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Chapter 4

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)

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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.