This section quantifies aliasing in the general case. This result is then used in the proof of the sampling theorem in the next section.
It is well known that when a continuous-time signal contains energy at a frequency higher than half the sampling rate , sampling at samples per second causes that energy to alias to a lower frequency. If we write the original frequency as , then the new aliased frequency is , for . This phenomenon is also called ``folding'', since is a ``mirror image'' of about . As we will see, however, this is not a complete description of aliasing, as it only applies to real signals. For general (complex) signals, it is better to regard the aliasing due to sampling as a summation over all spectral ``blocks'' of width .
Continuous-Time Aliasing Theorem
Proof: Writing as an inverse FT gives
The inverse FT can be broken up into a sum of finite integrals, each of length , as follows:
Let us now sample this representation for at to obtain , and we have
since and are integers. Normalizing frequency as yields
Introduction to Sampling