Transmit beampattern of basic FDA

Different from PA that transmits signals with identical frequency, there exists a small frequency increment in the carrier frequency across antennas in FDA. Consider a uniform linear array (ULA) consisting of M elements with spacing d. The signal radiated by the mth antenna can be represented as

(1)$$x_m(t) = e^{\,j2\pi f_mt},\;m = 0,\;1,\; \ldots, \;M-1,$$

where f _m is defined as

(2)$$f_m = f_0 + m\Delta f,$$

where f ₀ is the reference carrier frequency and Δf is the frequency increment. The overall signal observed at a far-field point with coordinate (r, θ) is

(3)$$X(t;\;r,\;\theta ) = \sum\limits_{m = 0}^{M-1} {x_m\left( {t-\displaystyle{{r_m} \over c}} \right)}, $$

where c is the speed of light in free space and r _m is the distance from the point to the mth antenna. Take far-field approximation r _m ≈ r − mdsinθ into consideration and substitute (1) and (2) into (3), we have

(4)$$\eqalign{X(t;\;r,\;\theta ) = &\sum\limits_{m = 0}^{M-1} {e^{\,j2\pi (\,f_0 + m\Delta f)[t-(r-md\sin \theta )/c]}} \cr &\approx e^{\,j2\pi f_0(t-r/c)}\sum\limits_{m = 0}^{M-1} {e^{\,j2\pi m\Delta f\,(t-r/c)}e^{\,j2\pi f_0md\sin \theta /c}},} $$

where r and θ are the range and azimuth angle of the point relative to the first antenna, respectively. Note that the term $e^{j2\pi \Delta fm^2d\sin \theta /c}$ has been neglected from (4) since Δf << f ₀. Thus, the array factor can be written as

(5)$$AF(t;\;r,\;\theta ) = \sum\limits_{m = 0}^{M-1} {e^{\,j2\pi m\Delta f\,(t-r/c)}e^{\,j2\pi f_0md\sin \theta /c}} $$

and accordingly the FDA transmit steering vector is given by

(6)$${\bf a}(r,\;\theta ) = \lsqb {1,\;e^{\,j\varphi}, \; \ldots, \;e^{\,j(M-1)\varphi}} \rsqb ^T,$$

where φ = 2πf ₀dsinθ/c − 2πΔfr/c and (·)^T denotes the transposition.

Figure 1 shows a basic FDA transmit beampattern distributed in the range–angle domain. The input parameters for this beampattern are: the carrier frequency is f ₀  =  10 GHz, the frequency increment is Δf  =  16 KHz, and the number of antennas is M = 10. It is clearly seen that the FDA transmit beampattern is a function of range and angle and has S-shape because of coupling, which can be properly exploited for joint range–angle estimation of targets.

Fig. 1. Transmit beampattern of FDA radar.

Signal model of FDA-MIMO radar

Now consider a co-located FDA-MIMO radar that contains M transmitting elements and N receiving elements. Compared with the basic FDA radar, the mth emitted signal is modified as

(7)$$x_m(t) = \phi _m(t)e^{\,j2\pi f_mt},\;0 \le t \le T,$$

where T is the signal duration, and ϕ _m(t) is the baseband modulation signal that satisfies the orthogonality condition, i.e.

(8a)$$\int_0^T {\phi _m(t)\phi _{{m}^{\prime}}^{^\ast} (t-\tau )dt = 0,\;m\ne {m}^{\prime},\;\forall \tau}, $$

(8b)$$\int_0^T {\phi _m(t)\phi _{{m}^{\prime}}^{^\ast} (t)dt = 1,\;m = {m}^{\prime}}, $$

where $(\cdot)^{^\ast}$ denotes the conjugation and τ is the time shift. Here, although the form of ϕ _m(t) is not defined in the theoretical derivation, specific modulation functions should be used in the numerical simulation. Actually, the radar performance can be affected by the waveform modulation so transmit waveform design and optimization for FDA-MIMO radar is also significant; however, this issue is beyond the scope of this paper.

