2016年7月24日日曜日

Theory of FM/PM demodulation using IQ demodulator ( preliminary )

Theory of FM/PM demodulation using IQ demodulator ( preliminary )
(C) Noboru, Ji1NZL JULY 22, 2016
I tried to configure FM/PM demodulation architectures by using IQ demodulator and MPU/DSP calculations.


1. Hardware architecture of IQ demodulator


 See at Fig. 1. It is the same as the IQ demodulator defined before for AM/SSB/CW modulated signal.
Fig.1 IQ demodulator for general purpose


2.  FM modulation : How to demodulate FM modulation signal


FM modulation voltage signal is expressed as the following equation (1).
Vfm=Vrf*cos(ωc*t+k*∫x(τ)dτ) …(1)
ψ(t) ≡ k*∫x(τ)dτ, integral τ: from 0 to t ( “t” is time variable by second unit) …(2)


Vosc=Vc*sin(ωc*t) …(3)
V2=Vc*sin(ωc*t-π/2) = -Vc*cos(ωc*t)…(4)


V3=Vfm*Vosc
   =Vrf*cos(ωc*t+ψ(t)) * Vc*sin(ωc*t)
   =Vrf*Vc*(1/2)*{ sin(ωc*t+ψ(t)+ωc*t) +sin(ωc*t-(ωc*t+ψ(t) )  }
   =(Vrf*Vc/2)*{ sin(2*ωc*t+ψ(t)) +sin(-ψ(t) }
   =(Vrf*Vc/2)*{ sin(2*ωc*t+ψ(t)) -sin(ψ(t) } …(5)


At (5), sin(2*ωc*t+ψ(t)) can be removed by LPF, then
V-i = (-Vrf*Vc/2)*sin(ψ(t)) …(6)


V4=Vfm*V2
   =Vrf*cos(ωc*t+ψ(t)) * ( -Vc*cos(ωc*t) )
   = -Vrf*Vc* cos(ωc*t+ψ(t)) * cos(ωc*t)
   = -Vrf*Vc*(1/2){ cos(ωc*t+ψ(t)+ωc*t) + cos(ωc*t+ψ(t) -ωc*t) }
   = (-Vrf*Vc/2){ cos(2*ωc*t+ψ(t)) + cos(ψ(t) } …(7)


At (7), cos(2*ωc*t+ψ(t)) can be removed by LPF, then
V-q = (-Vrf*Vc/2)*cos(ψ(t)) …(8)


Try to get Vrf from (6) and (8).
(V-i)^2  = {(-Vrf*Vc/2)*sin(ψ(t)}^2  = (Vrf*Vc/2)^2 * (sin(ψ(t))^2 …(9)
(V-q)^2 = {(-Vrf*Vc/2)*cos(ψ(t)}^2 = (Vrf*Vc/2)^2 * (cos(ψ(t))^2 …(10)
(9)+(10) is
(V-i)^2 + (V-q)^2 = (Vrf*Vc/2)^2 * (sin(ψ(t))^2 + (Vrf*Vc/2)^2 * (cos(ψ(t))^2
                       = (Vrf*Vc/2)^2 * {(sin(ψ(t))^2 +(cos(ψ(t))^2}
                       = (Vrf*Vc/2)^2 …(11)  ∵ sin(ψ(t))^2 +(cos(ψ(t))^2 = 1
Get root of (11)
√{(V-i)^2 + (V-q)^2} = Vrf*Vc/2 …(12)
MPU/DSP can calculate unknown Vrf from (12)
Vrf = 2*√{(V-i)^2 + (V-q)^2} /Vc …(13) -> (*[Note1}
Put calculated Vrf of (13) to (6), then get V-i.
V-i = ( - (2*√{(V-i)^2 + (V-q)^2} /Vc) *Vc/2)*sin(ψ(t)
   =  - (√{(V-i)^2 + (V-q)^2} ) *sin(ψ(t)) ,  ∵ “2” and Vc are cancelled here.
 
-V-i / ( √{(V-i)^2 + (V-q)^2} ) = sin(ψ(t))
∴ ψ(t) = arcsin( -V-i / ( √{(V-i)^2 + (V-q)^2} ) ) …(14)
from (2) ψ(t) ≡ k*∫x(τ)dτ, then
k*∫x(τ)dτ= arcsin( -V-i / ( √{(V-i)^2 + (V-q)^2} ) ) …(15)
∴ x(t) = (1/k)* d/dt { arcsin( -V-i / ( √{(V-i)^2 + (V-q)^2} ) ) } …(16)
(16) means the baseband signal voltage x(t) of FM signal of (1) is demodulated.
(Done)


3. PM modulation : How to demodulate PM modulation signal

PM modulation voltage signal is expressed as the following equation (17).
Vpm=Vrf*cos(ωc*t+k*Φ(t)) …(17)
ψ(t) ≡ k*Φ(t),  Φ(t) is baseband signal for phase modulation  …(18)
I can exchange ψ(t) ≡ k*Φ(t) instead of (2).
Then from (14), I get
ψ(t) = arcsin( -V-i / ( √{(V-i)^2 + (V-q)^2} ) ) …(14)’
From (18),
k*Φ(t) = arcsin( -V-i / ( √{(V-i)^2 + (V-q)^2} ) )
Φ(t) = (1/k)*arcsin( -V-i / ( √{(V-i)^2 + (V-q)^2} ) ) …(19)
(19) means the base band signal voltage Φ(t) of PM signal of (17) is demodulated.
(Done)


->[Note1]:

Received unknown variable “Vrf" can be calculated by MPU or DSP. This calculation by MPU or DSP in the equation (13) is some kind of equivalent function of so called “limiter amplifier” for the traditional analog FM receivers.