## Signals Energy of 2 signals - Integral limits correct?

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:

$$E_{z} => \int^{\infty}_{-\infty} {(x(t) + y(t))^{2}} dt => \int {(x(t) + y(t))^{2}}^{2} dt => \int (x^{2}(t)) + \int(y^{2}(t))dt + \int x(t)y(t)dt => E_{x} + E_{y}$$

because $$\int x(t)y(t)dt$$ = 0 because of integration by parts:

u = x(t) dv/dt = y(t)
u' = dx/dt, v = $$frac{y^{2}(t)}{2}$$

so $$x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}\frac{dx}{dt}}dt$$
$$x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}}dx$$
we can treat y^2(t) as a constant so:

$$x(t)\frac{y^{2}(t)}{2} - \int^{\infty}_{-\infty} {\frac{y^{2}(t)}{2}}dx$$
$$x(t)\frac{y^{2}(t)}{2} - } [{\frac{y^{2}(t)x}{2}}]^{\infty t}_{-\infty t}$$

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
 PhysOrg.com science news on PhysOrg.com >> King Richard III found in 'untidy lozenge-shaped grave'>> Google Drive sports new view and scan enhancements>> Researcher admits mistakes in stem cell study
 Recognitions: Homework Help Science Advisor The integral of x(t)y(t) isn't zero because of some bogus 'integration by parts' argument. It's zero because that's what 'orthogonal' means.
 Ofcourse! Execellent. May I ask, out of interest alone what the integral of x(t)y(t) with respect to t should be? Thanks Thomas

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