DFT or DFS: Are they the same thing?

On Aug 7, 3:38 pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote:
...
>
> Why does this all sound vaguely familiar?   ;)
>
> --
> Eric Jacobsen
> Minister of Algorithms
> Abineau Communicationshttp://www.abineau.com

Vaguely?

Dale B. Dalrymple

On 7 Aug, 23:14, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On Aug 7, 2:56&#4294967295;am, Rune Allnor <all...@tele.ntnu.no> wrote:

> > The different variants take
> > either discrete or continuous data as input, and the data are
> > of either finite or infinite extent.
>
> the simple rule to remember: discrete time means periodic in
> frequency. &#4294967295;and vise versa (because of duality in the FT).
>
> DFT (or DFS): &#4294967295; discrete &#4294967295; &#4294967295;periodic t; discrete &#4294967295; &#4294967295;periodic f
> DTFT: &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; discrete &#4294967295; aperiodic t; continuous &#4294967295;periodic f
> Fourier series: continuous &#4294967295;periodic t; discrete &#4294967295; aperiodic f
> continuous FT: &#4294967295;continuous aperiodic t; continuous aperiodic f
>
> did i miss a combination?

No, but you got the variants' defining characteristica wrong:

DFT (or DFS):   discrete     finite t; discrete     finite f
DTFT:           discrete   infinite t; continuous   finite f
Fourier series: continuous   finite t; discrete   infinite f
continuous FT:  continuous infinite t; continuous infinite f

Since the digital computer can only work with finite amounts
of discrete data, the DFT is the only variation of the FT that
can be analyzed by means of numerics.

The question then becomes how to map the other varaints onto
the DFT. That's the main purpose of the sampling theorem.
It lets people understand what conversions and trade-offs are
involved in mapping from a continuous t domain to the
discrete t domain.

This talk about extensions of the data, are similar attempts
to try and explain what modifications and tradeoffs are
involved when one does computations on finite amounts
of data where one ought to do computations on infinite
amounts of data.

It gives people a hook on how to understand the difference
between CT data and the DT representations: Nyquist said
that 'the DT representation is exact if the bandwidth of
the CT signal meets certain criteria.'

Both you and I know that the CT signals never satisfy the
Nyquist criterion exactly, so there are errors involved
when sampling the CT data.

This idea about periodic extendions of the data serve
the same purpose: It is the special case where the
spectrum of the infinite sequence or function is practically
indistinguishable from the spectrum of the finite-length
sequence or function.

The data never are periodic, but this periodic extension
idea gives a hook on how to understand the errors involved.

These matters are trivial when presented this way from the
outset. Unfortunately, I have seen no DSP textbooks that
emphasize this POV.

Rune

On 8 Aug, 01:36, dbd <d...@ieee.org> wrote:
> On Aug 7, 3:38 pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote:
> ...
> > Why does this all sound vaguely familiar? &#4294967295; ;)

> Vaguely?

It might have been before you started frequenting comp.dsp,
but there was a guy, Mr. Bean, who had some serious issues
with Dirac's delta functions.

Rune

Rune Allnor <allnor@tele.ntnu.no> wrote:
< On 7 Aug, 23:14, robert bristow-johnson wrote:

<> the simple rule to remember: discrete time means periodic in
<> frequency. ?and vise versa (because of duality in the FT).

<> DFT (or DFS): ? discrete ? ?periodic t; discrete ? ?periodic f
<> DTFT: ? ? ? ? ? discrete ? aperiodic t; continuous ?periodic f
<> Fourier series: continuous ?periodic t; discrete ? aperiodic f
<> continuous FT: ?continuous aperiodic t; continuous aperiodic f

<> did i miss a combination?
 
< No, but you got the variants' defining characteristica wrong:
 
< DFT (or DFS):   discrete     finite t; discrete     finite f
< DTFT:           discrete   infinite t; continuous   finite f
< Fourier series: continuous   finite t; discrete   infinite f
< continuous FT:  continuous infinite t; continuous infinite f

Finite isn't the same as periodic, but sometimes it is
good enough.  A finite sequence can be put through an
FFT followed by iFFT and, within rounding error, you
get the original sequence out again.  

