BPSK modulation
In BPSK (Binary Phase Shift Keying), each bit \((u_j \in \{0,1\}\) is mapped to a symbol.
\[
r_j = 1 - 2\,u_j
\quad\Longrightarrow\quad
r_j =
\begin{cases}
+1, & \text{if } u_j = 0 \\
-1, & \text{if } u_j = 1
\end{cases}
\]
The entire BPSK signal is written as:
\[
s(t)
= \sum_{j=1}^{N}
r_j \,\cos\bigl(2\pi f_c\,t\bigr)\,
\Pi\!\Bigl(\frac{t - jT_r}{T_r}\Bigr),
\]
where the rectangular function \(\Pi(z)\) is defined as:
\[
\Pi(z) =
\begin{cases}
1, & 0 \le z < 1,\\
0, & \text{otherwise}.
\end{cases}
\]
Symbols info
- \(\vec{u}\) – input bit sequence, where \(u_j \in \{0,1\}\)
- \(\vec{r}\) – BPSK symbols, where \(r_j \in \{+1,-1\}\)
- \(f_c\) – carrier frequency
- \(T_r\) – bit duration
- \(N\) – number of bits