The transmitted signals are reflected by the target located at (r, θ) and then recorded by the receiving array. The signal received by the nth antenna can be written as

(9)$$y_n(t) = \sum\limits_{m = 0}^{M-1} {\phi _m(t-\tau _{tr})e^{\,j2\pi f_m(t-\tau _{tr})}}, \;n = 0,1,\; \ldots, \;N-1,$$

where τ _tr is the total time delay corresponding to signal propagation from the mth transmitting element to the nth receiving element, expressed as

(10)$$\tau _{tr} = \displaystyle{{2r-md_T\sin \theta -nd_R\sin \theta} \over c},$$

where d _T and d _R are the element spacings of transmitting and receiving array, respectively. Substituting (10) into (9) yields

(11)$$\eqalign{y_n(t) = &\sum\limits_{m = 0}^{M-1} {\phi _m[t-(2r-md_T\sin \theta -nd_R\sin \theta )/c]e^{\,j2\pi f_m[t-(2r-md_T\sin \theta -nd_R\sin \theta )/c]}} \cr &\approx e^{\,j2\pi f_0(t-2r/c)}e^{\,j2\pi nd_R\sin \theta /\lambda _0} \cr &\times \sum\limits_{m = 0}^{M-1} {\phi _m[t-(2r-md_T\sin \theta -nd_R\sin \theta )/c]e^{\,j2\pi \Delta fm(t-2r/c)}e^{\,j2\pi md_T\sin \theta /\lambda _0}},} $$

where λ ₀  =  c/f ₀ is the carrier wavelength. Similar to (4), quadratic terms related to the element index have been neglected from (11).

After down converting and matched filtering, the output of the nth receiving element for the mth transmitting element is represented by

(12)$$y_{m,n}(r,\;\theta ) = \xi e^{\,j2\pi nd_R\sin \theta /\lambda _0}e^{-j4\pi \Delta fmr/c}e^{\,j2\pi md_T\sin \theta /\lambda _0},$$

where ξ is the complex scattering coefficient of the target.

In an FDA-MIMO radar, the range changing characteristic provided by FDA and the angle changing characteristic provided by MIMO are effectively combined, resulting in synchronous range and angle estimation. The schematic diagram of FDA-MIMO radar is illustrated in Fig. 2.

Fig. 2. Schematic diagram of FDA-MIMO radar.

Range-angle estimation using TR-FDA-MIMO radar

Now we present the TR-FDA-MIMO signal model and perform target range–angle estimation based on the proposed scheme. For expression conciseness, vector form is adopted in signal derivation, and meanwhile noise is taken into account.

Assume there are P incoherent targets located at (r _p, θ _p) in the far field with p = 0, 1, …, P−1. The transmitting signal vector is

(13)$${\bf X}(t) = \lsqb {x_0(t),\;x_1(t),\; \ldots, \;x_{M-1}(t)} \rsqb ^T$$

and the corresponding receiving signal vector is expressed as

(14)$${\bf Y}(t) = \sum\limits_{\,p = 1}^P {{\bf b}(\theta _p)\beta _p{\bf a}^T(r_p,\;\theta _p){\bf X}(t)} + {\bf n}_y(t),$$

where β _p is the complex scattering coefficient of the pth target. n_y(t) is the white Gaussian noise (WGN) vector with mean and variance being zero and σ ² respectively. b(θ) is the angle-dependent receiving steering vector which is given by

(15)$${\bf b}(\theta ) = \lsqb {1,\;e^{\,j\varphi_b},\; \ldots, \;e^{\,j(N-1)\varphi_b}} \rsqb ^T,$$

where φ _b = 2πd _Rsinθ/λ ₀. ${\bf a}(r,\;\theta )$ is the range–angle-dependent transmitting steering vector which is given by

(16)$${\bf a}(r,\;\theta ) = \lsqb {1,\;e^{\,j\varphi_a},\; \ldots, \;e^{\,j(M-1)\varphi_a}} \rsqb ^T,$$

where φ _a = 2πd _Tsinθ/λ ₀ − 4πΔfr/c.

