For digital systems, the upper limit of the frequency response is determined by the sampling frequency. The choice of sample sampling frequency in a digital system is based on the Nyquist–Shannon sampling theorem. This states that a sampled signal can be reproduced exactly as long as it is sampled at a frequency greater than twice the bandwidth of the signal, the Nyquist frequency. Therefore, a sampling frequency of 40 kHz is mathematically sufficient to capture all the information contained in a signal having frequency components less than or equal to 20 kHz. The sampling theorem also requires that frequency content above the Nyquist frequency be removed from the signal before sampling it. This is accomplished using anti-aliasing filters which require a transition band to sufficiently reduce aliasing. The bandwidth provided by the 44,100 Hz sampling frequency used by the standard for audio CDs is sufficiently wide to cover the entire human hearing range, which roughly extends from 20 Hz to 20 kHz. Professional digital recorders may record higher frequencies, while some consumer and telecommunications systems record a more restricted frequency range.