On the other hand, DCT works better for audio and video
compression because it doesn't have periodic boundary
conditions.  A boundary discontinuit in an FFT requires
high frequency terms that aren't needed for a DCT.
I might, then, agree with finite for DCT and DST, but
periodic for DFT.  
 
< Since the digital computer can only work with finite amounts
< of discrete data, the DFT is the only variation of the FT that
< can be analyzed by means of numerics.

As computers get bigger and faster, you can increase N,
in approximation of a continuous transform.  Even so,
you never reach the limit.
 
< The question then becomes how to map the other varaints onto
< the DFT. That's the main purpose of the sampling theorem.
< It lets people understand what conversions and trade-offs are
< involved in mapping from a continuous t domain to the
< discrete t domain.

One of the favorite problems in physics is the boundary
value differential equation.  The usual analytical solution
is to consider the solution as a sum of sines and cosines,
as restricted by the boundary conditions.  Sine and cosine
are solutions to one of the simplest second order differential
equations, and very common in real physics problems.
 
< This talk about extensions of the data, are similar attempts
< to try and explain what modifications and tradeoffs are
< involved when one does computations on finite amounts
< of data where one ought to do computations on infinite
< amounts of data.
 
< It gives people a hook on how to understand the difference
< between CT data and the DT representations: Nyquist said
< that 'the DT representation is exact if the bandwidth of
< the CT signal meets certain criteria.'

Is that what he said?  It has been a while, but I used
to have a copy of the paper.  His problem was how fast
he could send telegraph pulses through a band limited
cable.  (Especially high capacitance underwater cables.)
Conventiently, the problem of sending pulses through a
cable has the same solution as sending sampled signals
through a cable.
 
< Both you and I know that the CT signals never satisfy the
< Nyquist criterion exactly, so there are errors involved
< when sampling the CT data.
 
< This idea about periodic extendions of the data serve
< the same purpose: It is the special case where the
< spectrum of the infinite sequence or function is practically
< indistinguishable from the spectrum of the finite-length
< sequence or function.

I usually consider it the other way around, but the result
is the same either way.
 
< The data never are periodic, but this periodic extension
< idea gives a hook on how to understand the errors involved.

Periodic boundary conditions are used in many physics problems,
even those where the solution is not expected to be periodic,
but the solution is easier and the answer is useful.  As one
example, it works in solid state physics when one does not
need to consider the surface effects.  (As N increases an ever
smaller fraction of atoms are on the surface.)  
 
< These matters are trivial when presented this way from the
< outset. Unfortunately, I have seen no DSP textbooks that
< emphasize this POV.

-- glen

On 8 Aug, 10:29, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> Rune Allnor <all...@tele.ntnu.no> wrote:
>
> < On 7 Aug, 23:14, robert bristow-johnson wrote:
>
> <> the simple rule to remember: discrete time means periodic in
> <> frequency. ?and vise versa (because of duality in the FT).
>
> <> DFT (or DFS): ? discrete ? ?periodic t; discrete ? ?periodic f
> <> DTFT: ? ? ? ? ? discrete ? aperiodic t; continuous ?periodic f
> <> Fourier series: continuous ?periodic t; discrete ? aperiodic f
> <> continuous FT: ?continuous aperiodic t; continuous aperiodic f
>
> <> did i miss a combination?
>
> < No, but you got the variants' defining characteristica wrong:
>
> < DFT (or DFS): &#4294967295; discrete &#4294967295; &#4294967295; finite t; discrete &#4294967295; &#4294967295; finite f
> < DTFT: &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; discrete &#4294967295; infinite t; continuous &#4294967295; finite f
> < Fourier series: continuous &#4294967295; finite t; discrete &#4294967295; infinite f
> < continuous FT: &#4294967295;continuous infinite t; continuous infinite f
>
> Finite isn't the same as periodic, but sometimes it is
> good enough. &#4294967295;A finite sequence can be put through an
> FFT followed by iFFT and, within rounding error, you
> get the original sequence out again. &#4294967295;

Do you imply that the DFT can work with infinite
sequences? If so, you are plain wrong. The DFT only
works with finite sequences.

