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 .