# When is noise shaping useful?

Started by August 14, 2012
```On 8/14/12 9:42 PM, Robert Adams wrote:
> On Tuesday, August 14, 2012 8:59:01 PM UTC-4, robert bristow-johnson wrote:
>> On 8/14/12 6:41 PM, Vladimir Vassilevsky wrote:
>>
>>
>>>> However when you follow this through to it's logical conclusion, you end
>>>> up with using the same coefficients in your noise shaping
>>>> loop as you use in the recursive iir section, which turns out to be simply
>>>> double precision.
>>
>>> This is not the same as doing filter in double precision.
>>>
>>
>> yeah, Bob.  i agree with Vlad.  it's not the same.
>>
>
> Assume you have a 2nd order biquad using direct form 1. The only quantization error occurs at the output of the summer that combines the b0, b1, b2, a1, and a2 terms. The output contains the quantization shaped by 1/(1 + a1*z^-1 + a2*z^-2).
> So now lets take the low bits from that summer that we would have thrown away, and apply to 2 cascaded delays with coefficients aa1 and aa2 feeding back to the same summation node. Since we know that the quantization noise reponse has a peak at the pole frequency, we might want to have the aa1 and aa2 coefficients selected such that the error feedback 2-tap FIR has a corresponding dip to minimize the output quantization noise. If you work through the math you will discover that to whiten the output noise you should set aa1 = a1 and aa2 = a2. When this is done, it can be seen that you have really just applied double-precision math to the recursive portion of the IIR.
> I'll try to dig up the references on this, but it was well documented some years ago.

i agree that in the special case of the feedback coefs for noise shaping
matching the IIR feedback coefs, that the result is white and it's
equivalent to single times double precision (where you toss the least
significant word).  that's pretty clear if the quantization is always
rounding down (just dropping the bits and feeding back those dropped
bits, zero-extended).  it's just splitting the LS word and MS word and
applying the distributive property we learn in grade 7.

but noise shaping is more general than that.  the simplest noise
shaping, where whatever bits you drop are fed back zero-extended (with a
gain of 1) to the same quantization for the following sample, that is
not equivalent to just doing double-precision.  this method has been
called "fraction saving" by Randy Yates and is *very* inexpensive and
takes care of that limit-cycle problem where the IIR gets stuck on a
non-zero DC value when the input goes to zero.  in my opinion, *every*
fixed-point audio IIR should use this method at every quantization
point.  and Direct Form 2 should never be used, but DF1 still works
pretty good for me, unless there is a lot of coefficient modulation,
then some lattice structure is usually better.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```Robert

I think we agree. If you are custom designing your own hardware, using nois=
e shaping gives you the freedom to use a lower order or maybe fewer bits, r=
esulting in less hardware. However when you are coding on a commercial dsp,=
doing the double precision feedback may only take a few more cycles, and y=
ou have the hardware anyway so you might as well use it. But even on a comm=
ercial dsp I agree that 1st order shaping could be done with less overhead =
than full double precision.=20

Bob
```
```"Robert Adams" <robert.adams@analog.com> wrote in message

>1) Assume you have a direct form 1 biquad with a single adder that sums
>together all the feed-forward and feedback terms. This >adder will have
>extended resolution due to the multiplication of the data and the
>coefficients.

That is one particular case.
There could be more then one biquad in sequence and not necessarily
implemented as DF1.

>3) The traditional view of error shaping is that you view the feedback
>filter as a noise-shaping filter, and design it to minimize the >output
>noise.

Noise shaping is intended to minimize noise in the band of interest. It is
not the same as total noise.

>5) The last step is to realize that this is the same as double precision
>just applied to the recursive section of the filter, assuming
> that the coefficients in the recursive part of the biquad filter are the
> same as the coefficients used in the noise-shaping filter;

Yes, on assumption of noise from 0 to Nyquist and a single section DF1.

> I have read papers that indicate that if you want to minimize the rms
> noise, the best thing you can do is to whiten the noise spectrum, > so I
> think this scheme is optimal

The output noise of this scheme won't be white; it would be double precision
LSB noise shaped by recursive part of the filter.

>, and that other alignments of the noise-shaping filter give worse
>performance.

It depends.

> In other words, putting an infinite noise-shaping notch at the pole
> frequency may minimize the noise in that particular part
> of the spectrum, but you will pay the price in that the energy in other
> parts of the spectrum will become larger.

You can draw noise spectrum as log (energy) and log(frequency). If you dig a
hole at some place in that spectrum, you will have to put no less then the
same amout of ground on top at some other place in that spectrum.

DSP and Mixed Signal Consultant
www.abvolt.com

```
```Vlad

I agree with your points. I think you are agreeing with me that if the goal=
is to minimize the un-weighted noise from dc to pi then matching the coeff=
icients (=3D double precision)  is optimal. =20

With regards to the output noise, if you keep the full precision of the out=
put then you are correct that you still have truncation noise that is shape=
d by the poles, but much further down in amplitude.  I was assuming that yo=
u truncate to single precision on the way out in which case the truncation =
noise would swamp out the shaped noise, but this depends on how it is coded=
.

