[신호 및 시스템] Lec 06, 07 - Properties of LTI system, Eigenfunction

Precaution

본 게시글은 서울대학교 박성준 교수님의 신호 및 시스템 (26 Spring) 강의록입니다.

Properties of convolution

• Commutative property

\[x[n] * h[n] = h[n] * x[n]\] \[x(t) * h(t) = h(t) * x(t)\]

DT LTI system

• Make multiple copies of $x[n]$, give delays to the signal, add scalar multiplication with impulse reseponse of corresponding given delay, summation of all weighted delayed input is the output $y[n]$

• Distributive property

\[x[n] * (h_1[n] + h_2[n]) = x[n]* h_1[n] + x[n]*h_2[n]\]

• Associative property : Don’t need to care about the order of calculation

LTI systems without memory

• Application to impulse response :

\[\begin{aligned}h[n] = 0 \text{ for }n\neq 0 \\ h(t)= 0 \text{ for }t\neq 0 \end{aligned}\]

• Shape of impulse response must be impulse (since input of impulse response is impulse)

Causality of LTI systems

• Application to impulse response :

\[\begin{aligned}h[n] = 0 \text{ for }n< 0 \\ h(t)= 0 \text{ for }t< 0 \end{aligned}\]

Invertibility of LTI systems

• If LTI system is invertible, then its inverse system is also LTI and also $g(t) * h(t) = \delta(t)$
• Think of Invertibility of function. i.e. $y=cos(t), y=x^2$ is not invertible

Stability of LTI systems

• Sum of impulse response over time is finite, the system is BIBO stable

\[\sum_{k=-\infty}^\infty \vert h[k] \vert <\infty, \int_{-\infty}^\infty \vert h(\tau) \vert d\tau <\infty\]

• Proof :

\[\begin{aligned} &\text{if } \vert x[k] \vert < B \text{ for all } n \\ &\vert y[n]\vert = \vert \sum_{k=-\infty}^\infty h[k]x[n-k] \vert \leq \sum_{k=-\infty}^\infty \vert h[k]\vert \vert x[n-k]\vert \leq B \sum_{k=-\infty}^\infty \vert h[k]\vert \end{aligned}\]

• Therefore $\sum_{k=-\infty}^\infty \vert h[k]\vert <\infty$ is a sufficient condition for stability

\[\begin{aligned} \text{For any }\vert h[k]\vert \text{ such that } \sum_{k=-\infty}^\infty\vert h[k]\vert=\infty \\ \text{consider } x[n]=\begin{cases}0, & h[n]=0 \\ {h[-n] \over \vert h[-n] \vert} & \text{otherwise}\end{cases} \end{aligned}\] \[\begin{aligned}y[n] &= \sum_{k=-\infty}^\infty h[k]x[n-k] = \sum_{k=-\infty}^\infty h[k] {h[k-n] \over \vert h[k-n] \vert } \\ y[0] &= \sum_{k=-\infty}^\infty \vert h[k] \vert = \infty\end{aligned}\]

• Therefore $\sum_{k=-\infty}^\infty \vert h[k] \vert <\infty$ is a necessary condition

(Complex) sinusoidal input to LTI system

• Let $x_1(t) = e^{j\omega t}, x_2(t) = e^{j\omega (t+a)}$
• Due to linearity,

$y_2(t) = y_1(t+a) = e^{j\omega a} y_1(t)$ ← Plug in $t=0$

$y_1(a) = e^{j\omega a} y_1(0), \therefore y_1(t) = y_1(0) e^{j\omega t}$

Eigenfunctions of LTI system

• Nonzero function $\phi(t)$ such that $H{\phi(t)} = \lambda \phi(t)$ for some scalar eigenvalue $\lambda$,
• $\phi(t)$ : Eigenfunction, $\lambda$ : Eigenvalue for corresponding eigenfunction
• $\lambda = y(0)$ is the corresponding eigenvalue for $x(t) = e^{j\omega t}$