4.4 Lsi/lti systems

linear shift invariant system

A linear shift invariant system is one that is both:

• linear
• shift invariant

All systems

Characterizing lsi systems

1.

Since the system is linear , it can be represented as a matrix multiply ( ).

2.

Since $\mathscr{H}$ is shift invariant , it cannot be just any old matrix. Its values are highly constrained .

In particular we know that

Understanding conditions on matrixℋ for shiftInvariance

Recall .

Now shift $x$ down circularly $m$ units. If the system is SI then $y$ will also shift down circularly $m$ units. i.e. : $$C_{m}y=\mathscr{H}{C}_{m}x$$

Key: we want the value▪ in $C_{m}y$ to equal the· value in $y$ .

This implies that the rows of $\mathscr{H}$ must circularly shift as we shift $x$ and $y$ . i.e. : row $n+m$ of $\mathscr{H}$ is equal to the circular shift right of row $n$ of $\mathscr{H}$ by $m$ . i.e. : $$h_{n+m}^r=h_n^r{C}_m^T$$ i.e. : all rows of $\mathscr{H}$ are circular shifts of each other.

$\mathscr{H}$ is called a circulant matrix .

• each row is a circulary shifted version of the row above (right).
• each column is a circularly shifted version of the column to the left (down).

Also, row $n$ , column $k$ element of $\mathscr{H}$ is

Apply a 3-point moving average smoother to a signal $x=\begin{pmatrix}-2 & -1 & 0 & 1 & 2 & 1 & 0 & -1\\ \end{pmatrix}$ .

In ,

$y(n)=1x((n-1)\mod N)+2x(n)+3x((n+1)\mod N)$

$y=\begin{pmatrix}2 & 3 & 0 & 0 & 0 & 0 & 0 & 1\\ 1 & 2 & 3 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 2 & 3 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 2 & 3 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 2 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 2 & 3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 3\\ 3 & 0 & 0 & 0 & 0 & 0 & 1 & 2\\ \end{pmatrix}\left(\begin{array}{c}-2\\ -1\\ 0\\ 1\\ 2\\ 1\\ 0\\ -1\end{array}\right)=\left(\begin{array}{c}-8\\ -4\\ 2\\ 8\\ 8\\ 4\\ -2\\ -8\end{array}\right)$

The relationship between rows and columns of $\mathscr{H}$ :

$\mathbb{R}^4$

$\left(\begin{array}{c}y(0)\\ y(1)\\ y(2)\\ y(3)\end{array}\right)=\begin{pmatrix}2 & 3 & 0 & 1\\ 1 & 2 & 3 & 0\\ 0 & 1 & 2 & 3\\ 3 & 0 & 1 & 2\\ \end{pmatrix}\left(\begin{array}{c}x(0)\\ x(1)\\ x(2)\\ x(3)\end{array}\right)$

where the zeroth column, $\left(\begin{array}{c}2\\ 1\\ 0\\ 3\end{array}\right)$ , is the impulse response , $h(n)$ ( ).

In , the zeroth row, $\begin{pmatrix}2 & 3 & 0 & 1\\ \end{pmatrix}$ , is the time-reversed impulse response , $h(-k)$ .

Upshot for lsi systems

Summary: lsi systems and imuplse response

Given an LSI system ( ), we can characterize it by the impulse response , $h$ ( ).

How to get the impulse response?

4-point edge detector for 8-point signals in complex space ( ).

$h$ =

$\mathscr{H}$ =