The principle of frequency conversion by the Analog multiplier (アナログ乗算器による周波数変換の原理)

The principle of frequency conversion by the Analog multiplier

Following description is how the frequency conversion occurs by the Analog multiplier based on Math.

sin(A±B) = sinA*cosB ± cosA*sinB …(1)

cos(A+B) = cosA*cosB - sinA*sinB …(2)’

cos(A-B)  = cosA*cosB + sinA*sinB …(2)’’

(1), (2)’ and (2)’’ are known as formulas of trìgonométric(al) functions.

From (1), re-write (1) to,

sin(A + B) = sinA*cosB + cosA*sinB …(1)’

sin(A - B)  = sinA*cosB -  cosA*sinB …(1)’’

(1)’ + (1)’’ then,

sin(A+B)+ sin(A-B) = 2*sinA*cosB …(3)

From (3) then,

sinA*cosB = (1/2)*(sin(A+B)+ sin(A-B)) …(4)

Define A and B as follows,  

A ≡ ωc*t, B ≡ ωs*t …(5)

and re-write ωc and ωs to

ωc=2*π*fc, ωs=2*π*fs  …(6)

then (4) becomes

sin(ωc*t)*cos(ωs*t) = (1/2)*(sin(ωc*t+ωs*t)+ sin(ωc*t-ωs*t))

                             =(1/2)*( sin((ωc+ωs)*t)+ sin((ωc-ωs)*t) ) …(6)

Define amplitude of sin to Vc and cos to Vs here.

Then (6)*Vc*Vs becomes

{Vc*sin(ωc*t)} * {Vs*cos(ωs*t)} = (Vc*Vs/2)*(sin((ωc+ωs)*t)+ sin((ωc-ωs)*t)) …(7)

Re-write (7) by definition of angular frequency by (6).

{Vc*sin(2*π*fc*t)} * {Vs*cos(2*π*fs*t)}

= (Vc*Vs/2)*{sin(2*π*(fc+fs)*t)+ sin(2*π*(fc-fs)*t)} …(8)

This means multiplying  sine wave  (frequency=fc, amplitude=Vc)

and cos wave (frequency=fs, amplitude=Vs)

generates

the frequency (fc+fs) sine wave, (fc-fs) sin wave,

and their amplitudes become (Vc*Vs/2).

Add (2)’ and (2)’’ then get

cos(A+B) + cos(A-B) = 2*cosA*cosB

then,

cosA*cosB = (1/2)*(cos(A+B)+cos(A-B)) … (9)

from the definition of (5), (9) becomes

cos(ωc*t)*cos(ωs*t) = (1/2)*(cos(ωc*t+ωs*t)+cos(ωc*t-ωs*t))

                             = (1/2)*( cos((ωc+ωs)*t) +cos((ωc-ωs)*t)) ) …(10)

Apply (9)*Vc*Vs, then (9) becomes

{Vc*cos(ωc*t)}*{Vs*cos(ωs*t)} = (Vc*Vs/2)*( cos((ωc+ωs)*t) +cos((ωc-ωs)*t)) ) …(11)

Re-write (11) by definition of angular frequency by (6).

{Vc*cos(2*π*fc*t)}*{Vs*cos(2*π*fs*t)}

= (Vc*Vs/2)*( cos(2*π*(fc+fs)*t)) +cos(2*π*(fc-fs)*t)) ) …(12)

This means multiplying  cos wave  (frequency=ωc, amplitude=Vc)

and cos wave (frequency=fs, amplitude=Vs)

generates

the frequency (fc+fs) cos wave, (fc-fs) cos wave,

and their amplitudes become (Vc*Vs/2).

Reference and related documents :

[1] Analog multiplier / RF mixer reference

乗算器（ミキサー、周波数変換回路）　参考文献

http://ji1nzl-official.blogspot.jp/2015/10/blog-post_14.html

[2] Deference between heterodyne phenominon and beat phenominon

日本国内における (周波数変換/ヘテロダイン現象）と (ビート現象) 教育内容の誤り

http://ji1nzl-official.blogspot.jp/2016/06/blog-post.html

(C) Noboru, Ji1NZL Jun.21, 2016

Rev. 0.1; Corrected some mistypes on the Figure. Jul. 5, 2016
Rev. 0.2; Corrected some mistypes on the Figure. Sep.19, 2016
Rev. 1.0; Added related documents and reference. Sep.26, 2016

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