ABSTRACT
For a long time, the Shannon sampling theorem was the ruling paradigm in the signal processing community when it came to choosing sampling rates. The theorem, proved in 1949 by Claude E. Shannon, states that any function https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315154626/0b707827-3c3d-4599-b52b-18f9e5a66518/content/ch01_iequ_0001.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> f : ℝ → ℂ having limited bandwidth W (meaning that the support of the Fourier transform https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315154626/0b707827-3c3d-4599-b52b-18f9e5a66518/content/ch01_iequ_0002.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> f ̂ is contained in the interval https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315154626/0b707827-3c3d-4599-b52b-18f9e5a66518/content/ch01_iequ_0003.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> [ − W , W ] ) can be exactly reconstructed from its values https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315154626/0b707827-3c3d-4599-b52b-18f9e5a66518/content/ch01_iequ_0004.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> f ( n 2 W ) n ∈ ℤ . Put differently, sampling a function at a rate at least two times higher than the bandwidth of the function will provide enough information for perfect reconstruction of the signal of interest. The rate two times higher than the bandwidth of a signal is often referred to as the Nyquist rate.
