## Chapter 4

## 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)

## Step-by-Step

## 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)