Complex cepstrum of discrete Hartley and warped discrete Hartley filters

In this paper, we derive the theoretical complex cepstrum (TCC) of the discrete Hartley transform (DHT,[1]) and warped discrete Hartley transform (WDHT) filters. In our derivations, we start with the filter bank structure for the DHT, where each basis is represented by a finite impulse response (FIR) filter. The WDHT filter bank is obtained by substituting z(-1) in the DHT filter bank with a first-order all-pass filter having 0 as its coefficient. The value of the 0 controls the degree and nature of the frequency warping. Here, we choose the value of 13, such that the frequency warping follow the psychoacoustic bark scale. Using the filter bank structures, we first derive the transfer functions for the DHT and WDHT, and subsequently the TCC for each filter is computed. We analyze the DHT and WDHT filter transfer functions and the TCC by illustrating the corresponding pole-zero maps. Moreover, we also use the derived TCC expressions to compute the cepstral sequence for a synthetic vowel /aa/ and compare with the discrete cosine and warped-discrete cosine transform cepstral sequences derived in [2].