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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 the acronyms used in this topic.

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

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

These sections cover the main analytical expressions used in the berawgn function and Bit Error Rate Analysis app.

M-PSK

From equation 8.22 in [2],

Ps=1π0(M1)π/Mexp(kEbN0sin2[π/M]sin2θ)dθ

This expression is similar, but not strictly equal, to the exact BER (from [4] and equation 8.29 from [2]):

Pb=1k(i=1M/2(wi')Pi)

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

Pi=12π0π(1(2i1)/M)exp(kEbN0sin2[(2i1)π/M]sin2θ)dθ12π0π(1(2i+1)/M)exp(kEbN0sin2[(2i+1)π/M]sin2θ)dθ

For M-PSK with M = 2, specifically BPSK, this equation 5.2-57 from [1] applies:

Ps=Pb=Q(2EbN0)

For M-PSK with M = 4, specifically QPSK, these equations 5.2-59 and 5.2-62 from [1] apply:

Ps=2Q(2EbN0)[112Q(2EbN0)]Pb=Q(2EbN0)

DE-M-PSK

For DE-M-PSK with M = 2, specifically DE-BPSK, this equation 8.36 from [2] applies:

Ps=Pb=2Q(2EbN0)2Q2(2EbN0)

For DE-M-PSK with M = 4, specifically DE-QPSK, this equation 8.38 from [2] applies:

Ps=4Q(2EbN0)8Q2(2EbN0)+8Q3(2EbN0)4Q4(2EbN0)

From equation 5 in [3],

Pb=2Q(2EbN0)[1Q(2EbN0)]

OQPSK

For OQPSK, use the same BER and SER computations as for QPSK in [2].

DE-OQPSK

For OQPSK, use the same BER and SER computations as for DE-QPSK in [3].

M-DPSK

For M-DPSK, this equation 8.84 from [2] applies:

Ps=sin(π/M)2ππ/2π/2exp((kEb/N0)(1cos(π/M)cosθ))1cos(π/M)cosθdθ

This expression is similar, but not strictly equal, to the exact BER (from [4]):

Pb=1k(i=1M/2(wi')Ai)

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

Ai=F((2i+1)πM)F((2i1)πM)F(ψ)=sinψ4ππ/2π/2exp(kEb/N0(1cosψcost))1cosψcostdt

For M-DPSK with M = 2, this equation 8.85 from [2] applies:

Pb=12exp(EbN0)

M-PAM

From equations 8.3 and 8.7 in [2] and equation 5.2-46 in [1],

Ps=2(M1M)Q(6M21kEbN0)

From [5],

Pb=2Mlog2M×k=1log2Mi=0(12k)M1{(1)i2k1M(2k1i2k1M+12)Q((2i+1)6log2MM21EbN0)}

M-QAM

For square M-QAM, k=log2M is even, so equation 8.10 from [2] and equations 5.2-78 and 5.2-79 from [1] apply:

Ps=4M1MQ(3M1kEbN0)4(M1M)2Q2(3M1kEbN0)

From [5],

Pb=2Mlog2M×k=1log2Mi=0(12k)M1{(1)i2k1M(2k1i2k1M+12)Q((2i+1)6log2M2(M1)EbN0)}

For rectangular (non-square) M-QAM, k=log2M is odd, M=I×J, I=2k12, and J=2k+12. So that,

Ps=4IJ2I2JM×Q(6log2(IJ)(I2+J22)EbN0)4M(1+IJIJ)Q2(6log2(IJ)(I2+J22)EbN0)

From [5],

Pb=1log2(IJ)(k=1log2IPI(k)+l=1log2JPJ(l))

where

PI(k)=2Ii=0(12k)I1{(1)i2k1I(2k1i2k1I+12)Q((2i+1)6log2(IJ)I2+J22EbN0)}

and

PJ(k)=2Jj=0(12l)J1{(1)j2l1J(2l1j2l1J+12)Q((2j+1)6log2(IJ)I2+J22EbN0)}

Orthogonal M-FSK with Coherent Detection

From equation 8.40 in [2] and equation 5.2-21 in [1],

Ps=1[Q(q2kEbN0)]M112πexp(q22)dqPb=2k12k1Ps

Nonorthogonal 2-FSK with Coherent Detection

For M=2, equation 5.2-21 in [1] and equation 8.44 in [2] apply:

