The principle of SSB modulator / transmitter of “Weaver PSN method by using “IQ feeder” and “IQ modulator” (Weaver’s PSN method) (preliminary)

The principle of SSB modulator / transmitter of “Weaver PSN method 
by using “IQ feeder” and “IQ modulator”  (Weaver’s PSN method) (preliminary)

1. Architecture

“Weaver PSN method modulator / transmitter” has 3 functional blocks.

(A) “Baseband signal generator” … Eg. Microphone and/or with small signal amplifier(s)

(B) “IQ feeder for baseband signal” inputs baseband signal from “Baseband signal generator” , converts it IF frequency stage such as 10KHz to 40KHz and outputs two orthogonal voltage signals “I” and “Q”.

(C)“IQ modulator” inputs two port of orthogonal signals “I” and “Q” and outputs to 1 port of SSB modulated signal.

2. How it works

(A) Baseband signal generator

    Assume V1 is “baseband signal” to be input as audio signal. 
    (eg. We can use microphone to input V1.)

    V1 = Vs*sin(ωs*t) …(1)
    here ωs = 2πfs  …(2) 
          fs [Hz] is 0 to 20KHz Frequency of baseband signal on Audio frequency        ( 2.6KHz to 3.0KHz is preferred in the actual application as the maximum  
          frequency of baseband AF signal. )

(B) IQ feeder for baseband signal

OSC1 generates ωc1[rad*Hz] sin wave voltage. 
ωc1 is fixed frequency and OSC1 is used for first frequency conversion. 

Vosc1 = Vc1*sin(ωc1*t) …(3)
here ωc1 = 2πfc1 …(4) 
 fc [Hz] is set to IF frequency such as 40KHz to covert baseband frequency ω1  
          to ωc-ωs or ωc+ωs.

Shift Vosc1 signal to be -90 deg. by “-90 deg. Shifter” to get V2.

V2 = Vc1*sin(ωc1*t-π/2) = -Vc1*cos(ωc1*t) …(5)

Get V3 by multiplying V1 and Vosc1.

V3= V1*Vosc1 = (Vs*sin(ωs*t)) * (Vc1*sin(ωc1*t)) 
  = Vs*Vc1*sin(ωs*t) * sin(ωc1*t) 
    =Vs*Vc1*(1/2){ cos(ωc1*t+ωs*t) - cos(ωc1*t-ωs*t) }
    =(1/2)*(Vs*Vc1)*{ cos((ωc1+ωs)*t)) - cos((ωc1-ωs)*t)) } …(6)

Set cut frequency ωcut of LPF to be ωcut < ωc1, 
then cos((ωc1+ωs)*t) can be removed. (->*Note1)

Then get V4,
V4 = -(1/2)*(Vs*Vc1)*cos((ωc1-ωs)*t)) …(7)

...Output of “I” came out here.

Get V5 by multiplying V1 and V2.
V5= V1*V2 
= (Vs*sin(ωs*t)) * (-Vc1*cos(ωc1*t)) = -Vs*Vc1*sin(ωs*t) * cos(ωc1*t) 
    = -Vs*Vc1*(1/2){sin((ωc1*t+ωs*t) + sin((ωc1*t-ωs*t))}
    = -Vs*Vc1*(1/2){sin((ωc1+ωs)*t)  + sin((ωc1-ωs)*t))} …(8)

Set cut frequency ωcut of LPF to be ωcut < ωc1, 
then sin((ωc1+ωs)*t)  can be removed. (->*Note1’)

Then get V6
V6 = -(1/2)*Vs*Vc1*sin((ωc1-ωs)*t)) …(9)

... Output of “Q” came out here.

(C) IQ modulator

Input voltage of “I” and “Q” for “IQ modulator (C)” here are given 
as output of “IQ baseband feeder (B)”. 

OSC generates ωc2[rad*Hz] sin wave voltage.
Vosc2 = Vc2*sin(ωc2*t) …(10)
here ωc2 = 2πfc2 …(11) 
          fc2 [Hz] can be set to RF frequency of VFO such as 14MHz.

Shift Vosc2 signal (10) to be -90 deg. by “-90 deg. Shifter” and get V7. 

V7 = Vc2*sin(ωc2*t-π/2) = -Vc2*cos(ωc2*t) …(12)

Get V8 by multiplying V4 and V7.

