Can any one please explain me how the convolution of the following leads to delta(t).
To convolve these to function i considered delta(t+T) as x(t) and
delta(t-T) as the response. Then these two function overlaps only at t=-T and then the convolution should result in only delta(t+T). Then why delta(t) is the answer.
Your reasoning can bring you to the answer, but it's not spot on. Consider: $$ x(t) = \delta(t) $$ $$ h(t) = \delta(t-\tau) $$ $$ y(t) = x(t)*h(t) = \delta(t-\tau) $$
System output to an impulse at \$t = 0\$ is another impulse at \$t = \tau\$ (a time delay of \$\tau\$). If the system is LTI, then the following holds true:
$$ x(t) = \delta(t+\tau) $$ $$ h(t) = \delta(t-\tau) $$ $$ y(t) = x(t)*h(t) = \delta(t) $$
Because the impulse now happens at \$t = -\tau\$, system output now happens at \$t = 0\$.
Treat convolution as: fold, slide, multiply, add. If the impulse at \$t=\tau\$ is folded (about the vertical axis) it lies exactly coincident with the impulse at \$t=-\tau\$, and this initial position corresponds to \$t=0\$. The product of the two impulses is the unit impulse.
Subsequently sliding, ie for all other values of \$t\$, the multiplication of the two impulses is zero.
