Prove that, in DFT, Upsampling in time domain is equal to replication in frequency domain Given:
$x[n]$ is an $N$-point sequence whose DFT is $X[k]$
$$x[n]\xrightarrow{\mathcal{DFT}} X[k]$$
then,
Prove that: DFT of the same sequence after insertion of $(M-1)$ zeroes between successive samples is $M$ times repetition of DFT of original Sequence .
$$\begin{aligned}x\left[\frac{n}{M}\right]\xrightarrow{\mathcal{DFT}}\{&amp;X[0],0,0,\dots,0,\\&amp;X[1],0,0,\dots,0,\\&amp;X[2],0,0,\dots,0,\\&amp;\vdots\\&amp;X[N-2],0,0,\dots,0,\\&amp;X[N-1],0,0,\dots,0 \}\end{aligned}$$
My attempt:
$$ \mathcal{DFT} \left\{x\left[\frac{n}{M}\right] \right\}=\displaystyle\sum_{n=0}^{n=MN-1}x\left[\dfrac{n}{M}\right]W_{N}^{nk}$$
replacing $n\to{ nM}$
$$\text{LHS}=\displaystyle\sum_{n=0}^{n=N-\frac{1}{M}}x[n]W_{N}^{Mnk}\neq \big(X[k]\big)^M$$
I'm doing some mistake that is why i'm not getting $\text{RHS}=\big(X[k]\big)^M$.
Can anyone help me in proving this property?
Note:
$x[n]$  have period $ N$ so, $x[\frac{n}{M}]$ will be having period $MN$.
$W_{N}$ is twiddle factor $=N$ th root of unity$=\exp\left(-j\tfrac{2\pi}{N}\right) $
We start with
$$X[k] = \sum_{n=0}^{N-1} x[n] \cdot W_{N}^{nk}$$
Let's define a new sequence of $y[r]$ of length $R=M \cdot N$, where $R, M, N \in \mathbb{Z} &gt; 0$

$$y[r] = \begin{cases}
y[n \cdot M] = x[n] \qquad &amp; r = n \cdot M \\ 
\\ 0 \qquad &amp; \text{otherwise} \\
\end{cases}$$

Transform it
$$Y[k] = \sum_{r=0}^{R-1} y[r] \cdot W_{R}^{rk}$$

Now most of the samples of $y[r]$ are zero, so we can eliminate them from the sum

$$Y[k] = \sum_{r=0}^{R-1} y[r] \cdot W_{R}^{rk} = \sum_{n=0}^{N-1} y[n \cdot M] \cdot W_{R}^{nMk} = \sum_{n=0}^{N-1} x[n] \cdot W_{R}^{nMk} $$

Next we take a look at the twiddle factor
$$W_{R}^{nMk} = e^{-j\frac{2 \pi nMk}{R}} = e^{-j\frac{2 \pi nk}{N}} = W_{N}^{nk} $$
 and put this back into  the Fourier Transform

$$Y[k] =  \sum_{n=0}^{N-1} x[n] \cdot W_{N}^{nk} $$

That's pretty much it. Comparing to the first equation we see directly that
$$Y[k]=X[k]$$

We also see that $Y[k]$ is periodic in $N$ and not just in $R$. Since both $x[n]$ and $W_{N}^{nk}$ are periodic in $N$, so must be $Y[k]$.