Background knowledge

In an ESTMPA, as shown in Fig. 1, each antenna is connected to a reconfigurable power divider and two 0/π phase shifters, whereas one of the paths is connected to a π/2 fixed-phase shifter. The array factor of an N-element ESTMPA can be expressed as [Reference Yao, Wu and Fang15]

(1)$$AF( {\theta , \;t} ) = e^{\,j\omega _0t}\sum\limits_{n = 0}^{N-1} {I_n\cdot [ {U_n^I ( t ) + jU_n^Q ( t ) } ] \cdot e^{\,jKdn\sin \theta }} $$

where $U_n^I ( t )$ and $U_n^Q ( t )$ are switching functions for I/Q channels. The reconfigurable power divider has two states: “One-Path-On state” and “Two-Path-On” state. In the “One-Path-On state”, all energy is transferred to one channel and the other channel is closed, and $U_n^I ( t ) = {\pm} 1$, $U_n^Q ( t ) = {\pm} 1$. In the “Two-Path-On state”, energy is evenly divided between the two channels, and $U_n^I ( t ) = {\pm} \sqrt 2 /2$, $U_n^Q ( t ) = {\pm} \sqrt 2 /2$. The corresponding switching sequences in the two states are shown in Fig. 2. In addition, the 0/π phase shifters can manipulate the switching sequences to be “0 states” or “π states”, which control the amplitude of the sequence of positive states and negative states. ω ₀ = 2πf ₀ is the angular frequency of the primary signal, T _p = 1/f _p = 2π/ω _p is the modulation period, I _n is the excitation amplitude of the n-th element, K is the wavenumber, and θ is the angle measured from the broad-side direction of the array. Using the Fourier transform, (1) can be written as

(2)$$AF^{h, n} = e^{\,j( {\omega_0 + h\omega_p} ) t}\sum\limits_{n = 0}^{N-1} {I_n\cdot A_{h, n}\cdot e^{\,jKdn\sin \theta }} $$

where h represents the number of harmonics and $A_{h, n}^{}$ can be defined as (3)

(3)$$\eqalign{A_{h, n} = \left\{\matrix{\displaystyle{ 2 \over {\vert h \vert \pi }}\left[{\sin \left({\displaystyle{{3\pi } \over 8}} \right) + \left({\sqrt 2 -1} \right)\sin \left({\displaystyle{\pi \over 8}} \right)} \right]e^{{-}jh\pi ( {2t_{1n}^I + ( {1/3} ) } ) }, \; \quad {\rm if}\,h = 8k{\rm} + {\rm 1, \;}\,k\in {\mathbb Z} \cr 0 , \; {\rm others}} \right.}$$

Fig. 1. The system structure of ESTMPA.

Fig. 2. The modulation waveform of $U_n^I ( t )$ and $U_n^Q ( t )$ in one I/Q modulator.

Following the rules of the switching sequences, the modulation time is shown in Fig. 2, and their relation [Reference Yao, Wu and Fang15] can be written as

(4)$$\displaystyle{{{\tau }^{\prime}} \over {T_p}} = \displaystyle{3 \over 8}, \;\,\displaystyle{{{\tau }^{\prime \prime}} \over {T_p}} = \displaystyle{1 \over 8}, \;\,\displaystyle{{t_{1n}^Q } \over {T_p}} = \displaystyle{{t_{1n}^I } \over {T_p}}-\displaystyle{1 \over 4}, \;\,\displaystyle{{t_{2n}^Q -t_{1n}^Q } \over {T_p}} = \displaystyle{{t_{2n}^I -t_{1n}^I } \over {T_p}} = \displaystyle{1 \over 2}$$

(5)$$\displaystyle{{t_{1n}^{I + } -t_{1n}^I } \over {T_p}} = \displaystyle{{t_{2n}^{I + } -t_{2n}^I } \over {T_p}} = \displaystyle{{t_{1n}^{Q + } -t_{1n}^Q } \over {T_p}} = \displaystyle{{t_{2n}^{Q + } -t_{2n}^Q } \over {T_p}} = \displaystyle{{{\tau }^{\prime}-{\tau }^{\prime \prime}} \over {2T_p}}$$

where $t_{1n}^{I + }$, $t_{ 2n}^{I + }$, $t_{1n}^{Q + }$, and $t_{2n}^{Q + }$ represent the one-path switch-ON state starting instants, $t_{1n}^I$, $t_{1n}^Q$, $t_{2n}^I$, and $t_{2n}^Q$ represent the two-path switch-ON state starting instants. Following (4) and (5), i.e. corresponding to the modulation waveform shown in Fig. 2, a considerable number of harmonics (8k + 4 ± 1, k ∈ Z) in ESTMPA can be eliminated. According to equations (2) and (3), beamforming in the desired direction θ _d can be performed by designing $t_{1n}^I$, as follows

