A sinusoid of arbitrary phase and amplitude relative to a reference sinusoid may be generated if the in-phase (I) and quadrature (Q) components of the reference are available. In other words, the sinusoid

may be created from (the I component) and (the Q component) by forming the scaled sum with the proper choice of and . This represents a sinusoid which leads the reference sinusoid by degrees.

Start with the trig identity:

Multiply by Z:

Group terms on the right side:

Now it’s clear that

For example, to create a sinusoid which is .25 times the amplitude of the reference and leads it by 127 degrees, use

which is to say .