Bit Errors

Abstract

In the example of IQ modulation: We can see on the left graph that even with added AGWN noise, we can clearly classify every data point towards a certain value, however on the right side we have so much noise that we get data points that are on or over the boundary of certain bits, giving us BIT ERRORS, meaning that data transmitted gets lost, or worst incorrectly realised.

Bit Error Rate

We can quantify the rate of bit errors in a system without testing with the following formula

If provided with noise power ($n_0$) and energy per bit ($e_b$) the formula is as follows

$$\begin{array}{rl}\text{erfc(z)}=& 1-\frac{2}{\sqrt{\pi }}{\int }_0^{z}e-x^{2}dx\\ ⟹\text{BER}=& 0.5\text{erfc}(\sqrt{\frac{e_b}{n_0}})\end{array}$$

Or if we are provided with an SNR, $B=\text{Bandwidth}$ , $v_s=\text{Baud Rate}$, $m=\text{Modulation Depth}$

$$\text{BER}=0.5\text{erfc}(\text{SNR}\ast \frac{B}{v_s{\log }_{2}m})$$ $$\begin{array}{rl}\text{erfc(z)}=& 1-\frac{2}{\sqrt{\pi }}{\int }_0^{z}e-x^{2}dx\\ ⟹\text{BER}& =0.5\text{erfc}(\text{SNR}\ast \frac{B}{v_s{\log }_{2}m})\end{array}$$ $$\begin{array}{rl}\text{erfc(z)}=& 1-\frac{2}{\sqrt{\pi }}{\int }_0^{z}e-x^{2}dx\\ ⟹\text{BER}& =0.5\text{erfc}(\text{SNR}\ast \frac{B}{v_s{\log }_{2}m})\\ \text{BER}& =0.5\text{erfc}(\frac{\text{SNR}}{v_s{\log }_{2}8})\end{array}$$ $$\begin{array}{rl}{\log }_2(10^{\frac{SNR}{10}}+1)& =3\text{ ( }{2}^3=8\text{ )}\\ 10^{\frac{SNR}{10}}+1& =2^3\\ \frac{SNR}{10}+lo{g}_{10}1& =lo{g}_{10}8\\ SNR& \approx 10{\log }_{10}8\\ SNR& \approx 9dB\end{array}$$