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A cepstrum-based approach is proposed to design finite- and infinite-impulse-response (IIR) fractional-delay (FD) filters. The maximal-flatness criteria on frequency responses are formulated as a system of linear equations to solve the truncated complex cepstrum. The closed-form solutions to cepstrum sequences can be derived. Moreover, it is very attractive that the resultant cepstrum coefficients are directly proportional to the desired FD. Under a fixed filter order, the set of normalized complex cepstra needs to be computed once and stored, and the specific set for an arbitrary FD is obtained by simply multiplying the stored set with the delay value. According to this observation, we also design two kinds of tunable filter structures consisting of several linear-phase filters, in which it is more flexible to obtain better performance by adding the extra substructure without modifying the present one. Moreover, the tunable FD is simply controlled by a single parameter, and the usage of linear-phase filters saves half of the multipliers, largely reducing the cost of hardware implementation. In addition, we obtain an IIR all-pass filter with a wider useful band than that based on Thiran's design.