- #1
thomas49th
- 655
- 0
If signals x(t) and y(t) are orthogonal and if z(t) = x(t) + y(t) then
E_{z} = E_{x} + E_{y}:
Proof:
[tex] E_{z} => \int^{\infty}_{-\infty} {(x(t) + y(t))^{2}} dt
=> \int {(x(t) + y(t))^{2}}^{2} dt
=> \int (x^{2}(t)) + \int(y^{2}(t))dt + \int x(t)y(t)dt
=> E_{x} + E_{y}
[/tex]
because [tex]\int x(t)y(t)dt[/tex] = 0 because of integration by parts:
u = x(t) dv/dt = y(t)
u' = dx/dt, v = [tex]frac{y^{2}(t)}{2}[/tex]
so [tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}\frac{dx}{dt}}dt[/tex]
[tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}}dx[/tex]
we can treat y^2(t) as a constant so:
[tex]x(t)\frac{y^{2}(t)}{2} - \int^{\infty}_{-\infty} {\frac{y^{2}(t)}{2}}dx[/tex]
[tex]x(t)\frac{y^{2}(t)}{2} - } [{\frac{y^{2}(t)x}{2}}]^{\infty t}_{-\infty t}[/tex]
but the problem is that the limits were destined for integrating with respect to time. I'm not integrating with respect to x.
Any suggestions?
Thanks
Thomas
E_{z} = E_{x} + E_{y}:
Proof:
[tex] E_{z} => \int^{\infty}_{-\infty} {(x(t) + y(t))^{2}} dt
=> \int {(x(t) + y(t))^{2}}^{2} dt
=> \int (x^{2}(t)) + \int(y^{2}(t))dt + \int x(t)y(t)dt
=> E_{x} + E_{y}
[/tex]
because [tex]\int x(t)y(t)dt[/tex] = 0 because of integration by parts:
u = x(t) dv/dt = y(t)
u' = dx/dt, v = [tex]frac{y^{2}(t)}{2}[/tex]
so [tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}\frac{dx}{dt}}dt[/tex]
[tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}}dx[/tex]
we can treat y^2(t) as a constant so:
[tex]x(t)\frac{y^{2}(t)}{2} - \int^{\infty}_{-\infty} {\frac{y^{2}(t)}{2}}dx[/tex]
[tex]x(t)\frac{y^{2}(t)}{2} - } [{\frac{y^{2}(t)x}{2}}]^{\infty t}_{-\infty t}[/tex]
but the problem is that the limits were destined for integrating with respect to time. I'm not integrating with respect to x.
Any suggestions?
Thanks
Thomas