## 2 Analysis of OFDM-Induced IM2 Distortion Effects
When a OFDM blocker signal that has the total power of \(P_{in\_total}\) is fed to a receiver having \(II\!P_{2}\) and a power gain of \(G_{P}\), the input-referred IM2 distortion power \(P_{IIM2}\) (= \(P_{OIM2}-G_{P}\)) can be expressed by  
\[P_{IIM2}=2P_{in\_total}-II\!P_{2} \tag{2}\]  
If the total number of subcarriers is \(N_{sub}\), the power of each subcarrier \(P_{in\_sub}\) is given by  
\[P_{in\_sub}=P_{in\_total}-10 log(N_{sub}) \tag{3}\]  
Now, let us compute the total input-referred IM2 distortion power by integrating the IM2 PSD over the band of interest. [PERSON] et al. in their OFDM distortion analysis [PERSON] et al. (2019) derived an analytic expression of the IM2 PSD by assuming that IM2 tones at the baseband are uncorrelated with the original causative OFDM data. Although the exact PSD expression (Equation (12) of [PERSON] et al. (2019)) is not repeated here, we can find that the numerical integration of PSD can be simply carried out by collecting and adding the subcarrier PSD components at their center frequencies without a significant loss in generality and accuracy. Let us take an example of the 3 rd \(P_{IIM2}\) illustrated in Figure 2, which is the first IM2 tone that exists in the IF band. We can find that the 3 rd \(P_{IIM2}\) is created by (\(N_{sub}-3\)) of two subcarrier tones that are apart by 3\(f_{sub}\) in the original OFDM blocker signal. This observation can be generalized such that the \(k\)-th IM2 power \(P_{IIM2,th}\) can be obtained by \(k\)-th \(P_{IIM2}\) multiplied by the total number of the subcarrier pairs responsible for creating the \(k\)-th IM2 tone. It is expressed by  
\[10^{\frac{P_{IIM2,th}}{10}}=(N_{sub}-k)\cdot 10^{\frac{2P_{in\_sub}-II\!P_{2}}{10}} \tag{4}\]  
If the band of interest covers from the \(k\)-th IM2 tone through the \(k\)-th IM2 tone, then, the total input-referred IM2 power \(P_{IIM2,total}\) that exists within the band of interest can be calculated by summing (\(N_{sub}-k\)) of \(P_{IIM2,th}\) of (4) from the first \(k\)-th through the last \(k\)-th. It is expressed as follows  
\[10^{\frac{P_{IIM2,total}}{10}}=\sum_{k=k_{i}}^{k_{f}}\biggl{(}(N_{sub}-k)\cdot 1 0^{\frac{2P_{in\_sub}-II\!P_{2}}{10}}\biggr{)} \tag{5}\]  
Figure 2: IM2 distortions induced by the multi-tone OFDM blocker. Equation (5) can be written again in dB as follows  
\[P_{IIM2.total}=(2P_{in.sub}-IIP_{2})+10\log\Bigl{(}\sum_{k=k_{i}}^{k_{f}}(N_{sub}- k)\Bigr{)} \tag{6}\]  
By substituting \(P_{in.sub}\) of (6) with (3), Equation (6) can be arranged as  
\[P_{IIM2.total}=2\Bigl{(}P_{in.total}-P_{offset}\Bigr{)}-IIP_{2} \tag{7}\]  
where \(P_{offset}\) is given by  
\[P_{offset}=10 log\frac{N_{sub}}{\sqrt{\sum_{k=k_{i}}^{k_{f}}(N_{sub}-k)}} \tag{8}\]  
Equations (7) and (8) imply that the IM2 distortion power induced by the OFDM signal can be equivalently evaluated by the IM2 distortion power induced by the two-tone blocker signal as long as the two-tone blocker signal power is set to be lower than the original OFDM blocker signal power \(P_{in.total}\) by \(P_{offset}\). This finding greatly simplifies the simulation and characterization of the OFDM-induced IM2 distortion in an RF receiver. Involvement of a multi-carrier OFDM signal in RF circuit simulations will typically require a sophisticated OFDM signal modeling, a very complex numerical analysis method, and a long simulation time. Thus, it is very time-consuming and impractical to examine the OFDM-induced IM2 effects in RF circuit design. However, if the OFDM blocker signal can be replaced simply by an equivalent two-tone blocker signal by considering an offset parameter \(P_{offset}\), the whole simulation process will become much simpler and more convenient because we can use the conventional time- and frequency-domain circuit simulation methods. Thus, these analytic results enable us to evaluate the OFDM-induced IM2 effects very efficiently when only examining an equivalent two-tone blocker induced IM2 effects.