why the sampling proprety of dirac function gives this?

Question

In MuPAD I simplify:

simplify(2*t*dirac(t+1))

Which gives :

-2*dirac(t + 1)

I know that

f(t)*dirac(t-k)=f(k)*dirac(t)

so f(t)=2*t , dirac(t+1)=dirac(t-(-1)) so that means k=-1 So 2 * t * dirac(t+1) =2*(-1)*dirac(t) =-2*dirac(t) which is different than what MuPAD returns, -2*dirac(t + 1).

Where is my mistake?

Answer 1

Your assumption is wrong. It is not

f(t)*dirac(t-k)=f(k)*dirac(t)

but

f(t)*dirac(t-k)=f(k)*dirac(t - k)

The reason is simple: By pointwise multiplying with a function you cannot move the position at which the Dirac has its spike. If it was at t = k, it must remain at t = k after.