Abstract
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.
Highlights
In this paper, we are concerned with the design and analysis of perfect polyphase sequences
We explored the utility of the Zak transform for polyphase sequence (PPS) design focusing initially on the finite LFM chirp
We will show that time-frequency analysis of a single classical radar waveform—the LFM chirp—leads to more general results that are relevant to all polyphase sequences
Summary
We are concerned with the design and analysis of perfect polyphase sequences. The last three results rely, in part, on the discovery that the Zak space representation of a PPS can be expressed as a composition of modulation and permutation operators acting on the canonical chirp sequence. We will show that time-frequency analysis of a single classical radar waveform—the LFM chirp—leads to more general results that are relevant to all polyphase sequences While many of these sequences are traditionally associated with communications applications, they can be used in radar. The sparse and highly structured support of PPS waveforms in the time-frequency space can be used advantageously in both radar and communications applications These findings demonstrate that even though historically sequence/waveform design in the two fields progressed largely independently, the theoretical underpinnings are essentially the same, and a great deal of insight can be gained from juxtaposing ideas and results
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