# Getting wrong answer for problem involving transfer functions and Laplace transforms

I need help finding where I have made the error in this problem set by my control systems lecturer.

The Question: The output of a linear time invariant system for an input $r(t)$ equals $c(t)$. If the input signal is passed through a block with transfer function $e^{-s}$, and then applied to the system, what will the output, $y(t)$, be?

The answer should be $y(t) = c(t) \cdot u(t-1)$ but I get $y(t) = u(t) \cdot c(t-1).$

My working:

Transfer function one, $H_1(s)$, applies to the original system. Transfer function two, $H_2(s)$, applies to the system including the new block:

$$H_1(s) = \frac{C(s)}{R(s)}$$

$$H_2(s) = e^{-s} \cdot H_1(s) = \frac{e^{-s} \cdot C(s)}{R(s)} = \frac{Y(s)}{R(s)}$$

Therefore $Y(s) = e^{-s} \cdot C(s)$,

$y(t)$ is the inverse Laplace transform of $Y(s)$:

$$y(t) = u(t) \cdot c(t-1)$$

• Thank you Jakub for making my equations more readable :) – The Impossible Squish Mar 10 '17 at 10:11