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## Q. 4.6

OBTAINING A TRANSFER-DOMAIN FUNCTION MODEL FROM AN IMPULSE-RESPONSE MODEL.

Find the equivalent transform-domain transfer function representation of the following impulse-response model:

$y(t) = \frac{K_{t}}{\tau ^{2}} \int_{0}^{t}{e^{- (t – \sigma) / \tau} u(\sigma) d \sigma}$        (4.77)

## Verified Solution

Observe that the impulse-response function in this case is:

$g(t) = \frac{K_{t}}{\tau ^{2}} e^{-t / \tau}$        (4.78)

the Laplace transform of which is easily obtained as:

$g(s) = \frac{K}{(\tau s + 1 )^{2}}$        (4.79)

The required transfer function model equivalent of Eq. (4.77) is therefore:

$y(s) = \frac{K}{(\tau s + 1 )^{2}} u(s)$        (4.80)

In conclusion, we note that since we are concerned here with two fundamentally similar input/ output models having the same information content, all the transformations involved are unique.