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## Q. 4.5

IMPULSE-RESPONSE FORM OF A FIRST-ORDER TRANSFORM-DOMAIN TRANSFER FUNCTION.

Find the equivalent impulse-response representation of the following transformdomain transfer function model first given in Example 4.1:

$y(s) = g(s) u(s)$        (4.60a)

where

$g(s) = \frac{K}{\tau s + 1}$        (4.60b)

## Verified Solution

In this case, by Laplace inversion of g(s) we obtain the impulse-response function as:

$g(t) = \frac{K}{\tau} e^{-t/ \tau}$        (4.76)

so that the required impulse-response model is given by the convolution integral:

$y(t) = \frac{K}{\tau}\int_{0}^{t} {e^{- (t – \sigma) / \tau} u(\sigma) d \sigma}$        (4.75)

obtained by taking the inverse Laplace transform of the transfer function model given in Eq. (4.60) and using the convolution theorem of Laplace transforms stated earlier on in Eq. (4.22).

$L^{-1} \left\{f(s) g(s)\right\} = \int_{0}^{t}{g(t – \sigma) f(\sigma) d \sigma}$        (4.22)