Ps=Pb=Q(Eb(1Re[ρ])N0)

ρ is the complex correlation coefficient, such that:

ρ=12Eb0Tbs˜1(t)s˜2*(t)dt

where s˜1(t) and s˜2(t) are complex lowpass signals, and

Eb=120Tb|s˜1(t)|2dt=120Tb|s˜2(t)|2dt

For example, with

s˜1(t)=2EbTbej2πf1t, s˜2(t)=2EbTbej2πf2t

then

ρ=12Eb0Tb2EbTbej2πf1t2EbTbej2πf2tdt=1Tb0Tbej2π(f1f2)tdt=sin(πΔfTb)πΔfTbejπΔft

where Δf=f1f2.

From equation 8.44 in [2],

    Re[ρ]=Re[sin(πΔfTb)πΔfTbejπΔft]=sin(πΔfTb)πΔfTbcos(πΔfTb)=sin(2πΔfTb)2πΔfTbPb=Q(Eb(1sin(2πΔfTb)/(2πΔfTb))N0)

where h=ΔfTb.

Orthogonal M-FSK with Noncoherent Detection

From equation 5.4-46 in [1] and equation 8.66 in [2],

Ps=m=1M1(1)m+1(M1m)1m+1exp[mm+1kEbN0]Pb=12MM1Ps

Nonorthogonal 2-FSK with Noncoherent Detection

For M=2, this equation 5.4-53 from [1] and this equation 8.69 from [2] apply:

Ps=Pb=Q(a,b)12exp(a+b2)I0(ab)

where

a=Eb2N0(11|ρ|2), b=Eb2N0(1+1|ρ|2) 

Precoded MSK with Coherent Detection

Use the same BER and SER computations as for BPSK.

Differentially Encoded MSK with Coherent Detection

Use the same BER and SER computations as for DE-BPSK.

MSK with Noncoherent Detection (Optimum Block-by-Block)

The upper bound on error rate from equations 10.166 and 10.164 in [6]) is

Ps=Pb12[1Q(b1,a1)+Q(a1,b1)]+14[1Q(b4,a4)+Q(a4,b4)]+12eEbN0

where

a1=EbN0(134/π24),b1=EbN0(1+34/π24)a4=EbN0(114/π2),b4=EbN0(1+14/π2)

CPFSK Coherent Detection (Optimum Block-by-Block)

The lower bound on error rate (from equation 5.3-17 in [1]) is

Ps>KδminQ(EbN0δmin2)

The upper bound on error rate is

δmin2>min1iM1{2i(1sinc(2ih))}

where h is the modulation index, and Kδmin is the number of paths with the minimum distance.

PbPsk

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

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

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

Common Notation

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

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

γb=EbN0

Message length

K

Code length

N

Code rate

Rc=KN

Block Coding

This section describes the specific notation for block coding expressions, where dmin is the minimum distance of the code.

Soft Decision

For BPSK, QPSK, OQPSK, 2-PAM, 4-QAM, and precoded MSK, equation 8.1-52 in [1]) applies,

Pb12(2K1)Q(2γbRcdmin)

For DE-BPSK, DE-QPSK, DE-OQPSK, and DE-MSK,

Pb12(2K1)[2Q(2γbRcdmin)[1Q(2γbRcdmin)]]

For BFSK coherent detection, equations 8.1-50 and 8.1-58 in [1] apply,

Pb12(2K1)Q(γbRcdmin)

For BFSK noncoherent square-law detection, equations 8.1-65 and 8.1-64 in [1] apply,

