Reply by Martin Eisenberg●January 26, 20092009-01-26
jhealy wrote:
>>I don't know if this is it, but often a signal's spectrum is
>>stipulated to decay exponentially fast outside a certain
>>frequency interval. In that case, if you truncate outside that
>>interval, the integral of spectral magnitude over the tails
>>exists and is bounded by the largest (lowest-frequency) tail
>>magnitude times a constant (related to the decay rate).
> Martin, that sounds like exactly what I mean. Can you provide a
> reference for this?
No DSP reference, sorry. But integrals of exponentials
are Calculus 101 -- int B..inf M(f) df <= 1/r * M(B),
where M(f) <= a*exp(-r*f) for f > B.
Martin
--
Quidquid latine scriptum est, altum videtur.
Reply by jhealy●January 26, 20092009-01-26
>jhealy wrote:
>
>> I'm sure I've been told that there are known bounds on the mean
>> square error caused by the truncation of the Fourier transform
>> (analogue, preferably). Anyone know what I'm talking about?
>
>I don't know if this is it, but often a signal's spectrum is
>stipulated to decay exponentially fast outside a certain frequency
>interval. In that case, if you truncate outside that interval, the
>integral of spectral magnitude over the tails exists and is bounded
>by the largest (lowest-frequency) tail magnitude times a constant
>(related to the decay rate).
>
>
>Martin
Martin, that sounds like exactly what I mean. Can you provide a reference
for this?
Thanks again for your help everyone - I haven't been as clear in what I
was looking for as I would like, and your efforts are appreciated.
Reply by Fred Marshall●January 18, 20092009-01-18
jhealy wrote:
> I seem to have been characteristically unclear.
>
> F(k) is the function defined by the Fourier transform of f(x),
> obtained by an integral over a range from minus infinity to infinity
> (This is what I meant by analogue - continuous is the term I meant).
> If we change the limits to some finite range, we get a function which
> is increasingly unlike F(k). This is the error I'm looking for
> information on.
>
> I think Fred is talking about this when he mentions leakage or
> ripple. Can you point me in the direction of something on this Fred?
>
> As someone said, the equivalent case for the DFT depends on what
> algorithm you use, but I'm not interested in that right now.
>
> Regards,
> John
Well, the FT of a rectangular window or "gate" function is a sinc. The
"tails" of the sinc represent the leakage components.
When you multiply a time function by a gate function - that is, you limit
the extent of the nonzero record, then the frequency domain equivalent is
convolution by the sinc.
So, if you have a pure sinusoid of infinite extent, it will correspond to a
pair of Diracs in frequency. Then, when you time limit the sinusoid, you
also convolve the Diracs with a sinc in frequency. And that's what
introduces the leakage terms - the tails of the sinc.
The longer the time record, the narrower the sinc main lobe and the closer
together the "sidelobes" (using the antenna pattern analogy). But the
height of the peaks in the tails, the sidelobes, remains the same always as
long as the window is rectangular.
Is that what you were looking for?
Fred
Reply by Eric Jacobsen●January 16, 20092009-01-16
On Fri, 16 Jan 2009 15:44:42 -0500, Jerry Avins <jya@ieee.org> wrote:
>glen herrmannsfeldt wrote:
>> Gordon Sande <g.sande@worldnet.att.net> wrote:
>> (snip on naming of transforms)
>>
>>> As an undergraduate I learned about the Fourier Transform for unbounded
>>> continuous time. Then there was an extension for continuous periodic
>>> time to deal with rotating machinery with their discrete frequencies.
>>
>> Well, if you learn it that way...
>>
>> I knew about the Fourier series many years before undergrad, at
>> least that a square wave could be represented as an infinite
>> sum of sines. (But not yet how to find the appropriate sum.)
>>
>> Then my undergrad TA did a poor job of explaining the continuous
>> transform as the limit of the series as the period goes to infinity.
>>
>> Most likely I am not the only one who learned it that way.
>
>You aren't; I did too. What's more, the first harmonic analysis I did
>for real consisted of graphically calculating the lower (say through 7)
>harmonics generated in Class-B audio amplifiers using the transfer
>function taken from the tube's plate characteristic and its operating
>conditions. (Most AM transmitters of the time used no feedback in the
>final to suppress distortion.)
>
>>> As a mathematical curiosity there was its "paired" analysis of unbounded
>>> discrete time. The rotating machinery variant was called Fourier Series
>>> and the discrete time variant was labelled Fourier Sequences. The
>>> impression was that there was no good name with sequences being
>>> "something other than series" and a borrowed name from Taylor series
>>> and such. The secondary names seemed mostly to allow the several book
>>> chapters and sections to have distinct short titles.
