Nyquist–Shannon sampling theorem
![Example of magnitude of the Fourier transform of a bandlimited function](/Images/godic/202502/09/Bandlimited.svg5845.png")
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![X(f) (top blue) and XA(f) (bottom blue) are continuous Fourier transforms of two different functions, x(t) and xA(t) (not shown). When the functions are sampled at rate fs, 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 continuous pre-sampled functions are not. If these were audio signals, x(t) and xA(t) might not sound the same. But their samples (taken at rate fs) are identical and would lead to identical reproduced sounds; thus xA(t) is an alias of x(t) at this sample rate.](/Images/godic/202502/09/AliasedSpectrum5845.png")
![Spectrum, Xs(f), of a properly sampled bandlimited signal (blue) and the adjacent DTFT images (green) that do not overlap. A brick-wall low-pass filter, H(f), removes the images, leaves the original spectrum, X(f), and recovers the original signal from its samples.](/Images/godic/202502/09/ReconstructFilter5845.png")
In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals (often called "analog signals") and discrete-time signals (often called "digital signals"). It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.