**1.**

Since the system is *linear*,
it can be represented as a *matrix multiply*
(Figure 3).

**2.**

Since *shift invariant*, it cannot
be just any old matrix. Its values are *highly
constrained*.

In particular we know that

Inside Collection (Course): ECE 454 and ECE 554 Supplemental reading

Summary: Introduction to LSI/LTI systems.

- Definition 1: linear shift invariant system
- A linear shift invariant system is one that is both:
- linear
- shift invariant

All Systems |
---|

LSI systems are the bread 'n' budduh of DSP
(Figure 2).

Since the system is *linear*,
it can be represented as a *matrix multiply*
(Figure 3).

Since *shift invariant*, it cannot
be just any old matrix. Its values are *highly
constrained*.

In particular we know that

Recall Figure 3.

Now shift

Key: we want the value ▪ in

This implies that the rows of

- 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).

Circulant matrices are a special case of Toeplitz
matrices, which are constant along diagonals.
e.g.:
T = (
⋱ ⋱ ⋱ ⋱
⋱ ⋱ ⋱ ⋱
⋱ ⋱ ⋱ ⋱
⋱ ⋱ ⋱ ⋱
) = (
1 3 5 6
2 1 3 5
4 2 1 3
7 4 2 1
)
T
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
⋱
1
3
5
6
2
1
3
5
4
2
1
3
7
4
2
1
T n , k = t n − k
T
n
k
t
n
k

3-point smoother

Edge detector

Also, row

Apply a 3-point moving average smoother to a signal

The relationship between rows and columns of

Rows and columns run time in
*reverse order*!!!

Given an LSI system (Figure 3), we can
characterize it by the impulse response,

How to get the impulse response?

4-point edge detector for 8-point signals in complex space (Figure 10).

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