DFT Leakage

(snipped)

>>
>>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

Hi John,

   This entire subject of DFT leakage is, I think, 
one of the toughest concepts in DSP.  It's so 
unlike anything we had to deal with in the 
world of analog signals.

Your adept understanding of DFT leakage, John, 
explains why your company pays you so much!

This is good.

[-Rick-]