| Author |
Message |
robert bristow-johnson
Guest
|
 Posted:
Sat Jan 08, 2005 11:21 pm Post subject:
Re: Noise floor / FFT relationship |
|
|
in article 41de9b3f.71798156@news.sf.sbcglobal.net, Rick Lyons at
r.lyons@_BOGUS_ieee.org wrote on 01/07/2005 09:28:
| Quote: | On Mon, 03 Jan 2005 15:30:07 -0500, robert bristow-johnson
rbj@audioimagination.com> wrote:
a compliment from DSP gurus like you
two guys means a lot to me.
|
it (the compliment) is deserved. i don't peruse your book often enough.
one reason is that i start getting into something very interesting that i
never thought of (like now it's FSF) and "waste" about 4 or 5 hours reading,
thinking, and scribbling math about it. then my wife and/or kids start
getting mad at me and/or nag.
| Quote: | an old issue: we still need to talk about the DFT of all ones where N = an
even number. there was something not so kosher in your treatment of it as i
recall, but i have to familiarize myself with it again. i think there is a
phase problem or something because if N is even, you really cannot have a
symmetric spectrum about DC. there will always be at least one more "one"
on either the left or the right.
....
Not knowing what "kosher" means, I'm not sure what to say.
|
you need to spend some time in Brooklyn. those guys all dressed up in
black, wearing big long beards are neither Amish nor ZZ Top.
| Quote: | As you know, the DFT of an all-ones time sequence is a
sequence starting with a non-zero valued sample followed by
samples that are all zero-valued.
|
this was an unfinished discussion of a while ago. i have trouble with
3.13.3 "DFT of an All One Rectangular Function" when N is even and the
rectangle is considered to be symmetrical as depicted on Figure 3-31.
N even and "symmetrical" are mutually exclusive, IMO.
lemme ruminate about is some more and get back on it.
| Quote: | Although my understanding of symmetry has changed
recently. I now believe that there are two types of
symmetry.
|
not sure what that is.
best regards, Rick.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge." |
|
| Back to top |
|
 |
Jerry Avins
Guest
|
| Posted:
Sun Jan 09, 2005 6:14 am Post subject:
Re: Noise floor / FFT relationship |
|
|
robert bristow-johnson wrote:
...
| Quote: | N even and "symmetrical" are mutually exclusive, IMO.
|
Are you considering the cyclical nature of bandlimited signals? Even
without, abba is symmetric about a point between the two b's.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
|
| Back to top |
|
|
robert bristow-johnson
Guest
|
| Posted:
Sun Jan 09, 2005 7:57 am Post subject:
back to this dirchlet thingie (was: Noise floor / FFT relati |
|
|
in article 34beqvF484qn4U1@individual.net, Jerry Avins at jya@ieee.org wrote
on 01/08/2005 20:14:
| Quote: | robert bristow-johnson wrote:
...
N even and "symmetrical" are mutually exclusive, IMO.
Are you considering the cyclical nature of bandlimited signals?
|
i sure hope so.
| Quote: | Even without, abba is symmetric about a point between the two b's.
|
but then it's a size 2N DFT where
x[2*n] = 0
and
x[2*n+1] = 1
and then it's true that
X[k] = sin(pi*k)/sin(pi*k/N)
that X[k] repeats every 2*N rather than every N. but it *is* sorta
half-band symmetric. the first N values are the negative of the latter N.
for a size N DFT (N even) with
x[n] = 1 for all n
then
X[k] = exp(-pi*k/N)*sin(pi*k)/sin(pi*k/N)
or something like that. why this is periodic with period N (instead of 2N)
is because the complex phase factor flips the polarity every N samples
undoing the sign reversal of the "kernal". that's why i'm having some
trouble with Eq. (3-48) in Rick's book because it's correct for odd N but
can't be completely kosher for even N.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge." |
|
| Back to top |
|
|
jim
Guest
|
| Posted:
Mon Jan 10, 2005 4:07 am Post subject:
Re: Noise floor / FFT relationship |
|
|
Tim Wescott wrote:
| Quote: |
an old issue: we still need to talk about the DFT of all ones where N = an
even number. there was something not so kosher in your treatment of it as i
recall, but i have to familiarize myself with it again. i think there is a
phase problem or something because if N is even, you really cannot have a
symmetric spectrum about DC. there will always be at least one more "one"
on either the left or the right.
Well, you can have a symmetric spectrum if you remember that the DFT
spectrum is periodic -- the Fs/2 bin is also the -Fs/2 bin, so its
contents are automatically symmetric around 0.
|
But, you've completely missed the point. In order for the X[n] to be
symmetric x[n] need only be real. Apparently, a real all ones sequence
that is even in length violates some ancient religious code ergo such a
sequence must never be real and thus "you really cannot have a symmetric
spectrum about DC".
