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Analytical Expressions Used in berfading Function and Bit Error Rate Analysis App

This section covers the main analytical expressions used in the berfading function and the Bit Error Rate Analysis app.

Notation

This table describes the additional notations used in analytical expressions in this section.

Description Notation
Power of the fading amplitude rΩ=E[r2], where E[] denotes statistical expectation
Number of diversity branches

L

Signal to Noise Ratio (SNR) per symbol per branch

γ¯l=(ΩlEsN0)/L=(ΩlkEbN0)/L

For identically-distributed diversity branches,

γ¯=(ΩkEbN0)/L

Moment generating functions for each diversity branch

For Rayleigh fading channels:

Mγl(s)=11sγ¯l

For Rician fading channels:

Mγl(s)=1+K1+Ksγ¯le[Ksγ¯l(1+K)sγ¯l]

K is the ratio of the energy in the specular component to the energy in the diffuse component (linear scale).

For identically-distributed diversity branches,Mγl(s)=Mγ(s) for all l.

This table defines the additional acronyms used in this section.

Acronym Definition
MRCMaximal-ratio combining
EGCEqual-gain combining

M-PSK with MRC

From equation 9.15 in [2],

Ps=1π0(M1)π/Ml=1LMγl(sin2(π/M)sin2θ)dθ

From [4] and [2],

Pb=1k(i=1M/2(wi')P¯i)

where wi'=wi+wMi, wM/2'=wM/2, wi is the Hamming weight of bits assigned to symbol i,

P¯i=12π0π(1(2i1)/M)l=1LMγl(1sin2θsin2(2i1)πM)dθ12π0π(1(2i+1)/M)l=1LMγl(1sin2θsin2(2i+1)πM)dθ

For the special case of Rayleigh fading with M=2 (from equations C-18 and C-21 and Table C-1 in [6]),

Pb=12[1μi=0L1(2ii)(1μ24)i]

where

μ=γ¯γ¯+1

If L=1, then:

Pb=12[1γ¯γ¯+1]

DE-M-PSK with MRC

For M=2 (from equations 8.37 and 9.8-9.11 in [2]),

Ps=Pb=2π0π/2l=1LMγl(1sin2θ)dθ2π0π/4l=1LMγl(1sin2θ)dθ

M-PAM with MRC

From equation 9.19 in [2],

Ps=2(M1)Mπ0π/2l=1LMγl(3/(M21)sin2θ)dθ

From [5] and [2],

Pb=2πMlog2M×k=1log2M i=0(12k)M1{(1)i2k1M(2k1i2k1M+12)0π/2l=1LMγl((2i+1)23/(M21)sin2θ)dθ}

M-QAM with MRC

For square M-QAM, k=log2M is even (equation 9.21 in [2]),

Ps=4π(11M)0π/2l=1LMγl(3/(2(M1))sin2θ)dθ4π(11M)20π/4l=1LMγl(3/(2(M1))sin2θ)dθ

From [5] and [2]:

Pb=2πMlog2M×k=1log2M i=0(12k)M1{(1)i2k1M(2k1i2k1M+12)0π/2l=1LMγl((2i+1)23/(2(M1))sin2θ)dθ}

For rectangular (nonsquare) M-QAM, k=log2M is odd, M=I×J, I=2k12, J=2k+12, γ¯l=Ωllog2(IJ)EbN0,

Ps=4IJ2I2JMπ0π/2l=1LMγl(3/(I2+J22)sin2θ)dθ4Mπ(1+IJIJ)0π/4l=1LMγl(3/(I2+J22)sin2θ)dθ

From [5] and [2],

Pb=1log2(IJ)(k=1log2IPI(k)+l=1log2JPJ(l))PI(k)=2Iπi=0(12k)I1{(1)i2k1I(2k1i2k1I+12)0π/2l=1LMγl((2i+1)23/(I2+J22)sin2θ)dθ}PJ(k)=2Jπj=0(12l)J1{(1)j2l1J(2l1j2l1J+12)0π/2l=1LMγl((2j+1)23/(I2+J22)sin2θ)dθ}

M-DPSK with Postdetection EGC

From equation 8.165 in [2],

Ps=sin(π/M)2ππ/2π/21[1cos(π/M)cosθ]l=1LMγl([1cos(π/M)cosθ])dθ

From [4] and [2],

Pb=1k(i=1M/2(wi')A¯i)

where wi'=wi+wMi, wM/2'=wM/2, wi is the Hamming weight of bits assigned to symbol i,

A¯i=F¯((2i+1)πM)F¯((2i1)πM)F¯(ψ)=sinψ4ππ/2π/21(1cosψcost)l=1LMγl((1cosψcost))dt

For the special case of Rayleigh fading with M=2 and L=1 (equation 8.173 from [2]),

Pb=12(1+γ¯)

Orthogonal 2-FSK, Coherent Detection with MRC

From equation 9.11 in [2],

Ps=Pb=1π0π/2l=1LMγl(1/2sin2θ)dθ

For the special case of Rayleigh fading (equations 14.4-15 and 14.4-21 in [1]),

Ps=Pb=12L(1γ¯2+γ¯)Lk=0L1(L1+kk)12k(1+γ¯2+γ¯)k

Nonorthogonal 2-FSK, Coherent Detection with MRC

From equations 9.11 and 8.44 in [2],

Ps=Pb=1π0π/2l=1LMγl((1Re[ρ])/2sin2θ)dθ

For the special case of Rayleigh fading with L=1 (equations 20 in [8] and 8.130 in [2]),

Ps=Pb=12[1γ¯(1Re[ρ])2+γ¯(1Re[ρ])]

Orthogonal M-FSK, Noncoherent Detection with EGC

For Rayleigh fading, from equation 14.4-47 in [1],

Ps=101(1+γ¯)L(L1)!UL1eU1+γ¯(1eUk=0L1Ukk!)M1dUPb=12MM1Ps

For Rician fading from equation 41 in [8],

Ps=r=1M1(1)r+1eLKγ¯r/(1+γ¯r)(r(1+γ¯r)+1)L(M1r)n=0r(L1)βnrΓ(L+n)Γ(L)[1+γ¯rr+1+rγ¯r]nF11(L+n,L;LKγ¯r/(1+γ¯r)r(1+γ¯r)+1)Pb=12MM1Ps

where

γ¯r=11+Kγ¯βnr=i=n(L1)nβi(r1)(ni)!I[0,(r1)(L1)](i)β00=β0r=1βn1=1/n!β1r=r

and I[a,b](i)=1 if aib and 0 otherwise.

Nonorthogonal 2-FSK, Noncoherent Detection with No Diversity

From equation 8.163 in [2],

Ps=Pb=14πππ1ς21+2ςsinθ+ς2Mγ(14(1+1ρ2)(1+2ςsinθ+ς2))dθ

where

ς=11ρ21+1ρ2

See Also

Apps

Functions

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