>>
>> (snip)
>>
>>> The commmon names are more for convenience of engineering style
>>> cook books. Discussions over the correct name are about as serious as
>>> those over which is the better beer. As a grad student there was a
>>> local beer store that carried a nice Beck bock beer seasonally and some
>>> dark German beers the rest of the time so I never got into the discussions
>>
>> I agree, and am not expecting the names to change.
>>
>> But people expecting to use the FFT to transform non-periodic
>> data do need to be reminded of the difference.
>
>Oh? You mean that the FT assumes ...? :-)
>
>Jerry
> Gordon Sande <g.sande@worldnet.att.net> wrote:
> (snip on naming of transforms)
>
>> As an undergraduate I learned about the Fourier Transform for unbounded
>> continuous time. Then there was an extension for continuous periodic
>> time to deal with rotating machinery with their discrete frequencies.
>
> Well, if you learn it that way...
>
> I knew about the Fourier series many years before undergrad, at
> least that a square wave could be represented as an infinite
> sum of sines. (But not yet how to find the appropriate sum.)
>
> Then my undergrad TA did a poor job of explaining the continuous
> transform as the limit of the series as the period goes to infinity.
>
> Most likely I am not the only one who learned it that way.
You aren't; I did too. What's more, the first harmonic analysis I did
for real consisted of graphically calculating the lower (say through 7)
harmonics generated in Class-B audio amplifiers using the transfer
function taken from the tube's plate characteristic and its operating
conditions. (Most AM transmitters of the time used no feedback in the
final to suppress distortion.)
>> As a mathematical curiosity there was its "paired" analysis of unbounded
>> discrete time. The rotating machinery variant was called Fourier Series
>> and the discrete time variant was labelled Fourier Sequences. The
>> impression was that there was no good name with sequences being
>> "something other than series" and a borrowed name from Taylor series
>> and such. The secondary names seemed mostly to allow the several book
>> chapters and sections to have distinct short titles.
>
> (snip)
>
>> The commmon names are more for convenience of engineering style
>> cook books. Discussions over the correct name are about as serious as
>> those over which is the better beer. As a grad student there was a
>> local beer store that carried a nice Beck bock beer seasonally and some
>> dark German beers the rest of the time so I never got into the discussions
>
> I agree, and am not expecting the names to change.
>
> But people expecting to use the FFT to transform non-periodic
> data do need to be reminded of the difference.
Oh? You mean that the FT assumes ...? :-)
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by glen herrmannsfeldt●January 16, 20092009-01-16
Gordon Sande <g.sande@worldnet.att.net> wrote:
(snip on naming of transforms)
> As an undergraduate I learned about the Fourier Transform for unbounded
> continuous time. Then there was an extension for continuous periodic
> time to deal with rotating machinery with their discrete frequencies.
Well, if you learn it that way...
I knew about the Fourier series many years before undergrad, at
least that a square wave could be represented as an infinite
sum of sines. (But not yet how to find the appropriate sum.)
Then my undergrad TA did a poor job of explaining the continuous
transform as the limit of the series as the period goes to infinity.
Most likely I am not the only one who learned it that way.
> As a mathematical curiosity there was its "paired" analysis of unbounded
> discrete time. The rotating machinery variant was called Fourier Series
> and the discrete time variant was labelled Fourier Sequences. The
> impression was that there was no good name with sequences being
> "something other than series" and a borrowed name from Taylor series
> and such. The secondary names seemed mostly to allow the several book
> chapters and sections to have distinct short titles.
(snip)
> The commmon names are more for convenience of engineering style
> cook books. Discussions over the correct name are about as serious as
> those over which is the better beer. As a grad student there was a
> local beer store that carried a nice Beck bock beer seasonally and some
> dark German beers the rest of the time so I never got into the discussions
I agree, and am not expecting the names to change.
But people expecting to use the FFT to transform non-periodic
data do need to be reminded of the difference.
-- glen
Reply by Rune Allnor●January 16, 20092009-01-16
On 16 Jan, 15:32, Gordon Sande <g.sa...@worldnet.att.net> wrote:
> On 2009-01-16 08:05:08 -0400, Rune Allnor <all...@tele.ntnu.no> said:
> > On 16 Jan, 05:39, robert bristow-johnson <r...@audioimagination.com>
> > wrote:
> >> On Jan 15, 4:16�pm, Gordon Sande <g.sa...@worldnet.att.net> wrote:
>
> >>> On 2009-01-15 16:58:07 -0400, glen herrmannsfeldt <g...@ugcs.caltech.ed
> > u> said:
>
> >>>> The Fourier series is defined for periodic functions, integrating
> >>>> (or summing) over one period. �I believe that the FFT is misnamed
> >>>> (should be FFS) but it is a little late now.
