BER calculation

245813

BER calculation

Vahid Meghdadi

reference: Wireless Communications by Andrea Goldsmith

January 2008

1 SER and BER over Gaussian channel

1.1 BER for BPSK modulation

In a BPSK system the received signal can be written as: y=x+n(1) wherex2 fA;Ag,nCN(0;2) and2=N0. The real part of the above equation isyre=x+nrewherenre N(0;2=2) =N(0;N0=2). In BPSK constellationdmin= 2Aand bis dened asEb=N0and sometimes it is called

SNR per bit. With this denition we have:

b:=EbN 0=A2N

0=d2min4N0(2)

So the bit error probability is:

P b=Pfn > Ag=Z 1

A1p22=2ex222=2(3)

This equation can be simplied using Q-function as: P b=Q0 @sd

2min2N01

A =Qdminp2N0 =Qp2 b (4) where the Q function is dened as:

Q(x) =1p2Z

1 x ex22 dx(5)

1.2 BER for QPSK

QPSK modulation consists of two BPSK modulation on in-phase and quadrature components of the signal. The corresponding constellation is presented on gure

1. The BER of each branch is the same as BPSK:

P b=Qp2 b (6) 1

Figure 1: QPSK constellation

The symbol probability of error (SER) is the probability of either branch has a bit error: P s= 1[1Qp2 b ]2(7) Since the symbol energy is split between the two in-phase and quadrature com- ponents, s= 2 band we have: P s= 1[1Q(p s)]2(8) We can use the union bound to give an upper bound for SER of QPSK. Regard- ing gure 1, condition that the symbol zero is sent, the probability of error is bounded by the sum of probabilities of 0!1, 0!2 and 0!3. We can write: P sQ(d01=p2N0) +Q(d02=p2N0) +Q(d03=p2N0) (9) = 2Q(A=pN

0) +Q(p2A=p2N0) (10)

Since s= 2 b=A2=N0, we can write: P s2Q(p s) +Q(p2 s)3Q(p s) (11) Using the tight approximation of Q function forz0:

Q(z)1z

p2ez2=2(12) we obtain: P s3p2 se0:5 s(13) Using Gray coding and assuming that for high signal to noise ratio the errors occur only for the nearest neighbor,Pbcan be approximated fromPsbyPb P s=2. 2

Figure 2: MPSK constellation

1.3 BER for MPSK signaling

For MPSK signaling we can calculate easily an approximation of SER using nearest neighbor approximation. Using gure , the symbol error probability can be approximated by: P s2Qdminp2N0 = 2Q2AsinMp2N0 = 2Qp2 ssin(=M) (14)

This approximation is only good for high SNR.

1.4 BER for QAM constellation

The SER for a rectangular M-QAM (16-QAM, 64-QAM, 256-QAM etc) with sizeL=M2can be calculated by considering two M-PAM on in-phase and quadrature components (see gure 3 for 16-QAM constellation). The error probability of QAM symbol is obtained by the error probability of each branch (M-PAM) and is given by: P s= 1

12(sqrtM1)sqrtM

Q r3 sM1!! 2 (15) If we use the nearest neighbor approximation for an M-QAM rectangular con- stellation, there are 4 nearest neighbors with distancedmin. So the SER for high SNR can be approximated by: c(16) In order to calculate the mean energy per transmitted symbol, it can be seen thatE s=1M M X i=1A

2i(17)

3

Figure 3: 16-QAM constellation

ModulationPs(

s)Pb( b)BPSKPb=Qp2 b

QPSKPs2Qp

sPbQp2 b

MPSKPs2Qp2

ssinM

Pb2log

2MQp2 blog2MsinM

M-QAMPs4Q

q3 sM1 P b4log 2MQ q3 blog2MM1 Table 1: Approximate symbol and bit error probabilities for coherent modula- tion Using the fact thatAi= (ai+bi) andaiandbi2 f2i1Lgfori= 1;:::;L.

After some simple calculations we obtain:E

s=d2min2LL X i=1(2i1L)2(18)

For example for 16-QAM anddmin= 2 theE

s= 10. For 64-QAM anddmin= 2 theE s= 21.

