## 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)

Let

denote the samples of at uniform intervals of seconds, and denote its Discrete-Time Fourier Transform (DTFT) by

Then the spectrum of the sampled signal is related to the spectrum of the original continuous-time signal by

The terms in the above sum for are called aliasing terms. They are said to alias into the base band . Note that the summation of a spectrum with aliasing components involves addition of complex numbers; therefore, aliasing components can be removed only if both their amplitude and phase are known.
Proof: Writing as an inverse FT gives

Writing as an inverse DTFT gives

where denotes the normalized discrete-time frequency variable. 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

Since this is formally the inverse DTFT of written in terms of , the result follows.

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Sampling Theorem
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