For general FDA-MIMO radar, the output signal of the matched filter is

(17)$${\bf Y}_{MF} = E\lcub {{\bf Y}(t){\bf X}^{\it H}(t)} \rcub = \sum\limits_{\,p = 1}^P {\beta _p{\bf b}(\theta _p){\bf a}^T(r_p,\;\theta _p)} + {\bf N}_y,$$

where E{·} denotes the mathematical expectation, (·)^H denotes the conjugate transposition, and N_y is the matched filter output of n_y(t). For the proposed TR-FDA-MIMO radar, according to the principle of time reversal, the original received signals are time reversed, phase conjugated, and retransmitted by the receiving array to illuminate the targets again. The back-propagated signals will focus on the targets, be reflected, and recorded by the transmitting array. The retransmitted signal is represented as

(18)$${\bf Y}_{TR}(t) = {\bf Y}^{^\ast}(-t) = \sum\limits_{\,p = 1}^P {{\bf b}^{^\ast}(\theta _p)\beta _p^{^\ast} {\bf a}^H(r_p,\;\theta _p){\bf X}^{^\ast}(-t)} + {\bf n}_y^{^\ast} (-t)$$

thus the final received signal is

(19)$$\eqalign{{\bf Z}(t) = &\sum\limits_{\,p = 1}^P {{\bf a}_0(\theta _p)\beta _p{\bf b}^T(\theta _p){\bf Y}_{TR}(t)} + {\bf n}_z(t) \cr = &\sum\limits_{\,p = 1}^P {{\vert {\beta_p} \vert } ^2{\bf a}_0(\theta _p){\bf b}^T(\theta _p){\bf b}^{\rm {^\ast}}(\theta _p){\bf a}^H(r_p,\;\theta _p){\bf X}^{^\ast}(-t)} + {\bf n}(t),} $$

where a₀(θ) is the TR receiving array manifold. n_z(t) is also a WGN vector whose mean and variance are zero and σ ² respectively, and n(t) is the accumulated noise which is given by

(20)$${\bf n}(t) = \sum\limits_{\,p = 1}^P {{\bf a}_0(\theta _p)\beta _p{\bf b}^T(\theta _p){\bf n}_y^{^\ast} (-t)} + {\bf n}_z(t).$$

Note that a₀(θ) here actually acts as the receiving steering vector for TR signals, thus it is angle-dependent only and written as

(21)$${\bf a}_0(\theta ) = \lsqb {1,\;e^{\,j\varphi_{a_0}},\; \ldots, \;e^{\,j(M-1)\varphi_{a_0}}} \rsqb ^T,$$

where $\varphi _{a_0} = 2\pi d_T\sin \theta /\lambda _0$. Similar to (17), the matched filter output of the TR received signals is calculated as

(22)$${\bf Z}_{MF} = E\lcub {{\bf Z}(t){\bf X}^T(-t)} \rcub = N\sum\limits_{\,p = 1}^P {{\vert {\beta_p} \vert } ^2{\bf a}_0(\theta _p){\bf a}^H(r_p,\;\theta _p)} + {\bf N} = N{\bf A}_0{\bf \Lambda} {\bf A}^H + {\bf N},$$

where ${\bf N}$ is the matched filtering result of ${\bf n}(t)$, and ${\bf A}_0$, ${\bf \Lambda} $, and ${\bf A}$ are given by

(23a)$${\bf A}_0 = \lsqb {{\bf a}_0(\theta_1),\;{\bf a}_0(\theta_2),\; \ldots, \;{\bf a}_0(\theta_p)} \rsqb,$$

(23b)$${\bf \Lambda} = {\rm diag}\lpar {{\vert {\beta_1} \vert }^2,\;{\vert {\beta_2} \vert }^2,\; \ldots, \;{\vert {\beta_p} \vert }^2} \rpar,$$

(23c)$${\bf A} = \lsqb {{\bf a}(r_1,\;\theta_1),\;{\bf a}(r_2,\;\theta_2),\; \ldots, \;{\bf a}(r_p,\;\theta_p)} \rsqb,$$

respectively.

