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 |
| Field mapping operation from set to set, resulting in γ sequences x l (n), where l = 1,…,γ. For example, when, 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 |
| The interleaved version of x(n) with an interleaving factor R, l = 1,…,γ |
| 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 |
ϒ 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 blocks |
W | HHD H, an R × R block matrix of 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 |
| A complex valued array of R elements |
| A complex valued array of M/R elements |
| , an R × R matrix |
| block matrix of blocks |
| , an R × R block matrix of blocks |