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Fourier Transform (DFT), with Audio Applications --- Second
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*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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Aliasing of Sampled Continuous-Time SignalsThis section quantifies aliasing in the general case. This result is then used in the proof of Shannon’s 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 , then sampling at samples per second causes that energy to

aliasto 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 fundamental 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 .

Theorem.(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 calledaliasing terms. They are said toaliasinto thebase 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 theiramplitudeand phaseare 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

since and are integers.

Normalizing frequency as yields

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