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Hypothetical spectrum of a inadequately sampled bandlimited signal where the images overlap showing how aliases are created. X(f) (blue) and X_A(f) (blue) are continuous Fourier transforms of two different functions, x(t) and x_A(t) (not shown). When the functions are sampled at rate f_s, the images (green) are added to the original transforms (blue) when one examines the discrete-time Fourier transforms (DTFT) of the sequences. In this hypothetical example, the DTFTs are identical, which means the sampled sequences are identical, even though the original functions are not. If these were audio signals, x(t) and x_A(t) would not sound the same. But their samples (taken at rate f_s) would lead to identical reproduced sounds; thus x_A(t) is an alias of x(t) at this sample rate. In this example (of a bandlimited function), such aliasing can be prevented by increasing f_s such that the green images in the top figure do not overlap the blue portion. Created by Robert Bristow-Johnson for use in Wikimedia Commons. No Rights Reserved.

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