This paper proposes a systematic analysis and synthesis method for optimal design of filter for L1 adaptive output feedback controllers. In the L1 adaptive feedback structure, the low-pass filter plays an important role in the stability and transient performance of the closed-loop system. The filter design is also the key to the trade-off between the per- formance and robustness of the closed-loop system. Although, the filter design for the L1 adaptive architecture has been studied in previous papers, the need for a numerically tractable synthesis method is yet to be fulfilled. The key challenge in establishing a satis- factory trade-off scheme between performance and robustness in L1-adaptive control archi- tecture is the non-convex nature of the underlying optimization problem for filter design. We notice that the closed-loop L1 adaptive system converges to its corresponding linear reference system both in transient and steady-state in the presence of fast estimation rates. Taking advantage of this property, we discretize the closed-loop (linear) reference system of L1 adaptive controller and formulate the filter optimization in terms of robustness to time delay and performance both in transient and steady state. Illustrative examples are provided to validate the filter design methodology of this paper.