Aliasing of Sampled Signals
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
Let denote any continuous-time signal having a Fourier Transform (FT)
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