## 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]:

`$\frac{1}{4\pi \sigma }{\int }_{-\infty }^{\infty }\mathrm{exp}\left(-\frac{{\xi }^{2}}{2{\sigma }^{2}}\right){\int }_{\sqrt{2R}\left(1-2|\xi |\right)}^{\infty }\mathrm{exp}\left(-\frac{{x}^{2}}{2}\right)dxd\xi +\frac{1}{2\sqrt{2\pi }}{\int }_{\sqrt{2R}}^{\infty }\mathrm{exp}\left(-\frac{{x}^{2}}{2}\right)dx$`

where σ is the timing error, and R is the linear Eb/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]:

`$\frac{1}{\pi \sigma }{\int }_{0}^{\infty }\mathrm{exp}\left(-\frac{{\varphi }^{2}}{2{\sigma }^{2}}\right){\int }_{\sqrt{2R}\mathrm{cos}\varphi }^{\infty }\mathrm{exp}\left(-\frac{{y}^{2}}{2}\right)dyd\varphi$`

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