2015年9月13日日曜日

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”

I'm trying to understand how can I design FM/PM transmitters and receivers.

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.