How can I design Quadrature FM Demodulator / Detector ?

I’m thinking to design FM demodulation methods.

I’m reading this as my reference. on the internet.

“Quadrature FM Demodulator / Detector Tutorial”

http://www.radio-electronics.com/info/rf-technology-design/fm-reception/fm-quadrature-detector-demodulator.php

I'm trying to understand how can I design FM/PM transmitters and receivers.
If you notice something on my script here, please let me know your advice.

Calculation of i2 is something strange to me.

I'm afraid Current i2 should be corrected as voltage V3

because V3 = V1*V2 [V].

It is voltage V3[V] and is not current i3[A].

Because V3 is output of multiplied voltage of V1 and V2.

V1 = U1*sin(ωc*t+θ(t))

V2= U1*sin(ωc*t-π/2) = -U1*cos(ωc*t)

Then I get

V3 = V1*V1 = U1*sin(ωc*t+θ(t))* {-U1*cos(ωc*t)}

                  =-U1^2*sin(ωc*t+θ(t))*cos(ωc*t)

                 =-(U1^2/2)*{sin(2*ωc*t+θ(t))+cos(θ(t))}

I assume very high frequency  -(U1^2/2)*{sin(2*ωc*t+θ(t)) here can be removed by LPF(Low Pass Filter).

Hence I get

Vout = -(U1^2/2)*cos(θ(t)) … It is voltage after LPF.

I'm wandering  Voltage after LPF:

Vout= -(U1^2/2)*cos(θ(t)) can really become decorated FM modulation of V1.

Can I hear good audio signal of FM from  -(U1^2/2)*cos(θ(t))  ?

I’m wondering if F to V conversion circuit for Vout = -(U1^2/2)*cos(θ(t)) may be required.

And I'm afraid that Laplace transformation there is something strange for me.

Because reversed Laplace transformation is not calculated.

And I can not find any differential equations there that express transient voltage of demodulated voltage signal. It is strange for me. Because I cannot calculate it by reversed Laplace transformation without it.

How should I understand  this theory ?

Related theory is here. The situation here is the same as I show before. It is also strange theory for me.

A Quadrature Demodulator Tutorial

http://www.eetimes.com/document.asp?doc_id=1275839

Thanks for reading it.

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