To acquire the targets imaging spectra in the range–angle domain based on ${\bf Z}_{MF}$, we utilize the MUSIC algorithm that is commonly used in MIMO radar DOA estimation for favorable resolution performance. First, vectorization is performed to ${\bf Z}_{MF}$ to obtain the virtual data vector as follows

(24)$${\bf z} = {\rm vec}\lpar {{\bf Z}_{MF}} \rpar = N\sum\limits_{\,p = 1}^P {{\vert {\beta_p} \vert } ^2{\bf a}_{TR}(r_p,\;\theta _p)} + {\bf n},$$

where vec(·) denotes the vectorization and ${\bf n} = {\rm vec}\lpar {\bf N} \rpar $. ${\bf a}_{TR}(r,\;\theta )$ is the TR joint transmitting–receiving steering vector defined as

(25)$${\bf a}_{TR}(r,\;\theta ) = {\bf a}^{^\ast}(r,\;\theta )\otimes {\bf a}_0(\theta ),$$

with ⊗ denoting the Kronecker product. Next, the covariance matrix of ${\bf z}$ is solved and the eigenvalue decomposition (EVD) is conducted

(26)$${\bf R}_z = E\lcub {{\bf z}{\bf z}^H} \rcub = \lpar {{\bf U}_s \vert {\bf U}_n} \rpar {\bf \Sigma} \lpar {{\bf U}_s \vert {\bf U}_n} \rpar ^H,$$

where ${\bf U}_s$, ${\bf U}_n$, and ${\bf \Sigma} $ are signal subspace, noise subspace, and diagonal matrix of eigenvalues, respectively. At last, target range and angle estimation are realized by searching the spectral peak for each point in the discretized range–angle domain

(27)$$I(r,\;\theta ) = \lsqb {{\bf a}_{TR}^H (r,\;\theta ){\bf U}_n{\bf U}_n^H {\bf a}_{TR}(r,\;\theta )} \rsqb ^{-1}.$$

Simulation setup

In this section, we examine the performance of the proposed TR-FDA-MIMO radar by numerical simulations. Both single-target and multi-target cases are taken into account, and conventional FDA-MIMO results are provided for comparisons.

Simulation parameters are set as follows: the number of transmitting and receiving elements of the co-located radar are M = 8 and N = 6, respectively, the reference carrier frequency is f ₀ = 10 GHz, the frequency increment is Δf  =  30 KHz, the element spacing is set as d _T = d _R = (1/2)λ ₀ to avoid grating lobes in angle dimension, and the number of sampling snapshots is L = 50. The scope of range and angle domains is r ∈ [0, 5 km] and θ ∈ [−90°, 90°], respectively. The Hadamard matrix is used to generate orthogonal modulation signal for radar waveform transmission. Numerical simulations are conducted via MATLAB R2013b and results are exported through 200 Monte-Carlo simulations. Figure 3 shows the flow diagram of the simulation.

Fig. 3. Flow diagram of MATLAB simulation.

Simulation results

At first, we investigate the range–angle estimation for a single target located at (2 km, 30°) with the signal-to-noise ratio (SNR) being −10 dB. The range–angle imaging spectra given by FDA-MIMO and TR-FDA-MIMO radars are shown in Figs 4(a) and 4(b), respectively, where spectra intensities are normalized and depicted in the form of dB. It can be seen that both FDA-MIMO and TR-FDA-MIMO radars realize joint range and angle estimation, namely, unambiguous target localization. However, it is obvious that TR-FDA-MIMO radar has a more focused imaging spot. To make a more detailed comparison, the normalized imaging resolution in range and angle dimensions of the two methods are plotted, as shown in Fig. 5. It can be observed that by combining time reversal with FDA-MIMO radar, range and angle resolutions are both effectively enhanced, which means that more precise target detection can be achieved by the new approach.

