Please briefly explain how to use the unit impulse response

Please briefly explain how to use the unit impulse response h(t) of an LTI system to determine the system\'s frequency response function H(omega). In addition, describes the relationship between the Fourier transform of the input signal to the system and the Fourier transforms of the response of the system.

Solution

2 Ans) Consider a linear system characterised by an impulse response h(t), and hence also by a transfer function H(s) = L{h(t)}. Then the system response y(t) = [h*u](t) to a sinusoidal input u(t) = sin t of angular frequency rad/s is of the form y(t) = H()sin(t + ()). That is, the response y(t) is also sinusoidal at the same frequency of rad/s, but of a changed amplitude H() and with a phase shift () which both depend upon and are given by

H() = |H(j)|, () = ]H(j). In particular, since H(s) evaluated at s = j is given as H(j) = R()e j()

OR

For a linear and time invariant system with impulse response h(t), its frequency response H() = R()e j() is given as the Fourier transform of the impulse response: H() = F {h(t)} .

For a given signal f(t), its Fourier transform F() = F {f} is defined as F() = F {f(t)} =Integral(limits - to ) f(t)e^ jt dt.

2.Second Part Answer

Suppose that y(t) = [h*u](t). Then F {y(t)} = F {[h*u](t)} = F {h(t)} F {u(t)} . In the context of system frequency response, this principle is particularly important. It states that for a linear and time invariant system with impulse response h(t), then the spectrum U() = F {u(t)} of an input to that system is related to the spectrum Y () = F {y(t)} of the output of that system according to

Y () = H()U() where H() = F {h(t)} is the frequency response of the system.