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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 DE-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

See Also

Apps

Functions

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