# Equation for square law circuit modulator

For DSBFC AM (double side band full carrier amplitude modulation ) the message signal $$\m(t)\$$ must be multiplied by carrier maybe $$A_c\cos(\omega_c(t))$$ (For modulation)

This modulation is done in reality by using the non-linear characteristics of the diode, where

$$i(t)=av + bv^2$$

$$\v =\$$ applied voltage to diode

Here we apply

$$v =V_c(t)+V_m(t)$$ $$i(t)=a(V_c+_Vm)+b(V_c+V_m)^2$$ $$i(t)= aV_c +bV_m +bV_c^2+bV_m^2 +2V_cV_m$$

Here $$V_c=A_c\cos(\omega_c(t))$$

Thus $$i(t)=aA_c\cos(\omega_c(t))+bm(t)+b(A_c\cos(\omega_c(t)))^2 +bm(t)^2 +2m(t)A_c\cos(\omega_c(t))$$

After simplification

$$i(t)=aA_c\cos(\omega_c(t))+bm(t)+\frac{bA_c}{2} +\frac{bA_c\cos(2\omega_c(t))}{2} +bm(t)^2 +2m(t)A_c\cos(\omega_c(t))$$

In frequency domain I can clearly understand following components

$$f(\text{message frequency}),f_c,f_c+f,f_c-f,2f_c$$

but my book tells me there are additional components at $$\2f_m\$$.

Can someone help me understand where it is?

• $2f_m$ would be from the message frequency term $(m(t))^2$. e.g. $\cos^2(2\pi\ f_m t)$.
– AJN
Commented Oct 17, 2020 at 13:24
• What makes us assume my message signal is cos or sin. Couldn't it be anything maybe my voice or music? @AJN Commented Oct 17, 2020 at 13:26
• The book I am referring is electronic communication by frenkel. It actually ignores the higher order because they become very small.It just considered 2 @rpm2718 Commented Oct 17, 2020 at 13:37
• @NewtonNadar Good question. Multiplication of a signal with itself $(m(t))^2$ in time domain is represented in frequency domain as a convolution of the signal spectrum with itself $M(s) \circledast M(s)$. When a signal is convolved with itself, it becomes twice as wide; i.e., frequency content has values up to twice the the original value. Wikipedia. This is what books mentions as $2f_m$; even if signal was non sinusoidal.
– AJN
Commented Oct 17, 2020 at 13:38
• Thank you. That's the answer I needed(Does it mean convolution becomes multiplication in Frequency domain and multiplication becomes convolution in Frequency Domain?) .@AJN Commented Oct 17, 2020 at 13:40

This frequency comes from the message signal; specifically $$\(m_{(t)})^2\$$. Multiplication of a signal with another (or itself) in time domain is represented in frequency domain as a convolution. So the corresponding signal in frequency domain is
$$M(s) \circledast M(s)$$
If a signal with frequency contents from 0 to $$\f_{max}\$$ is convolved with itself, the resulting spectrum will have frequency content from 0 to $$\2f_{max}\$$.