# down sampling and a factor of 2 without loss of information

I am trying to understand the limits of down sampling on a signal, i.e. how far can I go without loss of information or overlapping. It says in the solution manual of the Oppenhiem and Schafer book (on a question where down sampling is 2) that

"There is no loss of information if X(e^(jw/2) ) and X( e^(j(w/2) - pi) ) do not overlap."

Does this mean that my original signal is periodic and when I down sample means that my frequency response X(e^jw) becomes X(e^(jw/2) ) ?

Please help,

• Can you give us more of the context. The full question from the book, if possible, would be ideal. – Jim Clay Dec 7 '14 at 19:25

## 1 Answer

If you down-sample without filtering, you will fold spectrum above half the new sample rate together with spectrum below half the new sample rate. If both those spectral bands (above and below) are sparse enough (big gaps in the spectral content) in the right places, then it is possible that when folded, there will be no overlap of the folded non-zero high band spectrum with the non-zero low band spectrum, thus no loss of information due to aliasing. If you know where the spectral gaps were, you could unfold the data (with some effort).

In the more typical case where those two spectrum bands (upper and lower) are not sparse enough and thus do overlap, the two spectrums be mixed (like mixing paint) when downsampling, thus incur an informationally lossy process.

• This makes perfect sense, but I had a hard time to translate it to equations.... – Mona Dec 7 '14 at 21:34