Notes for preparing IELTS

This post summaries notes from official guidence of IELTS.

Listening skills

Getting ready to listen

  • understanding the context
  • listening for specific details
  • using correct spelling
  • understanding numbers

Danger in performing phase shift with Python for a time series

Mathmatical expression

For scientific purpose, we sometime need to shift phase of time series with specfic angles. To acheive this, we can transform the time series into freqeuency domain with $$F(\omega) = \int_{-\omega}^{\omega} f(t) e^{-i \omega t} dt$$

where $F(\omega)$ as Fourier spectrum of signal $f(t)$. Thus, the phase shifted Fourier spectrum $F_{sh}$ is

$$F_{sh}(\omega) = F(\omega) * e^{-i \alpha}$$