time downsampling vs. frequency downsampling
Your second equation suggests that $y[n]$ is obtained by performing an $N$-point inverse DFT of an $M$-point DFT $X[k]$ of an $M$-point sequence $x[n]$.

The result can be interpreted as shifted copies of $x[n]$ are superimposed, causing possible aliasing in time, which depends on whether $ N \geq M$ or $N &lt; M$.

If $N \geq M$, there is no aliasing and $y[n]$ will be an $N$ point sequence whose first $M$ samples will be identical to $x[n]$ for $n=0,1,...,M-1$ and remaining $N-M$ samples will be zero-padded.

$$
y[n] = \begin{cases} { x[n] ~~~,~~~ n = 0,1,...,M-1 \\ ~~~~~0 ~~~,~~~ n= M,M+1,...N-1} \end{cases}
$$ 

Otherwise if $N &lt; M$, then $y[n]$ will be a time-aliased version of $x[n]$, as shifted copies of $x[n]$ would be overlapping each other. If you want any non-aliased samples in $y[n]$, then $N &gt; M/2$ in general and in such a case $y[n]$ could be defined as:

$$
y[n] = \begin{cases} { x[n]+x[n+N] ~~~, n = 0,1,...,M-N \\ x[n] ~~~~~~~~~~~~~~~~~~~~~,~~~ n= M-N+1,...,N-1 } \end{cases}
$$