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Rick Lyons wrote:
>On Sun, 03 Oct 2004 18:27:46 +1000, John Monro
><johnmonro@delete.optusnet.com.au> wrote:
>
>
>
>>Eric Jacobsen wrote:
>>(snip)
>>
>>
>>
>>>Sandeep,
>>>
>>>You're getting good answers from others, but I'll summarize a little
>>>bit. The interesting thing here is that there are multiple, valid,
>>>ways to explain this. Understanding all of them really helps, but
>>>initially just understanding one goes a long ways.
>>>
>>>The two fundamental explanations, both of which have already been
>>>described, go something like this:
>>>
>>>1. The sidelobes (aka, leakage energy) are there, you just can't see
>>>them because the frequency samples fall on exactly the zeros of the
>>>sinx/x frequency response sidelobes when the input sinewave has an
>>>exact integer number of cycles within the input window.
>>>
>>>
>>>
>>>
>>(snip)
>>
>>
>>
>>>2. The discontinuities at the end of a non-integer-cycle sequence
>>>require additional frequencies to create, so the additional Fourier
>>>Coefficients that are required start to appear. Only the case where
>>>there is no discontinuity in the input requires a single coefficient
>>>(this gets back to the periodic input assumption discussion).
>>>
>>>
>>>
>>>
>>(snip)
>>
>>Eric,
>>I agree completely with your second interpretation, but not with the
>>first, where you say that the sidelobes "are there."
>>Using Sandeep's original example of a set of signal examples making up
>>exactly five cycles of a sinusoidal wave, he found that the DFT yields
>>up only one component.( A complex component representing the amplitude
>>of the sin and cos components of a pure sinusoidal signal.) No suprises
>>here of course, but it is clear that there is simply no room for any
>>further components, as the original signal can be exactly reconstructed
>>
>>
>>from this one component.
>
>
>>Any further components 'between the bins' so to speak would imply that
>>the original signal was not a pure sinusoid. In fact, any components
>>'between the bins' would imply that the supposed pure sinusoidal signal
>>was in fact modulated by a signal that had a period longer than five
>>cycles of the original signal.
>>
>>Regards,
>>John
>>
>>
>
>Hi John,
>
> I thought Eric's post was very good! I agree Eric that there are
>two interpretations (explanations) of DFT "leakage".
>
>If you have a finite-length x(n) time sequence representing 5 cycles
>of a sinewave, you can perform the DFT on those x(n) samples
>to obtain the X(m) freq-domain samples. The question is, "What
>do those X(m) samples represent?" There are two different, but
>equivalent interpretations of the X(m) samples.
>
>Interpretation# 1: The X(m) sequence represent samples of the
>continuous Fourier transform (CFT) of the finite-length x(n) time
>sequence. Because x(n) is only five cycles of a sinewave
>(as opposed to an infinite number of cycles), its CFT is a
>sin(x)/xlike continuous function (a continuous curve). The DFT
>computes individual samples lying on that curve. In this case,
>the DFT sidelobes exist because of the sin(x)/x spectrum of
>a truncated (finite-length) x(n) sinusoid.
>
>
>Interpretation# 2: The X(m) sequence is the CFT of an
>infinitely long time sequence comprising infinite repetitions
>of the x(n) time sequence. Using this interpretation, some people
>say, "X(m) is the Fourier transform of a periodic version of x(n)".
>(What people should *never* say, however, is, "The DFT assumes
>it's input sequence is periodic.")
>In this case, the DFT sidelobes
>exist because of any discontinuity between the last sample
>of x(n), in one repetition, and the first sample of
>x(n) in the next repetition.
>
>As far as I know, both interpretations are valid, and
>equivalent to each other.
>
>Sheece! I hope the above makes some sense.
>[-Rick-]
>
>
Rick,
It does make sense, and I found it interesting to reflect further on the
two interpretations.
I did not like the idea of 'hidden' sidelobes in interpretation no.1,
however the sidelobes turn out to contain information that could be
used to reconstruct a version of the time-domain signal, and this would
appear as a short burst, preceded and followed by infinite periods of
silence.
On the other hand, in interpretation no. 2, the sampled CFT (the DFT)
contains information that could be used to reconstruct another verson of
the time-domain signal, and this signal would appear as the same short
burst of signal, but repeating over an over.
I prefer the second interpretation as it seems to me to relate more
closely to practical applications; the overlap / save FIR filter for
example. I do tend to agree that both interpretations are valid though.
I particularly liked Eric's remark:
If you only understand one of these points of view you can do
anything you need to do with DFTs/FFTs and be consistent and get
good results. Understanding both is even better. ;)
Thanks for an interesting explanation of the two points of view.
