Thesis: Chapter 2

CHAPTER 2

ERROR PERFORMANCE OF DIGITAL COMMUNICATION SYSTEM

Analytical Expressions of Probability of Error

for Various Modulation Schemes in Uncoded

AWGN Channel

Analytical Expressions of Probability of Error

for Various Modulation Schemes in Multipath

Fading Channel

Analytical Expressions of Probability of Error

for Various Modulation Schemes with Channel

Coded AWGN Channel

ANALYTICAL ANALYSIS OF

ERROR PERFORMANCE

Analytical Expressions of Probability of Error for Various Modulation Schemes in Uncoded AWGN Channel

BPSK

The symbol error rate and bit error rate of BPSK is given as (equation 5.2-57 from [20]):

Ps = Pb = Q

QPSK, OQPSK

The symbol error rate and bit error rate of QPSK and OQPSK is given as (equations 5.2-59 and 5.2-62 from [20]):

Symbol Error Rate, Ps = 2 Q

Bit Error Rate, Pb = Q

DE-BPSK

The symbol error rate and bit error rate of DE-BPSK is given as (equation 8.36 from [21]):

Ps = Pb = 2Q

2Q2

DE-QPSK, DE-OQPSK

The symbol error rate and bit error rate of DE-QPSK and DE-OQPSK is given as (equation 8.38 from [21]):

Symbol Error Rate, Ps = 4Q

8Q2

8Q3

4Q4

From equation 5 in [5]:

Bit Error Rate, Pb = 2Q

M-PSK, Precoded MSK with Coherent Detection

The symbol error rate and bit error rate of M-PSK, Precoded MSK with Coherent Detection is given as (from equation 8.22 in [21]):

Symbol Error Rate, Ps =

from equation 6.116 in [22]:

The following expression is very close, but not strictly equal, to the exact BER (from [25] and equation 8.29 from [21]):

where,

, is the Hamming weight of bits assigned to symbol i, and

Pi =

M-DPSK

The symbol error rate and bit error rate of M-DPSK is given as (from equation 8.84 in [21]):

Symbol Error Rate, Ps =

For M

4 is approximated by [23]:

The following expression is very close, but not strictly equal, to the exact BER [25]:

Pb =

where,

is the Hamming weight of bits assigned to symbol i, and

Ai = F

M-PAM

The symbol error rate and bit error rate of M-PAM is given as (from equations 8.3 and 8.7 in [21], and equation 5.2-46 in [20]):

Symbol Error Rate, Ps = 2

From [4]:

Bit Error Rate,

Pb =

M-QAM. For square M-QAM,

The symbol error rate and bit error rate of M-QAM for k = log2M is given as (equation 8.10 from [21], and equations 5.2-78 and 5.2-79 from [20]):

Symbol Error Rate, Ps = 4

From [4]:

Bit Error Rate,

Pb=

Nonorthogonal BFSK with Noncoherent Detection

The symbol error rate and bit error rate of this method is given as (from equation 5.4-53 in [20] and equation 8.69 in [21]):

Ps = Pb = Q

where, a =

, b =

Nonorthogonal BFSK with Coherent Detection

The symbol error rate and bit error rate of this method is given as (from equation 5.2-21 in [20] and equation 8.44 in [21]):

Ps = Pb = Q

is the complex correlation coefficient:

where

and

are complex lowpass signals, and

Eb =

For example:

dt =

where

= f1 – f2

= Re

(from equation 8.44 in [21], where h =

)

Orthogonal M-FSK with Coherent Detection

The symbol error rate and bit error rate of this method is given as (from equation 8.40 in [21] and equation 5.2-21 in [20]):

Ps = 1-

Pb =

Orthogonal M-FSK with Non-coherent Detection

The symbol error rate and bit error rate of this method is given as (from equation 5.4-46 in [20] and equation 8.66 in [21]):

Ps =

Pb =

MSK with Non-coherent Detection

The symbol error rate and bit error rate of this method is given as (from equations 10.166 and 10.164 in [22]):

Ps= Pb

where,

CPFSK Coherent Detection

The symbol error rate and bit error rate of this method is given as (from equation 5.3-17 in [20]):

Ps >

Upper bound:

where h is the modulation index, and

is the number of paths having the minimum distance.

