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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, 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=log2M

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

EbN0

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

EsN0=kEbN0

Bit error rate (BER)

Pb

Symbol error rate (SER)

Ps

Real part

Re[]

Floor, largest integer smaller than the value contained in braces

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

Function Mathematical Expression
Q function

Q(x)=12πxexp(t2/2)dt

Marcum Q function

Q(a,b)=btexp(t2+a22)I0(at)dt

Modified Bessel function of the first kind of order ν

Iν(z)=k=0(z/2)υ+2kk!Γ(ν+k+1)

where

Γ(x)=0ettx1dt

is the gamma function.

Confluent hypergeometric function

F11(a,c;x)=k=0(a)k(c)kxkk!

where the Pochhammer symbol, (λ)k, is defined as (λ)0=1, (λ)k=λ(λ+1)(λ+2)(λ+k1).

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-QPSKDifferentially encoded quadrature phase-shift keying
OQPSKOffset quadrature phase-shift keying
DE-OQPSKDifferentially encoded offset quadrature phase-shift keying
M-DPSKM-ary differential phase-shift keying
M-PAMM-ary pulse amplitude modulation
M-QAMM-ary quadrature amplitude modulation
M-FSKM-ary frequency-shift keying
MSKMinimum shift keying
M-CPFSKM-ary continuous-phase frequency-shift keying

These topics describe the analytical expressions used in error rate analysis:

See Also

Apps

Functions

Related Topics