# Frequency-Bounded Matching Strategy for Wideband LNA Design Utilising a Relaxed SSNM Approach
## 1 Introduction  
Low-noise amplifiers (LNAs) are critical components in modern radio frequency (RF) and microwave systems, typically serving as the first active stage in a receiver chain. Their performance directly impacts the overall system noise figure (NF), making LNA design essential for applications such as automotive radar, satellite communications, electronic warfare [1, 2, 3, 4, 5], and biomedical applications such as ultrasound imaging systems and short-range radio, etc., at low frequency (10-1000 MHz) [6, 7]. In addition to achieving an optimal trade-off between the gain and noise figure, many of these applications demand wideband operation. However, conventional LNA design methodologies often prioritise narrowband performance and rely on single-frequency matching strategies. Techniques such as source degeneration [8, 9] and drain-gate feedback [10, 11, 12, 13] have been widely adopted to bring the optimum noise and gain impedances closer together, enabling simultaneous signal and noise matching (SSNM). These feedback mechanisms help mitigate the effects of Miller capacitance (e.g., C\({}_{\text{gd}}\), C\({}_{\text{gg}}\)) at high frequencies, thereby improving amplifier stability and marginally reducing noise parameters. Nevertheless, excessive feedback can degrade small-signal gain. Furthermore, achieving ideal SSNM conditions at both ports is not feasible in a single-stage amplifier, making multistage topologies more suitable. In this context,Patrick et al. [14] proposed a systematic approach using mismatch charts to determine appropriate feedback for each stage, while [PERSON] et al. [15] introduced a method based on mismatch circles to realise SSNM. These works provide important foundations for SSNM-based design. Additionally, the influence of varying mismatch levels on gain has been explored in [16], and a graphical synthesis method on the load admittance plane has been proposed for optimised low-noise design [17]. Other efforts include multistage millimetre-wave LNAs using passive R-C-L shunt feedback to achieve wideband SNIM with high efficiency and low noise [18], although these techniques remain primarily focused on narrowband design. To address the challenges of broadband matching, several alternative circuit topologies have been explored. Transformer-based matching [19], filter-based matching [20], and forward combining techniques [21] have demonstrated extended bandwidth. Design incorporating a series inductor at drain terminal [22] and drain to source coupling (using coupled lines) [23] also achieved flat gain and a low noise figure. While these methods are effective, they often require complex matching structures and extensive use of computer-aided design (CAD) tools, limiting the designer's ability to control mismatch levels across multiple stages. An interdisciplinary approach in reference [24] integrates linear circuit theory, noise analysis, communication theory, and soft-computing methods, using particle swarm optimisation to synthesise frequency-dependent terminations via Darlington-based networks. Similarly, reference [25] proposes an analytical formulation for source reflection in distributed matching networks to achieve broadband gain flattening. This paper presents a systematic and practical strategy for designing wideband multistage LNAs using a relaxed SSNM framework. The approach introduces frequency-bounded mismatch regions derived from input-output mismatch circles plotted on the Smith chart. These graphical regions offer a visual and intuitive method for selecting source and load terminations across a wide frequency span, facilitating broadband impedance matching, a low noise figure, and unconditional stability, without relying on iterative CAD-based optimisation. The behaviour of input mismatch (IM) and output mismatch (OM) circles is analysed and extended into bounded regions that preserve geometric clarity while generalising their use to wideband, multistage LNA designs.