8.11 Removing the trend  8. Spectral Analysis  8.9 Aliasing

8.10 Effect of finite sample length

Because real samples are of finite length, frequency components contributing to the variance spectrum cannot be resolved if the components are too closely spaced. This effect can be illustrated by considering the observed series s_T(t) to be the product of a continuous signal s(t) defined for all time and a window function of the form

$$w(t) = 1 \ \ {\rm for}\ \vert t\vert\le T/2$$ (8.56)

 

$$w(t) = 0 \ \ {\rm for}\ \vert t\vert>T/2$$ (8.57)

 

The Fourier transform of this window function is

$$\tilde w(\nu) = T {{\sin(\pi\nu T)}\over{(\pi\nu T)}}$$ (8.58)

 

which has a maximum at $\nu=0$ and decays as $1/\nu$. The Fourier transform of s_T(t) is then

$$\tilde s_T(\nu) = \int_{-\infty}^\infty \tilde s(\nu^\prime)\tilde w(\nu-\nu^\prime)d\nu^\prime .$$ (8.59)

 

Because the transform of the window function $\tilde s_T(\nu)$ first falls to zero for $\nu T=1$, the effect of (8.59) is to mix together contributions from $\tilde s(\nu^\prime)$ for $\nu^\prime$varying over about $\pm 1/T$. A general guideline is that a series of length at least T is needed to resolve two contributions separated in frequency by 1/T.

   
 8.11 Removing the trend  8. Spectral Analysis  8.9 Aliasing