This invention describes a programmable, digital implementation to shift an arbitrary frequency, or various frequencies in various communication systems of the original signal in the frequency domain. The correspondent phase accumulation to perform the desired frequency shift per sampling instant is perfectly tracked by counting up or down a simple integer. Several arbitrary frequency shifters with different mathematical models are provided. The correspondent implementations with Look-Up-Tables (LUT) are derived for high-speed implementations without further calculations of the values of the sine and cosine functions every sampling instant. Furthermore, a simple shift-and-add phase rotation is described to replace the four required real multiplications. If the original complex signal contains only one-bit each from real part and imaginary part, a surprisingly simple implementation is derived and disclosed for the overall arbitrary frequency shift operation. Further simplifications are also disclosed to make this invention feasible for high sampling frequencies and small frequency drifts.