Semi-Experiments and the Modelling of Internet Traffic - SAMSI

Study Second Order Structure, Using Wavelet Spectra
Use the Discrete Wavelet Transform
Definition
Comparison of signal X(t) with family of functions ψ j,k via d X (j, k) =< X, ψ j,k >, where the
ψ j,k = 2 −j/2 ψ(2 −j t − k), are localised both in time and frequency.
Properties
• {d X (j, k), k ∈ Z} is stationary and short range dependent for fixed scale j,
• variance(j) = IE|d X (j, k)| 2
• For self-similar processes: IE|d X (j, k)| 2 = 2 jα E|d X (0, k)| 2 ,
• For LRD processes: IE|d X (j, k)| 2 ∼ 2 jα IE|d X (0, k)| 2 for large j.
Wavelet Spectrum Estimate (log-log plot):
log 2 [ 1 ∑
|d X (j, k)| 2 ]
n j
vs j
k
2