Hi Friends,
I have a signal data with real type, and want to Fourier transform it into the spectrum. To do this I use discrete fourier transform (dft) and discrete cosine transform (dct), respectively.
But the two FT methods give the spectrum with some difference in the location of the frequency peaks and also the intensity. According to my understanding, dct and dft do the same thing, the difference is dft use the input data with complex type data and dct use the real data, but this does not matter, dft can also just take the real part of the data. Then, the real part of the dft is a kind of dct.
The dct wiki suggests: "DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample."
Can anybody give some comments on this?
I just want to be sure that dct result is consistent with dft, i.e, shows less difference in the peak location and intensity for the obtained spectrum.
thanks
Firstly, your statement that the DFT uses input data which is complex is not correct. It can handle complex data, but can also be fed with real data. As the quote you give states: the DCT is equivalent for the DFT for real data with even symmetry. Due to the cyclic nature of the discrete transforms, this means that if you reverse the data array, you should have exactly the same sequence of numbers. If this is the case, then the DCT should give the same result as the DFT. If the reversed sequence isn't the same as the original sequence, then the data does not have even symmetry, and there is likely to be an even and an odd component. This means that the DFT will have real and imaginary components, with the imaginary component corresponding to the DFT of the component of the data with odd symmetry.
Given that you state that the DCT and DFT give different results, this leads to the conclusion that your data does not have even symmetry (as is nearly always the case with any dataset). This means that the DCT is not appropriate for analysing the data as the odd component of the data will not be represented.
Firstly, your statement that the DFT uses input data which is complex is not correct. It can handle complex data, but can also be fed with real data. As the quote you give states: the DCT is equivalent for the DFT for real data with even symmetry. Due to the cyclic nature of the discrete transforms, this means that if you reverse the data array, you should have exactly the same sequence of numbers. If this is the case, then the DCT should give the same result as the DFT. If the reversed sequence isn't the same as the original sequence, then the data does not have even symmetry, and there is likely to be an even and an odd component. This means that the DFT will have real and imaginary components, with the imaginary component corresponding to the DFT of the component of the data with odd symmetry.
Given that you state that the DCT and DFT give different results, this leads to the conclusion that your data does not have even symmetry (as is nearly always the case with any dataset). This means that the DCT is not appropriate for analysing the data as the odd component of the data will not be represented.
Basicly DCT is used for those processes where low frequency content will be emphasized. Such as in speech or image coding. For spectral analysis purposes however, DFT would yield a better tool and mapping its results to physical frequencies is so simple.
to add to Barry's answer, you can make the data symmetric by reflecting it at the endpoint, and taking the DFT and DCT, they should yield the same result now
Good point Vikas. In addition, you need to consider the difference in the data values at the start and the end of the dataset. If there is a significant difference, this will result in large amplitude high frequency signals being detected. There are several ways of overcoming this (e.g. detrending the data, using an apodisation function).
My last post wasn't particularly clear - apologies. The need to overcome any mismatch between the start and end of the dataset applies to applying the DFT or DCT to the original data. If you reflect the data about the y-axis then the new "data" is twice as long, and the start and end of this new "data" line up exactly. The problem is that the DFT and DCT assume that the data are cyclic (data loops from the end back to the start). You can analyse the effect of this by padding the data with zeroes (adding zeroes at the start or end of the dataset) and repeating the analysis.
In terms of your question to Vikas, in order to make the data suitable for using the DCT, the easiest way would be to follow Vikas' suggestion. This would ensure it has even symmetry, and the DCT can then be used. If you don't reflect the data, then you will likely have a component of the data with odd symmetry which cannot be represented by cosines. This would mean that the DCT will only be giving information about part of the data.
Note that this does depend on the DCT code you are using. It may be possible that this does the reflection for you - you should check.
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