-jim |
|
| Back to top |
|
|
Tim Wescott
Guest
|
| Posted:
Mon Jan 10, 2005 4:58 am Post subject:
Re: Noise floor / FFT relationship |
|
|
jim wrote:
| Quote: | Tim Wescott wrote:
an old issue: we still need to talk about the DFT of all ones where N = an
even number. there was something not so kosher in your treatment of it as i
recall, but i have to familiarize myself with it again. i think there is a
phase problem or something because if N is even, you really cannot have a
symmetric spectrum about DC. there will always be at least one more "one"
on either the left or the right.
Well, you can have a symmetric spectrum if you remember that the DFT
spectrum is periodic -- the Fs/2 bin is also the -Fs/2 bin, so its
contents are automatically symmetric around 0.
But, you've completely missed the point. In order for the X[n] to be
symmetric x[n] need only be real. Apparently, a real all ones sequence
that is even in length violates some ancient religious code ergo such a
sequence must never be real and thus "you really cannot have a symmetric
spectrum about DC".
-jim
|
Please give your references. "Violates some ancient religious code"
means that you or someone else has misunderstood something. Without the
reference nothing can be discussed.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
|
| Back to top |
|
|
Rick Lyons
Guest
|
| Posted:
Mon Jan 10, 2005 8:28 pm Post subject:
Re: Noise floor / FFT relationship |
|
|
On Sat, 08 Jan 2005 13:21:55 -0500, robert bristow-johnson
<rbj@audioimagination.com> wrote:
| Quote: | in article 41de9b3f.71798156@news.sf.sbcglobal.net, Rick Lyons at
r.lyons@_BOGUS_ieee.org wrote on 01/07/2005 09:28:
On Mon, 03 Jan 2005 15:30:07 -0500, robert bristow-johnson
rbj@audioimagination.com> wrote:
a compliment from DSP gurus like you
two guys means a lot to me.
it (the compliment) is deserved. i don't peruse your book often enough.
one reason is that i start getting into something very interesting that i
never thought of (like now it's FSF) and "waste" about 4 or 5 hours reading,
thinking, and scribbling math about it. then my wife and/or kids start
getting mad at me and/or nag.
an old issue: we still need to talk about the DFT of all ones where N = an
even number. there was something not so kosher in your treatment of it as i
recall, but i have to familiarize myself with it again. i think there is a
phase problem or something because if N is even, you really cannot have a
symmetric spectrum about DC. there will always be at least one more "one"
on either the left or the right.
...
Not knowing what "kosher" means, I'm not sure what to say.
you need to spend some time in Brooklyn. those guys all dressed up in
black, wearing big long beards are neither Amish nor ZZ Top.
As you know, the DFT of an all-ones time sequence is a
sequence starting with a non-zero valued sample followed by
samples that are all zero-valued.
this was an unfinished discussion of a while ago. i have trouble with
3.13.3 "DFT of an All One Rectangular Function" when N is even and the
rectangle is considered to be symmetrical as depicted on Figure 3-31.
N even and "symmetrical" are mutually exclusive, IMO.
lemme ruminate about is some more and get back on it.
Although my understanding of symmetry has changed
recently. I now believe that there are two types of
symmetry.
not sure what that is.
best regards, Rick.
|
Hi Robert,
years ago I thought I had a reasonable understanding
of what a symmetrical sequence was, but now I think
it's more complicated than I first thought.
I haven't "gathered" my thoughts on this issue, but
what made me realize that I have more to learn was
using a Hanning window. If you have a Hanning
sequence whose 1st & last samples are equal, then
that sequence looks "symmetrical" when you plot it
on paper. However, the DFT of that sequence does
*not* have all-zeros as its imaginary part.
So is that Hanning sequence really "symmetrical"?
Then again, if you have a Hanning window sequence
where the 1st and last samples are not equal, then
that sequence does not look "symmetrical" when
we plot it on paper. However, the DFT of that
second sequence does have all-zeros as its imaginary
part. So, ... is that second Hanning sequence
"symmetrical" or not?
I need to take some time out to see if I can
come up with sort of reasonable definition
of "symmetry" and then pass that definition
by you guys for your review.
[-Rick-] |
|
| Back to top |
|
|
Rick Lyons
Guest
|
| Posted:
Mon Jan 10, 2005 9:03 pm Post subject:
Re: back to this dirchlet thingie (was: Noise floor / FFT re |
|
|
On Sat, 08 Jan 2005 22:57:06 -0500, robert bristow-johnson
<rbj@audioimagination.com> wrote:
| Quote: | in article 34beqvF484qn4U1@individual.net, Jerry Avins at jya@ieee.org wrote
on 01/08/2005 20:14:
robert bristow-johnson wrote:
...
N even and "symmetrical" are mutually exclusive, IMO.
Are you considering the cyclical nature of bandlimited signals?
i sure hope so.