Fig. 4. Range–angle imaging spectra: (a) FDA-MIMO and (b) TR-FDA-MIMO.

Fig. 5. Resolution comparison between FDA-MIMO and TR-FDA-MIMO: (a) range resolution and (b) angle resolution.

Next, we study the multi-target case where two targets are located at (1 km, 30°) and (3 km, −40°), respectively, with SNR = −10 dB as well. The imaging results given by FDA-MIMO and TR-FDA-MIMO radars are demonstrated in Figs 6(a) and 6(b), respectively. As we can see, these two methods realize range–angle imaging for the two targets, indicating the multi-target detecting capacity of FDA-MIMO radar. But still, the new TR-FDA-MIMO radar provides a preferable imaging result. The imaging resolution in range and angle dimensions of the two methods are compared in Fig. 7, where the superiority of the proposed method is further revealed.

Fig. 6. Range–angle imaging spectra of two targets: (a) FDA-MIMO and (b) TR-FDA-MIMO.

Fig. 7. Resolution comparison between FDA-MIMO and TR-FDA-MIMO: (a) range resolution for the target located at (1 km, 30°), (b) angle resolution for the target located at (1 km, 30°), (c) range resolution for the target located at (3 km, −40°), and (d) angle resolution for the target located at (3 km, −40°).

Then, we consider the detection for three targets located at (2.5 km, −40°), (2.5 km, 15°), and (3 km, 30°), respectively, where two of them are located closely to each other. Figures 8(a) and 8(b) show the imaging spectra obtained with SNR = 10 dB which represents the low-noise case, while Figs 8(c) and 8(d) show the imaging spectra obtained with SNR = −15 dB which represents the high-noise case. It is observed that in the low-noise case, these two methods show excellent imaging results with TR-FDA-MIMO slightly outperforming FDA-MIMO. In the high-noise case, on the one hand, due to strong noise effects, the imaging performance of FDA-MIMO degenerates a lot, leading to a failure of resolving the two closely located targets. On the other hand, TR-FDA-MIMO still achieves accurate range–angle estimation for all the three targets in the strong noise condition. Moreover, the spatial half-power (−3 dB) profile of imaging spectra given by FDA-MIMO and TR-FDA-MIMO in the high-noise case is compared in Fig. 9, which clearly indicates the improvement of imaging performance brought by the proposed method.

Fig. 8. Range–angle imaging spectra for three targets: (a) FDA-MIMO with SNR = 10 dB, (b) TR-FDA-MIMO with SNR = 10 dB, (c) FDA-MIMO with SNR = −15 dB, and (d) TR-FDA-MIMO with SNR = −15 dB.

Fig. 9. The −3 dB profile of imaging spectra in the high-noise case. The blue thick profile corresponds to FDA-MIMO and the black thin profile corresponds to TR-FDA-MIMO.

At last, for robustness assessment, the root mean square error (RMSE) versus SNR curves of the two methods in range and angle estimation are plotted in Figs 10(a) and 10(b), respectively. It shows that the range and angle RMSEs of FDA-MIMO and TR-FDA-MIMO both reduce with the increase of SNR. As for range dimension, TR-FDA-MIMO has lower RMSE value when SNR is below 0 dB, and it has nearly the same estimation error with FDA-MIMO when SNR is above 0 dB. As for angle dimension, TR-FDA-MIMO has lower RMSE value at each SNR, and the stronger the noise, the more the TR-FDA-MIMO has the superiority. The comparison results indicate that the robustness to the noise also can be effectively improved by utilizing the proposed TR-FDA-MIMO radar.

Fig. 10. RMSE comparison between FDA-MIMO and TR-FDA-MIMO: (a) in range dimension and (b) in angle dimension.