I have to find The energy of a signal using Rayleigh th. the signal is $$ x(t) = A e^{-At } u(t) $$ assuming A>0

Using the classic definition of E , I found that it should be $$ \frac{A}{2} $$

Using Rayleigh I should do $$ \int_{-inf}^{inf} | \frac{A}{A+i2\pi f}|^2 df $$ This because I previously found X(f) From this I obtained $$\frac{A^2}{8 \pi^2 f} log (\sqrt (A^2 +4\pi^2f^2)) $$ with log from -infinite to infinite.

$$ log \sqrt x = log \frac{x}{2} $$ but now I have no idea how to continue , and if this is correct.. thank you

  • $\begingroup$ Why "Rayleigh"? $\endgroup$ – Matt L. Jan 12 '20 at 17:46

If I understand correctly, you want to verify the energy calculation in the frequency domain by computing the energy as



$$X(f)=\mathcal{F}\big\{x(t)\big\}=\frac{A}{A+i2\pi f}\tag{2}$$

From $(2)$ we get

$$|X(f)|^2=\frac{A^2}{A^2+(2\pi f)^2}=\frac{1}{1+\left(\frac{2\pi f}{A}\right)^2}\tag{3}$$

With $(3)$ and with the substitution $x=2\pi f/A$, the integral $(1)$ becomes



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.