Mixed microwave-digital and multi-rate approach for wideband beamforming applications using 2-D IIR beam filters and nested uniform linear arrays

Abstract

The presented work explores novel methods for synthesizing approximately frequency independent array factors at lower hardware complexity for wideband beamforming applications. The proposed approach employs 2-D infinite impulse response (IIR) digital beam filters together with nested uniform linear arrays (ULAs). The array is designed to have multiple levels of nesting. Each level of nesting consists of a ULA covering a temporal subband of the incident wideband signal. The use of nested arrays provides the required aperture size using a smaller number of elements compared to using a single ULA to capture the entire wideband signal. The use of different levels of nesting allows the operation of the digital processor for each sub-band at different clock rates. This is a hierarchical approach that saves both digital VLSI hardware and power consumption. The 2-D IIR digital beam filters that process each subband signal from each of the nested subarray achieves wideband beamforming. Simulations illustrate approximately frequency independent passbands as required in wideband beamforming.

Appendix

1.1 Review on 2-D IIR digital beam filters

The proposed method uses 2-D IIR beam filters in each subband. The beam filters have a line shaped passband in the 2-D spatio-temporal frequency domain (Madanayake and Bruton 2008) for directionally enhancing a propagating spatio-temporal uniform plane wave based on the direction of propagation. First-order 2-D IIR digital beam filters have an $\mathbf{z}$-domain transfer function (Madanayake and Bruton 2008)

$$\begin{aligned} T(z_x,z_{ct}) = \frac{ 1 + z_x^{-1} + z_{ct}^{-1} + z_x^{-1}z_{ct}^{-1} }{ 1 + b_{10}z_x^{-1} + b_{01}z_{ct}^{-1} + b_{11}z_x^{-1}z_{ct}^{-1} }, \end{aligned}$$

(7)

where the filter coefficients $b_{ij} = \frac{R+ (-1)^i (2L_x/{\varDelta }x) + (-1)^j (2L_{ct}/(c{\varDelta }T_s))}{R + (2L_x/{\varDelta }x) + (2L_{ct}/(c{\varDelta }T_s)) }~\text {with}~i+j \ne 0$. $2L_{x}/{\varDelta }x = \cos \theta$ and $2L_{ct}/(c{\varDelta }T_s) = \sin \theta$ sets the orientation $\theta$ of the passband ($L_x \ge 0$ is a spatial inductance, $L_{ct}\ge 0$ is a temporal inductance, and c is wave propagation speed). The beam passband oriented at $\theta$ in $\omega _x,~\omega _{ct}$ domain corresponds to directionally filtering a plane wave arriving at an angle $\psi$ where $\tan \theta = \sin \psi$ and $\omega _x,~\omega _{ct}$ refer to the spatial and wave speed normalized temporal frequencies. Variables $z_x,~z_{ct}$ correspond to the $\mathbf{z}$-domain transformations of discretized (x, ct).

Fig. 11

a Frequency response of the 2-D IIR beam filter, b frequency response when temporally oversampled by a factor of 2

Further, the termination $R>0$ sets the sharpness (i.e., selectivity) of the beam passband and ${\varDelta }T_s, ~{\varDelta }x$ corresponds to the reciprocals of temporal and spatial sampling frequencies, respectively. It is noted that the transfer function given in (7) is derived by applying the 2-D bilinear transform $s_k = \frac{1-z_k^{-1}}{1+z_k^{-1}}$, $k \in \{ x, ct \}$ to the continuous domain prototype 2-D transfer function

$$\begin{aligned} H_{IIR} = \frac{R}{R + L_x s_x + L_{ct}s_{ct}}, \end{aligned}$$

(8)

which corresponds to a resistively terminated 2-D passive prototype network (Bruton and Bartley 1985). Fig. 11a shows the normalized magnitude frequency response in 2-D frequency domain for the Nyquist square $-\pi< \omega _x< \pi , -\pi< \omega _{ct} < \pi$. Frequency warping, which is inherent in discrete digital systems synthesized from bi-linear transforms (Phadke 1999), results in the twisting of the beam-shaped passband towards the edges.

1.2 Array factor of a generic 2-D IIR beamfilter

Figure 12 illustrates the array factor of the 2-D IIR beamfilter given by (7). It should be noted that the plot has been drawn nullifying the effect warping by using pre-warping technique described in Sect. 4.1.

Fig. 12

Closed form array factor for the digital 2-D IIR beamfilter tuned for $30^\circ$