# Frequency Response

We have seen that if the input to an LTI system is a **complex exponential** signal* e* ∈ [*Time*→* Complex*] where for all *t *∈* Time*,

*e*(*t*) = exp(*j*ω*t*)
= cos(ω*t*) + *j* sin(ω*t*).

then the output can be written

*y*(* t*) = *H*(ω*
*) exp(*j*ω*t*)

where *H*(ω) is (possibly complex-valued) number that is a property of the system. *H*(ω) is called the frequency response at frequency ω.

Complex exponential inputs, however, are rather abstract. We
have seen that with audio signals, sinusoidal signals are intrinsically
significant because the human ear interprets the frequency of the sinusoid as
its **tone**. Note that a real-valued sinusoidal signal can be given as a
combination of exponential signals,

cos(ω*t*) = 0.5(exp(*j*ω*t*)
+ exp(− *j*ω*t*)).

Thus, if this is the input to an LTI system *H*, then the output will be

*y*(*t*) = 0.5(*H*(ω)
exp(*j*ω*t*) + *H*(−
ω) exp(− jω*t*)).

Many (or most) LTI systems are not capable of producing complex-valued outputs when the input is real, so this *y*(*t*) must be real. This implies that *H*(ω) = *H ^{*}*(− ω). Thus,

*y*(*t*) = Re{(*H*(ω)
exp(*j*ω*t*)}.

If we write *H*(ω) in polar form,

*H*(ω) = |* H*(ω)|exp(∠ *H*(ω))

then

*y*(*t*) = |*H*(ω)| cos(ω*t* + ∠ *H*(ω)).

Thus, *H*(ω) gives the gain |*H*(ω)| and phase shift ∠ *H*(ω) that a sinusoidal input with frequency ω experiences.