Performance of Sigma–Delta Quantizations in Finite Frames

In this paper, we consider sigma-delta (SD) quantization of geometrically uniform (GU) finite frames. In the first part, we prove that under some conditions, the variant I and II permutation modulation (PM) codes, first introduced by Slepian (1968, 1965), can belong to the class of GU frames. Then, we focus essentially on a subclass of a GU frame, namely, cyclic geometrically uniform (CGU) frame, family of frames containing finite harmonic frames (both in \BBC
M
and \BBR
M
). For first- and second-order SD quantizers, we establish that the reconstruction minimum squares error (MSE) behaves as [ 1/(
r

²
)] where
r
denotes the frame redundancy. This result is shown to be true both under the deterministic quantization model used in Benedetto (2004), Benedetto (2006), and Yilmaz (2001) as well as under the widely used additive white quantization noise assumption. For the widely used
L
th-order noise shaping filter
G
(
z
)=(1-
z

^-1
)
L
, we show that the MSE behaves as [ 1/(
r

²
)] irrespectively of the filter order
L
. More importantly, we prove also that in the case of tight and normalized CGU frame, when the frame length is too large compared to the filter order and under some conditions on the quantizer, the reconstruction MSE can decay as fast as
O
([ 1/(
r

²

L
+1)]). Finally, it is shown that degrees of freedom in CGU frames, when compared to harmonic frames, result in a smaller MSE, albeit MSE prop [ 1/(
r

²
)] in both cases.