A) MIMO RADAR

A MIMO radar system consisting of N transmitters and M receivers with amplitude distributions $\vec A_{Tx} $ and $\vec A_{Rx} $ (column vectors), respectively, is considered. As demonstrated by the example in Fig. 1(a), a transmitter/receiver pair can be represented by a virtual single-input single-output (SISO) element. The amplitude distribution $\vec A_{SISO} $ of the SISO array of maximum N × M elements can be constructed according to $\vec A_{SISO} = \vec A_{Tx} \;\ast \vec A_{Rx} $ with * being the convolution operator as shown in Fig. 1(b). As proven in [Reference Kirschner, Siart, Guetlein and Detlefsen7], a uniform quarter-wave (λ/4) spacing of the virtual elements, instead of λ/2, is required to avoid grating lobes as the SISO array works like a synthetic aperture radar (SAR) system with collocated movement of the transmitter and the receiver. Because of the coherent signal processing of the virtual array, the MIMO concept is also termed sum co-array.

Fig. 1.

Exemplary MIMO radar system with two transmitting and three receiving elements: (a) virtual element concept, (b) virtual array of six SISO elements, and (c) its corresponding co-array.

Various schemes, such as time division multiple access (TDMA), code division multiple access (CDMA), or frequency division multiple access (FDMA) orthogonal frequency division multiplexing (OFDM) [Reference Hanzo, Wong and Yee1], have been proposed in literature to construct the probing signals of MIMO systems. Despite of its limited update rate, the TDMA scheme is often preferred to CDMA or FDMA as it comes along with the lowest hardware complexity while offering full signal bandwidth. Thus, we choose a TDMA scheme for the measurements reported in Section V as a proof of concept.

B) Difference co-array

As proved in [Reference Hopperstad and Holm2] the detection capability of a linear array depends only on the relative and not the absolute positions of the sensors. Thus, in order to virtually fill the gaps in the SISO array, a difference co-array $\vec A_{co} $ of N _co elements as depicted in Fig. 1(c) can be constructed by means of the auto-correlation function of the SISO array, i.e. $\vec A_{co} = \vec A_{SISO} \ast \left( {\vec A_{SISO}} \right)^{\ast} $ with $^{\ast} $ denoting the complex conjugate. The aperture is thereby doubled.

If all array elements i with 1 ≤ N _co1 ≤ i ≤ N _co2 ≤ N _co fulfill the condition ${\rm \parallel} d_{i + 1} - d_i {\rm \parallel} \le \lambda /4$ , then the array length L can be defined as L = max{N _co2 − N _co1 + 1}. Here, d _i  (i = 1, …, N _co ) denotes the positions of the N _co elements in the co-array. The thinning rate is then determined by T = 1 − N + M/L.

A) Optimization procedure

The linear MIMO array arrangement with M receivers and N = 2N _T transmitters suggested in [Reference Pham and Jacob4] is sketched in Fig. 2. The transmit arrays are (initially) dense arrays with N _T elements each and displaced by a few wavelengths off broadside to reduce direct coupling with the Rx array. They are located on both sides of the receiver array and laterally offset by (initially) d _os  = λ/4. This is referred to as split configuration.

Fig. 2.

MIMO system configuration with a split dense Tx- (red) and a hybrid Rx-array (blue).

The receiver is composed of three sparse sub-arrays. Sub-array 1 consists of two periodically thinned sub-arrays 1a and 1b. Their $\lfloor \left( {M - 2k} \right)/2 \rfloor$ elements are d _uni  = kN _T λ/2 apart. They lie symmetrically at the position ± d _sub with d _sub  = d _uni or d _uni /2 if M is odd or even, respectively. In case M is odd, the last element of the sub-array 1 is located at the center of the receiver array. Sub-arrays 2 and 3 have k elements each. If these are spaced by N _T λ/2, a filled virtual SISO is obtained [Reference Pal and Vaidyanathan3]. To finalize the design of the initial configuration k has to be determined. As proposed in [Reference Pham and Jacob4], it is chosen as $k = \lfloor M/4 \rfloor$ as this maximizes the length [(k + 1)(M − 2k − 1) + 2k]N _T λ/4 of the virtual co-array.

