### 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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