# Effect in Fourier-domain when the signal is “zero-stuffed” in time-domain

Suppose $x[n]=[x_0,x_1, \ldots,x_{N-1}]$ and $y[n]=[x_0,0,x_1,0,\ldots,0,x_{N-1}]$. What is the relationship between the Fourier Transforms of both the signals?

When I try for some specific examples in MATLAB, I see that $Y[k]$ has an image of $X[k]$ but how do we show this mathematically?

Let's be more general and consider the insertion of $L-1$ zeroes between each sample. Mathematically, this can be written as $$y[m]= \begin{cases} x[n] & \text{if } m = nL\\ 0 & \text{otherwise} \end{cases}.$$ The DTFT of $y[n]$ is then given by $$Y(e^{j\Omega}) = \sum_{m=-\infty}^{\infty} y[m]e^{-jm\Omega}.$$ Because $y[m]$ is non-zero only when $m$ is a multiple of $L$, that is $m = nL$, this becomes $$Y(e^{j\Omega}) = \sum_{n=-\infty}^{\infty} x[n]e^{-jnL\Omega} = X(e^{jL\Omega}).$$ From this, you can see that the DTFT of $y$ is the DTFT of $x$ with a period $L$ times smaller (i.e. $2\pi/L$ instead of $2\pi$). Intuitively, this makes sense: you know that "expanding" a signal in the time-domain is equivalent to "squeezing" its spectrum in the frequency domain (see the time scaling property of the Fourier transform).