This crates provides a simple FM signal demodulator for use with software radio. It demodulates using phase difference approximation as described below.
Consider the classical equation  for an FM signal:
s(t) = a(t) cos(ωct + φ(t))
φ(t) = ω∆∫x(τ)dτ
where the integral is evaluated from 0 to t and x(t) is the modulating signal to be recovered.
Differentiating this gives
dφ(t) / dt = ω∆x(t)
x(t) = ω∆-1 dφ(t) / dt
Differentiation in continuous time is approximated by finite backward difference in discrete time, so
x(t) ≈ ω∆-1 (φ[t] - φ[t-1]) / T
Assuming a "normalized" period of T = 1, this becomes
x(t) ≈ w∆-1 (φ[t] - φ[t-1])
This requires the change in phase between the current and previous sampling instants, which can be computed from the corresponding I/Q samples. Given an FM signal s(t), the received I/Q sequence will have components
i(t) = a(t) cos φ(t)
q(t) = a(t) sin φ(t)
with each sample represented as
p(t) = i(t) + j q(t)
Evaluating the complex argument of this gives
arg(p(t)) = arctan[q(t) / i(t)] = arctan tan φ(t) = φ(t)
arg(p(t)) - arg(p(t-1)) = φ(t) - φ(t-1)
arg(p(t)p(t-1)*) = arg(p(t)) - arg(p(t-1)) = φ(t) - φ(t-1)
Combining all these results leads to the equation calculated at each sample:
x[t] = ω∆-1 arg(p[t]p[t - 1]*)
using angular frequency deviation ω∆ = 2π f∆ and the current and previous complex samples.
- "FM demodulation using a digital radio and digital signal processing", J.M. Shima, 1995.
Demodulates an FM signal using a phase difference approximation.