The main objectives of this thesis are to develop a computationally efficient realization method of adaptive filtering and to study the use of the developed ADF in a practical application. First, we discuss various realization methods of a class of FIR ADF``s including the fast transversal filter (FTF). Then, a computationally efficient frequency bin fast transversal filter (FBFTF) is introduced. By analysis of the computational complexity, we show that a dramatic reduction of multiplications can be achieved by the FBFTF as compared with the FTF. Also, we discuss the instability problem which can arise for an ADF with large order and some particular inputs. As an application of the FBFTF, we study a noise canceller system. Specifically, the performance of the FBFTF used in a noise canceller is investigated. It is shown that its performance is almost the same as that of the FTF, yet its computational complexity is reduced to one tenth to one thirtieth as compared to the FTF without degrading the performance.