Fundamental of the proposed system

Let us consider a monostatic radar with a linear array system. The linear array consists of N elements with the first element as the reference element. The inter-element spacing is d = λ ₀/2, where λ ₀ is the carrier wavelength of the reference element. The array structure is shown in Fig. 2.

Fig. 2. Array structure of the monostatic radar system.

The transmitted signal of each element is designed as:

(1)$$\left\{\matrix{s_n( t ) = \phi_n( t ) \exp \!( {\,j2\pi f_nt} ) {\rm , \;\ }t\in [ {{{-T} / 2}, \;{T / 2}} ] \hfill \cr f_n = f_0 + n \times \Delta f\,{\rm , \;\ }n = 0, \;1, \;\ldots , \;N-1 \hfill} \right.$$

where f ₀ is the carrier frequency of the reference element and f _n denotes the carrier frequency of the nth element. n × Δf denotes the frequency offset of the nth element and T is the pulse duration. The frequency offset satisfies Δf ≪ f ₀. Hence, the FDA system is a narrowband system. As mentioned before, when the pulse duration satisfies T ≥ [2(f ₀ − Δf)d]/(cΔf), the standard FDA can automatically scan the space. And under the premise of a narrowband system, the pulse duration can be designed as T ≥ 1/Δf approximately.

The baseband waveforms are designed as

(2)$$\left\{\matrix{\phi_n( t ) = \exp \!( {\,j\pi u_nt^2} ) , \;t\in [ {{{-T} / 2}, \;{T / 2}} ] \hfill \cr u_n = {{B_n} / T}, \;n = 0, \;1, \;\ldots , \;N-1 \hfill} \right.$$

where u _n denotes the chirp rate of the nth element and B _n denotes the bandwidth of the nth baseband waveform.

Suppose there is a target at point (r, θ) with the radial velocity v. Then, the accurate received signal model of the mth element can be derived as:

(3)$$\left\{\matrix{s_m( t ) = \sum\limits_{n = 0}^{N-1} {\exp [ {\,j\pi ( {u_n{( {t-\tau_{m, n}} ) }^2 + 2f_n( {t-\tau_{m, n}} ) } ) } ] } + {\bf n}( t ) \hfill \cr \tau_{m, n} = {{[ {2( {r-vt} ) -( {n + m} ) d\sin \theta } ] } / c} \hfill} \right.$$

where $t\in ( {{-T} / 2} + \tau _{m, n}, \;{T / 2} + \tau _{m, n}) , \;$ τ _m,n denotes the delay of each echo signal relative to the transmitted signal, c denotes the speed of light in free space, and ${\bf n}( t)$ represents the received additive white Gaussian noise with a mean of 0 and a variance of $\delta _n^2$.

Gui et al. [Reference Gui, Wang, Cui and So27, Reference Gui, Wang and Shao28] proposed general receiver structures for an FDA radar based on the maximum likelihood criterion. In these general receiver structures, each antenna channel consists of multiple demodulators with different carriers and several matched filters. Based on these structures, we introduce a time-varying beamforming chain in each antenna channel. And the proposed receiver architecture is shown in Fig. 3.

Fig. 3. Single-antenna receiver architecture with a time-varying beamforming chain.

In the proposed system, each receiving channel consists of an analog mixer, an analog-to-digital converter (ADC), N digital mixers, and N approximate matched filters. And the sampling rate of the ADC is f _s. In actual processing, the ideal matching filtering is impossible to achieve and the impulse compression performance is related to the time-bandwidth (TB) product of the signal. Kowatsch et al. [Reference Kowatsch, Stocker, Seifert and Lafferl29] studied the performance of an LFM pulse compression system for TB products between 10 and 400. In practical application, we can choose the appropriate TB product to meet the specific needs.

Then, the impulse response function of the m–nth matched filter is h _m,n(k) = exp [ − jπu _n(K − 1 − k)²], where k denotes the discrete time point and K denotes the total number of discrete points. Each channel can capture all the transmitted signals respectively, thanks to different chirp rates. The output signals of N beamformers are accumulated as y _m(k). Then, the output signals of the N receiving channels are accumulated as $y( k) = \sum\nolimits_{m = 0}^{N-1} {y_m( k) }$.

