Noise with sinusoidal trend

In this section, we study the effect of sinusoidal trends on the scaling properties of noisy signals. For a signal which is a superposition of correlated noise and sinusoidal trend, we find that based on the superposition rule (Appendix 7.2) the DFA rms fluctuation function can be expressed as

$$\left[F_{\rm\eta S}(n)\right]^2 = \left[F_{\rm\eta}(n)\right]^2 + \left[F_{\rm S}(n)\right]^2,$$ (10)

where $F_{\rm\eta S}(n)$ is the rms fluctuation function of noise with a sinusoidal trend, and $F_{\rm S}(n)$ is for the sinusoidal trend. First we consider the application of DFA-1 to a sinusoidal trend. Next we study the scaling behavior and the features of crossovers in $F_{\rm\eta \rm S}(n)$ for the superposition of correlated noise and sinusoidal trend employing the superposition rule [Eq.(10)]. At the end of this section, we discuss the results obtained from higher order DFA.