Calculating the complex cepstrum without phase unwrapping or integration

The complex cepstrum, as developed by Oppenheim [4] and Oppenheim, Schafer, and Stockham [5], originated from the pioneering paper of Bogert, Healy, and Tukey [3]. It is a nonlinear transformation from one signal space into another which converts two superposed signals into their sum. The cepstrum is usually computed by inverse Fourier transformation of the frequency domain complex logarithm of the original signal. The phase term or imaginary part of this complex logarithm presents a computational difficulty which is sometimes approached by integration of the derivative of the phase spectrum. An excellent approach to phase computation is the adaptive integration methodology as proposed by Tribolet [10]. Unfortunately, this technique can be computationally intensive. This note describes a simple technique for efficient computation of the complex cepstrum which requires no phase unwrapping or integration. The procedure is given in both one and two dimensions.