# Frequency-Bounded Matching Strategy for Wideband LNA Design Utilising a Relaxed SSNM Approach
## 4 Wideband LNA Design Strategy Using Modified SSNM Conditions  
A wideband design strategy is developed in this paper using relaxed SSNM conditions and frequency-bounded envelopes. The first step in wideband implementation, which is the focus of this work, involves decomposing the operating frequency band into discrete frequency points. For this study, a three-point decomposition is employed, as described in Section 3.2. While additional frequency points may improve accuracy, they also increase computational complexity. The second step is to select the proper size of the transistor and bias conditions based on the required noise figure and available gain. Subsequently, the appropriate L\({}_{\text{S}}\) for each stage is determined using mismatch charts (see Figures 2 and 3). The goal is to locate stable operating regions that satisfy gain-noise trade-offs and remain within the relaxed mismatch bounds. L\({}_{\text{S1}}\) = 70 pH, L\({}_{\text{S2}}\) = 60 pH and L\({}_{\text{S3}}\) = 70 pH was selected with gate bias voltages of 0.6 V, 0.65 V, and 0.7 V for stages 1, 2, and 3, respectively. The drain voltage (Vds) was set to 1.2 V for all stages. Next, it is necessary to define the design targets for each stage and synthesise the corresponding matching networks. This starts with synthesising the input matching network (IMN) of the first stage, as introducing input mismatch does not significantly impact its design. Following this, the output matching network (OMN) of the last stage is synthesised, which can be designed with or without deliberate mismatch, depending on the desired performance trade-offs. First stage: Minimum noise figure--IMN synthesis  
The total noise figure (NF\({}_{\text{CAS}}\)) of a cascaded system is given by [PERSON] formula [26]:  
\[\text{NF}_{\text{CAS}}\;=\;\text{NF1}\;+\;\frac{\text{NF2}-1}{\text{G1}}+\frac{ \text{NF2}-1}{\text{G1G2}}+\ldots\ldots. \tag{13}\]  
This expression highlights the predominant influence of the first-stage noise figure on overall system performance. However, the contribution of further stages can be understood by the available gain from each stage. [PERSON] and [PERSON] [27] introduced the noise measure (M) as a quantity to give an idea about the quality of the overall noise figure of a cascaded network. \[\text{M}=\frac{\text{F}-1}{1-\frac{1}{\text{G}}} \tag{14}\]  
where F is the noise factor and G is the available gain. Minimising the overall noise figure requires \"M\({}_{\text{N}-1}\) < M\({}_{\text{N}}\)\" across stages [28]. To synthesise IMN for minimum noise, we select three \(\Gamma_{\text{S}}\) values as follows: \(\Gamma_{\text{S11}}=\Gamma_{\text{OPT}}\) @\(f_{1}\), \(\Gamma_{\text{S12}}=\Gamma_{\text{OPT}}\) @\(f_{2}\), and \(\Gamma_{\text{S13}}=\Gamma_{\text{OPT}}\) @\(f_{3}\); match these three points to 50 \(\Omega\). This typically requires multiple matching elements instead of the classic two elements. We then introduce a deliberate mismatch at the first stage output, selecting a suitable \(\Gamma_{\text{L1 ref}}\) from mismatch region B1 or B2 (Figure 7b) and obtaining the corresponding OM values. Last stage: Maximum gain--OMN synthesis  
Instead of designing a second stage after the first one, the parameters of the third stage that yield the maximum gain are decided. Therefore, it is necessary to choose appropriate \(\Gamma_{\text{S3 ref}}\) for the third stage (ex., can be a point for max gain from constant gain circles) and select an appropriate mismatch, similar to the first stage. The procedure. Following this is then to choose a reference \(\Gamma_{\text{L3 ref}}\) from the mismatch region A1 or A2 and synthesize an output matching network to match the \(\Gamma_{\text{L31}}\), \(\Gamma_{\text{L32}}\), \(\Gamma_{\text{L33}}\) to \(\Gamma_{\text{L3 ref}}\) value. Middle stage: Relaxed SSNM based determination  
The middle stage is not directly designed to match 50 \(\Omega\); instead:  
* Derive its matching conditions from neighbouring stages via relaxed SSNM.