Joint design of transmit and receive array window functions via polynomial factorization

Abstract

Window functions are commonly used to balance the mainlobe width and sidelobe levels in beamforming applications. Traditionally, windows are selected independently for the transmit (Tx) and receive (Rx) side. However, in active systems like radar, medical ultrasound, and sonar, the same array may operate in both modes, and the convolution of the Tx and Rx windows determines the effective two-way response. Therefore, without jointly designing the Tx/Rx windows, the mainlobe and sidelobe characteristics do not fully benefit from the possibilities of a consolidated two-way approach. This paper presents a framework for jointly designing Tx/Rx window functions by factorization of any desired two-way effective aperture into separate Tx/Rx windows. To guide this factorization, we introduce the white noise gain product (WNGP), a metric quantifying the combined spatial filtering efficiency of the Tx/Rx pair. We then propose a root-allocation strategy for the factorization that maximizes this metric, enabling effective control over the two-way beampattern. Additionally, we derive a recursive formula for the optimal Tx/Rx windows that yield a uniform effective aperture. The approach is validated through simulations, showing improved sidelobe control and greater design flexibility compared to conventional windowing techniques.