رضا دیانت

Finding a new representation for functions is one of the beauties of mathematics. In addition to mathematical attractiveness, its applicable aspects are also considered, such as the Fourier transform that exhibits periodic and aperiodic functions on complex exponential bases in the frequency domain. It also has many applications in physics and engineering sciences. By changing in bases, it can be gotten a new view of the functions that some of them can be useful in applied sciences. Here, it will be discussed about two new representations for finite sequences called first and second Ramanujan representations. In both views, finite sequences are displayed on the integer-values bases known as Ramanujan's Sum and their applications will be studied in the context of signal processing. Recognizing Ramanujan Finite Transform and Ramanujan Periodic Transform is the main purpose of this thesis. Both can demonstrate the periodicity of some finite sequences, periodic or seemingly aperiodic that have been constructed of a combination of several periodic components. Here, the mathematical structure of the above mentioned transforms is investigated and their effected performance to Complex Cepstrum techniques and the Autoregressive processes will be challenged. In many cases, the hidden periods of a sequence are well characterized by the Ramanjan Periodic Transform and the energy diagram of the second representation components. But this method is sensitive to the signal length which considered as its disadvantage. Complex Cepstrum uses DFT in its structure. But this method does not work well if the sampled points met to zeroes on the unit circle. In Autoregressive processes, the parameters are calculated by using the Levinson algorithm. By this method, frequency values are estimated approximately, however acceptable results will be obtained. The advantage of Ramanujan's methods is its more precision and easy to the detection of basic and hidden periods.