Now, that finite sequence may or may not be a subset
of some infinite sequence. If it is, there are several
ways to relate the result of the DFT to the desired
(but uncomputable) result of the DTFT on the infinite
sequence. The periodic extension is one way. The
infinite pre- and post-window zero-pads is another.

...

> < The question then becomes how to map the other varaints onto
> < the DFT. That's the main purpose of the sampling theorem.
> < It lets people understand what conversions and trade-offs are
> < involved in mapping from a continuous t domain to the
> < discrete t domain.
>
> One of the favorite problems in physics is the boundary
> value differential equation. &#4294967295;The usual analytical solution
> is to consider the solution as a sum of sines and cosines,
> as restricted by the boundary conditions. &#4294967295;Sine and cosine
> are solutions to one of the simplest second order differential
> equations, and very common in real physics problems.

You need to separate between the application and the tool.
The FT, in all its guises, is a way of describing some
mathematical entity. Such mathematical entities exist
in several applications. The context you mention here is
the analytic study of PDEs.

In that context it often makes no sense to talk about
infinite extensions of the stated domain. Consider a guitar
string: There is no problems using the finite-domain
continuous domain FT to study it. Once one starts talking
about infinite extensions of the string, the whole analysis
breaks down: It makes no intuitive sense, since the physical
system under study is of finite extent, and even if one did
analyze strings of different lengths the distribution of
modes or harmonics would change.

In this context the FT on a finite domain is the one
to use.

> < This talk about extensions of the data, are similar attempts
> < to try and explain what modifications and tradeoffs are
> < involved when one does computations on finite amounts
> < of data where one ought to do computations on infinite
> < amounts of data.
>
> < It gives people a hook on how to understand the difference
> < between CT data and the DT representations: Nyquist said
> < that 'the DT representation is exact if the bandwidth of
> < the CT signal meets certain criteria.'
>
> Is that what he said? &#4294967295;It has been a while, but I used
> to have a copy of the paper. &#4294967295;His problem was how fast
> he could send telegraph pulses through a band limited
> cable. &#4294967295;(Especially high capacitance underwater cables.)
> Conventiently, the problem of sending pulses through a
> cable has the same solution as sending sampled signals
> through a cable.

I haven't read the original paper. It doesn't matter
how Nyquist phrased it; the above is the corollary of
his theorem that is the reason why his name is still
remembered.

> < Both you and I know that the CT signals never satisfy the
> < Nyquist criterion exactly, so there are errors involved
> < when sampling the CT data.
>
> < This idea about periodic extendions of the data serve
> < the same purpose: It is the special case where the
> < spectrum of the infinite sequence or function is practically
> < indistinguishable from the spectrum of the finite-length
> < sequence or function.
>
> I usually consider it the other way around, but the result
> is the same either way.

Really?

It does change the relation between causes and effects,
which in turn makes a certain difference about how to
understand the world:

Let's start with an arbitrary complex number, say, z = 0+i.

Is it periodic? Is it ambiguous? In the stated context
such questions don't even make sense.

However, if you write the number on polar form,

z = exp(i*pi/2 + i*n*2*pi)

then all of a sudden questions about periodic properties
and ambiguities with respect to n to make some sort of
sense. But only to a certain degree - the number z = i
is well-defined. It is the *context*, the polar notation,
that might cause people to ask the question.

Of course, once one starts to discuss complex-valued
analytic *functions*, these sorts of questions become
relevant wrt branch cuts, Riemann surfaces and so on.
But I started this example with a *number*, not a function.

As far as I am concerned, the question about periodic
properties of the DFT are of the same kind: Degenerate,
because different contexts are indiscriminatedly mixed.

