|WikiProject Mathematics||(Rated Start-class, Low-priority)|
The so-called "symmetric convolution operator" is mentioned but never mathematically defined.
The "most notable" advantage is described this way: "The implicit symmetry of the transforms involved is such that only data unable to be inferred through symmetry is required. For instance using a DCT-II, a symmetric signal need only have the positive half DCT-II transformed, since the frequency domain will implicitly construct the mirrored data comprising the other half." That amounts to a claim of computational efficiency. And yet there is no attempt to justify it in light of the renowned "N Log N" efficiency of the FFT-based algorithm it purports to replace. Furthermore, there is no other mention of "efficiency" in the article. That claim comes as a surprise at the end. The earlier claim is "easier to compute". But that is followed by the apparent contradiction: "Using these transforms to compute discrete symmetric convolutions is non-trivial since discrete sine transforms (DSTs) and discrete cosine transforms (DCTs) can be counter-intuitively incompatible for computing symmetric convolution".
- This article is kind of helpful if you, like I did, come here to look for some help in solving a specific problem. It is just not writen like a wikipedia article explaining the topic. The computational efficiency of the dct/dst is the same as for fft, that is NlogN. In fact, the fft may be used in calculating the cosine and sine tranforms (http://dx.doi.org/10.1109/TASSP.1980.1163351). The advantage is a factor of 2 in the size N of your data. If the functions to be convoluted are in some way symmetric, you really need only half of the data (or one fourth in two dimensions). The seemingly contradictive claim at the end means, that if you want to perform convolutions as simple multiplications in fourier-space, you have to choose the correct combination of forth and back transforms. The table provided in the article is therefore very helpful. It is probably taken from the quoted source (http://dx.doi.org/10.1109/78.295213). --188.8.131.52 (talk) 08:24, 22 August 2013 (UTC)