discrete fourier series and transform

On Sunday, December 30, 2012 5:31:27 PM UTC-8, robert bristow-johnson wrote:
> ..
> the basic fact that ... the sampling function is also a Fourier series:
>         +inf                       +inf
>      T  SUM{ delta(t - kT) }   =   SUM{ e^(j 2 pi n/T t) }
>        k=-inf                     n=-inf
> that means, whether you bandlimit the pre-sampled signal or not, that 
> turning this from a continuous-time signal, x(t), to a discrete-time 
> signal, x[n], causes the spectrum of the signal to be periodic with 
> period (in the frequency domain) of 1/T.  there is no avoiding that.

Yes, that's what we do, and what it results in.

Now for the fanasy:

> in this dispute with the periodicity deniers, i am simply applying the 
> very same reasoning when the continuous (but periodic) spectrum coming 
> out of the DTFT is, itself, sampled.  that necessarily causes the 
> time-domain signal (which was discrete, but not periodic before) to be 
> made into a periodic signal by the same, repeated shift-and-overlap-add 
> operation.  again, there is no avoiding that. 

We don't need to try to avoid it, it has never been done. The DTFT has an infinite number of inputs and ouputs, it can't be calculated on our data. And anti-aliasing and sampling, which is all we do, doesn't create sampling in the frequency domain. Our real world signals may contain noise (which is continuous in the frequency domain), deterministic non-periodic signals, periodic signals not periodic in an integer number of time samples (periodic in N). and periodic-in-N signals (that the DFT treats so well) and the time domain sampling process, which is what we do, doesn't calculate the DTFT or create frequency domain sampling of any of them. The only way we can get signals that are sampled in the frequency domain is to define, construct or assume them to be. You are welcome to do any of these, but the fascist of periodicity states that we must accept the assumption even when the data we have defined constructed or sampled is not periodic in N.

> ...
> or, they can just give into the Dark Side and accept the fact that the 
> DFT maps one discrete and periodic sequence of period N to another 
> discrete and periodic sequence of the same period in an invertible 
> manner.  but that would be the same as saying that the DFT is the same 
> as the Discrete Fourier Series.
> 
This last statement is true, because it states the assumption of periodicity in N. For the data we have that is not periodic in N, the DFT can still be calculated, but when the FT of the signal does not correspond to the uniform samples in frequency that can be calculated by the DFT, the DFT is unable to calculate the DTFT of our signals. Sometimes it is close enough to be useful to pretend the data was periodic in N (an assumption of periodicity denied to no one). Often we can make perfectly good use of the DFT output anyway precisely because we know that the signals are not periodic in N. An example of this is when we use the DFT coefficients to interpolate the frequency, amplitude and phase of a signal periodic, but not periodic at a frequency calculated by the DFT.

Dale B. Dalrymple

On Sunday, December 30, 2012 7:43:21 PM UTC-8, robert bristow-johnson wrote:
...
> On 12/23/12 4:15 PM, Eric Jacobsen wrote:
> 
> > I either don't have or can't find my copy of Oppenheim and Willsky,
> > but in other texts from Oppenheim, e.g., Digitial Signal Processing
> > and Discrete Time Signal Processing, periodic input sequences are used
> > for the analysis, but it is not stated (anywhere that I can find)
> > that this is necessary.
> 
> from my 1989 edition (this supports what you're saying, Eric):
> ...
> "We will bein by consider tihe Fourier series representation of periodic 
> 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.
> ..."
So, this is where O&S have stated that Ch 8 is based on the construction of sequence periodic in N and the application of the DFT to a period of that sequence.
> 
> chapter.section 8.6, p. 532: "In recasting Eqs. (8.11) and (8.12) [the 
> DFS analysis and synthesis definitions] in the form of Eqs. (8.61) and 
> (8.62) [the DFT analysis and synthesis definitions] for finite-length 
> sequences, we have not eliminated the *inherent* periodicity.  Was with 
> the DFS, the DFT X[k] is equal to samples of the periodic Fourier 
> transform X(e^jw), and if Eq. (8.62) is evalutated for values of n 
> outside the interval 0 <= n <= N-1, the result will not be zero but 
> rather a periodic extension of x[n].  The *inherent* periodicity is 
> *always* present.  Sometimes it causes us difficulty and sometimes we 
> can exploit it, but to totally ignore it is to invite trouble."

