I don't understand which signal I should use as a carrier in my excersise

Question

$\DeclareMathOperator{\sinc}{sinc}$ Given 3 signals

$x_1(t) = \sinc(t) , x_2(t) = 10\cos(2\pi f_2t)$ and $x_3(t) = 5\cos(2\pi f_1t)$,

where $f_2 = 100\text{ kHz}$ and $f_1 = 1\text{ kHz}$.

I must create the general form of the simple amplitude modulation AM ($S(t)= A_c\left[1+k_am(t)\right]\cos(2\pi f_ct)$) , the DSB-SC modulation ( $S(t)=A_cm(t)\cos(2\pi f_ct)$ ), the PM phase modulation and the FM frequency modulation, but I must use the sum of the 2 signals to create the information signal ( $m(t)$ )and the third must be the carrier.

What I can't understand is which one must be the carrier in every single case, the one with the highest frequency in our example $x_2(t)$ or the one with lowest?

Answer 1

The $\mathrm{sinc}$ function has the following Fourier Transform also shown below:

Here in your question the amplitudes do not matter much.

Consider the DSB-SC case:

We obviously cannot choose the $\mathrm{sinc}$ as our carrier since it does not have any modulating properties. We want something that will take our baseband (not necessarily, more appropriately lowpass) signal and put it at a higher available frequency band such that it can be easily transmitted without any error/loss of data.

To do this we could perhaps use $x_3(t) = 5\cos(2\pi f_1 t)$

This signal has the Fourier Transform as below:

$$5\mathrm{cos\:\omega_{0} t\overset{FT}{\leftrightarrow}\frac 52[\delta(f -f_{0})+\delta(f+f_{0})]} \quad \quad \tag{1}$$

where $\omega_0 =2\pi f_0$. Now if our $m(t) = \mathrm{sinc}(t) + 10\cos(2\pi f_2 t)$

then our resulting $S(t) = A_c(\mathrm{sinc}(t) + 10\cos(2\pi f_2 t))\cdot 5\cos(2\pi f_1 t)$ which can be written as:

$$ S(t) = \underbrace{25A_c(\mathrm{sinc}(t)\cos(2\pi f_1 t))}_{p_1(t)} + \underbrace{50A_c(\cos(2\pi f_2 t) \cos(2\pi f_1 t))}_{p_2(t)}$$

Remember that multiplication in time-domain is convolution in Fourier domain and that convolution with a shifted impulse merely shifts our signal corresponding to the shift of the impulse. Keeping these two things in mind, consider $p_1(t)$. We have a $\mathrm{rect}(\omega)$ (rectangle in the figure above) convolved with the two shifted impulses from $(1)$ where $f_0 = f_1 =1\texttt{kHz}$. This convolution will essentially place the $\mathrm{rect}$ at $\pm f_1$ in the frequency domain which looks something like this:

As you can see we have successfully placed a part of our message signal at a higher frequency. However, the task is not done. We still have $p_2(t)$ to analyze for which you can utilize the following identity:

$$ \cos(x)\cdot \cos(y) = 0.5 \cos(x + y) + 0.5\cos(x - y)$$

and then you can use $(1)$ to find the FT. If your analysis leads to overlapping between the Fourier Transforms of the different parts of your message signal then you can try a different carrier signal. This is how you are supposed to perform this analysis.

As per DSP.SE's rules it is discouraged to outright answer HW questions but I hope I have provided you with enough intuition that you can solve the rest of the exercise on your own.