The authors consider the problem of robust output estimation in the discrete-time setting. The uncertainty in the system is characterised by integral quadratic constraints with dynamic multipliers. The main technical contribution of this study is to derive two different synthesis conditions in terms of linear matrix inequalities (LMI) by two different approaches. The LMI condition derived by the first approach allows one to find the robust estimator and the performance certificate simultaneously, whereas the condition by the second approach has a smaller size but requires an additional step for recovering the estimator. The synthesis conditions are validated by numerical examples. The results verify the equivalence of the two approaches and show that the two-step approach is in fact computationally favourable when the system has large number of state variables. NomenclatureSymbols R, R n , R n×m and Z + are used to denote the sets of real numbers, n-dimensional real vectors, n × m real matrices and non-negative integers, respectively. Symbols I n and 0 n×m are used to denote n-dimensional identity matrix and n × m zero matrix, respectively. The subscripts are dropped when the dimension is evident from the text. Given a matrix M , the transposition and the conjugate transposition are denoted by M and M * , respectively. Given matrix M and N , we use diag(M , N ) to denote the block-diagonal matrix with M and N being the (1,1) block and the (2,2) block, respectively.