The Fourier Analysis for Stagger-Period Sequences, Its Applications

Stagger-period sequences are a kind of discrete-time sequences, but the Fourier analysis of the uniform-period, discrete-time sequences does not apply to a stagger-period sequence; this means that the uniform-period analytical conclusions would be misleading. In this chapter we first define the essential concepts related to the stagger-period sequence and the stagger-lag autocorrelation matrix; we propose the Fourier transform pair of the stagger-period deterministic sequence and its spectrum, and discuss properties related to the transform pair, such as the orthogonality of a complex staggered exponential sequence, extension of the spectral period, Toeplitz of the circularly stagger-lag matrix, the staggered Parseval’s theorem, etc.; we verify inverses of each other of this transform pair and derive the convergence condition of this Fourier transform. Then, another Fourier transform pair of the stagger-lag autocorrelation matrix and its power spectrum density, properties related to this transform pair, inverses of each other of this transform pair, and the convergence condition of this Fourier transform of power spectrum are also studied. During illustrating examples, the similarities and differences of the equations and properties between the uniform-period and stagger-period analyses are described. Two applications of the Fourier analysis: search of the best stagger periods and spectrum estimation of the stagger-period sequence, are also discussed in details later. In the end, the advantages and methodologies of this study are summarized. This chapter “The Fourier Analysis for Stagger-Period Sequences, its Applications” will open the first page of the stagger-period signal processing.