I have an assignment where I'm given the DFT of a sequence $x[n]$ as $X[k]=\{4,3,2,1,0,1,2,3\}$ and also $$y[n] = \left\{ \begin{array}[cc] xx[n/2] & \text{if n is even} \\ 0 & \text{otherwise} \end{array} \right. $$
and I'm supposed to find and sketch the DFT of $y[n]$.
So $y[n] = \{x[0], 0 x[1], 0 ... x[7], 0\}$ and it's not complicated to find $Y[k]$ if we know $x[n]$
I know how to use the definition of the DFT and IDFT to calculate $x[n]$ but it's a tedious task to do by hand, especially when the sequence is longer than a few items. Is there a quicker way to calculate the DFT and IDFT by hand without using a program like Matlab?