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Table 2 Useful functions for the analysis of the cases described in Section 4

From: DF-based sum-rate optimization for multicarrier multiple access relay channel

  (f 1 ( β [k])) −1 (f 2 ( β [k])) −1 (f 3 ( β [k])) −1 (f 4 ( β [k])) −1
Case 1 \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}{N}\) \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}{N}\) \(1+\frac {{\beta _{b}^{2}}[k]|h_{\text {bd}}[k]|^{2}P}{N+{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}\) \(1+\frac {{\beta _{b}^{2}}[k]|h_{\text {bd}}[k]|^{2}P}{N+{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}\)
Case 2 \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}{N}\) \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}{N}\) \(1+\frac {{\beta _{b}^{2}}[k]|h_{\text {br}}[k]|^{2}P}{N+{\beta _{a}^{2}}[k]|h_{\text {ar}}[k]|^{2}P}\) \(1+\frac {{\beta _{b}^{2}}[k]|h_{\text {bd}}[k]|^{2}P}{N+{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}\)
     \(+\frac {\beta _{\text {br}}^{2}[k]|h_{\text {rd}}[k]|^{2}P}{N}\)
Case 3 \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ar}}[k]|^{2}P}{N}\) \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}{N}\) \(1+\frac {{\beta _{b}^{2}}[k]|h_{\text {br}}[k]|^{2}P}{N+{\beta _{a}^{2}}[k]|h_{\text {ar}}[k]|^{2}P}\) 1+snr b [k]
   \(+ \frac {\beta _{\text {ar}}^{2}[k]|h_{\text {rd}}[k]|^{2}P}{N}\)   
Case 4 \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ar}}[k]|^{2}P}{N+{\beta _{b}^{2}}[k]|h_{\text {br}}[k]|^{2}P}\) \(1+\frac {{\beta _{a}^{2}}[k]|h_{\text {ad}}[k]|^{2}P}{N}\) \(1+\frac {{\beta _{b}^{2}}[k]|h_{\text {br}}[k]|^{2}P}{N}\) 1+snr b [k]
   \(+ \frac {\beta _{\text {ar}}^{2}[k]|h_{\text {rd}}[k]|^{2}P}{N}\)