In this paper, we consider the application of the frequency-response masking (FRM) technique in constructing the IIR approximately linear-phase half-band filters and Hilbert transformers. An efficient implementation structure, consisting of a FRM-based nearly all-pass filter and a pure delay term, has been derived. The FRM-based branch being composed of the recursive all-pass filter and the two polyphase components of FIR masking filters, exhibits nearly all-pass magnitude response and approximately linear phase. We are focused on the low-complexity implementation of the recursive all-pass filter. We propose a new simple approximation method, which we apply to design the approximately linear phase IIR filter with simple powers-of-two coefficients. The example half-band filters and Hilbert transformers show sharp magnitude responses and approximately linear phase.