We consider the achievable delay margin of a real rational and strictly proper plant,with unstable complex poles,by a linear time-invariant (LTI) controller.The delay margin is defined as the largest time delay such that,for any delay less than this value,the closed-loop stability is maintained.Drawing upon a frequency domain method,particularly a bilinear transform technique,we provide an upper bound of the delay margin,which requires computing the maximum of a one-variable function.Finally,the effectiveness of the theoretical results is demonstrated through a numerical example.