implies that one cannot at the same time localize the value of a function and its Fourier transform. To wit, the following inequality holds, Further mathematical uncertainty inequalities, including the above entropic uncertainty, hold between a function and its Fourier transform : Signal processing In the context of signal processing, and in particular time–frequency analysis, uncertainty principles are referred to as the Gabor limit, after Dennis Gabor, or sometimes the Heisenberg–Gabor limit. The basic result, which follows from "Benedicks's theorem", below, is that a function cannot be both time limited and band limited (a function and its Fourier transform cannot both have bounded domain)—see bandlimited versus timelimited. More accurately, the time-bandwidth or duration-bandwidth product satisfies where and are the standard deviations of the time and frequency energy or power (i.e. squared) representations respectively. The minimum is attained for a Gaussian-shaped pulse (Gabor wavelet) [For the un-squared Gaussian (i.e. signal amplitude) and its un-squared Fourier transform magnitude ; squaring reduces each by a factor .] Another common measure is the product of the time and frequency full width at half maximum (of the power/energy), which for the Gaussian equals (see bandwidth-limited pulse). Stated alternatively, "One cannot simultaneously sharply localize a signal (function ) in both the time domain and frequency domain (, its Fourier transform)". When applied to filters, the result implies that one cannot achieve high temporal resolution and frequency resolution at the same time; a concrete example are the resolution issues of the short-time Fourier transform—if one uses a wide window, one achieves good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite trade-off. Alternate theorems give more precise quantitative results, and, in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) time–frequency plane. In practice, the Gabor limit limits the simultaneous time–frequency resolution one can achieve without interference; it is possible to achieve higher resolution, but at the cost of different components of the signal interfering with each other. As a result, in order to analyze signals where the transients are important, the wavelet transform is often used instead of the Fourier. Discrete Fourier transform Let be a sequence of N complex numbers and its discrete Fourier transform. Denote by the number of non-zero elements in the time sequence and by