And this is the conclusion O&S draw from the application of the DFT to a sequence constructed as periodic in N.

For what O&S have to say when the DFT is applied to data that is not constructed to be periodic in N you will need to look at Ch 11 which discusses the result when the signal is not periodic at a frequency sampled by the DFT (Section 11.2.2) as well as stocastic and non-stationary signals. Hey everybody, go take a look at the whole serious presentation and decide what your own interpretation is.

Dale B. Dalrymple

On 12/30/2012 8:59 PM, robert bristow-johnson wrote:
> On 12/30/12 1:08 AM, rickman wrote:
>>
>> I'm not able to conduct the same test on your consciousness as I can on
>> my own. I can't see the meter on your device only what you tell me.
>
> no, you see that my power supply appears to be exactly the same model as
> yours. and you see that yours is a fixed 15 volt supply. and you see
> that the resistor connected to my terminals have yellow, violent, and
> red stripes, just like the resistor connected to your terminals.
>
> so if your meter tells you that there are 3 mA flowing in your resistor
> and then the Nazi drill instructor comes in, points to your neighbor's
> lab station (i guess that's my lab station), and demands that you tell
> him if there is enough information to know approximately what the
> current is (in my 4.7K resistor) and what that approximate current would
> be, and the Nazi drill instructor tells you that if your answer is
> wrong, you'll be taken out into the courtyard and shot, what will you
> say? that you don't know?
>
> maybe the firing squad is just a figure of imagination. you can't tell
> if it's real.
>
> see, Rickman, you need to apply as tough critical thinking to your own
> position as you think you're applying to the others.

You don't get the fact that what I can observe of your test bench is not 
the same as what I can observe of mine.  I'm looking out from the inside 
and know what I feel.  I can't know anything of what you feel, only how 
you react.  Maybe your meter is defective or maybe your resistor is out 
of tolerance, or maybe none of that is really what it looks like and the 
meter read out is controlled in some way that it totally different from 
mine.  I have no way to tell.

I think this analogy has very limited value in this case as everything 
on both benches are external to the observer.  That is the difference 
between me observing myself and me observing you.


>> That's just what I would expect an automaton to say.
>
> and that's what I would expect the other automaton (who is deluded into
> thinking it's special) to say. you have to turn the whole critical
> skepticism around toward yourself. apply the same critical thinking to
> yourself that you apply toward the "others" that you think are just
> stimulus and not real.

No, "turning" anything is of no value to *me* because I can't know 
*anything* of what is going on inside your observation.  It is not a 
matter of thinking, it is a matter of what I can observe and what I can't.


>> That is not direct evidence.
>
> so was the case for Copernicus. but what is the *most* reasonable thing
> to believe? that we are at the center of the universe when there is no
> astronomical evidence that we are anywhere special? is that the
> reasonable conclusion?

I'm not talking about what I *believe*, I'm talking about what I can prove.


>> It just an opinion based on very little.
>
> no it's not. none of us have direct evidence. and the fact that you
> *think* you have consciousness is not evidence***. you could be an
> automaton that *thinks* it has consciousness. this is the point that
> Daniel Dennett makes
>
> *** Now, some of us, believe that experiential evidence accounts for
> more than Dennett would grant. but if you are willing to accept "I think
> therefore I am" as evidence that you are a being, where are you willing
> to stop with the evidence you "perceive" (using your word) with your
> senses?
>
> unless you think you're living in some kinda Truman Show (did you see
> that movie), you have to account for what your senses tell you. if you
> say that you have no proof that what your senses *apparently* convey to
> you (because they might not be real), then your problem is much deeper.
> if you are at that philosophical position, you have no way of knowing
> you're not an automaton programmed with the illusion of consciousness.
>
> but if you get past that sorta sophism, it's pretty hard to conclude
> that these other biological beings that look sorta like you look (head,
> eyes, two arms, two legs) and interact with you in a predictable manner
> that might be how you might interact with others, if you get to that
> point and conclude that the biological being that seems to encase your
> consciousness is qualitatively so much different than the others that
> you can't even verify to yourself that they also have consciousness, i
> think you need to start thinking like Copernicus.

