Problem A real sequence satisfies
for all
, where
. Let
be the length of the
th interval. Show that there exists a unique
such that
for all
.

Fig. 1. Nonlinearly mapped intervals (pdf version)
Remark This is a problem that arises from nonlinear quantization in AAC. There, MDCT coefficient is power-compressed to
before linear quantization. Then we face the problem of choosing boundaries
for the
th quantization interval where all
are quantized to
and de-quantized to
, such that the de-quantization noise
is minimized statistically. Of course, the solution to this problem depends on the underlying probability distribution of
. In case of uniform distribution, the de-quantized value
should be at the center of the
th interval, or
. With this restriction, all the boundaries
are uniquely determined by
. But the length of the interval
, as revealed by numerical experiments, almost always undulates and at the same time, asymptotically increases to
as
(
), for a randomly chosen
. Then, is there a
to make the length of the interval monotonically increase? The answer is yes, and this
is uniquely determined by
, as shown in the following.
Proof Let us denote the difference of th and
th intervals’ lengths by
. We claim that
for all
. If so, to ensure
for all
, then we need to show that there exists a unique
such that both
and
exist and are non-negative. First, by the relation
, we have
. Furthermore, let
, then
where and the last equation is due to recursive application of Rolle’s theorem (first about
, then about about
). Since
for
when
, we have
thus
. Next, consider the limit of
:
where the last equation is due to for
. Thus, the desired
, if it ever exits, must ensure
. Conversely, if with some
,
, then
for all
.
Now, if suffices to show there exists a unique such that
. Let
, then
. We claim that
converges as
. In that case, we shall have
where and the last equation above follows from
when
. Therefore, the unique
will be
. It remains to show that
does converge.
Let us investigate the difference between and
:
where and the last equation is also due to recursive application of Rolle’s theorem. (First about
, then about
, and finally about
.) Therefore, we have
, which implies that
converges.
This completes the proof.