Asymptotic quantization error of continuous signals and the quantization dimension

Extensions of the limiting qnanfizafion error formula of Bennet are proved. These are of the form
D_{s,k}(N,F)=N^{-\beta}B
, where
N
is the number of output levels,
D_{s,k}(N,F)
is the
s
th moment of the metric distance between quantizer input and output,
\beta,B>0,k=s/\beta
is the signal space dimension, and
F
is the signal distribution. If a suitably well-behaved
k
-dimensional signal density
f(x)
exists,
B=b_{s,k}[\int f^{\rho}(x)dx]^{1/ \rho},\rho=k/(s+k)
, and
b_{s,k}
does not depend on
f
. For
k=1,s=2
this reduces to Bennett's formula. If
F
is the Cantor distribution on
[0,1],0<k=s/ \beta=\log 2/ \log 3<1
and this
k
equals the fractal dimension of the Cantor set
[12,13]
. Random quantization, optimal quantization in the presence of an output information constraint, and quantization noise in high dimensional spaces are also investigated.