What is the unit-power information signal?

Question

I am learning signal processing for my subject and have some problems with the unit-power information signal. I refer to this paper: https://arxiv.org/pdf/1906.03949.pdf

In this paper, they formulate the received signal as: $y=h_{sd} \sqrt{p}s+n$ with $h_{sd}$ is the channel, $p$ is the transmit power, $s$ is the unit-power information signal, and $n$ is the noise.

What is the unit-power information signal? As I understand, the unit-power signal is the signal that has the power of 1. However, how can I formulate it (e.g., using Matlab)? For example, with BPSK signal, the signal is formulated as:

$s_1(t) = A_c \cos(2\pi f_ct)$ and $s_0(t)=A_c \cos(2\pi f_ct + \pi)$ for bit 1 and 0, respectively.

With a given transmit power $p$, how can I turn $s_1(t)$ and $s_0(t)$ into unit-power signal?

Is there any suggestion or reference?

Answer 1

The power of a deterministic signal $s(t)$ is given by

$$P_s=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Ts^2(t)dt\tag{1}$$

A unit power signal is obtained by normalizing $s(t)$ by the root of its power:

$$\hat{s}(t)=\frac{s(t)}{\sqrt{P_s}}\tag{2}$$

In the case of $s(t)=A\cos(2\pi f_0t)$ you obtain from $(1)$

$$P_s=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^TA^2\cos^2(2\pi f_0t)dt=\frac{A^2}{2}\tag{3}$$

So the corresponding unit power signal is

$$\hat{s}(t)=\frac{\sqrt{2}}{A}s(t)=\sqrt{2}\cos(2\pi f_0t)\tag{4}$$