Talk:Cepstrum
From Wikipedia, the free encyclopedia
Lots of sources claim that the cepstrum is FT->log->IFT, while others say "not only is the cepstrum FT->log->FT, but FT->log->IFT is wrong", whereas I don't see FT->log->IFT sources claiming that "FT->log->FT" is wrong. What is it? -Jay Kominek
The two methods are functionally equivalent. In the real case, when we use the IFT there is more data in the end result; however half of it is redundant due to symmetry. The IFT will double the number of points, while the FT will halve the number of points. So, using the IFT will yeild twice the frequency resolution. In this respect, using the IFT is equivalent to using the FT with zero padding to double the length of the signal. Using the IFT may also be desirable for purposes of mathematical analysis, which is probably why most DSP references prefer the IFT. From the intuitionists' perspective is that using the FT 'makes more sense', however using the IFT is certainly NOT wrong. In fact, it is always possible to interchange IFT and FT operators, as long as you know what you're doing. ;) --andy
- Andy, this is misleading. The only difference between the Fourier transform and the inverse Fourier transform (or DFT and inverse DFT) is the sign of the exponent in the transform (and possibly the normalization), not the number of outputs. I think you are confusing this with the fact that there are often specialized versions of the DFT (FFT) for real inputs, which only store half of the complex outputs, because the other half is redundant (and conversely, there are specialized inverse FFTs that take conjugate-symmetric outputs to real inputs), but this is an optimization of the implementation, and does not change the logical mathematical transform being computed. —Steven G. Johnson 19:44, Apr 21, 2004 (UTC)
I removed the following note (in bold) from the article: "There are many ways to calculate the cepstrum, some of them need a phase-warping algorithm, others do not. (fixme which one one is?)" Dori 21:16, Nov 18, 2003 (UTC)
- Many texts incorrectly state that the process is FT->log->IFT, i.e. that the cepstrum is the "inverse Fourier transform of the log of the spectrum". This is not the definition given in the original paper, but unfortunately is widespread.
Note that taking the IFT is equivalent to an FT if you have a purely real signal as input and you are computing the "real cepstrum" (log of magnitudes), which gives a real-symmetric signal as input to the final IFT or FT. So, maybe the IFT version is simply a generalization of the original definition to complex signals or complex cepstrums, and is not "incorrect?"
I don't know much about cepstrums (although I have a lot of experience with FFTs), but this article strikes me as bit weak from a mathematical standpoint. —Steven G. Johnson 19:44, Apr 21, 2004 (UTC)
"...unfortunately is widespread." it's more than widespread, it's almost the only used definition in the signal processing community. By the way, the definition in the mel-frequency domain (MFCC) use a DCT (discrete cosinus transform) instead of IFFT or FFT.Celsius813 17 May 2005
---
This is a joke, right? "saphe-cracking" is the give away. Nominate for deletion, as well as anything that links here. --Wtshymanski 21:48, 20 Dec 2004 (UTC) My mistake...this really is legitimate, though in my defense Wikipedia is the first place I've ever heard of this. --Wtshymanski 18:31, 21 Dec 2004 (UTC)
---
Shouldn't the first sentence be "A cepstrum (pronounced "kepstrum") is the result of taking the Fourier transform (FT) of the logarithm of the decibel spectrum as if it were a signal."?
- please get a WP username and sign your posts. anyway, i think the sentence could be improved but (even though i don't use the term), the "decibel spectrum" has already had a logarithm applied to it. you don't want to apply the log to the log of the spectrum. r b-j 03:08, 11 January 2006 (UTC)
Hey...I have read about cepstrums as homomorphic systems. So, if 2 signals are related in the time domain by a non-linear combination as a convolution..taking the FFT would result in the multiplication of the signals in the ffrequency domain. A log maps the multiplication into the addition domain. Any filtering action can now be performed to remove/smooth any of the signals. But the signals still remain in the frequency domain and an inverse FFT would result in us getting the transformed signal in the time domain. I think an IFFT is more appropriate. But this is the only place where I have seen this definition. --S.Sriram 11:28, 25 January 2006 (UTC)
Is the etymology section some kind of joke? It's completely useless and should be taken out. One part of a sentence is all that is needed to explan that cepstrum is a rearrangement of spectrum.
Contents |
[edit] "Correctness" of cepstrum definition
I know this has been discussed above, but a lot of the above discussion is old, and I don't want my contribution to be lost, especially since it concerns an edit I made. Moreover, I'm going to take this from a slightly different angle: what really makes one definition of "cepstrum" more "correct" than the other? Yes, FFT -> log -> FFT was the original definition, but FFT -> log -> IFFT is obviously useful to many people, and is what a lot of people mean when they say "cepstrum". Why can't it be both? Why does the original definition have to be the One True Definition? This is obviously a somewhat subjective matter. Because of this, I decided that it is not NPOV to declare that a given definition is "correct" and have reworded the article accordingly. - furrykef (Talk at me) 18:11, 27 March 2006 (UTC)
[edit] "Real cepstrum"
- The real cepstrum uses the logarithm function defined for real values, while the complex cepstrum uses the complex logarithm function defined for complex values also.
I'm not quite sure what this means. Is it saying that the real cepstrum is real FFT -> log -> real FFT? In other words, throwing away the imaginary part after the first FFT as well as the second? I've also seen "real cepstrum" defined as the complex cepstrum with the real components thrown away at the end; which is it? - furrykef (Talk at me) 18:21, 27 March 2006 (UTC)
- Phase unwrapping does not involve "throwing away" or ignoring either real or imaginary components. It is the treatment of rotating the vector angle of each real+imaginary pair of values to zero angle, where the imaginary part becomes zero. Only phase information is ignored, the new real component alone keeps magnitude information. Cuddlyable3 15:38, 4 March 2007 (UTC)
[edit] Absolute value
Also, I've also seen a cepstrum defined such that the absolute value is taken before the logarithm. This is not mentioned anywhere in the article; which way is it? Or is it another matter of debate like whether or not taking the IFFT can be considered "correct"? - furrykef (Talk at me) 18:51, 27 March 2006 (UTC)
- The absolute value of a real+complex value pair is what is given by phase unwrapping.Cuddlyable3 15:51, 4 March 2007 (UTC)
[edit] Added convolution note
The property that convolutions turn into additions is very important and is indeed a large part of the motivation behind the cepstral domain in the first place. I'm not too good at the math latex stuff so maybe someone can fix that up. --Speedplane 05:14, 5 March 2007 (UTC)
