In this paper, we present a unified framework based on integral quadratic constraints for analyzing the convergence of distributed push-pull based optimization algorithms for directed graphs. Our framework provides numerical upper bounds on linear convergence rates of existing distributed push-pull based algorithms when local objective functions are strongly convex and smooth and directed graphs are strongly connected. Moreover, we propose a new distributed optimization algorithm for directed graphs and show that the proposed framework can also be applied to establish its linear convergence rate. The theoretical results are illustrated and validated via numerical examples.