On 8/3/13 7:48 PM, dvsarwate wrote:
> Ah, Robert, Robert, to be so young
> and yet so
> ossified (or did I mean to say O&Ssified?)in
> your thinking.....
>
> As Rick Lyons pointed out to you, O&S said
>
> "Although several points of view can be taken toward the
> derivation and interpretation of the DFT representation
> of a finite-duration sequence, we have CHOSEN (emphasis
> added) ....
>
> and so it is not O&S who are denying the
> validity of other points of view; in fact
> they are admitting upfront that other
> viewpoints are possible.
they are conceding that other viewpoints exist. not that they are correct.
but i am not depending on O&S. they just agree with me about the math.
explicitly. i just find it astonishing when people deny that O&S say
that the periodicity is always there, and that to ignore that fact can
lead to trouble. they're clear about that.
alls i am saying is, whether you explicitly periodically extend the N
samples passed to the DFT (which is saying x[n+N] = x[n] for all n) or
not, the value that comes after x[N-1] is the value of x[0].
the input to the DFT: {x[0], x[1], x[2] ... x[N-1]} is not just an
unordered collection of values. i.e. suppose N = 8 and the input {x[0],
x[1], x[2] ... x[7]} is {4, 2, -1, -2, 0, 3, 6, 5}. now the order in
that is important. the DFT is going to treat that input sequence
differently than it will {0, 6, 4, -2, 5, -1, 3, 2}, even though they
are the same values, just in a different order. so in the original
input, it *matters* that x[1]=2 comes right after x[0]=4, and that
x[6]=6 comes right after x[5]=3.
so this is what i am saying, what O&S is saying, and what the math is
saying in the context of the DFT:
in every manner that it is true that x[6]=6 comes right after x[5]=3, so
also does x[0]=4 comes right after x[7]=5. in the same way that 6 comes
after 3, so also does 4 have the very same relationship with the 5
sitting next to it. to deny that is to deny the math of the DFT. you
wouldn't be denying what the DFT linearity property says, but you *are*
denying the shifting and convolution theorems and that is not a matter
of interpretation. you cannot simply interpret that away because the
math doesn't let you.
SO... i think (this is preference, not a theorem or a fact claim) the
simplest and most compact way to state this is that "x[N] = x[0]" and,
in general "x[n+N] = x[n]". but you can also say that there *is* no
x[N] or any other sample with index above N-1 or with index below 0.
but if you do that, you must replace x[n] in general with x[n mod_N]. i
do not know of another notational convention to express that.
now lacking a third notational convention, then, if you're talking about
the DFT of length N, you *must* choose one of those two. and both are
periodic extensions. the "x[n+N] = x[n]" is explicit about the periodic
extension and the "x[n mod_N]" is implicit, but they say the same thing.
if you deny that you need one or the other, then you're mathematically
incorrect unless the only property of the DFT you care about is
linearity. if you are using the DFT and any other theorem regarding the
DFT, then you *must* say either that "x[n+N] = x[n]" or use "x[n mod_N]"
or your math will be wrong. i (and O&S) say that this is "periodic
extension". you can call it a radish or something else, but it doesn't
change the mathematical fact.
> It is natural
> for them to prefer their own chosen viewpoint
> over others, but why shouldn't they? After
> all, they have put in considerable effort in
> writing a textbook from that viewpoint, and
> there is no reason to expect them to be
> neutral in this matter! But, as even you admit
>
> "Rick, O&S are recognizing that there are
> other viewpoints (and i quoted that in 2011..."
>
> I don't see where O&S are DENYING the validity
> of other viewpoints whether in diplomatically
> acceptable terms or not; it is YOU who are
> continually insisting that the O&S viewpoint
> is the _only_ correct one and that all who
> use other viewpoints are in a state of mortal
> sin. You are insisting that
>
> "Thou shalt have no other gods before O&S"
no, i am insisting that O&S say that the input to the DFT is
periodically extended. and Dale (and perhaps Eric and Rick, i dunno)
say that O&S do not say that. that's astonishing, because O&S are
explicit about it. with words like "the inherent periodicity is always
present". they're not saying that the inherent periodicity is present
only for those who recognize the inherent periodicity. and neither do i
say that. but i think that some others on this forum (perhaps you) *do*
say that taking another point of view allows one to avoid this inherent
periodicity and i am saying that taking such a position is simply
mathematically incorrect. and that the math is clear.
> is a commandment that must be accepted by
> all even though all the evidence indicates
> that this commandment does not issue from
> O&S themselves. So, I will disregard your
> annotations and additions to the received
> wisdom from O&S and interpret their sayings
> according to what **I** read in their text,
> and what you say they do.
>
> Dilip Sarwate
>
> P.S. Since this post has gotten various
> non-dsp overtones, let me, as a determined
> polytheist, remind you that atheists differ
> from monotheists such as yourself by only
> one in the number of gods whose existence
> they deny.
i hadn't at all brought issues of faith into this, either directly or
metaphorically. early on (back in the 90s), i think it was Eric that
first referred to *my* citing of O&S as "quoting O&S scripture" and i
thought that was kinda cute so i sorta ran with it. but only
metaphorically and only tongue-in-cheek.
the only god in this is Mathematics. and, in this metaphor, the Word of
God are mathematical theorems. and it is a mathematical fact that x[0]
comes after x[N-1] in the DFT in the same way that x[5] comes after
x[4]. that's a fact claim, and if you deny it or even say that it's
only an interpretation or a viewpoint; something less than a
mathematical fact, i say you're wrong. and that the mathematics say
you're wrong. and that O&S agree (but what do they know?).
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge."