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Sparse representations of sequences facilitate signal processing tasks in many radar, sonar, communications, and information hiding applications. Previously, conditions for the construction of a compactly supported finite Zak transform of the linear FM chirp were investigated. It was shown that the discrete Fourier transform of a chirp is, essentially, a chirp, with support similar to the support of the time-domain signal. In contrast, the Zak space analysis produces a highly compactified chirp, with support restricted to an algebraic line. Further investigation leads to relaxation of the original restriction to chirps, permitting construction of a wide range of polyphase sequence families with ideal correlation properties. This paper contains an elementary introduction to the Zak transform methods, a survey of recent results in Zak space sequence design and analysis, and a discussion of the main open problems in this area.

Construction of Sparse Representations of Perfect Polyphase Sequences in Zak Space with Applications to Radar and Communications