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The Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample rate must be at least twice the bandwidth of the signal to avoid aliasing.
Let F be any sampling method, i.e. a linear map from the Hilbert space of square-integrable functions to complex space . In our example, the vector space of sampled signals C n {\displaystyle \mathbb {C} ^{n}} is n -dimensional complex space.
A simple illustration of aliasing can be obtained by studying low-resolution images. A gray-scale image can be interpreted as a function in two-dimensional space. An example of aliasing is shown in the images of brick patterns in Figure 5. The image shows the effects of aliasing when the sampling theorem's condition is not satisfied.
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics, which refers to a set of such values ...
Rounding real numbers to integers is an example. The Nyquist–Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal. In practice, the sampling frequency is often significantly higher than this. [8]
An early breakthrough in signal processing was the Nyquist–Shannon sampling theorem. It states that if a real signal's highest frequency is less than half of the sampling rate, then the signal can be reconstructed perfectly by means of sinc interpolation. The main idea is that with prior knowledge about constraints on the signal's frequencies ...
A PCM stream has two basic properties that determine the stream's fidelity to the original analog signal: the sampling rate, which is the number of times per second that samples are taken; and the bit depth, which determines the number of possible digital values that can be used to represent each sample.
Nyquist–Shannon sampling theorem; S. Schwartz–Zippel lemma; Shannon–Hartley theorem; Shannon's source coding theorem This page was ...