Page 111 - ISCI’2017

The channel capacity with uniformly distributed noise half as much again as the
(45)
Gaussian channel capacity calculated for equipotent signal and noise!

Now let’s use the geometric method, discussed in Sec. 2, to determine channel capacity with
uniformly distributed noise. To that end, we compare the geometric representation and the

characteristics of uncertainty spheres of (Fig. 2), within which the signal points are shifted by the

action of normal and uniform noise. Let’s introduce the concept of normalized (to the dimension of
n
the signal space ) displacement of a signal point under the influence of noise:

 1 n  12
− for Gaussian noise r =  ξ n i  ∑ 2 ; (46)
n
 n i1 = 
− for uniformly distributed noise

12
 1 n 
r =  u ξ i  ∑ 2 ; (47)
u
 n i1= 
1,
where n, u, iξ ξ ∈ [ ] n – random value i -th coordinates of additive noise for a normal and uniform
i i
noise respectively. The probability distribution densities of these quantities are determined by the

formulas

1  ξ n  2
f ξ ( ) n = exp −   ; (48)
2N   2N  
π

 1 12 N , при u ξ ∈ − 3N, 3N ;  ⋅ ⋅
⋅
f ( ) u ξ =    (49)

 0, при ξ> 3 N .
n
⋅

Normalized radii of uncertainty spheres r n and r u for two noise distributions, under
consideration, are determined by the mathematical expectation of random variables (46) and (47),

which are the functions of random summands having the PDF (48) and (49), and "delineation degree"
of the spheres determined by their dispersion [ ] и [ ] . The analytical result for normal noise
Dr
Dr
u
n
is known [9]:
2N  n1 n  
+ 
[ ] =
r = n Mr n n   Γ  2  Γ   2 ; (50)


 2   n1 n   2  
+ 
[ ] =
Dr n N 1−   Γ   Γ   . (51)
 n   2   2   
Analytical calculation of similar numerical characteristics for the uniformly distributed noise r u

and [ ] is difficult because a multidimensional compositional PDF of a random variable (47) is a
Dr
u
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