Question

The following signal, f(t), is periodic. Over the interval t = [0,1], f(t) is proportional to an exponential (f(t)~e^at (you need to specify ‘a’ and the DC offset). Find the RMS value of the following signal. Begin by writing an equation for f(t).

The following signal, f(t), is periodic. Over the interval t = [0,1],f(t) is proportional to an exponential(t)-e" (you need to specify a and the DC offset). Find the RMS value of the following signal. Begin by writing an equation for f(t). f(t) - f(t) « eat 2 -1.5 -1 -0. 5 0 05 1 1.5 2

Answer 1

Given : $f(t)\alpha e^{at}$

At t=0

$\rightarrow 0=1+c$

> C = - 1

At t = 1

$\rightarrow a = 2.3979$

Therefore, the equation of f(t) is given as

$\Rightarrow f(t) = e^{2.3979t}-1$

RMS of the signal is given as $f_{RMS} =\sqrt{ \frac{1}{T}\int_0^T [f(t)]^2 dt}$

$\rightarrow f_{RMS} =\sqrt{\int_0^1 (e^{2.3979t}-1)^2dt}$

$\rightarrow f_{RMS} =\sqrt{\int_0^1 (e^{4.7958t}-2e^{2.3979t}+1)dt}$

$\rightarrow f_{RMS} =\sqrt{\int_0^1 e^{4.7958t}dt-2\int_0^1 e^{2.3979t}dt+\int_0^1dt}$

$\rightarrow f_{RMS} =\sqrt{\frac{1}{4.7958}(e^{4.7958}-1)-2\times \frac{1}{2.3979} (e^{2.3979}-1)+1}$

$\rightarrow f_{RMS} =4.2049$