V8= V4*V7 = (-(1/2)*(Vs*Vc1)*cos( (ωc1-ωs)*t)) ) * (-Vc2*cos(ωc2*t)) 
                  = (1/2)*Vs*Vc1*Vc2*cos( (ωc1-ωs)*t)) ) * cos(ωc2*t) …(13)


Get V9 by multiplying V6 and Vosc2.
V9 = V6*Vosc2 = (1/2)*Vs*Vc1*cos((ωc1-ωs)*t)) * Vc2*sin(ωc2*t)
    =  (1/2)*Vs*Vc1*Vc2*sin((ωc1-ωs)*t)) * sin(ωc2*t) …(14)

When “Adder” used before Vout,
Vout = V8+V9 =  (1/2)*Vs*Vc1*Vc2*cos((ωc1-ωs)*t)))*cos(ωc2*t)      
                        + (1/2)*Vs*Vc1*Vc2*sin((ωc1-ωs)*t)) * sin(ωc2*t)
= (1/2)*Vs*Vc1*Vc2*(cos((ωc1-ωs)*t)))*cos(ωc2*t) 
  + sin((ωc1-ωs)*t))*sin(ωc2*t)
= (1/2)*Vs*Vc1*Vc2*cos( (ωc1-ωs)*t - ωc2)*t) )
= (1/2)*Vs*Vc1*Vc2*cos( (ωc1-ωc2)-ωs)*t )  … here cos() is even function then 
= (1/2)*Vs*Vc1*Vc2*cos( (ωc2-ωc1)+ωs)*t ) …(15)

  … This (15) means USB generated as output signal Vout. 
     (15) is USB signal and Carrier ωc2-ωc1 signal voltage is removed here
      and receivers have to set tuning Carrier frequency to be ωc2-ωc1.
            -> (*Note 2)

When “Subtracter” used before Vout,

Vout = V9-V8 = (1/2)*Vs*Vc1*Vc2*sin((ωc1-ωs)*t)) *sin(ωc2*t)    
             - (1/2)*Vs*Vc1*Vc2*cos((ωc1-ωs)*t)) *cos(ωc2*t)
 = -(1/2)*Vs*Vc1*Vc2*{cos((ωc1-ωs)*t-ωc2)*t)-sin((ωc1-ωs)*t))*sin(ωc2*t) }  
 = -(1/2)*Vs*Vc1*Vc2*({cos( ((ωc1-ωs)+ωc2))*t )  
 = -(1/2)*Vs*Vc1*Vc2*({cos( ((ωc2+ωc1) -ωs))*t ) …(16)

 … This (16) means LSB generated as output signal Vout. 
    (16) is LSB signal and Carrier ωc2+ωc1 signal voltage is removed here
    and the receiver has to set running Carrier frequency to be ωc2+ωc1.
          -> (*Note 3)

“Weaver’s PSN method” by using “IQ baseband feeder” and “IQ modulator” 
is proved to generate SSB modulated signal ( USB or LSB).

-> (*Note1, Note1’) : 
DSB(Double sided band) is generated by the Multiplier 
on the “IQ feeder” and  LPF can remove USB band theoretically. 

However Characteristics of “Shape factor” and “signal group phase delay” 
on the IF signal band is very important to design the LPF.

Otherwise, unwanted signal may remain on the IF signal line V4 or V6. 
This subject is derived from the principle of the “Weaver’s PSN method” 
but DSP filter’s characteristics can solve this subject.
(Analog type of LPF or HPF may be difficult to solve it.)

HPF can be used instead of LPF on this method as the same principle.

-> (*Note 2)(*Note3) ; 
    SSB receivers have to their tuning frequency to become shifted  
    carrier by Vosc1. 
    This subject can solve easily by the tuning frequency adjustment  
    at SSB receivers side.

(C) Noboru, Ji1NZL, Jun.29, 2016

Rev.0.1 : 
Add missing variable "Vc1" in the equation (16) (mistypes) Jul.5, 2016
(No change of the theory itself. I'm sorry for it.)

Rev.0.1A: 
Changed variable ωc2 to ωc1 in the equation (16) (mistypes) Jul.6,2016
(No change of the theory itself. I'm sorry for it.)

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