1.5 conclusion

The approximations or exact values for SER has the following form: P s( s)MQp M s (19) whereMandMdepend on the type of approximation and the modulation type. In the table 1 the values forMandMare semmerized for common modulations. We can also note that the bit error probability has the same form as for

SER. It is:

P

BER calculation

Vahid Meghdadi

reference: Wireless Communications by Andrea Goldsmith

January 2008

1 SER and BER over Gaussian channel

1.1 BER for BPSK modulation

In a BPSK system the received signal can be written as: y=x+n(1) wherex2 fA;Ag,nCN(0;2) and2=N0. The real part of the above equation isyre=x+nrewherenre N(0;2=2) =N(0;N0=2). In BPSK constellationdmin= 2Aand bis dened asEb=N0and sometimes it is called

SNR per bit. With this denition we have:

b:=EbN 0=A2N

0=d2min4N0(2)

So the bit error probability is:

P b=Pfn > Ag=Z 1

A1p22=2ex222=2(3)

This equation can be simplied using Q-function as: P b=Q0 @sd

2min2N01

A =Qdminp2N0 =Qp2 b (4) where the Q function is dened as:

Q(x) =1p2Z

1 x ex22 dx(5)

1.2 BER for QPSK

QPSK modulation consists of two BPSK modulation on in-phase and quadrature components of the signal. The corresponding constellation is presented on gure

1. The BER of each branch is the same as BPSK:

P b=Qp2 b (6) 1

Figure 1: QPSK constellation

The symbol probability of error (SER) is the probability of either branch has a bit error: P s= 1[1Qp2 b ]2(7) Since the symbol energy is split between the two in-phase and quadrature com- ponents, s= 2 band we have: P s= 1[1Q(p s)]2(8) We can use the union bound to give an upper bound for SER of QPSK. Regard- ing gure 1, condition that the symbol zero is sent, the probability of error is bounded by the sum of probabilities of 0!1, 0!2 and 0!3. We can write: P sQ(d01=p2N0) +Q(d02=p2N0) +Q(d03=p2N0) (9) = 2Q(A=pN

0) +Q(p2A=p2N0) (10)

Since s= 2 b=A2=N0, we can write: P s2Q(p s) +Q(p2 s)3Q(p s) (11) Using the tight approximation of Q function forz0:

Q(z)1z

p2ez2=2(12) we obtain: P s3p2 se0:5 s(13) Using Gray coding and assuming that for high signal to noise ratio the errors occur only for the nearest neighbor,Pbcan be approximated fromPsbyPb P s=2. 2

Figure 2: MPSK constellation

1.3 BER for MPSK signaling

For MPSK signaling we can calculate easily an approximation of SER using nearest neighbor approximation. Using gure , the symbol error probability can be approximated by: P s2Qdminp2N0 = 2Q2AsinMp2N0 = 2Qp2 ssin(=M) (14)

This approximation is only good for high SNR.

1.4 BER for QAM constellation

The SER for a rectangular M-QAM (16-QAM, 64-QAM, 256-QAM etc) with sizeL=M2can be calculated by considering two M-PAM on in-phase and quadrature components (see gure 3 for 16-QAM constellation). The error probability of QAM symbol is obtained by the error probability of each branch (M-PAM) and is given by: P s= 1

12(sqrtM1)sqrtM

Q r3 sM1!! 2 (15) If we use the nearest neighbor approximation for an M-QAM rectangular con- stellation, there are 4 nearest neighbors with distancedmin. So the SER for high SNR can be approximated by: c(16) In order to calculate the mean energy per transmitted symbol, it can be seen thatE s=1M M X i=1A

2i(17)

3

Figure 3: 16-QAM constellation

ModulationPs(

s)Pb( b)BPSKPb=Qp2 b

QPSKPs2Qp

sPbQp2 b

MPSKPs2Qp2

ssinM

Pb2log

2MQp2 blog2MsinM

M-QAMPs4Q

q3 sM1 P b4log 2MQ q3 blog2MM1 Table 1: Approximate symbol and bit error probabilities for coherent modula- tion Using the fact thatAi= (ai+bi) andaiandbi2 f2i1Lgfori= 1;:::;L.

After some simple calculations we obtain:E

s=d2min2LL X i=1(2i1L)2(18)

For example for 16-QAM anddmin= 2 theE

s= 10. For 64-QAM anddmin= 2 theE s= 21.

1.5 conclusion

The approximations or exact values for SER has the following form: P s( s)MQp M s (19) whereMandMdepend on the type of approximation and the modulation type. In the table 1 the values forMandMare semmerized for common modulations. We can also note that the bit error probability has the same form as for

SER. It is:

P