[origion](https://www.mathworks.com/help/comm/ref/comm.fmdemodulator-system-object.html)

Algorithms

A frequency-modulated passband signal, Y(t), is given as

Y(t)=Acos(2πfct+2πfΔt0x(τ)dτ) ,

where:

• A is the carrier amplitude.

• fc is the carrier frequency.

• x(τ) is the baseband input signal.

• fΔ is the frequency deviation in Hz.

The frequency deviation is the maximum shift from fc in one direction, assuming |x(τ)| ≤ 1.

A baseband FM signal can be derived from the passband representation by downconverting the passband signal by fc such that

ys(t)=Y(t)e−j2πfct=A2[ej(2πfct+2πfΔt0x(τ)dτ)+e−j(2πfct+2πfΔt0x(τ)dτ)]e−j2πfct=A2[ej2πfΔt0x(τ)dτ+e−j4πfct−j2πfΔt0x(τ)dτ] .

Removing the component at -2fc from yS(t) leaves the baseband signal representation, y(t), which is given as

y(t)=A2ej2πfΔt0x(τ)dτ.

The expression for y(t) can be rewritten as y(t)=A2ejϕ(t) ,, where ϕ(t)=2πfΔt0x(τ)dτ. Expressing y(t) this way implies that the input signal is a scaled version of the derivative of the phase, ϕ(t).

To recover the input signal from y(t), use a baseband delay demodulator, as this figure shows.

Subtracting a delayed and conjugated copy of the received signal from the signal itself results in this equation.

w(t)=A24ejϕ(t)e−jϕ(t−T)=A24ej[ϕ(t)−ϕ(t−T)] ,

where T is the sample period. In discrete terms,

wn=w(nT),wn=A24ej[ϕn−ϕn−1] , andvn=ϕn−ϕn−1 .

The signal vn is the approximate derivative of ϕn such that vn ≈ xn.