Analytical Expressions of Probability of Error for Various Modulation Schemes in Multipath Fading Channel

M-PSK with MRC

The symbol error of M-PSK with MRC in multipath fading channel is given as (from equation 9.15 in [21]):

Ps =

From [25] and [21]:

where,

is the Hamming weight of bits assigned to symbol i, and

Pi=

BPSK

The bit error rate of BPSK in multipath fading channel is given as (from equations C-18, C-21, and Table C-1 in [22]):

Pb =

where,

If L=1:

Pb =

DE-BPSK with MRC

The symbol error rate and bit error rate of DE-BPSK with MRC in multipath fading channel is given as (from equations 8.37 and 9.8-9.11 in [21]):

Ps = Pb =

M-PAM with MRC

The symbol error rate of M-PAM with MRC in multipath fading channel is given as (from equation 9.19 in [21]):

Ps =

The bit error rate of M-PAM with MRC in multipath fading channel is given as (from [4] and [21]):

Pb =

M-QAM with MRC.

The symbol error rate of M-QAM with MRC in multipath fading channel is given as (from equation 9.21 in [21]):

Ps =

The bit error rate of M-QAM with MRC in multipath fading channel is given as (from [4] and [21]):

Pb =

Orthogonal BFSK, Coherent Detection with MRC

The symbol and bit error rate of this method in multipath fading channel is given as (from equation 9.11 in [21]):

Ps = Pb =

Nonorthogonal BFSK, Coherent Detection with MRC.

The symbol and bit error rate of this method in multipath fading channel is given as (equations 9.11 and 8.44 in [21]):

Ps = Pb =

Analytical Expressions of Probability of Bit for Various Modulation Schemes with Coded AWGN Channel

Block Coding

Soft Decision

BPSK, QPSK, OQPSK, PAM-2, QAM-4, and precoded MSK

Bit error rate of transceiver comprising of block coding method with these modulation methods is given as (equation 8.1-52 in [20]):

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

Bit error rate of transceiver comprising of block coding method with these modulation methods is given as

BFSK, coherent detection

Bit error rate of transceiver comprising of block coding method with BFSK is given as (from equations 8.1-50 and 8.1-58 in [20]):

DPSK

Bit error rate of transceiver comprising of block coding method with BFSK is given as

Hard Decision

General linear block code

Bit error rate of transceiver comprising of linear block coding method with Hard decision is given as (from equations 4.3, 4.4 in [3], and 12.136 in [22]):

t =

Hamming code

Bit error rate of transceiver comprising of Hamming coding method with Hard decision is given as (equations 4.11, 4.12 in [3], and 6.72, 6.73 in [21]):

Pb =

(24, 12) extended Golay code

Bit error rate of transceiver comprising of extended Golay coding method with Hard decision is given as (equation 4.17 in [3], and 12.139 in [22]):

where

is the average number of channel symbol errors that remain in corrected N-tuple when the channel caused m symbol errors (table 4.2 in [3]).

Reed-Solomon code with N = Q1 = 2q1

Bit error rate of transceiver comprising of Reed-Solomon coding method with Hard decision is given as

for FSK (from equations 4.25, 4.27 in [3], 8.1-115, 8.1-116 in [20], 8.7, 8.8 in [21], and 12.142, 12.143 in [22]),

Pb =

otherwise

Pb =

where h is an integer (equation 1 in [26]):

then symbol error rate is given as, Ps = 1

where s is the symbol error rate (SER) in an uncoded AWGN channel.

For example, for BPSK, M=2 and Ps = 1

Convolutional Coding

Soft Decision.

Bit error rate of transceiver comprising of convolutional coding method with soft decision decoding is given as (from equations 8.2-26, 8.2-24, and 8.2-25 in [20], and equations 13.28 and 13.27 in [22]):

Pb <

with transfer function

T(D, N) =

where

is the exponent of N as a function of d.

Results for BPSK, QPSK, OQPSK, PAM-2, QAM-4, precoded MSK, DE-BPSK, DE-QPSK, DE-OQPSK, DE-MSK, DPSK, and BFSK are obtained as:

P2(d) =

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

P2(d) =

Hard Decision

Bit error rate of transceiver comprising of convolutional coding method with soft decision decoding is given as (from equations 8.2-33, 8.2-28, and 8.2-29 in [20], and equations 13.28, 13.24, and 13.25 in [22]):

Pb<