DFT of signal with mod

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My question is about a real signal : $x[n]$ ,$n=0....N-1$ and DFT $X[k]$ also with $N$ length. I know that $X[k]=$$X[((-k))_N]$ and $x[n]$=$x[((-n))_N] $. now, there is a signal : $y[n]$=$x[((n-1))_{N/2}]$. Im asking for matrix $A$ that $Y=AX$ , while $Y$ is $Y[k]$ vect and $X$ is $X[k]$ vect. first,I tried to find the DFT by definition:
$Y[k]=$$\sum_{n=0}^{N-1} x[((n-1))_{N/2}] e^{\frac{-j2\pi kn}{N}}$ . I tried to find a matrix but I dont know how to continue from this point.

  • $[((n ))_N]$ means $n $ modulo N

    Thanks for help