Regards,
John
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Rick Lyons wrote:<br>
<blockquote type="cite" cite="mid415fe1c0.3450781@news.sf.sbcglobal.net">
<pre wrap="">On Sun, 03 Oct 2004 18:27:46 +1000, John Monro
<a class="moz-txt-link-rfc2396E" href="mailto:johnmonro@delete.optusnet.com.au"><johnmonro@delete.optusnet.com.au></a> wrote:
</pre>
<blockquote type="cite">
<pre wrap="">Eric Jacobsen wrote:
(snip)
</pre>
<blockquote type="cite">
<pre wrap="">Sandeep,
You're getting good answers from others, but I'll summarize a little
bit. The interesting thing here is that there are multiple, valid,
ways to explain this. Understanding all of them really helps, but
initially just understanding one goes a long ways.
The two fundamental explanations, both of which have already been
described, go something like this:
1. The sidelobes (aka, leakage energy) are there, you just can't see
them because the frequency samples fall on exactly the zeros of the
sinx/x frequency response sidelobes when the input sinewave has an
exact integer number of cycles within the input window.
</pre>
</blockquote>
<pre wrap="">(snip)
</pre>
<blockquote type="cite">
<pre wrap="">2. The discontinuities at the end of a non-integer-cycle sequence
require additional frequencies to create, so the additional Fourier
Coefficients that are required start to appear. Only the case where
there is no discontinuity in the input requires a single coefficient
(this gets back to the periodic input assumption discussion).
</pre>
</blockquote>
<pre wrap="">(snip)
Eric,
I agree completely with your second interpretation, but not with the
first, where you say that the sidelobes "are there."
Using Sandeep's original example of a set of signal examples making up
exactly five cycles of a sinusoidal wave, he found that the DFT yields
up only one component.( A complex component representing the amplitude
of the sin and cos components of a pure sinusoidal signal.) No suprises
here of course, but it is clear that there is simply no room for any
further components, as the original signal can be exactly reconstructed
</pre>
</blockquote>
<pre wrap=""><!---->>from this one component.
</pre>
<blockquote type="cite">
<pre wrap="">Any further components 'between the bins' so to speak would imply that
the original signal was not a pure sinusoid. In fact, any components
'between the bins' would imply that the supposed pure sinusoidal signal
was in fact modulated by a signal that had a period longer than five
cycles of the original signal.
Regards,
John
</pre>
</blockquote>
<pre wrap=""><!---->
Hi John,
I thought Eric's post was very good! I agree Eric that there are
two interpretations (explanations) of DFT "leakage".
If you have a finite-length x(n) time sequence representing 5 cycles
of a sinewave, you can perform the DFT on those x(n) samples
to obtain the X(m) freq-domain samples. The question is, "What
do those X(m) samples represent?" There are two different, but
equivalent interpretations of the X(m) samples.
Interpretation# 1: The X(m) sequence represent samples of the
continuous Fourier transform (CFT) of the finite-length x(n) time
sequence. Because x(n) is only five cycles of a sinewave
(as opposed to an infinite number of cycles), its CFT is a
sin(x)/xlike continuous function (a continuous curve). The DFT
computes individual samples lying on that curve. In this case,
the DFT sidelobes exist because of the sin(x)/x spectrum of
a truncated (finite-length) x(n) sinusoid.
Interpretation# 2: The X(m) sequence is the CFT of an
infinitely long time sequence comprising infinite repetitions
of the x(n) time sequence. Using this interpretation, some people
say, "X(m) is the Fourier transform of a periodic version of x(n)".
(What people should *never* say, however, is, "The DFT assumes
it's input sequence is periodic.")
In this case, the DFT sidelobes
exist because of any discontinuity between the last sample
of x(n), in one repetition, and the first sample of
x(n) in the next repetition.
As far as I know, both interpretations are valid, and
equivalent to each other.
Sheece! I hope the above makes some sense.
[-Rick-]
</pre>
</blockquote>
<br>
Rick,<br>
It does make sense, and I found it interesting to reflect further on
the two interpretations. <br>
<br>
I did not like the idea of 'hidden' sidelobes in interpretation no.1,
however the sidelobes turn out to contain information that could
be used to reconstruct a version of the time-domain signal, and this
would appear as a short burst, preceded and
followed by infinite periods of silence. <br>
On the other hand, in interpretation no. 2, the sampled CFT (the DFT)
contains information that could be used to reconstruct another verson
of the time-domain signal, and this signal would appear as the same
short burst of signal, but repeating over an over.<br>
<br>
I prefer the second interpretation as it seems to me to relate more
closely to practical applications; the overlap / save FIR filter for
example. I do tend to agree that both interpretations are valid though.<br>
<br>
I particularly liked Eric's remark:<br>
<blockquote>If you only understand one of these points of view you can
do anything you need to do with DFTs/FFTs and be consistent and get
good results. Understanding both is even better. ;)<br>
</blockquote>
Thanks for an interesting explanation of the two points of view.<br>
<br>
Regards, <br>
John<br>
<blockquote><br>
<br>
<br>
</blockquote>
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