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Ron N.
Guest
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 Posted:
Fri Dec 16, 2005 12:55 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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abariska@student.ethz.ch wrote:
| Quote: | Jerry Avins wrote:
abariska@student.ethz.ch wrote:
...
You claim that the infinitely long sequence
b[n] = sinc(n + 1/7),
for all n, is linear-phase. That could be true. Do you have an idea for
a proof?
Over all n, sinc(n) is symmetric, hence linear phase. Sinc(n + 7) has a
pure delay added to it. Pure delay doesn't upset linear phase. Over the
range -18 <= n <= 32, sinc(n +7) is also linear phase. It's symmetry
about the center that counts.
But the delay in b[n] is fractional, 1/7 sample, and the resulting
windowed and sampled sinc does not have linear phase response. It is
interesting that with infinite many samples, strict symmetry of the
coeffcients is not needed anymore for linear phase.
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Not only infintely many samples, but infinite precision as well,
are needed to get linear phase out of a non-integer delay of
a sinc.
It's very a useful fact that symmetric coefficient sets do not need
infinite width or precision to have exact perfect linear phase (as
long as the truncation and quantization is symmetric as well,
of course).
IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M |
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Bob Cain
Guest
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| Posted:
Fri Dec 16, 2005 1:02 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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abariska@student.ethz.ch wrote:
| Quote: | Jerry Avins wrote:
abariska@student.ethz.ch wrote:
...
You claim that the infinitely long sequence
b[n] = sinc(n + 1/7),
for all n, is linear-phase. That could be true. Do you have an idea for
a proof?
Over all n, sinc(n) is symmetric, hence linear phase. Sinc(n + 7) has a
pure delay added to it. Pure delay doesn't upset linear phase. Over the
range -18 <= n <= 32, sinc(n +7) is also linear phase. It's symmetry
about the center that counts.
But the delay in b[n] is fractional, 1/7 sample, and the resulting
windowed and sampled sinc does not have linear phase response. It is
interesting that with infinite many samples, strict symmetry of the
coeffcients is not needed anymore for linear phase.
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Regardless of the number of samples, it isn't symmetry of the
coefficients that causes linear phase but rather symmetry of the
continuous function that is reconstructed from the samples by a band
limiting brick wall lowpass. Thus a sinc delayed by any fraction of a
sample is still symmetric because the underlying continuous function is.
One way to see why symmetry gives linear phase is to consider the
logically equivalent filtering of a signal in the forward direction by
the leading half of the filter (evenly symmetric for ease of
visualization) and in the reverse direction by the reverse of the
trailing half. Any frequency dependent phase shift induced in the
forward direction is exactly canceled by that induced in the reverse
direction because the two filters are identical after reversal of one of
them.
Bob
--
"Things should be described as simply as possible, but no simpler."
A. Einstein |
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Guest
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| Posted:
Fri Dec 16, 2005 1:06 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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Ron N. wrote:
| Quote: | Not only infintely many samples, but infinite precision as well,
are needed to get linear phase out of a non-integer delay of
a sinc.
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Well, once you have that InfinyDrive (TM) computer to store the
infinite number of samples, I don't think you are going to worry much
about having to store an infinite number of digits for each sample :-).
| Quote: | It's very a useful fact that symmetric coefficient sets do not need
infinite width or precision to have exact perfect linear phase (as
long as the truncation and quantization is symmetric as well,
of course).
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Good point.
Regards,
Andor |
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robert bristow-johnson
Guest
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| Posted:
Fri Dec 16, 2005 1:16 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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abariska@student.ethz.ch wrote:
| Quote: | robert bristow-johnson wrote:
does that do it for you, Abariska?
In case you hadn't noticed, "abariska" stands for Andor Bariska. It's
my old and inactive student account (go spam that) :-).
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ooops. sorry Andor. currently i am spending about 5% of the time that
i used to hanging out here. i am not looking over the threads
completely so i miss context. sorry.
| Quote: | Sounds good to me. The truncated sinc in Randy's example was not
bandlimited,
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any sequence can be hooked up to a sequence of bandlimited sinc()
functions. but Randy's truncated sinc() sequence might not have
perfectly satisfied that symmetry about "d" condition necessary.
anyway i was only responding to your infinite sequence
| Quote: | and the aliasing when sampling messed up the phase
linearity. I think Randy's proof outline was somewhere on along your
lines as well.
What do you think of that theorem that I posted earlier from this
paper:
http://0xdc.com/paper.pdf
(at the bottom of the first page).
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it (the IEEE proof), to me, looks exactly like the argument i made.
they say "Omega" and i say "2*pi*f". otherwise they're the same. oh,
all right, their's was a little more general than mine. if you say
that Randy's proof is the same as mine and i say that mine was the same
as Clements & Pease, then doesn't that mean that Randy's is the same as
theirs? i didn't read the whole paper nor the whole thread.
BTW, i like this paper. gonna have to put it somewhere to keep.
| Quote: | I have a feeling that anti-symmetric linear-phase sequences are not
covered in the statement of the proof. This would make the "iff" claim
false.
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but i think you could make it happen, in the same way as is done for
the regular symmetric sequences.
r b-j |
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Bob Cain
Guest
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| Posted:
Fri Dec 16, 2005 9:16 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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Bob Cain wrote:
| Quote: | One way to see why symmetry gives linear phase is to consider the
logically equivalent filtering of a signal in the forward direction by
the leading half of the filter (evenly symmetric for ease of
visualization) and in the reverse direction by the reverse of the
trailing half.
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Scratch that. It's wrong.
Bob
--
"Things should be described as simply as possible, but no simpler."
A. Einstein |
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