9.6 An Expression for BER with Gaussian Noise.pdf

An Expression for BER with Gaussian Noise

The Q-function f v(v) •

•

Suppose that v is Gaussian with m = 0 and σ = 1. The probability that v is greater than a particular value T is given by the Qfunction

Q(T)  P[v  T]

•

•

Q(T) = P[v>T]

P[v≤T]=1-Q(T)

There is no closed-form expression for Q(T). Its value must be found numerically, e.g. –

from tables, or

–

the MATLAB function qfunc(T)

Properties

P[v  T]  1  Q(T) Q(0)

1

m = 0,

σ

=1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1-Q(T) 0.2 P[v≤T] 0.1 0 -3 -2 -1

v

T

P[v>T] Q(T) 0

1

2

3

Probabilities for Other Gaussians If y is Gaussian with my ≠ 0 and σy = 1,

•

f v

f y

0.2

mv



0

my



2





1



1



0 -5

0

5

y

1

v y

0.4 f v 0.2 f y 0

-5

0

mv



0

my



0





1



2



5

y

1 Q(T)

Q(T-2) Q(T) 0.5

0.5 Q(T/2)

0

T

•

T P[y > T]= Q   

P[y > T] = Q T - my 0.4

If y is Gaussian with my = 0 and σy ≠ 1,

0

T

v y

In general, if y is Gaussian with my ≠ 0 and σ y ≠ 1,

 T - my P[y > T]= Q    y 

  

Expressions for Pe0 and Pe1 •

If IN = 0, there is an error if – OUT = 1 – The noise pushes y above T

f(y if IN=0) Pe0

Pe0  P[y  T if IN  0]

 T  rmin   Q   σ  •

If IN = 1, there is an error if – OUT = 0 – The noise pushes y below T

Pe1  P[y  T if IN  1]

 T  rmax   1  Q  σ  

rmin

T

rmax

y

f(y if IN=1)

Pe1

T

y

Predicting BER  If 0 and 1 input bits are equally likely

BER  12 Pe0  12 Pe1  12 Q



T rmin σ

  12 1  Q

f(y if IN=0)



T rmax σ



f(y if IN=1)

½

Pe0

½ Pe1

rmin

T

rmax

rmin

y ½f(y if IN=1)

½f(y if IN=0)

BER

rmin

T

rmax

y

T

rmax

y