Round off spectrum spikes in 1/2 LSB rounding when performing fixed point filtering

Even when a filter's output is less than one, internal results can exceed it. That's the primary need for wide accumulators.

Jerry
-- 
Engineering is the art of making what you want from things you can get.

On Apr 5, 9:13&#4294967295;pm, Jerry Avins <j...@ieee.org> wrote:
> Even when a filter's output is less than one, internal results can exceed it. That's the primary need for wide accumulators.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.

From what I know, DF I has a property that any internal overflow will
be wrapped back to the correct final value due to properties of 2-
complement. Do correct me if I've mistaken.

> On Apr 5, 9:13&#4294967295;pm, Jerry Avins <j...@ieee.org> wrote:
>
> > Even when a filter's output is less than one, internal results can exceed it. That's the primary need for wide accumulators.

that's the primary use for guard bits on the left.  but by "double
wide accumulator" i meant the extra guard bits on the right which
eliminate any quantization error from the multiply and multiply-
accumulate instructions.


On Apr 5, 8:46 pm, Zhi Ping Ang <angzhip...@gmail.com> wrote:
>
> From what I know, DF I has a property that any internal overflow will
> be wrapped back to the correct final value due to properties of 2-
> complement. Do correct me if I've mistaken.

some of this needs correction.

this is the fact from what you are referring:  for twos-complement
(doesn't matter if it's DF1 or DF2 or some other form), **if** you
knew in advance that the final value would not overflow the bounds for
your word and if there were intermediate sums (before the final sum)
that *did* exceed the bounds, if the intermediate value *wraps* around
rather than saturate, then that overflow does not hurt you.  but you
have to insure that there will be later additions that will bring the
value back within bounds.  otherwise wrap-around overflow is much
worse than the clipping overflow that most DSPs do.

but most DSPs have guard bits on the left so you can have the best of
both worlds; if the end result *does* exceed the bounds, the DSP will
know that and will (if instructed to) saturate or clip the value,
which is distortion but *much* better than wrap-around distortion.
and if the end result does not exceed the bounds, the DSP will know
that and will return the correct unclipped value.

now the problem with the DF2 and fixed-point is, *if* your biquad
filter has highly resonant poles (that means poles that are getting
rather close to the unit circle), then internally the signal will get
boosted by many dB at frequencies in the neighborhood of the resonant
frequency and then get cast back into single-width words to be used in
the latter half of the biquad section where the feedforward
coefficients (that are determined by the zeros and the overall gain)
scale it.  so the danger of the DF2 is that the signal is boosted and
clipped by the poles before the zeros (that are likely very close to
the poles to kinda "tame them down") can do anything about it.  once
there is clipping there is nothing that the zeros can do about the
clipping, even though they will likely reduce the amplitude of the
whole signal along with the clipped part.

with the DF1, the effect of the zeros come before the poles, so the
signal (or the part of it close to the resonant frequency) will get
reduced in amplitude by the zeros before the poles boost it.  now *if*
you had to cast the intermediate word back to a single-word width,
then the DF1 would have a corresponding flaw to the DF2, but instead
of clipping, it would be the poles boosting the quantization noise
added to the weakened (by the zeros) signal.

but with a double-wide accumulator *and* using DF1, then there is no
quantization of the intermediate signal coming from the feedforward
subsection.  so you avoid internal clipping (assuming that there is no
overall clipping of the biquad filter) *and* you don't have the
resonance-boosted quantization noise.  also, there is only 1
quantization point per biquad section with DF1 and a double-wide
accumulator whereas the DF2 would have 2 sources of quantization per
biquad section.

in addition (as i've illustrated with the little code example), you
can perform "fraction saving" a.k.a. "first-order noise shaping with a
zero at z=1" which will kill any error or limit-cycling at DC (and
steers error noise and/or limit-cycling to the Nyquist frequency where
we might be more tolerant of it).  and this fraction saving is easy
and cheap to do.

