The previous work for the filter design considers uncorrelated white measurement noise disturbance. For more complex correlated noise disturbance, the conventional adaptive filter results in biased estimates. To overcome this problem, we introduce a linear prefilter to whiten the correlated noise (i.e., colored noise) for obtaining the unbiased estimate of the filter weight. Moreover, the design of some adaptive filters mainly focuses on the integer-order optimization methods. However, compared with the integer-order-based adaptive algorithms, the fractional-order-based algorithms show better performance. Thus, this letter develops a new gradient approach for the adaptive filter design based on the fractional-order derivative and a linear filter. Finally, the simulation results are provided from the system identification perspective for demonstrating the performance analysis of the proposed algorithms.