[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)=A2ej?(t)?,, where??(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,??(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)=A24ej?(t)e?j?(t?T)=A24ej[?(t)??(t?T)]?,

where?T?is the sample period. In discrete terms,

wn=w(nT),wn=A24ej[?n??n?1]?,?andvn=?n??n?1?.

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