Wednesday, September 19, 2018

fourier transform - Aliasing after downsampling




Let me start with time domain representation of the original signal xn=2N1k=0Xkej2πnk2N where 2N is number of time/frequency samples while n and k are time and frequency indices, respectively. Downsampling by factor of M is essentially multiplying the time domain signal with the comb function which selects every M-th sample while others are set to zero, i.e.,


ˇxn=xnIII(n;M)


where comb function III(n;M)=2NM1m=0δ(nmM)=2NM1m=0ej2πnmM. In other words downsampled signal is equal to ˇxn=2N1k=02NM1m=0Xkej2πnk2Nej2πnmM=2N1k=0Xk2NM1m=0ej2πn(k2N+mM)



  • Are described steps correct up to now?

  • I assume that at one point 2N1k=0 should become (2N/M)1k=0 since by downsampling I am actually reducing max "visible" frequency.This combined with summation over m should ,I suppose, mimic aliasing but I am having problems mathematically showing that. So if somebody could make this a little bit more clear to me.


p.s. I know there a couple of topics related with downsampling and aliasing but although I've read them really carefully I was not able to grasp it completely.




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