(6)$$\displaystyle{{t_{1n}^I } \over {T_p}} = \displaystyle{{nKd( {\sin \theta_d} ) /\pi - 1/3} \over 2}$$

Principle of HBF in automotive radar signal processing

Figure 3 shows the HBF algorithm's basic operation and signal processing steps. As can be seen, using the beam adder and beam subtractor, the left and right beams generate sum and difference beams. The left and right beams are then modified to their c power by the beam indexer, and a beam subtracter and vector logarithm are used to generate the hyper beam for DFT/FFT. Last but not least, are the moving target detection and constant false alarm rate techniques. The proposed system differs from existing radar signal processing methods in that the received signal is processed using HBF. The hyper beam can offer improved angular resolution for future signal processing since it has a shorter beamwidth and lower sidelobe than the normal beam. As a result, the percentage of false alarms may be decreased, which is particularly beneficial for radar detection.

Fig. 3. The signal processing process and principle of hyper beamforming algorithm.

Figure 4 depicts the HBF block diagram of an ESTMPA-based automobile radar receiver. Two subarrays make up the ESTMPA. There are N items in each of the left and right subarrays, which are differentiated by Rn and Ln (n = 1, 2,…, N), and the distance between them is specified as a wavelength. The left and right beams are produced by the left and right subarrays, respectively, and the array signal is obtained using two analog-to-digital converters. Additionally, signal processing and signal control may be put into practice.

Fig. 4. Automotive radar receiver block diagram using hyper beamforming with STMPA.

The derivations are then provided. If the reference point is placed in the midst of two subarrays, then (

) and (7b) may be used to describe the array factors of the left subarray and right subarray, correspondingly

(7a)

(7b)$$AF_l^{h, n} = \left\{\matrix{e^{\,j( {\omega_0 + h\omega_p} ) t}\sum\limits_{n = 0}^{L_N-1} {I_n\cdot } \displaystyle{ 2 \over {\vert h \vert \pi }}\left[{\sin \left({\displaystyle{{3\pi } \over 8}} \right) + \left({\sqrt 2 -1} \right)\sin \left({\displaystyle{\pi \over 8}} \right)} \right]\cdot e^{\,jh\pi ( {2t_{1n}^I + ( {1/3} ) } ) }\cdot e^{{-}jKd( {n{\rm + \ }( {1/2} ) } ) \sin \theta } , \cr \quad \;{\rm if}\,h = 8k{\rm} + {\rm 1, \;}\,k\in {\mathbb Z} \cr 0, \;\,{\rm others} } \right.$$

Referring to (6), the start time of the array element in each subarray can be obtained using the following equation

(8a)$$\displaystyle{{t_{1n}^{R, I} } \over {T_p}} = \displaystyle{1 \over 2}[ {nKd( {\sin \theta_d + 0.5} ) /\pi - 1/3} ] $$

(8b)$$\displaystyle{{t_{1n}^{L, I} } \over {T_p}} = \displaystyle{1 \over 2}[ {-nKd( {\sin \theta_d + 0.5} ) /\pi - 1/3} ] $$

where $t_{1n}^{R, I}$ and $t_{1n}^{L, I}$ denote the start time of right and left subarray, respectively. Then the other switching times can be obtained according to (4) and (5). After the above formulas are obtained, the sum beam and difference beam can be obtained as

(9)$$AF_S = \vert {AF_l^{h, n} } \vert + \vert {AF_r^{h, n} } \vert $$

(10)$$AF_D = \vert {AF_l^{h, n} -AF_r^{h, n} } \vert $$

Then the hyper beam can be obtained as

(11)$$AF_H = [ {AF_S^c -AF_D^c } ] ^{1/c}$$

where c stands for the hyper confidence, a parameter that influences the beamwidth and sidelobe of the hyper beam and typically falls between 0.2 and 1 [Reference Ram, Mandal, Kar and Ghosal20]. In the meantime, beamforming may call for maintaining a consistent beamwidth in each direction, which is very readily accomplished in HBF, as will be covered in the next subsection.

