# What is the total impulse response in a system with feedback interconnection?

Let's assume that we have a system with a typical feedback interconnection where the output is given by the following equation:

$$y(t) = \left(x(t) - z(t)\right) \star h_{1}(t) \tag{1}$$

where: $$z(t) = y(t) \star h_{2}(t) \tag{2}$$

So, is $h_{1}(t)$ the system's total impulse response or the convolution between $h_{1}(t)$ and $h_{2}(t)$ after replacing $z(t)$ in the equation $(1)$ while using the associative property?

It is easier to work in the $s$-domain:
$$Z=YH_2$$ $$Y=(X-Z)H_1$$ Hence, $$Y=(X-YH_2)H_1=XH_1-YH_1H_2\Rightarrow Y(1+H_1H_2)=XH_1$$ Therefore, $$H(s)=\frac{Y(s)}{X(s)}=\frac{H_1(s)}{1+H_1(s)H_2(s)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse Laplace transform.