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Table 1 Summary of the article notations

From: Map-invariant spectral analysis for the identification of DNA periodicities

{A,C,G,T}, the field of DNA nucleotides

{0,1}, the field of Voss binary elements

A general field of complex valued elements

F↦D

Field mapping operation from set to set, resulting in γ sequences x l (n), where l = 1,…,γ. For example, whenD=V,F↦D results in γ = 4 binary sequences, namely: x A (n),x C (n),x G (n), and x T (n)

x l (n)

A discrete time sequence of length N whose elements belong to the mapped field

x l (n)

The nth window of length M, extracted from x l (n), l = 1,…,γ

x Ì‚ l (n)

The interleaved version of x(n) with an interleaving factor R, l = 1,…,γ

X l (Rn, M R )

The ST-DFT of x l (n), generated using a sliding window of length M and a window shift of length R

Ï’ v (n)

[X A (n) X C (n) X G (n) X T (n)]T, the array of the four-based ST-DFTs

Ï’ d (n)

[X1(n) X2(n) … X γ (n)]T, the array of the γ-based ST-DFTs

X lr (n)

The rth filtered polyphase component of X l (n), where r = 0,1,…,R − 1 and l = 1,…,γ

S v (n)

The DNA spectrum computed by adding the magnitude squared of the ST-DFT of the four-based sequences

S d (n)

The DNA spectrum computed by adding the magnitude squared of the ST-DFT of the γ-based sequences

Γ l (n)

[Xl 0(n) Xl 1(n) … Xl,R−1(n)]T, the array of the R filtered polyphase components X lr (n), r = 0,1,…,R − 1 and l = 1,…,γ

I γ

An identity matrix of size γ × γ

C

An array of length R whose elements are equally spaced on the unit circle

h

An array of length M/R whose elements are all equal to one

D

C⋆CT, an R × R matrix

H

I R  ⊗ hT, an R × R block matrix of M R ×1 blocks

W

HHD H, an R × R block matrix of M R × M R blocks

A,b

The affine transformation matrices of size γ × 4 and γ × 1, respectively, that map the four-based sequences into the γ-based sequences.

B

AHA, a 4 × 4 matrix

C ~

A complex valued array of R elements

h ~

A complex valued array of M/R elements

D ~

C ~ ⋆ C ~ T , an R × R matrix

h ~

I R ⊗ h ~ T R × R block matrix of M R ×1 blocks

W ~

H ~ H D ~ H ~ , an R × R block matrix of M R × M R blocks