Pb122K122dmin1exp(12γbRcdmin)i=0dmin1(12γbRcdmin)i1i!r=0dmin1i(2dmin1r)

For DPSK,

Pb122K122dmin1exp(γbRcdmin)i=0dmin1(γbRcdmin)i1i!r=0dmin1i(2dmin1r)

Hard Decision

For general linear block code, equations 4.3 and 4.4 in [9], and 12.136 in [6] apply,

Pb1Nm=t+1N(m+t)(Nm)pm(1p)Nmt=12(dmin1)

For Hamming code, equations 4.11 and 4.12 in [9] and 6.72 and 6.73 in [7] apply

Pb1Nm=2Nm(Nm)pm(1p)Nm=pp(1p)N1

For rate (24,12) extended Golay code, equations 4.17 in [9] and 12.139 in [6] apply:

Pb124m=424βm(24m)pm(1p)24m

where βm is the average number of channel symbol errors that remain in corrected N-tuple format when the channel caused m symbol errors (see table 4.2 in [9]).

For Reed-Solomon code with N=Q1=2q1,

Pb2q12q11Nm=t+1Nm(Nm)(Ps)m(1Ps)Nm

For FSK, equations 4.25 and 4.27 in [9], 8.1-115 and 8.1-116 in [1], 8.7 and 8.8 in [7], and 12.142 and 12.143 in [6] apply,

Pb1q1Nm=t+1Nm(Nm)(Ps)m(1Ps)Nm

otherwise, if log2Q/log2M=q/k=h, where h is an integer (equation 1 in [10]) applies,

Ps=1(1s)h

where s is the SER in an uncoded AWGN channel.

For example, for BPSK, M=2 and Ps=1(1s)q, otherwise Ps is given by table 1 and equation 2 in [10].

Convolutional Coding

This section describes the specific notation for convolutional coding expressions, where dfree is the free distance of the code, and ad is the number of paths of distance d from the all-zero path that merges with the all-zero path for the first time.

Soft Decision

From equations 8.2-26, 8.2-24, and 8.2-25 in [1] and 13.28 and 13.27 in [6] apply,

Pb<d=dfreeadf(d)P2(d)

The transfer function is given by

T(D,N)=d=dfreeadDdNf(d)dT(D,N)dN|N=1=d=dfreeadf(d)Dd

where f(d) is the exponent of N as a function of d.

This equation gives the results for BPSK, QPSK, OQPSK, 2-PAM, 4-QAM, precoded MSK, DE-BPSK, DE-QPSK, DE-OQPSK, DE-MSK, DPSK, and BFSK:

P2(d)=Pb|EbN0=γbRcd

where Pb is the BER in the corresponding uncoded AWGN channel. For example, for BPSK (equation 8.2-20 in [1]),

P2(d)=Q(2γbRcd)

Hard Decision

From equations 8.2-33, 8.2-28, and 8.2-29 in [1] and 13.28, 13.24, and 13.25 in [6] apply,

Pb<d=dfreeadf(d)P2(d)

When d is odd,

P2(d)=k=(d+1)/2d(dk)pk(1p)dk

and when d is even,

P2(d)=k=d/2+1d(dk)pk(1p)dk+12(dd/2)pd/2(1p)d/2

where p is the bit error rate (BER) in an uncoded AWGN channel.

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

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

Timing Synchronization Error

To compute the BER for a communications system with a timing synchronization error, the bersync function uses this formula from [13]:

14πσexp(ξ22σ2)2R(12|ξ|)exp(x22)dxdξ+122π2Rexp(x22)dx

where σ is the timing error, and R is the linear Eb/N0 value.

Timing Synchronization Error

To compute the BER for a communications system with a carrier synchronization error, the bersync function uses this formula from [13]:

1πσ0exp(ϕ22σ2)2Rcosϕexp(y22)dydϕ

where σ is the phase error R is the linear Eb/N0 value.

See Also

Apps

Functions

Related Topics