Checking for interference equation - two wave mixing

[watertreader]

Homework Statement

Not too sure if my problem falls under this category. Hope I have done all formatting correctly

Basically I am seeking some confirmation on my maths derivation and would like to enquire why there is discrepancies when we expressed the result in Fourier Spectrum.

Homework Equations

The fundamental equation I am working on is
$$\left|Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right|^{2}$$

By expanding out the equation, I would get
$$\left|Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right|^{2}$$
$$=\left(Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right)\left(Z_{1}+Z_{2}exp\left(-jw_{c}t\right)\right)^{*}$$

$$=\left|Z_{1}\right|^{2}+ \left|Z_{2}\right|^{2}+Z_{1}Z_{2}^{*}exp\left(jw_{c}t\right) +Z_{2}Z_{1}^{*}exp\left(-jw_{c}t\right)$$

Using $$AB^{*}+A^{*}B = 2\Re\left(AB^{*}\right)$$,
$$=\left|Z_{1}\right|^{2}+\left|Z_{2}\right|^{2}+2Re\left(Z_{1}Z_{2}^{*}exp\left(jw_{c}t\right)\right)$$

Let's Call this equation 1,

Expanding out, we have
$$=\left|Z_{1}\right|^{2}+ \left|Z_{2}\right|^{2}+2\left[Re\left(Z_{1}Z_{2}^{*}\right)cos\left(w_{c}t\right)-Im\left(Z_{1}Z_{2}^{*}\right)sin\left(w_{c}t\right)\right]$$

For the 3rd Equation, we use another property,
$$Acos\left(w_{c}t\right)-Bsin\left(w_{c}t\right)$$
$$= \sqrt{A^{2}+B^{2}}cos\left(w_{c}t + \theta\right)$$

we have the following 3rd equation
$$=\left|Z_{1}\right|^{2}+ \left|Z_{2}\right|^{2}+2Re\left|Z_{1}Z_{2}^{*}\right|cos\left(w_{c}t+\theta\right)$$

Hope my derivation is correct

The Attempt at a Solution

However, if we convert them by applying Fourier Transform, they seem to be unequal

$$F\left[ Eq1\right] = 2F\left(Re\left( Z_{1}Z_{2}^{*}\right)\right)\otimes \delta\left(f+f_{c}\right)$$

$$F\left[ Eq2\right] = \frac{2}{2}F\left(Re\left( Z_{1}Z_{2}^{*}\right)\right)\otimes \left[\delta\left(f+f_{c}\right)+\delta\left(f-f_{c}\right) \right] +\frac{2}{j}F\left(Im\left( Z_{1}Z_{2}^{*}\right)\right)\otimes \left[\delta\left(f+f_{c}\right)-\delta\left(f-f_{c}\right) \right]$$

$$F\left[ Eq3\right] = 2F\left(\left| Z_{1}Z_{2}^{*}\right|\right)\otimes \left[\delta\left(f+f_{c}+\theta\right) + \delta\left(f-f_{c}+\theta\right)\right]$$

 

[Valkarie]

I am not sure why the result of the Fourier Transform is not consistent. Can someone please help me with this?