### Continuous-Time Aliasing Theorem

Let denote any continuous-time signal having a Fourier Transform (FT)

*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

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

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Reconstruction from Samples--The Math