Convolution-like change of variables

[Taraborn]

Homework Statement

Hi, this is not homework exactly, I'm doing some exercises as part of my personal study. I'm analizing linear invariant systems and I'm stuck in an apparently trivial step, please, help. I have these integrals:

Homework Equations

integral( x(tau)*dtau, from -infinity to t) - integral ( x(tau)*dtau, from -infinity to t-T). I must prove that the former expression is the same as integral( x(t)-x(t-T), from -infinity to t).

The Attempt at a Solution

I have manipulated the expression a little, tried the change of variables s=tau+T; ds=dtau and now I have -integral(x(s-T)ds, from -infinity to t) + integral(x(tau)dtau, -infinity to t) but I don't know how continue. I'm pretty sure it's really simple, but there's something I'm missing or something that I don't fully understand about dummy variables and so. Help, please.

 

[Zone Ranger]

Homework Statement

Hi, this is not homework exactly, I'm doing some exercises as part of my personal study. I'm analizing linear invariant systems and I'm stuck in an apparently trivial step, please, help. I have these integrals:

Homework Equations

integral( x(tau)*dtau, from -infinity to t) - integral ( x(tau)*dtau, from -infinity to t-T). I must prove that the former expression is the same as integral( x(t)-x(t-T), from -infinity to t).

The Attempt at a Solution

I have manipulated the expression a little, tried the change of variables s=tau+T; ds=dtau and now I have -integral(x(s-T)ds, from -infinity to t) + integral(x(tau)dtau, -infinity to t) but I don't know how continue. I'm pretty sure it's really simple, but there's something I'm missing or something that I don't fully understand about dummy variables and so. Help, please.

I assume you want

\int_{-\infty}^t x(\tau)d\tau-\int_{-\infty}^{t-T} x(\tau)d\tau=\int_{-\infty}^t [x(\tau)- x(\tau-T)]d\tau

from where you are stuck...let s=\tau and combine the integrals