Generating signal in a different frequency band than another signal but transmitted at the same time

Question

I have a signal $x(t)$ of bandwidth $W_1$ transmitted over carrier frequency $f_1$, and another signal $i(t)$ of bandwidth $W_2$ transmitted over carrier frequency $f_2$. The main lobe of both signals' spectrums don't overlap. They are transmitted on the same time. So the passband received signal can be expressed as

$$r(t)=x(t)+i(t)+z(t)$$

and the corresponding baseband signal is (relative to the carrier frequency $f_1$)

$$\tilde{r}(t)=\tilde{x}(t)+\tilde{i}(t)e^{j2\pi(f_2-f_1)t}+\tilde{z}(t)$$

where $\sim$ over a signal means the corresponding baseband signal, and $z(t)$ is additive white Gaussian noise process of zero mean and power spectral density $N_0$.

Although $\tilde{i}(t)e^{j2\pi(f_2-f_1)t}$ doesn't actually interfere with $\tilde{x}(t)$, its spectrum is assumed to appear in the power spectrum density of $\tilde{r}(t)$ at the output of the analog-to-digital converter by sampling the received signal $\tilde{r}(t)$ at a sampling rate $f_s$, where the frequency range of the double-sided PSD is $[-\frac{f_s}{2},\frac{fs}{2}]$.

I want to model the above system in MATLAB, however I have doubts on how to generate $\tilde{i}(t)$. If $\tilde{i}(t)$ interfered with $\tilde{x}(t)$, I would define the signal-to-noise ratio (SNR) and signal-to-interference ratio (SIR) and the received signal can be written as

$$y(t)=\sqrt{\mathtt{SNR}}\left[x(t)+\sqrt{\mathtt{SIR}^{-1}}i(t)\right]+n(t)$$

where $n(t)$ is additive white Gaussian noise process of zero mean and power unity.

In my case, can I write the received signal as

$$y(t)=\left[\sqrt{\mathtt{SNR}_x}x(t)+n_x(t)\right]+\left[\sqrt{\mathtt{SNR}_i}i(t)+n_i(t)\right]$$

where $\mathtt{SNR}_x$ and $\mathtt{SNR}_i$ are the SNR of the signals $x(t)$ and $i(t)$, respectively, and $n_x(t)$ and $n_i(t)$ are additive white Gaussian processes of zero mean and normalized power?

EDIT

Can I write the passband received signal as

$$r(t)=\Re\left\{\left[x(t)+n_x(t)\right]e^{j2\pi f_1t}+\left[i(t)+n_i(t)\right]e^{j2\pi f_2t}\right\}$$

such that when I find the baseband signal with respect to $f_1$ I get

$$y(t)=x(t)+n_x(t)+e^{j2\pi (f_2-f_1)t}\left[i(t)+n_i(t)\right]$$

assuming $n_x(t)$ has a normalized [unit] power over bandwidth $W_1$, while $n_i(t)$ has a normalized power over bandwidth $W_2$?

In practice, both signal will experience the same additive noise sample at the front-end receiver, that's where my confusion is!

Answer 1

I still don't fully understand the question but I'll take a swing at it: Let's start with the initial definition

$$r(t) = x(t) + i(t) + z(t) \tag{1}$$

If we want to write this in a power normalized form we get

$$r(t) = a\frac{x(t)}{x_{RMS}} + b\frac{i(t)}{i_{RMS}} + c\frac{z(t)}{z_{RMS}} $$

where $x_{RMS} = \sqrt{<x^2(t)>}$ and $<>$ the mean value operator.

$a,b,c$ are gain values that you can adjust as needed and expressed in terms of SNR, SIR, etc as you see fit. This will depend on how exactly you define these quantities. The power of white noise is a function of bandwidth which is in turn a function of the sample rate. Personally I would define SNR only over the bandwidth of the signal, since any reasonable receiver would use a band pass to filter out the rest. But that depends on the goals of your model.