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)
Step-by-Step
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.