## Analytical Expressions and Notations Used in BER Analysis

This topic covers the analytical expressions and notations for the theoretical analysis used in the BER functions (`berawgn`, `bercoding`, `berconfint`, `berfading``berfit`, `bersync`), Bit Error Rate Analysis app, and Bit Error Rate Analysis Techniques topic.

### Common Notation

This table defines the notations used in the analytical expressions in this topic.

Description Notation
Size of modulation constellation

M

Number of bits per symbol

`$k={\mathrm{log}}_{2}M$`

Energy per bit-to-noise power-spectral-density ratio

`$\frac{{E}_{b}}{{N}_{0}}$`

Energy per symbol-to-noise power-spectral-density ratio

`$\frac{{E}_{s}}{{N}_{0}}=k\frac{{E}_{b}}{{N}_{0}}$`

Bit error rate (BER)

`${P}_{b}$`

Symbol error rate (SER)

`${P}_{s}$`

Real part

`$\mathrm{Re}\left[\cdot \right]$`

Floor, largest integer smaller than the value contained in braces

`$⌊\cdot ⌋$`

This table describes the terms used for mathematical expressions in this topic.

Function Mathematical Expression
Q function

`$Q\left(x\right)=\frac{1}{\sqrt{2\pi }}\underset{x}{\overset{\infty }{\int }}\mathrm{exp}\left(-{t}^{2}/2\right)dt$`

Marcum Q function

`$Q\left(a,b\right)=\underset{b}{\overset{\infty }{\int }}t\mathrm{exp}\left(-\frac{{t}^{2}+{a}^{2}}{2}\right){I}_{0}\left(at\right)dt$`

Modified Bessel function of the first kind of order $\nu$

`${I}_{\nu }\left(z\right)=\sum _{k=0}^{\infty }\frac{{\left(z/2\right)}^{\upsilon +2k}}{k!\Gamma \left(\nu +k+1\right)}$`

where

`$\Gamma \left(x\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-t}{t}^{x-1}dt$`

is the gamma function.

Confluent hypergeometric function

`${}_{1}F{}_{1}\left(a,c;x\right)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_{k}}{{\left(c\right)}_{k}}\frac{{x}^{k}}{k!}$`

where the Pochhammer symbol, ${\left(\lambda \right)}_{k}$, is defined as ${\left(\lambda \right)}_{0}=1$, ${\left(\lambda \right)}_{k}=\lambda \left(\lambda +1\right)\left(\lambda +2\right)\cdots \left(\lambda +k-1\right)$.

This table defines acronyms for common digital modulation schemes.

Acronym Definition
M-PSKM-ary phase-shift keying
DE-M-PSKDifferentially encoded M-ary phase-shift keying
BPSKBinary phase-shift keying
DE-BPSKDifferentially encoded binary phase-shift keying
QPSKQuaternary phase-shift keying
DE-OQPSKDifferentially encoded offset quadrature phase-shift keying
M-DPSKM-ary differential phase-shift keying
M-PAMM-ary pulse amplitude modulation