Gate 2016 EE (Set 1) Q9

9. The value of$$ \int_{-\infty}^{+\infty} e^(-t) \delta (2t-2)dt$$  where  $$\delta (t)$$ is the Dirac delta function, is:——–

Solution:

Using sifting property of Impulse function,

$$\int_{-\infty }^{+\infty }f(t)\delta (t-t_0 )dt=f(t)|t=t_0$$

$$\int_{-\infty }^{+\infty }e^(-1) \delta (2t-t)dt=> \int_{-\infty }^{+\infty }e^(-t) (1/2)\delta (t-1)dt$$ at$$t_0=1$$

$$e^(-t)/2$$ at $$t=1=>e^(-1)/2=>1/2e$$

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