Reducing computational complexity of time delay estimation method using frequency domain alignment

In this paper, we consider the estimation of time delays between multiple waveforms which are delayed forms of a single waveform. We use a previously defined cost function whose minimization is achieved through applying linear phase shift operators to the discrete Fourier transforms (DFTs) of the waveforms. The optimal phase shift operators result in the least differences between the phase shifted DFTs of the waveforms in the frequency domain. The time delays associated with the optimal phase shift operators become the optimal time delays between these waveforms. We demonstrate that the matrix form of the cost function is symmetric and has all zero diagonal entries. Therefore, by using these two features, we achieve a considerable reduction in the computational complexity of the optimization problem without losing accuracy. Performance investigation with six noisy speech waveforms shows that this procedure is very accurate and computationally efficient even under very noisy conditions.