We present a simplified Fourier-transform process, called the orthogonal frequency Fourier transform (OFFT). Conventional divide-and-conquer techniques, such as the fast Fourier transform (FFT), reduce the number of operations in a Fourier transform and simplify at least some of the complex-valued terms (i.e. twiddle factors). The FFT reduces the number of multipliers, which account for much of the chip area and power consumption in digital VLSI design. The OFFT and inverse OFFT exploit orthogonal frequency relationships to replace multiplications with simpler sampling and adding operations. Specifically, the OFFT replaces twiddle factors with step functions, which are superpositions of harmonic sinusoids. The resulting transform is adapted to add samples that are selected relative to at least one periodic step function, thus eliminating all complex multiplications. In phase and quadrature phase OFFT processing may be performed. OFFTs can be combined with pass-band sampling to simultaneously perform filtering, down conversion, and demodulation. Inverse OFFTs combined with pass-band filters can be used to provide up conversion of multi-carrier signals. Since OFFTs are substantially less complex than FFTs, OFFT processing is applicable to digital radio systems where there are considerable constraints on power consumption and chip size. The OFFT is particularly useful for processing multi-carrier transmission protocols in wireless communications, such as Carrier Interferometry, Orthogonal Frequency Division Multiplexing, and Multi-carrier Code Division Multiple Access, which are quickly gaining favor over single-carrier protocols. OFFT algorithms can process a greater number of carriers and provide lower complexity compared to FFTs.