Rune

Rune Allnor <allnor@tele.ntnu.no> wrote:
< On 8 Aug, 10:29, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
<> Rune Allnor <all...@tele.ntnu.no> wrote:
(snip)
<> < DFT (or DFS): ? discrete ? ? finite t; discrete ? ? finite f
<> < DTFT: ? ? ? ? ? discrete ? infinite t; continuous ? finite f
<> < Fourier series: continuous ? finite t; discrete ? infinite f
<> < continuous FT: ?continuous infinite t; continuous infinite f

<> Finite isn't the same as periodic, but sometimes it is
<> good enough. ?A finite sequence can be put through an
<> FFT followed by iFFT and, within rounding error, you
<> get the original sequence out again. ?
 
< Do you imply that the DFT can work with infinite
< sequences? If so, you are plain wrong. The DFT only
< works with finite sequences.

A periodic function (continuous or discrete) is fully defined
when its value is given over a finite interval.  The DFT, 
unlike the DST and DCT, has periodic boundary conditions.
I like the explanation in Numerical Recipes for the differences
between DFT, DST, and DCT.
 
< Now, that finite sequence may or may not be a subset
< of some infinite sequence. If it is, there are several
< ways to relate the result of the DFT to the desired
< (but uncomputable) result of the DTFT on the infinite
< sequence. The periodic extension is one way. The
< infinite pre- and post-window zero-pads is another.
 
< ...
(snip)
 
<> One of the favorite problems in physics is the boundary
<> value differential equation. ?The usual analytical solution
<> is to consider the solution as a sum of sines and cosines,
<> as restricted by the boundary conditions. ?Sine and cosine
<> are solutions to one of the simplest second order differential
<> equations, and very common in real physics problems.
 
< You need to separate between the application and the tool.
< The FT, in all its guises, is a way of describing some
< mathematical entity. Such mathematical entities exist
< in several applications. The context you mention here is
< the analytic study of PDEs.

The technique is independent of the Fourier transform.
The solution to a linear differential equation can be
described as the sum of solutions to the homogeneous equation,
plus a particular solution to the inhomogeneous equation.

It does happen that for a large number of real problems the
solutions to the differential equation are sines and cosines.
 
< In that context it often makes no sense to talk about
< infinite extensions of the stated domain. Consider a guitar
< string: There is no problems using the finite-domain
< continuous domain FT to study it. Once one starts talking
< about infinite extensions of the string, the whole analysis
< breaks down: It makes no intuitive sense, since the physical
< system under study is of finite extent, and even if one did
< analyze strings of different lengths the distribution of
< modes or harmonics would change.

One usually starts out with the ideal string with ideal 
boundary conditions.  Much of the design of musical instruments
is compensating for the less than idea materials that they are
built with.  In the case of a taut string, one starts with the
equation for an infinite string and adds boundary conditions
for a finite length string.
 
< In this context the FT on a finite domain is the one
< to use.

You don't need an FT at all.  Write the differential equation, and
find the set of solutions to that equation.  Then determine which
solutions are applicable to the given problem.  For strings and
uniform diameter tubes, they are sines and cosines.  
For drum heads, they are (cylindrical) Bessel functions.
For cones (oboe and bassoon if I remember) they are spherical
bessel functions.  
 
(snip on Nyquist)
 
< I haven't read the original paper. It doesn't matter
< how Nyquist phrased it; the above is the corollary of
< his theorem that is the reason why his name is still
< remembered.

You should read it.  It is interesting to see how different
things were not so long ago.
 
<> < Both you and I know that the CT signals never satisfy the
<> < Nyquist criterion exactly, so there are errors involved
<> < when sampling the CT data.

<> < This idea about periodic extendions of the data serve
<> < the same purpose: It is the special case where the
<> < spectrum of the infinite sequence or function is practically
<> < indistinguishable from the spectrum of the finite-length
<> < sequence or function.

<> I usually consider it the other way around, but the result
<> is the same either way.
 
< Really?
 
< It does change the relation between causes and effects,
< which in turn makes a certain difference about how to
< understand the world:

(snip)

-- glen 

Rune Allnor <allnor@tele.ntnu.no> wrote:
(snip)

<> < DFT (or DFS): ? discrete ? ? finite t; discrete ? ? finite f
(snip)
 
< Do you imply that the DFT can work with infinite
< sequences? If so, you are plain wrong. The DFT only
< works with finite sequences.

