## Expand description

This crates provides a simple FM signal demodulator for use with software radio. It demodulates using phase difference approximation as described below.

### §Theory

Consider the classical equation [1] for an FM signal:

s(t) = a(t) cos(ω

_{c}t + φ(t))

with

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)

so

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) +

jq(t)

Evaluating the complex argument of this gives

arg(p(t)) = arctan[q(t) / i(t)] = arctan tan φ(t) = φ(t)

so

arg(p(t)) - arg(p(t-1)) = φ(t) - φ(t-1)

Applying the complex identities arg(uv) ≡ arg(u) + arg(v) (mod (-π,
π]) and
arg(u^{*}) = -arg(u),

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.

### §References

- “FM demodulation using a digital radio and digital signal processing”, J.M. Shima,

## Structs§

- Demodulates an FM signal using a phase difference approximation.