>
> >>> It is the correct Fourier Transform over its index group. It is
> >>> just that the classical mathematical physics never bothered with
> >>> sampled and periodic time. They did continuous unbounded, sampled
> >>> unbounded and continuous periodic time. Leaving out the only case
> >>> that could actual be subject to arithmetical procedures which
> >>> is the DFT and its efficient version called the FFT.
>
> >> the only context i have ever heard of the FFT is as the DFT, as Gordon
> >> says the FFT is an efficient method to compute the DFT. �never heard
> >> of the "FFS".
>
> > "Fast Fourier Series"? The "Fourier Series" is, I believe, the
> > conventional name for the series computed from a continuous function
> > of finite duration. Extending it to be used with infinite-duration
> > periodic signals is a more or less ad hoc practical adaption, on a
> > par with using the DFT to estimate the spectra of infinite-length
> > discrete sequences.
>
> As an undergraduate I learned about the Fourier Transform for unbounded
> continuous time. Then there was an extension for continuous periodic
> time
Don't have the books available, but as I remember it, we were taught
the
Fourier series in the oposite order: First for infinite periodic
continuous
functions, then - in a different class, on differential equations -
for
continuous functions on an finite domain. Years later, on DSP, the
discrete
variations came in. It was years before I understood that the
different
variants of the Fourier transform were in fact different. Can't point
to specific dates, but it was at least five, maybe ten, years from
I had my first class on the Fourier transform till I started to
see the whole picture. Or at least get the sense there was a bigger
picture.
> Discussions over the correct name are about as serious as
> those over which is the better beer.
Beg to differ. If one had a good education, as you seems to have,
where the terms are introduced one by one in a sensible, cohesive
way, the discussion might seem ludacris. My experience is that
one sees the difference between the different variants of the
FT only after close scrutiny. Insight is only achived after
verbalizing the differences between the variants. If at all.
Rune
Reply by Gordon Sande●January 16, 20092009-01-16
On 2009-01-16 08:05:08 -0400, Rune Allnor <allnor@tele.ntnu.no> said:
> On 16 Jan, 05:39, robert bristow-johnson <r...@audioimagination.com>
> wrote:
>> On Jan 15, 4:16�pm, Gordon Sande <g.sa...@worldnet.att.net> wrote:
>>
>>> On 2009-01-15 16:58:07 -0400, glen herrmannsfeldt <g...@ugcs.caltech.ed
> u> said:
>>
>>>> The Fourier series is defined for periodic functions, integrating
>>>> (or summing) over one period. �I believe that the FFT is misnamed
>>>> (should be FFS) but it is a little late now.
>>
>>> It is the correct Fourier Transform over its index group. It is
>>> just that the classical mathematical physics never bothered with
>>> sampled and periodic time. They did continuous unbounded, sampled
>>> unbounded and continuous periodic time. Leaving out the only case
>>> that could actual be subject to arithmetical procedures which
>>> is the DFT and its efficient version called the FFT.
>>
>> the only context i have ever heard of the FFT is as the DFT, as Gordon
>> says the FFT is an efficient method to compute the DFT. �never heard
>> of the "FFS".
>
> "Fast Fourier Series"? The "Fourier Series" is, I believe, the
> conventional name for the series computed from a continuous function
> of finite duration. Extending it to be used with infinite-duration
> periodic signals is a more or less ad hoc practical adaption, on a
> par with using the DFT to estimate the spectra of infinite-length
> discrete sequences.
As an undergraduate I learned about the Fourier Transform for unbounded
continuous time. Then there was an extension for continuous periodic
time to deal with rotating machinery with their discrete frequencies.
As a mathematical curiosity there was its "paired" analysis of unbounded
discrete time. The rotating machinery variant was called Fourier Series
and the discrete time variant was labelled Fourier Sequences. The
impression was that there was no good name with sequences being
"something other than series" and a borrowed name from Taylor series
and such. The secondary names seemed mostly to allow the several book
chapters and sections to have distinct short titles.
As a graduate student one quickly learned that FTs are associated with
groups. The approximation of one FT by another is really an issue of
how well one indexing group captures the properties of another. The
discussion seems more rational (and much less heated) when posed in
terms of how the indexing groups approximate each other. Much better
to avoid statments like "Fourier Series are only an approximate form
of Fourier Transform" when the intent is to notice that periodic
time is only like unbounded time when the period is quite long.
