Discrete time Fourier transform on decimated signals

Question

If I have a signal $x[n]$ and its decimated version, $y[n]=x[2n]$, is there a known expression for the DTFT of $y[n]$, $Y(\theta)$, as a function of $X(\theta)$?

Thanks,

Answer 1

Yes, for decimation by an integer factor $M$, the DTFT of the decimated signal is given by

$$Y(e^{j\theta})=\frac{1}{M}\sum_{m=0}^{M-1}X\left(e^{j(\theta-2\pi m)/M}\right)$$

which shows that the resulting spectrum is an expanded and aliased version of the original spectrum.