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It is easier to work in the $z$$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, $$H(z)=\frac{Y(z)}{X(z)}=\frac{H_1(z)}{1+H_1(z)H_2(z)}$$$$H(s)=\frac{Y(s)}{X(s)}=\frac{H_1(s)}{1+H_1(s)H_2(s)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse $z$-TransformLaplace transform.

It is easier to work in the $z$-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, $$H(z)=\frac{Y(z)}{X(z)}=\frac{H_1(z)}{1+H_1(z)H_2(z)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse $z$-Transform.

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.

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It is easier to work in the $z$-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, $$H(z)=\frac{Y(z)}{X(z)}=\frac{H_1(z)}{1+H_1(z)H_2(z)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse $z$-Transform.