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I came across this formulation of the probability that the achievable data rate is greater than the target data rate for the NOMA system.

$$\Pr\left[\frac{1}{2}\log_2\left(1+\text{SINR}\right)>R_{\text{target}}\right],$$

where $\text{SINR} = \frac{\alpha_2|h|^2\beta}{\alpha_1|h|^2\beta+1}$, $\beta=\frac{P}{N_0}$, $\alpha_1$ and $\alpha_2$ are power allocation coefficients, and $h \sim\mathcal{CN}(0,\sigma^2)$. $\mathcal{CN}(\cdot)$ denotes the complex Gaussian distribution.

Substituting $\text{SINR}$, we get

$$\Pr\left[\frac{1}{2}\log_2\left(1+\frac{\alpha_2|h|^2\beta}{\alpha_1|h|^2\beta+1}\right)>R_{\text{target}}\right].$$

And after some manipulations, we obtain

$$\Pr\left[\left(\alpha_2-\left(2^{2R_{\text{target}}}-1\right)\alpha_1\right)|h|^2\beta>2^{2R_{\text{target}}}-1\right].$$

Let $\gamma =2^{2R_{\text{target}}}-1$. Then

$$\Pr\left[\left(\alpha_2-\gamma \alpha_1\right)|h|^2\beta>\gamma\right].$$

I have understood the steps until here. The final expression obtained is

$$\begin{cases} e^{\frac{\gamma}{\alpha_2-\gamma\alpha_1}\frac{1}{\beta\sigma^2}}, & \text{for }\gamma<\frac{a_2}{a_1}\\ 0, & \text{for } \gamma>\frac{a_2}{a_1} \end{cases} $$

I cannot figure out how the final expression was derived. Also, why do we have two cases of the expression, and is $\gamma$ the $\text{SINR}$ in linear form?

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I cannot figure out how the final expression was derived.

$h \sim\mathcal{CN}(0,\sigma^2)$ means $h=h_r+jh_i$ with $h_r$ and $h_i$ are iid $\mathcal{N}(0,\sigma^2/2)$ and $j^2=-1$.

Then, $\frac{\sqrt{2}}{\sigma}h_r$ and $\frac{\sqrt{2}}{\sigma}h_i$ are iid $\mathcal{N}(0,1)$ and, therefore, $z=h_r^2+h_i^2=\frac{2}{\sigma^2} |h|^2$ is a chi squared distributed random variable with $k=2$ degrees of freedom.

As the CDF for $k=2$ is $F(z;k=2)=1-e^{-z/2}$, the last expression is the complementary CDF of $z=\frac{\gamma}{\alpha_2-\gamma\alpha_1}\frac{2}{\beta\sigma^2}$.

Also, why do we have two cases of the expression,

We have to distinguish the two cases because if $\alpha_2-\gamma\alpha_1 < 0$, $\Pr\left[|h|^2 \geq 0 > \textrm{a negative value}\right] = 1$.

and is $\gamma$ the SINR in linear form?

This is your definition $\gamma =2^{2R_{\text{target}}}-1$ for a given $R_{\text{target}}$. It is what it is defined.

You can interpret it in many ways. For example, if you see it as $R_{\text{target}} = \frac{1}{2}\log_2\left(1+\gamma\right)$, then $\gamma$ is the linear SNR of an AWGN channel that has the (Shannon discrete model) capacity that equals to $R_{\text{target}}$ (note that this is SNR, not SINR as there is not interference in this model).

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  • $\begingroup$ This was super helpful. I still have a small question. Where did the $2$ in $\frac{2|h|^2}{\sigma^2}$ come from? My understanding is that $z$ must be equal to $\frac{|h|^2}{\sigma^2}$ $\endgroup$
    – nashynash
    Oct 29, 2021 at 12:15
  • $\begingroup$ @nashynash I have updated my answer. If you are satisfied with the answer, please mark it "accepted" so that the question can be closed. $\endgroup$
    – AlexTP
    Oct 29, 2021 at 12:23
  • $\begingroup$ Thank you very much once again! Truly appreciate it. $\endgroup$
    – nashynash
    Oct 29, 2021 at 12:25
  • $\begingroup$ By the way, referring to your last line, I think it is the $\text{SINR}$ as the $\alpha_1|h|^2\beta$ term in the denominator of the aforementioned definition of the $\text{SINR}$ is the superpositioned signal of another user. This is because, in NOMA, the base station superimposes the data (signal) of multiple users in the power domain but uses the same time and frequency resource. In this case, the data of two users were superimposed. Maybe, I wasn't clear on that. $\endgroup$
    – nashynash
    Oct 30, 2021 at 5:21
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    $\begingroup$ @nashynash that phrase refers to $\gamma =2^{2R_{\text{target}}}-1$. Specifically, given a $R_{\text{target}}$, you look at all AWGN channels for the one that has capacity $C=R_{\text{target}}$, and in that specific AWGN channel, the SNR equals to $\gamma$, no interference. Whether this interpretation makes sense, whether this $\gamma$ has the same nature as your SINR, whether $\gamma$ is comparable to your SINR, I have no idea. But you must be careful when you conclude something with information-theoretical expressions. $\endgroup$
    – AlexTP
    Oct 30, 2021 at 7:09

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