Elsa CHOI

Elsa CHOI

DT LTI Systems and Convolution

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LTI Systems

  • An LTI system is a linear time-invaiant system, that is
  1. Linear:
    graph LR
A("x[n]")-->B("System S")
B-->C("y[n]")

    graph LR
A("r[n]") --> B("System S") 
B-->C("s[n]")

    graph LR
A("x[n]+r[n]") --> B("System S") 
B-->C("y[n]+s[n]")
  2. Time-invaiant:
    graph LR
A("x[n-n0]") --> B("System S") 
B-->C("y[n-n0]")

convolution sum

\begin{equation}\label{convolution} y[n]=\sum\limits_{k=-\infty}^{\infty}x[k]h[n-k]\triangleq (h*x)[n] \end{equation}

unit pluse signal $\delta[n]$

\[\delta[n]=\Bigg\{ \begin{aligned} &1 &\text{when} \quad &n=0 \\ &0 &\text{when} \quad &n\neq 0 \end{aligned}\]

\begin{equation}\label{convolution identity} x[n]=\sum\limits_{k=-\infty}^{\infty}x[k]\delta[n-k] \end{equation}

LTI system and convolution sum

Let $h[n]=y[n], x[n]=\delta[n]$, then $y[n]$ can be written as a convolution sum:

\[y[n]=\sum\limits_{k=-\infty}^{\infty}h[k]x[n-k]\]

We will represent this LTI system as following digram:

graph LR
A("x[n]") --> B("LTI h[n]") 
B-->C("y[n]")

properties of convolution sum

  • Commutativity
\[(x*h)[n]=(h*n)[n]\]
  • Associativity
\[(x*(h_1*h_2))[n]=((x*h_1)*h_2)[n]\]
  • Distributivity
\[(x*(h_1+h_2))[n]=(x*h_1)[n]+(x*h_2)[n]\]
  • Shift property let $\hat x [n]=x[n-n_0]$
\[(\hat x * h)[n]=(x*h)[n-n_0]\]
  • Identity
\[(x*\delta)[n]=x[n]\]

Causal System

  • Def: An LTI shstem is causal iff
\[h[n]=0 \quad \text{if } \quad n<0\]

Memoryless System

  • Def: An LTI shstem is memoryless iff
\[h[n]=c\delta[n]\]

Stable System

  • Def: An LTI shstem is stable iff
\[\sum\limits_{n=-\infty}^{\infty} |h[n]| < \infty\]

Complex Eigenfunctions

\[x[n]=x_R[n]+jx_I[n]\] \[y[n]=\sum\limits_{k=-\infty}^{\infty}x_R[k]h[n-k] +j \sum\limits_{k=-\infty}^{\infty}x_I[k]h[n-k]\]

Hence,

\[\begin{aligned} \text{Re}\{y[n]\}&=\text{Re}\{(x*h)[n]\}=(x_R*h)[n] \\ \text{Im}\{y[n]\}&=\text{Im}\{(x*h)[n]\}=(x_I*h)[n] \end{aligned}\]

Example: If $x[n]=e^{j \omega_0 n}$,then

\[y[n]=\sum\limits_{k=-\infty}^{\infty} h[k]e^{j \omega_0 (n-k)} =\sum\limits_{k=-\infty}^{\infty} h[k] e^{j \omega_0 (n)}e^{j \omega_0 (-k)}\]

Say,

\[H(\omega_0)=\sum\limits_{k=-\infty}^{\infty} h[k] e^{-j \omega_0 (k)}\]

then,

\[y[n]=H(\omega_0)e^{j \omega_0 (n)}\]

Example: If $x[n]=\cos(\omega_0 n)=\text{Re}{ e^{j\omega_0 n } }$, then

\[y[n]=\text{Re}\{ H(\omega_0) e^{j\omega_0 n } \}\]

Remark: CT LTI Systems and Convolution is similar to DT one, just replace $\sum$ to $\int$.

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