Theory of FM/PM modulation using IQ modulator( preliminary )

Theory of FM/PM modulation using IQ modulator( preliminary )

 (C) Noboru, Ji1NZL JULY 21, 2016

I tried to configure FM/PM modulation circuit architectures from FM/PM equations
as my top-down design method.

1. FM modulation : How to generate FM modulation signal

FM modulation voltage signal is expressed as the following equation (1).

Vfm=Vc*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)

From (1) and (2), I get

Vfm=Vc*cos(ωc*t+ψ(t))

     =Vc*{cos(ωc*t)*cos(ψ(t)) - sin(ωc*t)*sin(ψ(t)) }

     =Vc*{ cos(ωc*t)*(-sin(ψ(t)-π/2) ) - cos(ωc*t-π/2)*sin(ψ(t)) }

     = {-Vc*cos(ωc*t)}*sin(ψ(t)-π/2) }  + {-Vc*cos(ωc*t-π/2)*sin(ψ(t)) } …(3)

At the equation (3), I get the following functions from (D-1) to (D-5) that they realize the electric works introduced from the equation (1).

 

(D-1) -Vc*cos(ωc*t)} can be defined as a VFO or an oscillator of frequency

of ωc[rad*Hz], and the inverted amplitude voltage is -Vc[V].

(D-2) -Vc*cos(ωc*t-π/2) can be defined as the phase shifted -π/2 [rad] of a VFO or an oscillator that is predefined in (D-1).

(D-3) ψ(t) can be defined as the integral signal of base band signal k * x(t)  .

(D-4) sin(ψ(t)) is the value of sine function’s value of ψ(t).

(D-5) sin(ψ(t)-π/2) is -π/2 phase shifted signal of (D-4).

From the definitions from (D-1) to (D-5), the block diagram of the FM modulator can be written as Fig.1 by using a IQ modulator.

Fig1. FM modulator used an IQ modulator

Here, the architecture of FM modulation circuit is introduced from the equation (1).

(Done)

2. PM modulation : How to generate PM modulation signal

PM modulation voltage signal is expressed as the following equation (4).

Vpm=Vc*cos(ωc*t+ψ(t)) …(4)

ψ(t) ≡ k*Φ(t), Φ(t) is assumed as phase value of baseband signal …(5)

As the equation (4) is the expressed as same way of (1), then I get

Vpm=Vc*cos(ωc*t+ψ(t))

     = { -Vc*cos(ωc*t)*sin(ψ(t)-π/2) } + {-Vc*cos(ωc*t-π/2)*sin(ψ(t)) } …(6)

The equation (6) means I can simply define ψ(t) ≡ k*Φ(t) instead of FM case of (2).

At the equation (6),

The equation (6) introduces the block diagram of PM modulation circuit as the Fig.2.

Fig.2 PM modulator used an IQ modulator

Fig.2 is simply the diagram that is same as the FM modulator without an integral circuit.

The architecture of PM modulation circuit is introduced from the equation (6).

(Done)

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