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In this paper decorrelating detector we have the decorrelating detector. With $K$ users, the correlation matrix of the signature waveform is $R=S^TS$ with $S=[s_1,...,s_K]$. Moreover $size(R)=K \times K$

This is my question: In the article (p $679$), we have: $d_1=\sum_{i=1}^K [R^{-1}]_{1,i}s_i$. But, how can we make this product, the elements do not have the same dimensions? In fact, $size(R^{-1})=K \times K$ and $size(s_i)=N \times 1$ with $N \neq K$

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It all works out because $[\mathbf{R}^{-1}]_{1, i}$ is just a number. It is the number in row $1$ column $i$ of the matrix $\mathbf{R}^{-1}$. So the product $[\mathbf{R}^{-1}]_{1, i}\mathbf{s}_i$ returns a vector of length $N \times 1$. See the line after Equation 10.

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