Maintain the constant beamwidth

Since the direction of beamforming may change in practice, the beamwidth can be kept constant by adjusting the confident c. The standard to split the independent beamspace is considered to be an amplitude of −3 dB of the extreme value of the beampattern, which means the following equation should be satisfied

(12)$$ {AF_H} \vert _{\theta \in [ {-\pi /2, \pi /2} ] } = \displaystyle{{{ {AF_H} \vert }_{\theta = \theta _0}} \over {\sqrt 2 }}$$

substituting (11) into (12), we can get

(13)$$[ {AF_S^c -AF_D^c } ] ^{1/c}\vert {_{\theta \in [ {-\pi /2, \pi /2} ] } } = \displaystyle{{{ {AF_H} \vert }_{\theta = \theta _0}} \over {\sqrt 2 }}$$

Through the above formula, the constant beamwidth can be guaranteed by using different c for different θ ₀. For example, c varies with θ ₀ as shown in Fig. 5 to keep constant beamwidth in a four-element array. As can be seen, the variation range of c is between 0.25 and 0.6 in order to maintain the beamwidth.

Fig. 5. The coefficient c versus θ ₀ in a four-element array.

Beamforming

The suggested system's beamforming capability is implemented in this subsection. For comparison, the beam produced by ESTMPA is employed. Assume that N is 8 and that the beam direction is set in 30°. Figure 6(a) displays the hyper beam that was produced. Figures 6(b) and 6(c) demonstrate the switching patterns of the left subarray and right subarray, respectively. It can be observed that the hyper beam's sidelobe is smaller and its beamwidth is narrower.

Fig. 6. (a) Radiation pattern; (b) time switching sequences of right subarray (t1_RX denotes the switching time); (c) time switching sequences of left subarray (t1_LX denotes the switching time).

Beamwidth

The beamwidth is a crucial factor to consider while evaluating the beam's performance. The suggested system is compared to the beam produced by a typical phased array. Coefficient c is the primary factor impacting the beamwidth and SLL in HBF. SLL and beamwidth may be decreased for an eight-element array, as shown in Fig. 7(a), by lowering c. Multiple incidence directions should be taken into consideration while adjusting c, due to the constant beamwidth that may be needed in practice. If beamwidth is fixed, the beamwidth will shrink as c increases. The lowest beamwidth, less than 0.2°, occurs when θ and c are adjusted to 0° and 0.3, as illustrated in Fig. 7(a). The beamwidth and SLL continue to grow as c rises. Figure 7(b) illustrates how c affects beamwidth. It may be observed that the beamwidth widens as c increases or as θ rises. The results of testing first null beamwidth (FNBW) and constant beamwidth performance are displayed in Table 1. It is clear that the beamwidth may be kept constant with great accuracy in each pointing direction after using (13) to modify coefficient c. Additionally, the normal beam's beamwidth is larger than the hyper beam's and varies with θ. These findings demonstrate that the hyper beam's beamwidth may be maintained constant in all directions. The FNBW, however, remains constant, indicating that the HBF approach just adds processing to the data already received and does not obtain the signal from a different source.

Fig. 7. (a) Radiation pattern of hyper beam with different c; (b) beamwidth versus c in different θ.

Table 1. Comparison of hyper beam and conventional beam

SLL

SLL has a great influence on the performance of beamforming, so in this subsection, we evaluate the SLL of the system in many aspects. Figure 8(a) shows that when the number of array elements is constant, the SLL of the hyper beam is only related to c, and the influence of different incident angles on SLL is negligible. It indicates that the SLL of the hyper beam inherits very strong stability while c keeps unchanged. In addition, as the number of array elements increases, the SLL of the hyper-beam increases slightly, as shown in Fig. 8(b). This is because when the beamwidth is constant, the greater the number of array elements, the larger the value of c will be, resulting in an increase in SLL. In a practical system, a constant beam width helps to keep the performance of the system constant in each direction. When the beamwidth is constant, the SLL of the hyper-beam formed by the proposed system decreases with the increase of θ. However, compared with conventional beams, the SLL of the proposed system decreased by 13–40 dB. In practical situations, a beam with such a low SLL can guarantee the high performance of communication or radar sensing.

Fig. 8. (a) SLL versus c in different θ; (b) SLL versus θ with varies N.

Signal transmission

In this subsection, a linear frequency modulation (LFM) signal is used as an example to show the advantage of ESTMPA compared to traditional TMA in signal transmission. The bandwidth of the transmitted signal cannot exceed the modulation frequency in traditional TMA. By applying ESTMPA to the automotive radar, the bandwidth of the transmitted signal can be expanded by eight times compared with the traditional TMA. Suppose a TMLA and an ESTMPA, both with eight elements, are tested in the time domain. F _p is set as 15 MHz. Figure 9 shows the comparison of the transmitted signal between the traditional TMA and ESTMPA. It can be found that ESTMPA has a larger frequency interval due to the elimination of more harmonics, which can be applied to signal transmission with higher bandwidth. Wider bandwidth means higher resolution, which will help improve the resolution of automotive radar.

Fig. 9. LFM signal in $\theta = 10^\circ$ transmission by (a) traditional TMA (B = 15 MHz); (b) the proposed system (B = 120 MHz).