ASN7 - Sampled half of signal

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Can you retrieve an original signal from a sample of range ${\frac {1}{2}}f>f_{s}>f\!$?

A rule of thumb for proper sampling is to sample range slightly greater than the range of the signal. Suppose however,you sampled only half of the signal can you do something to recover the whole original signal again?

1. Lets imagine that from our the result of our sampling we measured the positive and the negative part of the signal seperately. [Assume the original signal was centered at zero such that $f_{tot}=f_{-}+f_{+}]$

So we have repeated on the negative axis the -f and the positive axis +f.

The trick now is to re-sample at a frequency that will place the negative and positive half side by side.

2. Find the right frequency and sample again.

The result of this second sampling are a number of copies of the original signal $nf_{tot}$ .

3. finally, to get just one copy of the original signal apply a band pass filter.