# Power and Energy spectral densities of a linear system

Here is a schematic of my desired circuit. I want to calculate Power and Energy that the load dissipates during 0< t<8 You can see my attempts:

1. How can find load's energy in Frequency domain for desired time domain interval (0 < t < 8) ?
2. Did I obtain power correctly in frequency domain?
• Are you sure you have written the question correctly? It seems like $H(f)=1$ probably for $|f|<f_0$ and $H(f)=0$ otherwise. It is somehow meaningless at the moment. – msm Dec 5 '16 at 9:03
• Thanks for your response.Actually I think my system is an Ideal wire where y(t)=x(t) and h(t) should be equal to delta function – Ehsan Zakeri Dec 5 '16 at 9:50

## 1 Answer

The relationships, with $x$ the input voltage to the network, $y$ the output voltage from the network, $R$ the load resistance, $p(t)$ the instantaneous power dissipated at the load, and $E$ the energy dissipated at the load and $t_0=8$

$$Y(f)=H(f)X(f)\\ y(t)=\mathscr{F}^{-1}\{Y(f)\}\\ p(t)=\frac{y^2(t)}{R}\\E\bigg|_{t=0}^{t_0}=\int_0^{t_0}p(t)dt=\int_0^{t_0}\frac{y^2(t)}{R} dt$$

IF the cable or network is a lossless cable (!), the example turns trivial: $$y(t)=x(t)\\ p(t)=\frac{x^2(t)}{R}\\ E\bigg|_{t=0}^{t_0}=\int_0^{t_0}\frac{x^2(t)}{R} dt$$

• Thanks hypfco for your contribution and response ,as you can see in the first section of my attempts I use your equation to obtain Energy in time domain but my question is how can find Energy in frequency domain while 0 < t<8 , is it possible without going to time domain obtain Energy in frequency domain for specific time duration ? and was my attempts correct to obtain power in frequency domain? – Ehsan Zakeri Dec 5 '16 at 13:39