10 GHz, f = 20 kHz and M = N = 10, Figure 11 shows that FDA radar can suppress the mainlobe interference and detect the desired targets successfully, while phased-array radar cannot localize the target due to its range-independent transmit-receive target response.

IV. DISCUSSIONS AND FUTURE WORKS
FDA radar can adaptively update its range-dependent beampattern in both range and azimuth dimensions to suppress space-time-range-dependent interferences. However, in the FDA radar signal processing algorithm, when the noise and interference are Gaussian, the unconstrained maximum likelihood (ML) estimate of the signal covariance matrix is actually the sample variance matrix, but it is difficult to obtain the required covariance matrix of noise and interference in the FDA radar due to its range, angle and time-dependent response. This nonstationarity results in a limited amount of snapshot data available for the processing that may be VOLUME 10, 2022 insufficient for an algorithm to coverage. A potential solution is to adopt subspace tracking algorithms such as projection approximation and fast approximated power iteration. Reiterative minimum mean square error algorithm has the capability of simultaneously adaptation in slow-time and fasttime to enhance the sensitivity by suppressing Doppler and range sidelobes, but its large computation complexity due to multidimensional adaptivity must be significantly reduced for practical engineering applications. Reduced rank techniques can reduce the rank associated with the interference-plusnoise covariance matrix. Many rank reduction methods in the literature can achieve the performance near or better than their full rank counterparts but with reduced sample support and computational load. Particularly, suboptimal fast-time STAP algorithms to suppress multipath interferences [23], while preserving the final radar detection performance, should be developed.
The basic implementation using monochromatic assumption and equally spaced elements is generally considered in the FDA literature. Many more fertile areas remain for further exploration. Linearly increasing frequency offsets and a uniform weighting vector are often assumed in existing FDA papers. However, they may be not the best choice for specific applications. Two-dimensional FDA allows threedimensional beam-steering in the azimuth and elevation dimensions, which pattern will be extremely unique and affords additional properties to develop operational strategies. Amplitude weighting can be employed for FDA beamforming techniques, which may be also useful for rejecting multipath interferences, but more investigations are required.
Furthermore, directional modulation can be achieved for FDA. One simple approach is to divide the whole array into multiple subarrays, each of which is steered to a particular position. The frequency progression can be used across the subarrays, instead of across the elements as the basic FDA. Since the frequency progression produces both angle-dependent and time-variant beampattern, we can utilize the frequency progression in vertical dimension to alleviate the range layover effects [45]. The essence is to construct an effective and efficient mapping between the actual collection location and the apparent collection location. This is an interesting future research topic.
Another future work is to optimize the frequency offsets. It is likely that the optimized frequency allocation is actually dependent on the physical scenarios. Several methods have been studied to optimally design the frequency offsets for FDA radar (see [12] and the references wherein). It is expected that the sample support data homogeneity can be considerably increased for the targets off-boresight in azimuth, as the isodop shape is better accentuated by the FDA beampattern. It is necessary to investigate possible benefits of adaptive frequency offsets control in FDA radar systems.
Additionally, besides varying apparent angle, FDA also exhibits interesting non-uniform plane wave. This is because that, for a given range, the FDA planes of constant phase and constant amplitude do not coincide, which generates different orientations in the propagation. According to [45], the variation in FDA radar signal amplitude and phase may produce a set of apparent collection locations that are nonuniformly spaced and appear to alter the radar platform's actual moving track. An additional processing step is thus required between data collection and parameter estimation to formulate a mapping from the actual collection locations to the apparent radar collection locations. The purpose is to match the phase history collected at the actual collection locations for the FDA radar transmitted signal. In addition, the FDA radar assigned frequency offsets should operate precisely, but it is not an easy task in practical system implementation. Robust processing algorithm is necessary to address the effects of unideal frequency offsets.

V. CONCLUSION
This paper studied FDA auto-scanning beam characteristics including range and angle coupling, beam auto-scanning rate and integral array again. Both theoretical analysis and simulation results verify that FDA produces rather different array beam characteristics from traditional phasedarray. Several potential application such as RF stealth or RF localization deception to encounter unintended interceptor, range-ambiguous clutter suppression and mainlobe interference suppression were presented to exploit FDA auto-scanning transmit beam and range-dependent transmitreceive channel response. In addition, technical challenges in space-time adaptive processing and possible future works were discussed. Certainly, it may be not possible to replace phased-array with FDA totally, as phased-arrays are prevalent today. Nevertheless, FDA can be regarded as an extension of phased-array using distinct baseband waveforms for different array elements, which can be employed to compensate the disadvantage of phased-array in range-independence and MIMO in loss of transmit array gain. It is thus believed that FDA is a promising array technique that will play an important role in radar applications.