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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)