# Frequency Response of Feedback Systems

Consider the feedback composition with two LTI systems:

Assume the frequency response of *S*_{1}
is *H*_{1},
of *S*_{2}
is *H*_{2},
and of *S* is *H*. Then assume that

*x* = exp(*i *ω
*t*).

The output must be

*y *= *H*(ω)*x*

Since this is itself a complex exponential, it must be true that

*z* = *H*_{2}(ω)*y
= **H*_{2}(ω)*H*(ω)*x*

Hence

*u* = *x* −
z* = x* − *H*_{2}(ω)*H*(ω)*x*
= (1* − H*

_{2}(ω)

*(ω))*

*H*

*x*which is also a complex exponential. Since *y* = *H*_{1}(ω)*u*,
it must be that

*y* = *H*_{1}(ω)(1*
− H*

_{2}(ω)

*(ω))*

*H*

*x*Since *y *= *H*(ω)*x*,

*H*(ω)*x
= H*

_{1}(ω)(1 −

*H*_{2}(ω)

*(ω))*

*H*

*x*Eliminate *x* and solve for *H* to get

*H*(ω)
=

*H*_{1}(ω)/(1 −

*H*_{2}(ω)

*H*_{1}(ω))

when (1 −* H*

_{2}(ω)

*H*_{1}(ω)) is not zero. This gives the frequency response of the feedback system in terms of those of the component systems.