With the premise of a narrowband system (f ₀ ≫ (N × Δf), f ₀ ≫ max (B _n)), the equivalent output signal model of the ADC can be approximately derived as:

(4)$$\eqalign{s_m( k ) & = {\bf n}( k ) + \exp ( {-j2\pi f_0\tau_0} ) \sum\limits_{n = 0}^{N-1} {\{ {\phi_n( {k-\tau_0} ) } } \cr & \quad \times {\exp ( {\,j\pi [ {2n\Delta fk + ( n + m) \sin \theta -2n\Delta f\tau_0-2f_dk} ] } ) } \} } $$

where τ ₀ = 2r/crepresents the reference time delay, and f _d = 2f ₀v/c denotes the Doppler frequency. Then, the output signal of the m–nth digital mixer can be derived as:

(5)$$\eqalign{s_{m,n} (k) & = {\exp [-j2\pi \left(\,f_{d}k + f_{0}\tau _0 \right)]} \cr & \bigg\{\exp [\,j \pi u_n{(k - \tau_0)}^2 + j\pi (n + m)\sin \theta -j2\pi n\Delta f\tau _0] \right. \cr & + \sum \limits_{n^{\prime} = 0,n^{\prime}\ne n}^{N-1} {\exp [\,j\pi u_{n^{\prime}} (k - \tau_0)}^2 + j2\pi (n^{\prime} - n) \cr & \left. {\Delta f\,(k -\tau_0) + j\pi (n^{\prime} + m)\sin \theta]} \bigg\} + {\bf n}(k)} $$

Then, the envelope of the output signal of the m–nth matched filter can be derived as:

(6)$$\eqalign {s_{m,n}^{\prime} \left(k \right) & = \exp [-j2\pi f_0\tau _0 + j2\pi f_0(n + m)d \sin \theta / c - j2\pi n\Delta f\tau_0] \cr & {\rm rect} \left({\displaystyle {k - \tau_0} \over {2T}} \right) \left \vert {\displaystyle{\sin \pi \left( f_d + u_nk \right)T} \over {\pi \left(\,f_d + u_nk \right)}} \right \vert \cr & + {\rm Envelope} \Bigg( {\sum \limits_{n^{\prime} = 0}^{N-1} h_{m,n^{\prime}}} (k) \ast {\sum \limits_{n^{\prime} = 0,n^{\prime} \ne n}^{N-1}} \exp [\,j \pi u_{n^{\prime}}(k - \tau_0)^2 \cr & + j2\pi (n^{\prime}-n) \Delta f {(k - \tau_0) + j \pi (n^{\prime} + m) \sin \theta ]} \Bigg) + {\bf n} (k)} $$

where * denotes convolution operation. To effectively accumulate the filtered outputs, the complex weighting factor is designed as ω _m,n(k′) = exp (j2πnΔfk′ − jπf ₀(n + m)sinθ), where k′ denotes the discrete time of the output of matched filter. Therefore, the weighting factor is time-varying. When a radar system is designed, the value of ω _m,n(k′) at a certain moment is determined. Thereby, we can calculate and store ω _m,n(k′) in advance, and then call the corresponding value when beamforming.

The proposed waveform design method

Based on the relationship between the TB product and the pulse compression performance reported in [Reference Kowatsch, Stocker, Seifert and Lafferl29], we can choose the appropriate TB value according to the needs of the actual application. Since the baseband waveforms are not mutually orthogonal, the second addend in (6) cannot be ignored. Then, the high side lobe level may occur. In order to reduce the impact of this addend, we apply the ABC algorithm to find the appropriate chirp rate u _n of each element. And u _n is designed as u _n = Δf × η _n/T, where η _n is the weighting factor that needs to be optimized. The determination of the distribution value of η _n can be considered as a minimization problem of the cost function Ψ given by (7), where SLL_max represents a function that can find the highest sidelobe level. And the SLL_max function can be implemented by finding the global submaximal value. The $s_{m, n}^{\rm \prime\prime } ( k )$ is derived with the assumption that there is a stationary target at point (r′′, θ′′) and $\tau _0^{\rm \prime\prime } = 2r{\rm \prime\prime }/c$. H denotes the Heaviside step function, and SLL_d represents the desired sidelobe level:

