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Let G be a finite abelian group. If f : G &#x2192; C is a nonzero function with Fourier transform f, the Donoho-Stark uncertainty principle states that |supp(f)||supp(f&#x0302;)| &#x2265; |G|. The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, if f : G &#x2192; F is a non-zero function from G to a field F, and if f has a Fourier transform f&#x0302;, the sharpened uncertainty principle states that |supp(f)||supp(f&#x0302;)| &#x2265; |G|+|supp(f)|-|H(supp(f))|, where H(supp(f)) is the stabilizer of supp(f) in G.

The Shift Bound for Abelian Codes and Generalizations of the Donoho-Stark Uncertainty Principle