# "Best" energy efficiency via capacity of uncoded BPSK AWGN

I have numerically approximated the capacity of an uncoded, binary uniform input, real AWGN BPSK channel versus the SNR using mutual information and entropy, via Gauss-hermite quadrature. My task is to examine the capacity plotted versus $$\frac{E_b}{N_0}$$, and answer three questions.

1. "What do you notice?" I guess that means that the curve saturates at the limits, and therefore there is no achievable information rate greater than one, and at a certain SNR, there is no benefit to further increasing signal power.
2. Whether information can be transmitted "arbitrarily efficiently", which I assume means with arbitrarily low $$\frac{E_b}{N_0}$$. It seems obvious that no, one can't, because I can see that the capacity tends to zero for $$\frac{E_b}{N_0} \rightarrow 0$$.
3. How one can achieve "best" energy efficiency.

The third question especially confuses me. Wouldn't this actually be a "soft" trade-off between power consumption and capacity? What would be a good target value for the capacity then? Or is there actually a way to determine an optimal $$\frac{E_b}{N_0}$$ value?

I am quite new to wireless comms, so apologies if this may appear trivial.

The plot in question:

• Hello. What do you mean by "arbitrarily low error probability greater than one" as probability is lower than or equal to 1 by usual definitions? For other questions, first define "best energy efficiency". Maybe I don't understand what you want to do. If I were you, I would use the classic energy efficiency reasoning : first fix the rate $R$ I want to transmit, next choose the lowest $Eb/N0$ that the channel capacity $C(Eb/N0)=R$, then the power derived from such $Eb/N0$ is the efficient scheme. Commented May 25 at 8:34
• Such scheme can be achieved theoretically by random coding, can be approached practically by codes like turbocodes, LDPC, and polar codes. Commented May 25 at 8:37
• "arbitrarily low error probability greater than one" is a typo as I wrote the question a bit late at night. I edited it to fix the mistake. "first define "best energy efficiency"": This is the term my university instructors used in the task. I don't know what they mean either and couldn't find any clear definition online. Commented May 25 at 13:43