Signal Space Analysis of BASK, BFSK, BPSK, And QAM on Mac

Signal Space Analysis of BASK, BFSK, BPSK, and QAM on Mac

30. Signal Space Analysis of BASK, BFSK, BPSK, and QAM The vector-space representation of signals and the optimum detection process which chooses the signal closest to the received signal is particularly useful in signal design and in probability-of-error calculations. The material presented herein includes the error performance of binary amplitude-shift keying (BASK), binary frequency-shift keying (BFSK), binary phase-shift keying (BPSK), and M-ary quadrature amplitude modulation (QAM) signals. Error Performance of Binary ASK A binary amplitude-shift keying (BASK) signal can be defined by  A cos(2π fct ), s(t) =  0,

0≤t≤T

(30.1)

elsewhere

where A is a constant, fc is the carrier frequency, and T is the bit duration. It has a power P = A2/2, so that A = 2P . Thus equation (30.1) can be written as

s(t)

 2 P cos(2π fct ), =  0,  2 cos(2π fct ),  PT =  T 0, 

0≤t≤T elsewhere 0≤t≤T elsewhere

 2 cos(2π fct ),  E =  T 0, 

0≤t≤T

(30.2)

elsewhere

where E = PT is the energy contained in a bit duration. Figure 30.1 shows the signal constellation diagram of BASK signals and the conditional probability density functions associated with the signals. Figure 30.1

BASK signal constellation diagram and the conditional probability density functions associated with the signals.

The energy contained in a bit duration is E = d2

(30.3)

and so

30.1

Signal Space Analysis of BASK, BFSK, BPSK, and QAM on Mac

d=

E

(30.4)

It is assumed that the noise is additive white Gaussian noise (AWGN) with a two-sided power spectral density of n 0 /2, zero mean and fixed variance σ 2 = n 0 /2. In the presence of AWGN, the conditional probability density function of φ1 assuming that s0 is transmitted is

1

f(φ1 / 0) =

2πσ 2

φ2 − 1 2 e 2σ

and the probability of error given that s0 is transmitted is ∞

∫

P e0 =

f(φ1/0)dφ1 =

φ1 = d / 2

∞

∫

1

2 φ1 = d / 2 2πσ

e

φ12 − 2σ 2

dφ1

(30.5)

Similarly, the probability of error given that s1 is transmitted is

φ1 = d / 2

∫

P e1 =

f(φ1/1)dφ1 =

φ1 = d / 2

∫

−∞

−∞

(φ − d )2 − 1 1 2σ 2 dφ1 = P e1 e 2πσ 2

where

1

f(φ1 /1) =

2πσ 2

(φ − d )2 − 1 2σ 2 e

(30.6)

Let p0 be the probability of sending s0 and p 1 be the probability of sending s1 . For equally likely transmission of binary signals, we have p 0 = p 1 = 0.5. The average probability of error is given by Pe

= p0 Pe0 + p1 Pe1 = P e0 ∞ =

∫

1

2 φ1 = d / 2 2πσ

e

φ12 − 2σ 2

dφ1

30.2

(30.7)

Signal Space Analysis of BASK, BFSK, BPSK, and QAM on Mac

φ12 φ1 dφ1 2 Let y = . Then y = and d y = . Substituting y and dy into 2σ 2σ 2σ 2 equation (30.7), we get ∞ Pe

= y=

1 e-y2 dy π

∫

d 2 2σ

2 1 = 2[ π

∞

∫

e-y2 dy ]

y 1 = 2 erfc( d ) 2 2σ 1 = 2 erfc( d ) 2 n0 E 1 ) = 2 erfc( 4n0

(30.8)

where the complementary error function is

erfc (x) =

2 π

∞

∫

e-α 2 d α

(30.9)

x

σ2 = n0/2 for AWGN, and d =

E.

Error Performance of Binary PSK A binary phase-shift keying (BPSK) signal can be defined by s(t) = +A cos 2πfct,

0