nal, 'T 
the v Probality 
e distribution 
: = of signal 
e Clean signal Noisy signal values 
S 
e ee — 
rm > [7177 
e Oo 
the 2 
the e 
luce «c J 
an 
pe op s > 
y, 1) 2 3 1 
a en 1 0 Area is proportional to 
3 T + probality of being in | 
0 0 ee a À the first level above correct | 
|] € 0 ; : | 
© + + -+ — Area is proportional to 
EA m Ya ie me - probality of being in 
the d 0 1 0 the correct level 
O fp il ————————Ááám—À—)—oiÜ 
"o 0 T Area is proportional to 
nce v ah [s2+12142<QL probality of being in 
Most the first level below correct 
significant 
) bit 
fig. 6.1 
For a clean signal contaminated with noise there will be a nonzero 
probability of assigning an incorrect digital level in the analogue 
the digital conversion ( after Billingsley, 1975 |. 
The probability distribution of signal and noise can be found by 
convolving the distribution of the signal with that of the noise. The 
probability of correct digitizing is found by integrating the joint 
distribution between limits representing the quantization interval 
boundaries. A graphic representation of these probabilties as a 
function of B is given in fig. 6.2. 
rom The analysis can be carried further ( Fulz, 1965) to provide for 
red determination of quantization within + N levels of the correct value. 
nal This probability is given by the expression 
te 
e n pe CEN) z (N«1) erf((n«1)a]-N erf(Na)-(exp(-N a? )-expL- (N«1)^a7 )) /a/v 
to 
( 6.1 ) 
ps. where a=quantizing step size/o/2 and N denotes the number of levels 
ven from the correct level. If N=0 the expression reduces t the 
nal probability of correct classification and erf(x) - 2//w f exp(-t ) dt 
we is evaluated from 0 to x. 
of 
bi- 
tep 
149