The commmon names are more for convenience of engineering style
cook books. Discussions over the correct name are about as serious as
those over which is the better beer. As a grad student there was a
local beer store that carried a nice Beck bock beer seasonally and some
dark German beers the rest of the time so I never got into the discussions
of Schlitz (Milwaukee) vrs Smidts (Philly). It was a chemistry grad
student friend who found the good beer store although the late night
beer and pasta joint we went to did not have the imports.
>> but, Gordon, i really agree with Glen that the DFT is identical in
>> every respect to the DFS. �two different names for the same thing.
>
> Depends on what they are. If the Fourier Series is indeed the
> series computed from the finite-duration continuous function,
> it is *not* the same as the DFT.
>
>> yes, the DFT can be thought of as the DTFT of a finite length sequence
>> that is zero-padded and the DTFT is then sampled at equally spaced
>> points on the unit circle. �it's true, but i think the most
>> fundamental definition of the DFT is that it maps a periodic sequence
>> with period N in the "time" domain to another periodic sequence with
>> the same period N in the reciprocal "time" domain.
>
> It *can* do that, but it's not necessary. The DFT is well-defined
> in its own right. The problem occurs when one, as Gordon says,
> use it to compute some quantity that *ought* tho have been
> computed with one of the other variations of the Fourier transform.
>
> Rune
Reply by Rune Allnor●January 16, 20092009-01-16
On 16 Jan, 05:39, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On Jan 15, 4:16�pm, Gordon Sande <g.sa...@worldnet.att.net> wrote:
>
> > On 2009-01-15 16:58:07 -0400, glen herrmannsfeldt <g...@ugcs.caltech.edu> said:
>
> > > The Fourier series is defined for periodic functions, integrating
> > > (or summing) over one period. �I believe that the FFT is misnamed
> > > (should be FFS) but it is a little late now.
>
> > It is the correct Fourier Transform over its index group. It is
> > just that the classical mathematical physics never bothered with
> > sampled and periodic time. They did continuous unbounded, sampled
> > unbounded and continuous periodic time. Leaving out the only case
> > that could actual be subject to arithmetical procedures which
> > is the DFT and its efficient version called the FFT.
>
> the only context i have ever heard of the FFT is as the DFT, as Gordon
> says the FFT is an efficient method to compute the DFT. �never heard
> of the "FFS".
"Fast Fourier Series"? The "Fourier Series" is, I believe, the
conventional name for the series computed from a continuous function
of finite duration. Extending it to be used with infinite-duration
periodic signals is a more or less ad hoc practical adaption, on a
par with using the DFT to estimate the spectra of infinite-length
discrete sequences.
> but, Gordon, i really agree with Glen that the DFT is identical in
> every respect to the DFS. �two different names for the same thing.
Depends on what they are. If the Fourier Series is indeed the
series computed from the finite-duration continuous function,
it is *not* the same as the DFT.
> yes, the DFT can be thought of as the DTFT of a finite length sequence
> that is zero-padded and the DTFT is then sampled at equally spaced
> points on the unit circle. �it's true, but i think the most
> fundamental definition of the DFT is that it maps a periodic sequence
> with period N in the "time" domain to another periodic sequence with
> the same period N in the reciprocal "time" domain.
It *can* do that, but it's not necessary. The DFT is well-defined
in its own right. The problem occurs when one, as Gordon says,
use it to compute some quantity that *ought* tho have been
computed with one of the other variations of the Fourier transform.
Rune
Reply by robert bristow-johnson●January 16, 20092009-01-16
On Jan 15, 4:16�pm, Gordon Sande <g.sa...@worldnet.att.net> wrote:
> On 2009-01-15 16:58:07 -0400, glen herrmannsfeldt <g...@ugcs.caltech.edu> said:
>
> > The Fourier series is defined for periodic functions, integrating
> > (or summing) over one period. �I believe that the FFT is misnamed
> > (should be FFS) but it is a little late now.
>
> It is the correct Fourier Transform over its index group. It is
> just that the classical mathematical physics never bothered with
> sampled and periodic time. They did continuous unbounded, sampled
> unbounded and continuous periodic time. Leaving out the only case
> that could actual be subject to arithmetical procedures which
> is the DFT and its efficient version called the FFT.
the only context i have ever heard of the FFT is as the DFT, as Gordon
says the FFT is an efficient method to compute the DFT. never heard
of the "FFS".
but, Gordon, i really agree with Glen that the DFT is identical in
every respect to the DFS. two different names for the same thing.
yes, the DFT can be thought of as the DTFT of a finite length sequence
that is zero-padded and the DTFT is then sampled at equally spaced
points on the unit circle. it's true, but i think the most
fundamental definition of the DFT is that it maps a periodic sequence
with period N in the "time" domain to another periodic sequence with
the same period N in the reciprocal "time" domain.
r b-j