We formulate and solve the design of perfect reconstruction (PR) analysis/synthesis filter banks by direct optimization of the structured polyphase factorization. The unknowns are the sparse folding matrix coefficients and the parameters of the zero-delay/maximum-delay stages. Because the mapping from these parameters to the prototype frequency response is nonlinear, we adopt numerical optimization (gradient-based or quasi-Newton) on a weighted frequency-domain objective. Practical recipes are given for extracting the prototype from the cascade, constructing loss functions, setting constraints (e.g., \(\det (\cdot )=-1\) submatrix conditions), initialization, and implementing the loop in Python.