r b-j

On Apr 6, 11:06=A0am, robert bristow-johnson <r...@audioimagination.com>
wrote:
> > On Apr 5, 9:13=A0pm, Jerry Avins <j...@ieee.org> wrote:
>
> > > Even when a filter's output is less than one, internal results can ex=
ceed it. That's the primary need for wide accumulators.
>
> that's the primary use for guard bits on the left. =A0but by "double
> wide accumulator" i meant the extra guard bits on the right which
> eliminate any quantization error from the multiply and multiply-
> accumulate instructions.
>
> On Apr 5, 8:46 pm, Zhi Ping Ang <angzhip...@gmail.com> wrote:
>
>
>
> > From what I know, DF I has a property that any internal overflow will
> > be wrapped back to the correct final value due to properties of 2-
> > complement. Do correct me if I've mistaken.
>
> some of this needs correction.
>
> this is the fact from what you are referring: =A0for twos-complement
> (doesn't matter if it's DF1 or DF2 or some other form), **if** you
> knew in advance that the final value would not overflow the bounds for
> your word and if there were intermediate sums (before the final sum)
> that *did* exceed the bounds, if the intermediate value *wraps* around
> rather than saturate, then that overflow does not hurt you. =A0but you
> have to insure that there will be later additions that will bring the
> value back within bounds. =A0otherwise wrap-around overflow is much
> worse than the clipping overflow that most DSPs do.

This has been guaranteed. I used the method of dividing the numerator
coefficients by the absolute sum of the impulse response, so that the
output y[n] is in the range +/- 1 given x[n] is in the range of +/- 1,
but at the expense of my filter gain. The chances of a +/- 1 output
happening is also very remote, as the input signs have to match with
the signs of the impulse response.

>
> but most DSPs have guard bits on the left so you can have the best of
> both worlds; if the end result *does* exceed the bounds, the DSP will
> know that and will (if instructed to) saturate or clip the value,
> which is distortion but *much* better than wrap-around distortion.
> and if the end result does not exceed the bounds, the DSP will know
> that and will return the correct unclipped value.
>
> now the problem with the DF2 and fixed-point is, *if* your biquad
> filter has highly resonant poles (that means poles that are getting
> rather close to the unit circle), then internally the signal will get
> boosted by many dB at frequencies in the neighborhood of the resonant
> frequency and then get cast back into single-width words to be used in
> the latter half of the biquad section where the feedforward
> coefficients (that are determined by the zeros and the overall gain)
> scale it. =A0so the danger of the DF2 is that the signal is boosted and
> clipped by the poles before the zeros (that are likely very close to
> the poles to kinda "tame them down") can do anything about it. =A0once
> there is clipping there is nothing that the zeros can do about the
> clipping, even though they will likely reduce the amplitude of the
> whole signal along with the clipped part.
>
> with the DF1, the effect of the zeros come before the poles, so the
> signal (or the part of it close to the resonant frequency) will get
> reduced in amplitude by the zeros before the poles boost it. =A0now *if*
> you had to cast the intermediate word back to a single-word width,
> then the DF1 would have a corresponding flaw to the DF2, but instead
> of clipping, it would be the poles boosting the quantization noise
> added to the weakened (by the zeros) signal.

I do not get the explanation about the effect of poles and zeros
coming before or after one another. Aren't we not multiplying inputs
and previous outputs with filter coefficients? I do know that the
coefficients affects the locations of the poles and zeros, but just
can't make the logical link.

>
> but with a double-wide accumulator *and* using DF1, then there is no
> quantization of the intermediate signal coming from the feedforward
> subsection. =A0so you avoid internal clipping (assuming that there is no
> overall clipping of the biquad filter) *and* you don't have the
> resonance-boosted quantization noise. =A0also, there is only 1
> quantization point per biquad section with DF1 and a double-wide
> accumulator whereas the DF2 would have 2 sources of quantization per
> biquad section.
>
> in addition (as i've illustrated with the little code example), you
> can perform "fraction saving" a.k.a. "first-order noise shaping with a
> zero at z=3D1" which will kill any error or limit-cycling at DC (and
> steers error noise and/or limit-cycling to the Nyquist frequency where
> we might be more tolerant of it). =A0and this fraction saving is easy
> and cheap to do.
>

I should say I'm pretty impressed with fraction saving. I've changed
my filter implementation and now it exhibits a far stable magnitude
spectra as compared to previous designs which does not use fraction
saving. But the problem of spikes still appear in the spectra. Upon
inspection of the impulse response, the output goes like this (an
actual example of a filter with 16-bit fractional precision): {0,
3.05e-5, 3.05e05, 0, -3.05e-5 -3.05e-5, 0, 3.05e-5, 3.-5e-5...}. These
spikes are much more pronounced for bandpass filters with center
frequencies towards fs/4.

I generated filter coefficients usign MATLAB. I suspect it is the way
MATLAB double floating point values are converted into fixed point
format. Are there any sort of guidelines about how floating points
should be rounded to give fixed point filter coefficients?

> r b-j