I know that in audio there was a proposal for physchoacoustic noise shaping=
that was called Sony super-bit-mapping or something like that. I believe i=
t came from a guy at the University of Waterloo.  In this case the goal was=
to minimize the physchoacoustic weighted noise and I believe his shaping s=
ection was fairly high order.  So that is an example of shaping the quantiz=
ation noise to meet a specific non-white target spectrum, although this was=
not in the context of an iir filter.

Bob
```
```On 8/19/12 12:58 PM, Robert Adams wrote:
>
> I agree with your points. I think you are agreeing with me that if the goal is to minimize the un-weighted noise from dc to pi then matching the coefficients (= double precision)  is optimal.
>

yeah, but Bob, that's the case with, say, simple dither.  the dither of
the lowest unweighted mean-square is flat or white dither.

now you remember that Gerzon-Craven result using information theory.
the integral of log(S/N) adds to the bit rate and noise shaping always
trades this log(S/N) off with the case of minimum noise power being the
case when it's flat.

but the point is we trade off a quantization noise power for audibility
of quantization noise.  the minimum audible noise is not likely the same
as the minimum power noise.

> With regards to the output noise, if you keep the full precision of the output then you are correct that you still have truncation noise that is shaped by the poles, but much further down in amplitude.  I was assuming that you truncate to single precision on the way out in which case the truncation noise would swamp out the shaped noise,

what does this mean, Bob?  what is the difference between the (net)
truncation noise and "shaped noise"?  the truncation noise is always
shaped, even if you're doing nothing other than truncation, no?

> but this depends on how it is coded.
>
> I know that in audio there was a proposal for physchoacoustic noise shaping that was called Sony super-bit-mapping or something like that. I believe it came from a guy at the University of Waterloo.  In this case the goal was to minimize the physchoacoustic weighted noise and I believe his shaping section was fairly high order.  So that is an example of shaping the quantization noise to meet a specific non-white target spectrum, although this was not in the context of an iir filter.

yup.  and they even had (or proposed) SBM-2 which was dynamically
adjusted like compressed audio quantization error is.  but this is not
compressed.  it's just dynamically-shaped quantization noise and
intended to be dynamically shaped in such a way to be minimally audible.
perhaps the best way to put (or "master") 32-bit music onto 16-bit red
book CDs.

you can kinda emulate it with MATLAB by segmenting the audio and
computing (with FFT) the magnitude of the spectrum and adding a
specially weighted copy of that to something that looks like the 0-dB
curve of the Fletcher-Munson curve (or something more modern).  then
define a bunch of feedback coefs to hit that target spectrum.  the noise
would be "pumping", but if you synchronize it well (you have to delay
the audio a little to line it up), it should pump up and down with the
audio energy on a frequency band-by-band basis.  if the audio goes down
to zero, the noise spectrum becomes shaped like the Fletcher-Munson
curve so should be minimally audible in that case.

On 8/19/12 9:50 AM, Vladimir Vassilevsky wrote:
>
>> , and that other alignments of the noise-shaping filter give worse
>> performance.
>
> It depends.

yeah, i'm not sure that all non-flat alignments are worser.  but i
agree, in a filter where the quantized signals are fed back, you have to
include the shaping the poles do.  you would also have to do that for
regular non-shaped quantization (dithered or not).

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```Sorry for using confusing terminology.  What I meant was that it is common =
to use double precision in the feedback path, but just use the high accumul=
ator bits as the final filter output. In this case the output noise is domi=
nated by simple truncation noise, or you could add dither before the trunca=
tion to make things nicer. The truncation that happens in processing the lo=
w order bits is still shaped by the filter poles, but unless the filter Q i=
s extreme it's unlikely it will rise to the level where it dominates the qu=
antization noise introduced by the output truncation.=20

Bob
```
```On Tue, 14 Aug 2012 10:17:47 -0700, benjamin.couillard wrote:

> Hi everyone,
>
> I've implemented a 6-order IIR in an FPGA (technically an order 2 with
> scattered look-ahead with pole-zero cancellation, check
> http://www.ece.umn.edu/users/parhi/SLIDES/chap10.pdf).
>
> I've tried using noise shaping (1st-order) to get away with using less
> bits in my feedback loop (as I did sucessfully in a 1st-order IIR) but
> it only added oscillation to the output. So basically, I was wondering
> are there way to use noise-shaping with high-order IIR filters? Can
> anyone suggest a good reference on the subject?
>
> Thanks
>
> Benjamin

You do realize that a 1st-order filter with noise shaping (or fraction
saving, whatever you want to call it) is going to give you an oscillation?

Where did you apply your noise shaping?  What form of filter did you use
(or, alternately, what is your difference function?)  Noise shaping (or
fraction saving, or whatever you want to call it) is a nonlinear
operation, so the filter topology matters a lot.

I would be inclined to try breaking the filter into a 1st-order low-pass
filter and an integrator, with feedback to implement the resonant pole:

in -- + --> H1(z) ---> H2(z) --o-->
A                        |
'--------- z^-1 ---------'

where H1(z) = (1-d)z/(z-d), H2(z) = kz/(z-1), and d and k are adjusted to
give the desired response (you should be able to get an arbitrary
response this way).  Then do noise shaping at both H1 and H2.

I couldn't guarantee that you'd get less oscillation, but it may fly.

--
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com
```