Mathematical Expressions and Notations Used in BER Analysis

This topic defines and describes the notations and mathematical expressions used in the analytical expressions and the theoretical analysis used by the BER functions (berawgn, bercoding, berconfint, berfadingberfit, 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	$$k={\log }_{2}M$$
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	$$\mathrm{Re}[\cdot ]$$
Floor, largest integer smaller than the value contained in braces	$$\lfloor \cdot \rfloor$$

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

Function	Mathematical Expression
Q function	$$Q(x)=\frac{1}{\sqrt{2\pi }}{\displaystyle \underset{x}{\overset{\infty }{\int }}\exp (-t^2/2)dt}$$
Marcum Q function	$$Q(a,b)={\displaystyle \underset{b}{\overset{\infty }{\int }}t\exp \left(-\frac{t^2+a^2}{2}\right)I_0(at)dt}$$
Modified Bessel function of the first kind of order $$\nu$$	$$I_{\nu }(z)={\displaystyle \sum _{k=0}^{\infty }\frac{{\left(z/2\right)}^{\upsilon +2k}}{k!\Gamma (\nu +k+1)}}$$ where $$\Gamma (x)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-t}{t}^{x-1}dt}$$ is the gamma function.
Confluent hypergeometric function	$${}_{1}F{}_1(a,c;x)={\displaystyle \sum _{k=0}^{\infty }\frac{{(a)}_k}{{(c)}_k}\frac{x^k}{k!}}$$ where the Pochhammer symbol, $${(\lambda )}_k$$, is defined as $${(\lambda )}_0=1$$, $${(\lambda )}_k=\lambda (\lambda +1)(\lambda +2)\cdots (\lambda +k-1)$$.

This table defines acronyms for common digital modulation schemes.

Acronym	Definition
M-PSK	M-ary phase-shift keying
DE-M-PSK	Differentially encoded M-ary phase-shift keying
BPSK	Binary phase-shift keying
DE-BPSK	Differentially encoded binary phase-shift keying
QPSK	Quaternary phase-shift keying
DE-QPSK	Differentially encoded quadrature phase-shift keying
OQPSK	Offset quadrature phase-shift keying
DE-OQPSK	Differentially encoded offset quadrature phase-shift keying
M-DPSK	M-ary differential phase-shift keying
M-PAM	M-ary pulse amplitude modulation
M-QAM	M-ary quadrature amplitude modulation
M-FSK	M-ary frequency-shift keying
MSK	Minimum shift keying
M-CPFSK	M-ary continuous-phase frequency-shift keying