Even without, abba is symmetric about a point between the two b's.
but then it's a size 2N DFT where
x[2*n] = 0
and
x[2*n+1] = 1
and then it's true that
X[k] = sin(pi*k)/sin(pi*k/N)
that X[k] repeats every 2*N rather than every N. but it *is* sorta
half-band symmetric. the first N values are the negative of the latter N.
for a size N DFT (N even) with
x[n] = 1 for all n
then
X[k] = exp(-pi*k/N)*sin(pi*k)/sin(pi*k/N)
or something like that. why this is periodic with period N (instead of 2N)
is because the complex phase factor flips the polarity every N samples
undoing the sign reversal of the "kernal". that's why i'm having some
trouble with Eq. (3-48) in Rick's book because it's correct for odd N but
can't be completely kosher for even N.
|
Hi,
as far as I can tell, my Eq. (3-48) is correct.
If you plug an N-point all-ones sequence into the
DFT equation you get a finite-length summation
of a geometric series. If you convert that
series to a "closed form" equation you obtain
a DFT of a "phase factor" times a "magnitude
factor", or:
DFT = X(m) = Phase(m) times sin(pi*m)/sin(pi*m/N)
where "m" is the freq-domain index. For any
non-zero integer value of m the DFT's X(m) is
equal to zero. When m = 0, then X(0) is
indeterminant. But using the Marquis de L'Hopital
Rule we can determine that X(0) = N.
At least that's how I see it.
[-Rick-] |
|
| Back to top |
|
|
Peter K.
Guest
|
| Posted:
Mon Jan 10, 2005 9:44 pm Post subject:
Re: back to this dirchlet thingie (was: Noise floor / FFT re |
|
|
r b-j wrote:
] why this is periodic with period N (instead of 2N)
] is because the complex phase factor flips the polarity every N
] samples undoing the sign reversal of the "kernal". that's why
] i'm having some trouble with Eq. (3-48) in Rick's book because
] it's correct for odd N but can't be completely kosher for even N.
I think the non-kosher part you're coming across is clear
if you look at Figure 3-31.
If N is even, Rick's n_0 = (N-1)/2 means that the samples
start at a half-index offset... so that the whole sequence
of 1s is symmetric about 0, but the samples are at
+/- 0.5, +/- 1.5, +/- 2.5 etc. etc.
That's how the imaginary component manages to disappear:
the imaginary components cancel out pair-wise.
Ciao,
Peter K. |
|
| Back to top |
|
|
Bob Cain
Guest
|
| Posted:
Tue Jan 11, 2005 12:10 am Post subject:
Re: Noise floor / FFT relationship |
|
|
Rick Lyons wrote:
| Quote: | I need to take some time out to see if I can
come up with sort of reasonable definition
of "symmetry" and then pass that definition
by you guys for your review.
|
Isn't it simply whether the sinc reconstructed analog signal
is symmetric within the block regardless of length?
Bob
--
"Things should be described as simply as possible, but no
simpler."
A. Einstein |
|
| Back to top |
|
|
Rick Lyons
Guest
|
| Posted:
Tue Jan 11, 2005 11:41 pm Post subject:
Re: Noise floor / FFT relationship |
|
|
On Mon, 10 Jan 2005 11:10:07 -0800, Bob Cain
<arcane@arcanemethods.com> wrote:
| Quote: |
Rick Lyons wrote:
I need to take some time out to see if I can
come up with sort of reasonable definition
of "symmetry" and then pass that definition
by you guys for your review.
Isn't it simply whether the sinc reconstructed analog signal
is symmetric within the block regardless of length?
Bob
|
Hi,
humm, ... you seem to be looking at this
"symmetrical sequence" issue from a very different
viewpoint than I am.
You may have a "good point" here, but I don't
understand exactly what a "block" is with regard to
an analog signal.
It hadn't occurred to me to consider analog signals
in any way when trying to understand symmetrical
discrete sequences.
[-Rick-] |
|
| Back to top |
|
|
Bob Cain
Guest
|
| Posted:
Wed Jan 12, 2005 12:58 am Post subject:
Re: Noise floor / FFT relationship |
|
|
Rick Lyons wrote:
| Quote: | On Mon, 10 Jan 2005 11:10:07 -0800, Bob Cain
arcane@arcanemethods.com> wrote:
Rick Lyons wrote:
I need to take some time out to see if I can
come up with sort of reasonable definition
of "symmetry" and then pass that definition
by you guys for your review.
Isn't it simply whether the sinc reconstructed analog signal
is symmetric within the block regardless of length?
Bob
Hi,
humm, ... you seem to be looking at this
"symmetrical sequence" issue from a very different
viewpoint than I am.
You may have a "good point" here, but I don't
understand exactly what a "block" is with regard to
an analog signal.
It hadn't occurred to me to consider analog signals
in any way when trying to understand symmetrical
discrete sequences.
|
Yeah, after I sent it I realized I shouldn't have used the
word analog. What I really mean is the continuous sinc
reconstruction, real or complex. Substitude the word
"sequence" for "block".
My bias says that what is represented by a discrete sequence
is always the continuous sinc reconstruction of it but
perhaps that is too narrow a view.
Bob
--
"Things should be described as simply as possible, but no
simpler."
A. Einstein |
|
| Back to top |
|
|
|
|
|
|
FAQ
Memberlist
Register
Profile
Log in to check your private messages
Log in