Note that periodic boundary conditions does not say anything
about the function outside that range.  It only restricts the
function to have the same value, and often some derivatives,
at two points.  (Or sets of points for functions of more than
one variable.)

Some solid state physics problems involve a cube of atoms
with periodic boundary conditions, such that the function
(and some derivatives) are equal on opposing faces.  There
is no suggestion that the wave function continue outside
the cube.  

A second order differential equation usually requires two 
boundary conditions.  Those can be the function value at
two points, the first derivatives at two points, the value
and derivative at one point, the value at one point and the 
derivative at a different point.  With periodic boundary
conditions, they are that the function has the same value
at two points (one condition), and that the derivative has
the same value at two points (second condition).  

All that is completely separate from the fact that one
might be interested in Fourier transforms on the functions.

-- glen

On Aug 8, 1:03&#4294967295;am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 7 Aug, 23:14, robert bristow-johnson <r...@audioimagination.com>
> wrote:
>
> > On Aug 7, 2:56&#4294967295;am, Rune Allnor <all...@tele.ntnu.no> wrote:
> > > The different variants take
> > > either discrete or continuous data as input, and the data are
> > > of either finite or infinite extent.
>
> > the simple rule to remember: discrete time means periodic in
> > frequency. &#4294967295;and vise versa (because of duality in the FT).
>
> > DFT (or DFS): &#4294967295; discrete &#4294967295; &#4294967295;periodic t; discrete &#4294967295; &#4294967295;periodic f
> > DTFT: &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; discrete &#4294967295; aperiodic t; continuous &#4294967295;periodic f
> > Fourier series: continuous &#4294967295;periodic t; discrete &#4294967295; aperiodic f
> > continuous FT: &#4294967295;continuous aperiodic t; continuous aperiodic f
>
> > did i miss a combination?
>
> No, but you got the variants' defining characteristics wrong:
>
> DFT (or DFS): &#4294967295; discrete &#4294967295; &#4294967295; finite t; discrete &#4294967295; &#4294967295; finite f
> DTFT: &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; discrete &#4294967295; infinite t; continuous &#4294967295; finite f
> Fourier series: continuous &#4294967295; finite t; discrete &#4294967295; infinite f
> continuous FT: &#4294967295;continuous infinite t; continuous infinite f

well, i beg to differ that i'm the one who got it wrong.  a Fourier
series describes or models a function of finite support rather than a
periodic function?  when you were replacing "periodic" with "finite"
above, i think some kinda warning bell should have been going off in
your head when you did it to Fourier series.

you said to Glen:

> Do you imply that the DFT can work with infinite
> sequences? If so, you are plain wrong. The DFT only
> works with finite sequences.

it works with a finite data set.  or, using your semantics, a finite
number of elements.  however, even with a finite number of elements,
the question remains as to the nature (as in the word "natural") of
the data outside of that finite domain.  is there any justification to
imagine that this finite set naturally extends beyond that domain, and
if so, how?

a bandlimited periodic function has infinite support, but it is fully
describable with a finite number of elements.  just because we pass to
the DFT operation a finite number of elements (as we must with a
finite computer doing the number crunching) does not mean that the
finite number of elements corresponds to a truncated function of some
kind.

now for this bandlimited periodic function, in the frequency domain,
these finite number of elements appear truncated (with zeros appended)
because it's bandlimited.  but it's still continuous-time so although
in the frequency domain it is obviously fully described by a finite
data set, it doesn't appear to be so in the continuous time domain.
but we can sample it (because it's bandlimited).  once we do that, it
is clear in both domains that a finite set of N numbers fully
describes our bandlimited periodic function in either domain.  but it
should also be clear that in both domains there is periodicity.  the
DFT is passed one set of N numbers (that fully describes the periodic
function in the time domain) and converts that set to another set of N
numbers (that fully describes the periodic function in the frequency
domain).

> Since the digital computer can only work with finite amounts
> of discrete data, the DFT is the only variation of the FT that
> can be analyzed by means of numerics.

we agree about that.  even the fastest computer will take until
eternity to crunch an infinite set of arbitrary numbers.

