Linear Time-Invariant (LTI) systems

The linear time-invariant (LTI) system is an example of linear shift-invariant systems, with the stimulus $s (t)$ and the response $r (t)$ .

The LTI systems have:

linearity = superposition principle

f_{L T I} (\sum s_{i} (t)) = \sum f_{L T I} (s_{i} (t)) = \sum r_{i} (t)

time-invariance = output is independent of time shift
since:

r (t) = f_{L T I} (s (t))

then:

r (t - t_{0}) = f_{L T I} (s (t - t_{0}))

Eigenfunctions of LTI systems

Eigenfunction

It is similar to eigenvectors: it is the eigenfunctions in a function space.

Df = \lambda f$$ where f is the eigenfunction of D. ### Eigenfunctions of LTI systems are sin and cos functions If $s_E(t)$ is the eigenfunction and H is the eigenvalue, there should be theConvolution: $$s_E(t) * h(t) = H s_E(t)

then the solution is:

s_{E} (t) = e^{i ω t} = e^{i 2 π f t} = c o s (2 π f t) + i s i n (2 π f t)

Correlation and LTI systems

Auto-correlation of the output signal $r$ in LTI systems (i.e. $r$ and $s$ are real-valued):

because $r (τ) = s (τ) * h (τ)$ ,

φ_{r r} (τ) = r (τ) ⋆ r (τ) = r (- τ) * r (τ) = s (- τ) * h (- τ) * s (τ) * h (τ)

which means:

φ_{r r} (τ) = φ_{s s} (τ) * φ_{h h} (τ) $ $ (W i e n - L e e R e l a t i o n) w i t h W i e n e r - L h i n c h i n T h e o r e m, f o r p o w e r s p e c t r a l d e n s i t y : $ $ | R (f) |^{2} = | S (f) |^{2} | H (f) |^{2}

Cross-correlation between input $s$ and output $r$

φ_{s r} (τ) = s (τ) ⋆ r (τ) = s^{*} (- τ) * r (τ) = s (- τ) * r (τ) = s (- τ) * s (τ) * h (τ) = φ_{s s} * h (τ)

F T {φ_{s r} (τ)} = F T {φ_{s s} * h (τ)}

with Wiener-Khinchin Theorem:

Φ_{s r} = | S (f) |^{2} H (f)

System identification is based on this, i.e. to determine the $H (f)$ :
The transfer function of an LTI system ( $H (f)$ ) can be determined from the power spectrum of the input signal ( $S (f)$ ) and the cross spectrum between input and output signals ( $Φ_{s r}$ ).

LTI system with white noise

The output of LTI system with white noise signals is:

φ_{r r} (τ) = φ_{h h} (τ) * φ_{s s} (τ) = φ_{h h} (τ) * N_{0} δ (τ) = N_{0} φ_{h h} (τ)

with FT:

Φ_{r r} (f) = N_{0} | H (f) |^{2}