Transfer Function of LTI system
A continuous-time LTI system’s transfer function can be defined via the Fourier or Laplace transforms. Further more, the LTI system’s transfer function can only be defined with zero initial circumstances. The transfer function of the LTI system is described in detail in s – domain as well as in frequency domain as follows:
Transfer function
In Frequency Domain
Assuming that the initial conditions are zero, the ratio of the output signal’s Fourier transform to the input signal’s Fourier transform is the transfer function of an LTI system.
Mathematically, transfer function in frequency domain is defined as
H(ω) = Y(ω) / X(ω) ;
suppose if H(ω) is complex then this will be written in the magnitude and phase,
H(ω) = |H(ω)| ejθ(ω) ;
Magnitude response is defined as the |H(ω)| and the phase response θ(ω) = ∠H(ω)
Frequency response of the output,
Y(ω)= X(ω) . H(ω)
Output magnitude = |X(ω)| |H(ω)|
output phase = ∠Y(ω) = ∠H(ω) + ∠X(ω)
In S Domain
When the initial conditions are zero, the ratio of the output signal’s Laplace transform to the input signal’s Laplace transform is the transfer function of the LTI system. Alternatively, when the beginning circumstances are disregarded, the transfer function is defined as the ratio of output to input in the s-domain.
the mathematically explained as the,
H(s) = Y(s) / X(s)
here, The inverse Laplace transform of the transfer function yields the impulse response ℎ(t) of the LTI system in the s-domain.
h(t) = L-1 [H(s)]
Once an LTI system’s transfer function in the s-domain H(s) is known, determining the transfer function in the frequency domain H(s) only requires changing s in jω.
H(ω) = H(s) |s = jω
jω -> in complex domain.
LTI System
Systems that are both linear and time-invariant are known as linear time-invariant systems, or LTI systems for short. When a system’s outputs for a linear combination of inputs match the outputs of a linear combination of each input response separately, the system is said to be linear. Time-invariant systems are ones whose output is independent of the timing of the input application. Long-term behavior in a system is predicted using LTI systems. The term “linear translation-invariant” can be used to describe these systems, giving it the broadest meaning possible. The analogous term in the case of generic discrete-time (i.e., sampled) systems is linear shift-invariant.
Table of Content
- LTI System
- Types
- Properties
- Transfer Function
- Convolution
- Sampling Theorem
- Nyquist Rate