In the following, the individual positions of the N _T elements in the transmitter and of the k elements in the receiver sub-arrays 2 (or 3) as well as the offset d _os between Tx- and Rx arrays are optimized to achieve the largest possible virtual co-array with no gaps. Constraints, such as the minimum element spacing, are denoted by Ω and taken into account in the procedure. The constrained minimization problem is formulated as

(1) $$\mathop {\min} \limits_{\vec d_{Tx}, d_{os}, \vec d_{Rx}, d_{uni}, d_{sub}} \left[ {\displaystyle{{N + M} \over L}} \right]\;{\rm subjected \,to}\;\left[ {\vec d_{Tx}, \vec d_{Rx}} \right] \in {\rm \Omega} $$

with $\vec d_{Tx} $ and $\vec d_{Rx} $ denoting the position vector of the N _T elements in the transmitter and that of the k elements in sub-arrays 2 and 3, respectively. As a result of the optimization, these sub-arrays become non-uniform in general.

A three-step procedure is proposed as outlined in Fig. 3. The pattern is first optimized with respect to $\vec d_{Tx} $ and d _os by means of a GA [Reference Haupt8] to fulfill the constraints in the transmitter array. Then, the same optimizer is applied to d _os and $\vec d_{Rx} $ with the receiver constraints. Lastly, the optimum configuration can be found by applying the pattern search to d _uni , d _sub and d _os . The search is conducted on a λ/4 grid. This configuration can be used as an initial array arrangement, and all three steps can be performed again to achieve a better solution. If an improved configuration with larger virtual aperture is found in the n ^th attempt (L _n >L _n−1), the optimization procedure performs the next attempt with an updated configuration. Otherwise the procedure is stopped and the last configuration is used. It was found that in most cases a solution to (1) is obtained after the first iteration, so that the loop is processed only twice.

Fig. 3.

Optimization procedure.

As a result, the optimization cost and time are dramatically decreased compared to a one-step optimization as the number of variables is reduced from (N + M) to ( $\lfloor M/4 \rfloor + N/2 + 5$ ). Because of the statistical property of the genetic optimizer, the whole procedure should be repeated several times to obtain the best configuration. The constraints can be slightly modified in the process as demonstrated in the examples below. One should note, however, that because of the complexity of the problem the solution most likely represents only a local minimum. In these cases, the redundancy of the array is not minimum, albeit is sufficiently low, in general.

B) Examples

To validate the concept, the synthesis procedure is assessed in MATLAB. A MIMO Radar system with ten transmitters (N _T  = 5) and 13 receivers (M = 13) is examined. This yields k = 3 elements for the receiver sub-arrays 2 and 3 and a uniform spacing of d _uni  = 15λ/2 for the seven elements of the receiver sub-array 1. All antennas are considered as isotropic radiators.

As shown in Fig. 4(a), the nested structure is used as the initial system configuration with a large virtual aperture and a thinning rate exceeding 95%. However, redundancy is still observed in the virtual co-array, and the configuration can thus be optimized.

Fig. 4.

MIMO Radar system with ten transmitters (N _T  = 5) and 13 receivers (M = 13) and its co-array: (a) initial configuration, (b) only the transmitters are optimized, (c) only the receivers are optimized, (d) the whole non-uniform array is optimized without (system C0) and with element spacing constraint of (e) λ (system C1), and (f) 4 λ (system C2).

If the genetic optimizer is applied only to either the transmitter (optimization 1) or the receiver (optimization 2), the virtual aperture increases by more than 52% (Fig. 4(b)) and 125% (Fig. 4(c)), respectively. If the whole array is optimized, then the aperture can be increased by nearly 147% as illustrated in Fig. 4(d). The thinning rate then exceeds 98%.

In addition, system configurations with a minimum element spacing constraint of Δd = λ (system C1) and Δd = 4 λ (system C2) are shown in Fig. 4(e) and 4(f), respectively. The aperture of system C2 is 9% smaller than that of the configuration without constraint, whilst the aperture of system C1 is nearly 6% larger. The thinning rate varies between 97.9 and 98.2%. This observation indicates that for large N and M the search space is reduced when constraining the element spacing even to small values. This, in turn, improves the convergence of the GA. Thus, the design procedure can be performed for several Δd (Δd ≥ Δd _min ) to yield the best array arrangement.

A) Range estimator

The range of the objects is estimated by Fourier transform [Reference Stoica and Moses12]. The 1D-FFT spectrum can be defined as

(2) $$Z_q \left( r \right) = \sum\limits_{k = 0}^{K - 1} z_q \left( k \right) \cdot \exp \left( { - j2\pi rk/K} \right).$$

Here, Z _q and z _q denote the received signals of the q ^th virtual SISO element in the rth range bin and at the k ^th normalized frequency (k = 0, …, K − 1), respectively. Besides, q = nm (q = 1, …, NM) denotes the SISO element of the n ^th transmitter (n = 1, …, N) and the m ^th receiver (m = 1, …, M).