> The question then becomes how to map the other varaints onto
> the DFT. That's the main purpose of the sampling theorem.

i do not believe that is the main purpose of the sampling theorem.
with Beanie and so many others, i have gone over a zillion times what
(in my opinion) is the main purpose of the sampling (and
reconstruction) theorem.  without repeating myself (i wish Google
Groups would succeed in searching out these old posts), i will just
say that the sampling theorem needs no assumption of finite support
(but the DFT does).

> It lets people understand what conversions and trade-offs are
> involved in mapping from a continuous t domain to the
> discrete t domain.

but the DFT is not just about discrete t.  it's about a finite set of
data.  the DTFT is also about discrete t (which causes it to be
*periodic*, not just finite in the f domain).  but in both the t and f
domains, the DTFT is dealing with an infinite set of data (the curious
thing is that it is countably infinite in the t domain and appears to
be uncountably infinite in the f domain - i haven't figgered that out
yet, but i am not concerned).

and our disagreement is about what it does with this finite set of
data and how theorems associated with the DFT deal with this finite
set of data.  i continue to maintain that these theorems (like
convolution and shifting, that involve N multiplications in the other
domain) act as though, or *assume*, that the data is periodically
extended beyond the original limits of indices; 0 to N-1.

> This talk about extensions of the data, are similar attempts
> to try and explain what modifications and tradeoffs are
> involved when one does computations on finite amounts
> of data where one ought to do computations on infinite
> amounts of data.
>
> It gives people a hook on how to understand the difference
> between CT data and the DT representations: Nyquist said
> that 'the DT representation is exact if the bandwidth of
> the CT signal meets certain criteria.'
>
> Both you and I know that the CT signals never satisfy the
> Nyquist criterion exactly, so there are errors involved
> when sampling the CT data.

well, i am not going into the really anal-retentive and existential
issues about whether my sine wave is really bandlimited because
someone next year is gonna turn off the signal generator.  for the
purposes of discussion, i am fully happy to assume or treat this sine
wave as having started before the beginning of time and extending on
to eternity.  then that conceptual CT signal can satisfy Nyquist
exactly.

> This idea about periodic extensions of the data serve
> the same purpose: It is the special case where the
> spectrum of the infinite sequence or function is practically
> indistinguishable from the spectrum of the finite-length
> sequence or function.

this is not the reasoning behind "this idea about periodic
extensions".  the idea behind it is that is what the DFT math is going
to do to your data whether you think of it as periodic or not, and if
you don't understand and accept that mathematical fact, you might get
burned when you do things with your DFT such as convolution.

> The data never are periodic, but this periodic extension
> idea gives a hook on how to understand the errors involved.
>
> These matters are trivial when presented this way from the
> outset. Unfortunately, I have seen no DSP textbooks that
> emphasize this POV.

O&S, although they obfuscate it a bit in their exhaustive (and wordy)
text, they *do* emphasize the correct (IMO) POV.  at the beginning of
their DFT chapter (ch 8, pp 514 in my edition):

"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 to base our presentation on the relationship
between periodic sequences and finite-length sequences.  We will begin
by considering the Fourier series representation of periodic
sequences.  While this representation is important in its own right,
we are most often interested in the application of Fourier series
results to the representation of finite-length sequences.  We
accomplish this by constructing a periodic sequence for which each
period is identical to the finite-length sequence.  As we will see,
the Fourier series representation of the periodic sequence corresponds
to the DFT of the finite-length sequence.  Thuse our approach is to
define the Fourier series representation for periodic sequences and to
study the properties of such representations.  The we repeat
essentially the same derivations assuming that the sequence to be
represented is a finite-length sequence.  This approach to the DFT
emphasizes the fundamental inherent periodicity of the DFT
representation and ensures that this periodicity is not overlooked in
applications of the DFT."

one can only hope.

