Editing Discrete Fourier transform (section)

==== Deterministic uncertainty principle ====
There is also a well-known deterministic uncertainty principle that uses signal sparsity (or the number of non-zero coefficients).<ref name=Donoho/> Let <math>\left\|x\right\|_0</math> and <math>\left\|X\right\|_0</math> be the number of non-zero elements of the time and frequency sequences <math>x_0,x_1,\ldots,x_{N-1}</math> and <math>X_0,X_1,\ldots,X_{N-1}</math>, respectively. Then,
:<math>N \leq \left\|x\right\|_0 \cdot \left\|X\right\|_0.</math>
As an immediate consequence of the [[Arithmetic–geometric mean|inequality of arithmetic and geometric means]], one also has <math>2\sqrt{N} \leq \left\|x\right\|_0 + \left\|X\right\|_0</math>. Both uncertainty principles were shown to be tight for specifically chosen "picket-fence" sequences (discrete impulse trains), and find practical use for signal recovery applications.<ref name=Donoho/>