The search for weak periodic signals in time series data is an active topic of research. Given the fact that rarely a real world dataset is perfectly periodic, this paper approaches this problem in terms of data mining, trying to discover weak periodic signals in time series databases, when no period length is known in advance. In existing time series mining algorithms, the period length is user-specified. We propose an algorithm for finding approximate periodicities in large time series data, utilizing autocorrelation function and FFT. This algorithm is an extension to the partial periodicity detection algorithm presented in a previous paper of ours. We provide some mathematical background as well as experimental results.