## 1 Introduction  
[PERSON] _et al_. [1] developed fast DCT-domain convolution for time/spatial-domain multiplication using DCT type-2. They exploited symmetry and orthogonality for the fast algorithm. Logo-keying operation in the spatial domain can be done in the DCT domain for compressed image/video editing. However, they have not derived the convolution from the symmetric-convolution multiplication (SCM) property of discrete trigonometric transforms (DTTs), which will be the focus of this paper. Time-domain symmetric convolution for a linear phase filtering application has been developed by [PERSON] [2] and is called the symmetric-convolution multiplication property of DTTs. Based on the SCM property, [PERSON] _et al_. [3] have developed a symmetric convolution for linear phase FIR filtering. Filter coefficients consist of symmetric and asymmetric parts. Based on those earlier works, [PERSON] _et al_. [4] have finally developed fast circular convolution using DTTs. The input sequences to be convolved need not be symmetric or asymmetric. Thus fast DCTsand DSTs can be used instead of the FFTs for FIR filtering. Generalized fast convolution using numerous transforms as well as the DFT and DTTs has been studied by [PERSON] _et al._ in [5]. In this paper we show that swapping the forward and inverse transforms in the SCM property [2] yields a multiplication-convolution (MSC) property. That is, the convolution of transformed sequences gives the same results as the forward transform after element-by-element multiplication of the data sequences. The necessary scaling factor \(M\) for these new properties has been described in the third line below Eq. (20) in [2], saying \"it is possible to swap the usage of the forward and inverse transform; in that case, an _extra scaling factor_ may be required\". Here \(M\) is the size of the generalized DFT (GDFT) when a DTT is derived from an \(M\)-point GDFT. We can get 40 types of MSCs corresponding to 40 types of SCMs in Tables 6, 7 of [2]. In (1) of [6], the formulation of the SCM for the unitary matrices is presented, with only the exception of some of the 40 options mentioned by [PERSON] [2], because they cannot be expressed with the assumed tools. Here, we fill that gap. Consider the following matrix relationship between convolutional/unnormalized DTTs (denoted with lower-case subscript like C\({}_{1e}\)) and unitary DTTs (denoted with capital-letter subscript like C\({}_{1E}\)). \[ \begin{array}{ll}\mbox{C}_{1e}=\sqrt{2N}\;{\rm A}_{1}^{-1}\;\mbox{C}_{IE}\, \mbox{A}_{1}&\mbox{C}_{2e}=\sqrt{2N}\;\;{\rm A}_{2}^{-1}\mbox{C}_{1 IE}\\ \mbox{C}_{3e}=\sqrt{2N}\;\mbox{C}_{1 IIE}\,\mbox{A}_{2}&\mbox{C}_{4e}=\sqrt{2N} \;\mbox{C}_{1 VE}\\ \mbox{S}_{1e}=\sqrt{2N}\;\mbox{S}_{1E}&\mbox{S}_{2e}=\sqrt{2N}\;\mbox{A}_{3} ^{-1}\mbox{S}_{1 IE}\\ \mbox{S}_{3e}=\sqrt{2N}\;\;\mbox{S}_{1 IIE}\,\mbox{A}_{3}&\mbox{S}_{4e}=\sqrt{2N }\;\mbox{S}_{1 VE}\\ \mbox{C}_{1o}=\sqrt{2N\!-\!1}\;\;{\rm A}_{2}^{-1}\mbox{C}_{IO}\,\mbox{A}_{2}& \mbox{C}_{2o}=\sqrt{2N\!-\!1}\;\;{\rm A}_{2}^{-1}\mbox{C}_{1 IO}\,\mbox{A}_{4} \\ \mbox{C}_{3o}=\sqrt{2N\!-\!1}\;\;{\rm A}_{4}^{-1}\,\mbox{C}_{1 IIO}\,\mbox{A}_{2 }&\mbox{C}_{4o}=\sqrt{2N\!-\!1}\;\;{\rm C}_{1 VO}\\ \mbox{S}_{1o}=\sqrt{2N\!-\!1}\,\mbox{S}_{1O}&\mbox{S}_{2o}=\sqrt{2N\!-\!1}\, \mbox{S}_{1 IIO}\\ \mbox{S}_{3o}=\sqrt{2N\!-\!1}\,\mbox{S}_{1 IIO}&\mbox{S}_{4o}=\sqrt{2N\!-\!1}\;\; {\rm A}_{4}^{-1}\;\mbox{S}_{1 VO}\,\mbox{A}_{4}\end{array}  \tag{1}\]  
Here the fact that scalars commute with matrices is used for C\({}_{3e}\) and others.