I am learning the double (Brown) exponential smoothing. Comparing to the simple exponential smoothing, the Brown exponential smoothing smooths the output sequence $Y_t$ twice, such that \begin{align} S^{(1)}_t &= \alpha Y_t + (1-\alpha)S^{(1)}_{t-1}\\ S^{(2)}_t &= \alpha S^{(1)}_t + (1-\alpha)S^{(2)}_{t-1} \end{align} I tried to derive its equivalent smoothing equations via level and trend functions \begin{align} L_t & = 2S^{(1)}_t - S^{(2)}_t\\ T_t &= \alpha \left(S^{(1)}_t - S^{(1)}_{t-1}\right) + (1-\alpha)T_{t-1} \end{align} where $Y_{t+1} = L_t + T_t + e_{t+1} = S^{(1)}_t + \frac{1}{\alpha} T_t + e_{t+1}$.
But I failed, can anyone help me to derive smoothing equations from the double exponential smoothing? many thanks!