The recursive inverse (RI) adaptive algorithm, was shown to have comparable performance to that of the well-known recursive-least-squares (RLS) algorithms but with reduced computational complexity. Although the RI algorithm provides significant performance, it suffers from low convergence rate in some situations where a relatively low initial step-size is required. In this paper, we propose a new RI algorithm that applies a discrete wavelet transform (DWT) to the input signal. This transformation reduces the self-correlation of the input signal which, in turn, overcomes the low convergence rate of the RI algorithm when a relatively small initial step-size is used. The performance of the proposed algorithm (DWT-RI) is compared to those of the RI, DWT-RLS and DWT normalized least-mean-square (DWT-NLMS) algorithms in additive white Gaussian noise (AWGN) environment in a noise cancellation setting. The simulations show that the proposed algorithm has a superior convergence rate compared to the other algorithms.