Analytical Expressions Used in bersync Function and Bit Error Rate Analysis App

Analytical Expressions Used in `bersync` Function and Bit Error Rate Analysis App

This section covers the main analytical expressions used in the bersync function and the Bit Error Rate Analysis app.

Timing Synchronization Error

To compute the BER for a communications system with a timing synchronization error, the bersync function uses this formula from [13]:

$\frac{1}{4 π σ} \int_{- \infty}^{\infty} \exp (- \frac{ξ^{2}}{2 σ^{2}}) \int_{\sqrt{2 R} (1 - 2 | ξ |)}^{\infty} \exp (- \frac{x^{2}}{2}) d x d ξ + \frac{1}{2 \sqrt{2 π}} \int_{\sqrt{2 R}}^{\infty} \exp (- \frac{x^{2}}{2}) d x$

where σ is the timing error, and R is the linear E_b/N₀ value.

Carrier Synchronization Error

To compute the BER for a communications system with a carrier synchronization error, the bersync function uses this formula from [13]:

$\frac{1}{π σ} \int_{0}^{\infty} \exp (- \frac{ϕ^{2}}{2 σ^{2}}) \int_{\sqrt{2 R} \cos ϕ}^{\infty} \exp (- \frac{y^{2}}{2}) d y d ϕ$

where σ is the phase error R is the linear E_b/N₀ value.