In Laplace transform, multiplication by e^-at in time domain becomes

Add elements individually.

The probability that first ball is white is and the probability the second ball is white is . The probability that both are white is .

.

For causal output must depend upon Present and past not on future.

£f(t) =

£^-1F(s) = f(t)

£[a f₁(t) + bf₂(t)] = aF₁(s) + bF₂(s)

where

£[f(t - T)] = e^-sT F(s)

£[e^-at f(t)] = F(s + a)

Initial value theorem

Final value theroem

Convolution Integral

where t is dummy variable for t.

Because to design band pass or band stop, transfer function of must be second order.

The simplest method to find the ROC. Put denominator to greater than zero.

For an odd function f(- x) = - f(x) Hence, only sine terms.

H(z) = 6 + z^-1 - z^-2, solve it by considering H(z) = 0 z = 1/3, -1/2 in H(z) only numerator.

Hence z = 1/3, - 1/2 will be zero, and if zero lies inside the unit circle, system will be of minimum phase.

det.|A| = 3(8 - 12) = - 12.