Gaussian white noise is commonly assumed for data for date in fields such as electrical and electronic engineering. However, optimal systems designed under this assumption experience severe performance degradation in colored noise environments, necessitating effective whitening. Based on the theory of random signal whitening filtering, this paper proposes an adaptive design method for linear-phase finite impulse response (FIR) filters to achieve stable and practical colored noise whitening. First, the mainstream existing whitening filtering methods are briefly summarized and classified into two categories: orthogonal basis-based and transform domain-based approaches. Subsequently, leveraging the covariance matrix whitening principle of orthogonal basis-based methods, it is shown that the key whitening matrix possesses a Toeplitz structure and Hermitian property. Due to the convergence of its elements, it is shown that both the row and column vectors of this matrix possess approximate whitening capability. Furthermore, utilizing the convergence of the approximate whitening vectors, a suitable threshold is set for truncation to automatically determine the filter length. The proposed filter, named the approximate whitening transversal filter (AWTF), is a type of linear-phase FIR filter that can be adaptively designed directly from noise samples. It offers advantages including absolute stability, simple structure, zero phase distortion, and low computational complexity. In addition, the issue of signal detection after whitening filtering is discussed, and a signal detection scheme based on generalized matched filtering is proposed. Simulations are conducted using colored noise generated by autoregressive (AR) models. The results verify the convergence of the approximate whitening vectors and demonstrate that the AWTF can effectively whiten colored noise. When combined with generalized matched filtering, the resulting bit error rate (BER) is essentially consistent with that of traditional optimal methods. Finally, the application of the AWTF in a sky-wave radar system is analyzed. Measured data processing experiments indicate that the radio frequency interference (RFI) conforms to the noise assumption in this paper, and the AWTF can effectively suppress RFI.