The range of the objects is allocated to range bins noted as r = r ₁, …, r _T for T detected peaks. The joint range/DOA estimation problem is transformed into T-independent DOA-estimation problems in range bin r _t  (t = 1, …, T).

B) DOA estimator

A combination of difference co-array and MUSIC is proposed to solve the DOA-detection problem [Reference Pham and Jacob4]. First, the co-array of the SISO array signals $\vec Z(r_t ) = \left[ {Z_1 (r_t ), \ldots, Z_{N \cdot M} (r_t )} \right]^T $ in range bin r _t is constructed and renormalized to form a uniform virtual co-array of L elements with the amplitude distribution $\vec A_{co} = \left[ {x_1, \ldots, x_L} \right]^T $ .

Because of the non-linear processing path, the correlation between the sources deteriorates the image. To deal with this problem in the MUSIC spectrum, a forward–backward spatial smoothing technique as described in [Reference Pillai and Kwon11] is applied to the response of the virtual co-array. The effective aperture is however equal to the aperture P of the sub-arrays, which is limited by $\lfloor 3L/4 \rfloor$ as proved in [Reference Pillai and Kwon11]. The method can be summarized as follows:

• • The virtual co-array of L elements is divided into (L − P + 1) overlapping sub-arrays of $P\;( \le \lfloor 3L/4 \rfloor$ ) elements as depicted in Fig. 6. The amplitude distributions $\vec y_{fp} $ and $\vec y_{bp} $ of the pth sub-array in forward/backward directions are

  (3) $$\eqalign{\vec y_{fp} & = \left[ {x_p, \ldots, x_{p + P - 1}} \right]^T, \;{\rm and}\; y_{bp} \cr & = \left[ {x_{L - p + 1}^\ast, \ldots, x_{L - p + P + 2}^{\ast}} \right]\comma}$$
  with T being the transpose operator.

• • The estimated correlation matrix ${\bf \hat R}$ is

  (4) $$\eqalign{{\bf \hat R} = & \displaystyle{1 \over {2\left( {{\rm L} - {\rm P} + {\rm 1}} \right)}}\sum\limits_{p = 1}^{L - P + {\rm 1}} \vec y_{fp} \vec y_{fp}^H + \vec y_{bp} \vec y_{bp}^H \cr = & \left[ {{\bf E}_S {\bf E}_N} \right]{\rm diag}\left[ {} \right.\underbrace {\Lambda _1, \ldots, \Lambda _Q} _{{\bf \vec \Lambda} _S}, \underbrace {\Lambda _{Q + 1}, \ldots, \Lambda _P} _{{\bf \vec \Lambda} _N} \left. {} \right]\left[ {{\bf E}_S {\bf E}_N} \right]^H} $$
  with H being the Hermitian operator and the subscripts S and N denoting the signal and the noise subspace, respectively. If the number Q of targets within the current range bin is known, the vectors ${\bf \vec \Lambda} _S $ and ${\bf \vec \Lambda} _N $ consist of Q and P–Q eigenvalues of the matrix ${\bf \hat R}$ with Λ₁ > · · · >Λ _Q >Λ _Q+1 > · · · >Λ _P while $\left[ {{\bf E}_S {\bf E}_N} \right]$ contains the corresponding eigenvectors.

• • With $\vec a = [1,\exp \left( { - j\pi \sin (\theta )} \right), \ldots, \exp \left( { - j(P - 1)\pi \sin (\theta )} \right)]^T $ being the DOA vector, the estimated angular positions $\hat \theta $ of the far-field sources can be determined as peaks in the MUSIC spectrum:

  (5) $$\Psi \left( \theta \right) = \displaystyle{1 \over {\vec a^H (\theta ){\bf E}_N {\bf E}_N^H \vec a(\theta )}}.$$

Fig. 6.

Forward/backward spatial smoothing scheme.

The computational effort can be further reduced by applying other high-resolution spectrum estimation methods such as ESPRIT or Root-MUSIC instead of MUSIC after the spatial smoothing step. However, the estimation accuracy decreases slightly in scenarios having radar signals with lower signal-to-noise-ratio (SNR). More information can be found in [Reference Waweru, Konditi and Langat13].