(7)$$\left\{ \eqalign{\Psi & = \left\{ {SLL_{max}\left( {\left\vert {\sum\limits_{m = 0}^{N-1} {\sum\limits_{n = 0}^{N-1} {\omega _{m,n}({k}^{\prime}){{s}^{\prime \prime}}_{m,n}\left( k \right)*h_{m,n}\left( k \right)} } } \right\vert} \right)-SLL_d} \right\} \cr & \times H\left\{ {SLL_{max}\left( {\left\vert {\sum\limits_{m = 0}^{N-1} {\sum\limits_{n = 0}^{N-1} {\omega _{m,n}(k{\rm ^{\prime}})s_{m,n}^{\rm \prime\prime } \left( k \right)*h_{m,n}\left( k \right)} } } \right\vert} \right)-SLL_d} \right\} \cr & s_{m,n}^{\rm \prime\prime } \left( k \right) = \exp [-j2\pi \left( {\,f_dk + f_0\tau _0^{\rm \prime\prime } } \right)] \cr & \left\{ {\exp [\,j\pi u_n{(k-\tau _0^{\rm \prime\prime } )}^2 + j\pi (n + m)\sin \theta {\rm \prime\prime }-j2\pi n\Delta f\,{{\tau }^{\prime \prime}}_0]} \right. \cr & + \sum\limits_{n{\rm ^{\prime}} = 0,n{\rm ^{\prime}}\ne n}^{N-1} {\exp [\,j\pi u_{{n}^{\prime}}{(k-\tau _0^{\rm \prime\prime } )}^2 + j2\pi (n{\rm ^{\prime}}-n)} \cr & \left. {\Delta f\,(k-\tau _0^{\rm \prime\prime } ) + j\pi (n{\rm ^{\prime}} + m)\sin \theta {\rm \prime\prime }]} \right\} \cr & \omega _{m,n}(k{\rm ^{\prime}}) = \exp (\,j2\pi n\Delta fk{\rm ^{\prime}}-j\pi f_0(n + m)\sin \theta )} \right.$$

Inspired by Yao et al. [Reference Yao, Wu and Fang30, Reference Yao, Rocca, Wu, Massa and Fang31], the optimization process based on the ABC algorithm is shown in Table 1.

Table 1. Optimization process of ABC algorithm

The ABC algorithm is a global optimization algorithm based on the intelligent foraging behavior of honey bee swarm. Besides, it can be efficiently used for multivariable, multimodal function optimization. In the ABC model, the colony consists of three groups of bees: employed bees, onlookers, and scouts. It is assumed that there is only one artificial employed bee for each food source. Employed bees go to their food source and come back to hive and dance on this area. The employed bee whose food source has been abandoned becomes a scout and starts to search for finding a new food source. Onlookers watch the dances of employed bees and choose food sources depending on dances. Then the honey bee swarm can accomplish tasks through social cooperation. The first half of the swarm consists of employed bees, and the second half constitutes the onlooker bees. Therefore, the optimization starts from randomly distributed initial values x _i,n (i = 1, 2, …, SN/2, n = 0, 1, …, N − 1, where SN is the size of the bee colony). And updates x _i,n at the qth iteration (q = 1, 2, …, Q, where Q is the preset maximum number of iterations). The greedy selection criterion is as follows. First, we compare the probability value of the old solution x _i,n and the new solution v _i,n based on (8). Then, replace x _i,n by v _i,n for the next iteration and update η _n = v _i,n as a possible solution, if p(v _i,n) > p(η _i,n); otherwise, retain x _i,n and update $\eta _n{\rm} = x_{i, n}$.

The probability value p _i is evaluated using (8), where fit(x _i,n) is the fitness function value of Ψ(x _i,n):

(8)$$\left\{\matrix{\,p( x_{i, n}) = fit( x_{i, n}) /\sum\limits_i^{{{SN} / 2}} {\,fit( x_{i, n}) } \hfill \cr fit( x_{i, n}) = \displaystyle{1 \over {1 + \Psi ( x_{i, n}) }} \hfill} \right.$$