Question : For an energy signal x(t) with energy Ex, show that the energy of x(-

Question

Question : For an energy signal x(t) with energy Ex, show that the energy of x(-t) is also Ex.


My problem : I do not understand the process of substituting -t=t1, dt=-dt, and inverting the bounds. Please explain these to me.

The energy of x ( - t ) is given by. E= |x( - t)|2 dt Substitute - t = t1 in the above equation then dt = - dt1 and the limits are from infinity to - infinity E = - |x( - t1)|2 (- dt1) E = |x( - t1)|2 dt1 E = |x( - t1)|2 dt1 Now substitute t1 =t in the above equation then dt1 = dt and the limits are from - infinity to infinity. E = |x( - t)| dt

Explanation / Answer

when we substitute -t=t1 ot t= -t1 then differentating it on both sides we get dt = -dt1

initially t is variable now we have replaced it with -t1

t was lying between -infinity and +inifinity initially now we replaced it with -t1

so -t1 should be lying between -infinity and +inifinity

or t1 lying between +infinity and -inifinity i.e reverse the bounds

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