So what happens if, as it's very likely to happen in real life,
the input is not bounded to a known interval A, B.
Well, we have two choices.
The first one is to clip sample.
So if the sample is smaller than A, we set it to A, and if it's greater than B,
we set it to B.
In this case, we introduced linear distortion in the signal and
this is really the principle behind distortion boxes for guitars.
So Interesting but not necessarily what we want to hear all the time.
Alternatively we can use a saturation curve to smoothly map
the input onto the desired range.
This is closer to the saturation characteristic of all the electronic
components such as tubes that are praised by audiophiles for
their lack of audible distortion.
We can plot the clipping and saturation occurence by taking for
instance the interval minus one one.
And we can see that the clipping curve would map
values outside of the range to the edge of the range.
Whereas the situation curve is linear around zero.
And then tapers off asymptotically.
If the input is not uniform, we can still use a uniform quantizer and
accept an hour of penalty.
For instance, if the input is Gaussian,
it can be shown that the mean square error has this form.
It depends now of course on the variants of the input.
But even for
input signals with unit variants it would be larger than square over 12.
Alternatively if we know the exact probability distribution function for
the input.
We can use the Lloyd-Max algorithm to design an optimal quantizer for the input.
Alternatively a common practice in audio signal processing is the use of
companders.
The idea is that samples from signals such as speech or music will be generally small
in amplitude with the occasional excursion into the outer reaches of the range.
So for instance, this is a mu law compander commonly used in radio.
And you see that small values, say between here and here,
get allocated to a wide number of quantization levels.
Whereas the rest of the range will share the other half.
So there's a, so say, 10% of the range will get 50% of
the quantization level, and the remaining 90% will get the remaining 50%.
So, we have more precision for the values that we know to be more probable.
Paolo PrandoniLecturer
Martin VetterliProfessor
