An efficient algorithm for computing the frequency response of discrete-time systems described by rational transfer functions is presented. The algorithm is simple, fast, recursive, and can be used for equally or unequally spaced frequencies. Based on an initial expansion of the system transfer function to a novel Jacobi-type trigonometric continued fraction, the algorithm proposed permits all operations to be performed by real arithmetic, guarantees real results, saves a number of operations, and produces accurate results. The algorithm is easily programmable and needs only 2nN real multiplications/divisions for evaluating the frequency response of an nth-order system at N different frequencies.