The fact that Dennett says something doesn't make to true.  If an 
automaton were programmed with the *illusion* of consciousness, it means 
to me that it is not conscious and therefore has no thoughts, only 
programmed actions.  So that could not be me who thinks.  If I were an 
automaton that was *programmed* to -have- consciousness, how would that 
be different from "actually" having it?  In that case I say I have 
consciousness and my position still holds.  I don't maintain I am not an 
automaton, I maintain that I am conscious because I know I am.  In fact, 
I expect I am not much more than a biochemical automaton.


>> Do you follow Isaac Asimov's "Three Laws of Robotics"?
>
> the only science fiction i consume are in movies. from 2001 to Alien to
> Star Wars. i'm practically illiterate.

Then I'll keep my distance from you as you may cause me harm or allow 
harm to come to me by inaction.

Rick

On 12/31/12 4:26 AM, dbd wrote:
> On Sunday, December 30, 2012 7:43:21 PM UTC-8, robert bristow-johnson wrote:
> ...
>> On 12/23/12 4:15 PM, Eric Jacobsen wrote:
>>
>>> I either don't have or can't find my copy of Oppenheim and Willsky,
>>> but in other texts from Oppenheim, e.g., Digitial Signal Processing
>>> and Discrete Time Signal Processing, periodic input sequences are used
>>> for the analysis, but it is not stated (anywhere that I can find)
>>> that this is necessary.
>>
>> from my 1989 edition:
>> chapter.section 8.0, p 514: .
>> "We will begin by considering the Fourier series representation of periodic
>> 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.   Thus our approach
>> is to define the Fourier series representation for the periodic sequence
>> and to study the properties of such representation.  Then 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 thes periodicity is not overlooked in
>> applications of the DFT."
>
> So, this is where O&S have stated that Ch 8 is based on the construction of
> sequence periodic in N and the application of the DFT to a period of that
> sequence.

okay, it also simply says what it simply says.

>>
>> chapter.section 8.6, p. 532: "In recasting Eqs. (8.11) and (8.12) [the
>> DFS analysis and synthesis definitions] in the form of Eqs. (8.61) and
>> (8.62) [the DFT analysis and synthesis definitions] for finite-length
>> sequences, we have not eliminated the *inherent* periodicity.  As with
>> the DFS, the DFT X[k] is equal to samples of the periodic Fourier
>> transform X(e^jw), and if Eq. (8.62) is evaluated for values of n
>> outside the interval 0 <= n <= N-1, the result will not be zero but
>> rather a periodic extension of x[n].  The *inherent* periodicity is
>> *always* present.  Sometimes it causes us difficulty and sometimes we
>> can exploit it, but to totally ignore it is to invite trouble."
>
> And this is the conclusion O&S draw from the application of the DFT to a
> sequence constructed as periodic in N.

no, it's not in sections 8.1 or 8.2 or 8.3 (about the DFS).  it's in 
section 8.6 titled "Fourier representation of finite duration sequences: 
the Discrete Fourier Transform.  it's about what the title says it's 
about and it says about the topic simply what it says.  it says, even in 
the context where periodicity was not assumed in the outset, that the 
periodicity is "inherent" and "always present".

>
> For what O&S have to say when the DFT is applied to data that is not
> constructed to be periodic in N you will need to look at Ch 11 which
> discusses the result when the signal is not periodic at a frequency sampled
> by the DFT (Section 11.2.2) as well as stochastic and non-stationary signals.

it doesn't matter whether the data was sampled from physical process and 
windowed to get x[n] for  0 <= n < N or if that data is generated from 
some deterministic definition (say a sine wave), or if that data is 
generated from a RNG or some random process.  the spectrum returned by 
the DFT is discrete and represents exactly what that data is as 
periodically extended.  doesn't matter what the source is.

> Hey everybody, go take a look at the whole serious presentation and decide
> what your own interpretation is.

i second the motion.  before deciding whether or not the DFT inherently 
imposes periodicity onto the input sequence passed to it, we should get 
people to weigh in on what O&S say.  and (after fixing misspellings) i 
represented it accurately.

i would like to see how people who read English text normally would 
interpret words like "inherent" and "always".

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."