r b-j

On Aug 8, 2:49 pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
> ...
>
.> O&S, although they obfuscate it a bit in their exhaustive (and
wordy)
.> text, they *do* emphasize the correct (IMO) POV.  at the beginning
of
.> their DFT chapter (ch 8, pp 514 in my edition):
.>
.> "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 to base our presentation on the
relationship
.> between periodic sequences and finite-length sequences.  We will
begin
> by considering the Fourier series representation of periodic
> sequences.  While this representation is important in its own right,
> we are most often interested in the application of Fourier series
> results to the representation of finite-length sequences.  We
> accomplish this by constructing a periodic sequence for which each
> period is identical to the finite-length sequence.  As we will see,
> the Fourier series representation of the periodic sequence corresponds
> to the DFT of the finite-length sequence.  Thuse our approach is to
> define the Fourier series representation for periodic sequences and to
> study the properties of such representations.  The we repeat
> essentially the same derivations assuming that the sequence to be
> represented is a finite-length sequence.  This approach to the DFT
> emphasizes the fundamental inherent periodicity of the DFT
> representation and ensures that this periodicity is not overlooked in
> applications of the DFT."
>
> one can only hope.
>
> r b-j

So there are two kinds of people, those like O&S who consider the
periodic to finite sequence correspondence as one very useful
interpretation among several valid interpretations of the DFT and
those who are too inflexible to admit the existence of other
interpretations even at the insistence of their own chosen references.

Dale B. Dalrymple

On Aug 8, 10:18&#4294967295;pm, dbd <d...@ieee.org> wrote:
>
> So there are two kinds of people, those like O&S who consider the
> periodic to finite sequence correspondence as one very useful
> interpretation among several valid interpretations of the DFT and
> those who are too inflexible to admit the existence of other
> interpretations even at the insistence of their own chosen references.

i remember seeing some interesting graffiti that said, "There are
really only two kinds of people, those who write on walls and those
who don't."

dichotomies are sorta fun.  male vs. female.  us vs. them.  "yer
either fer us or yer fer them terrorists."  remember that?

dichotomy is fun, simple.  *sometimes* accurately reflecting reality.
but not always.  life (or DSP) is not always split into two mutually
exclusive and exhaustive sets.  sometimes there is a third or fourth
way.

nonetheless, the mathematics doesn't give a rat's ass about
dichotomy.  it just is what it is.  mathematical facts (a.k.a.
theorems) say what they say without regard to what kinds of people
there are.

sure, another way to look at the DFT is to truncate an infinite
sequence for n<0 and n>=N and send it to the DTFT.  by "truncate" we
mean to set those samples to zero by definition (so that those zeroed
samples are definition and do not comprise any of the N elements of
information, they cannot be merely "undefined" because those terms in
the DTFT summation must be zero to not contribute to the sum).  in
this first step, we still have a continuous set of frequency-domain
data from -pi to +pi (and it repeats, but don't worry about that for
the moment).  that continuous frequency-domain data knows that only
the samples from 0 to N-1 might be non-zero and it knows that the
other samples are zero.  no periodicity yet.

but the DFT does not return a continuous function from -Nyquist to
+Nyquist.  it samples that with N uniformly spaced samples.  just like
ideal uniform sampling in the time domain causes the spectrum to be
repeated and overlap-added forever in the frequency domain, the same
is the case if you swap time and frequency.  the DFT returns N samples
of the DTFT and those N values repeat (because the DTFT repeats in the
frequency domain), but not only that - because they are N discrete
samples, that action of sampling causes periodic repetition in the
time domain.  when you throw away the information between the samples
in the frequency domain, you necessarily cause periodic repetition and
overlapping in the time domain.  that's just a mathematical fact (that
comes from the sampling theorem).

other interpretations are fine as long as they are mathematically
valid.  is there a single, mathematically valid interpretation of the
DFT that shows that when the samples are shifted, either to the left
or to the right, that the samples that are shifted in from the right
or from the left are zero?  or undefined?  or *anything* other than
the samples you would get from being periodically extended?  can you
name a single theorem that has that?  sometimes it doesn't matter.
the linearity theorem doesn't shift the data from right or left.  but
when it *does* matter (this is what makes my claim about periodicity
meaningful, it is *falsifiable*) is there *ever* a time when zeros or
something else different from the periodic extension is shifted in?
please provide an example, then my claim is disproven (falsified).

r b-j