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

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!

• Note that the "So (the received signal can…)" in your first paragraph is not an implication in the logical sense: you can add up any two signals, no matter whether they spectrally overlap or not. Commented Oct 5, 2022 at 9:01
• It's really not clear what you're sampling, at which rate. I understand your sampling rate is $f_s$, but that would mean that the representable frequency range is $[-f_s/2;+f_s/2]$, and not centered around $f_1$. Did you forget to mention a mixer? What's the bandwidth of your anti-aliasing filter? Commented Oct 5, 2022 at 9:03
• @MarcusMüller OK, let's consider the baseband signals by shifting the frequency of the signals by $f_1$ so that the the main lobe of the $x(t)$ is centered around 0, while that of $i(t)$ around $f_2-f_1$ assuming $f_2>f_1$. What do you mean exactly when you say "It's really not clear what you're sampling, at which rate"? At the front-end receiver I have a received signal $r(t)$ and I sample this signal at rate $f_s$. I assume that the intended receiver of $x(t)$ is not the same at the intended receiver of $i(t)$. Commented Oct 5, 2022 at 10:06
• can you please do this clarification in the question text? what is not clear is exactly what I wrote, at which point you're sampling what. You're just claiming you're sampling something. Commented Oct 5, 2022 at 10:07
• @MarcusMüller You are right, when I read it again it wasn't clear. I edited the question Commented Oct 5, 2022 at 10:11

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 $$x_{RMS} = \sqrt{}$$ 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.

• I want to get the PSD of the received signal immediately after analog-to-digital converter. My question is how can I generate the signals in MATLAB when they are not interfering, i.e., the main lobe of $i(t)$ doen't overlap with the main lobe of the $x(t)$. Commented Oct 6, 2022 at 7:39
• why don't you just create the (bandlimited) base band signals and modulate them up ? Commented Oct 6, 2022 at 11:25
• My issue is with the scaling. What should I scale each signal with to be accurate in the presence of a normalized noise? Commented Oct 6, 2022 at 11:27
• I thought I answered that: just normalize all signals to unity power and then scale to your desired target SNR (in whatever way you define this). It's not clear to me how exactly you define "normalized noise" in this context since you are looking at a much larger bandwidth then you would need for the signal of interest. So this depends on what exactly you want to do with the results Commented Oct 6, 2022 at 13:52
• I would like to detect if there is a signal in the PSD spectrum of the received signal outside the main lobe of $x(t)$. You're exactly right, the problem is with noise. If they signals interfered, I can write the received signal in baseband (relative to the carrier frequency $f_1$ as $$r(t)=\sqrt{\mathrm{SNR}}\left[x(t)+\sqrt{\mathrm{SIR}}i(t)\right]+z(t)$$ where all the signals $x(t)$, $i(t)$ and $z(t)$ are normalized to unit power. What is not clear to me, is how to write this equation if $i(t)$ doesn't interfere with $x(t)$. Do I need to split the noise signal, or I can keep it as it is? Commented Oct 7, 2022 at 8:07