azay
Oct17-11, 05:18 AM
I read in various articles about Prony analysis that
{\displaystyle \sum_{i=1}^{L}}A_{i}e^{\sigma_{i}t}cos(2\pi f_{i}t+\phi_{i})
Using Euler theorem this equals:
{\displaystyle \sum_{i=1}^{L}}A_{i}e^{\sigma_{i}t}\left(\frac{e^{ j2\pi f_{i}t}e^{j\phi_{i}}}{2}+\frac{e^{-j2\pi f_{i}t}e^{-j\phi_{i}}}{2}\right)
={\displaystyle \sum_{i=1}^{L}}\left(\frac{A_{i}e^{(j2\pi f_{i}+\sigma_{i})t}e^{j\phi_{i}}}{2}+\frac{A_{i}e^ {(-j2\pi f_{i}+\sigma_{i})t}e^{-j\phi_{i}}}{2}\right)
is supposed to be equal to
{\displaystyle \sum_{i=1}^{N}}\frac{A_{i}}{2}e^{j\phi_{i}}e^{(σ_{ i}+j2\pi f_{i})t}
Why?
{\displaystyle \sum_{i=1}^{L}}A_{i}e^{\sigma_{i}t}cos(2\pi f_{i}t+\phi_{i})
Using Euler theorem this equals:
{\displaystyle \sum_{i=1}^{L}}A_{i}e^{\sigma_{i}t}\left(\frac{e^{ j2\pi f_{i}t}e^{j\phi_{i}}}{2}+\frac{e^{-j2\pi f_{i}t}e^{-j\phi_{i}}}{2}\right)
={\displaystyle \sum_{i=1}^{L}}\left(\frac{A_{i}e^{(j2\pi f_{i}+\sigma_{i})t}e^{j\phi_{i}}}{2}+\frac{A_{i}e^ {(-j2\pi f_{i}+\sigma_{i})t}e^{-j\phi_{i}}}{2}\right)
is supposed to be equal to
{\displaystyle \sum_{i=1}^{N}}\frac{A_{i}}{2}e^{j\phi_{i}}e^{(σ_{ i}+j2\pi f_{i})t}
Why?