Connexions

You are here: Home » Content » Discrete-time Systems

Recently Viewed

This feature requires Javascript to be enabled.

Discrete-time Systems

Module by: Mark A. Davenport. E-mail the author

We begin with the simplest of discrete-time systems, where X=CNX=CN and Y=CMY=CM. In this case a linear operator is just an M×NM×N matrix. We can generalize this concept by letting MM and NN go to , in which case we can think of a linear operator L:2(Z)2(Z)L:2(Z)2(Z) as an infinite matrix.

Example 1

Consider the shift operator Δk:2(Z)2(Z)Δk:2(Z)2(Z) that takes a sequence and shifts it by kk. As an example, Δ1Δ1 can be viewed as the infinite matrix given by

y - 1 y 0 y 1 = 0 0 1 0 0 1 0 0 1 0 0 x - 1 x 0 x 1 y - 1 y 0 y 1 = 0 0 1 0 0 1 0 0 1 0 0 x - 1 x 0 x 1
(1)

Note that Δk2=1Δk2=1 (for any kk and pp) since the delay doesn't change the norm of xx. The delay operator is also an example of a linear shift-invariant (LSI) system.

Definition 1

An operator L:2(Z)2(Z)L:2(Z)2(Z) is called shift-invariant if L(Δk(x))=Δk(L(x))L(Δk(x))=Δk(L(x)) for all x2(Z)x2(Z) and for any kZkZ.

Observe that Δk1(Δk2(x))=Δk1+k2(x)Δk1(Δk2(x))=Δk1+k2(x) so that ΔkΔk itself is an LSI operator.

Lets take a closer look at the structure of an LSI system by viewing it as an infinite matrix. In this case we write y=Hxy=Hx to denote

y - 1 y 0 y 1 = | | | h - 1 h 0 h 1 | | | x - 1 x 0 x 1 y - 1 y 0 y 1 = | | | h - 1 h 0 h 1 | | | x - 1 x 0 x 1
(2)

Suppose we want to figure out the column of HH corresponding to h0h0. What input xx could help us determine h0h0? Consider the vector

x = 0 1 0 , x = 0 1 0 ,
(3)

i.e., x=δ[n]x=δ[n]. For this input y=Hx=h0y=Hx=h0. What about h1h1? Δ1(x)=δ[n-1]Δ1(x)=δ[n-1] would yield h1h1. In general Δk(x)=δ[n-k]Δk(x)=δ[n-k] tell us the column hkhk. But, if HH is LSI, then

h k = H ( Δ k ( δ [ n ] ) ) = Δ k ( H ( δ [ n ] ) ) = Δ k ( h 0 ) h k = H ( Δ k ( δ [ n ] ) ) = Δ k ( H ( δ [ n ] ) ) = Δ k ( h 0 )
(4)

This means that each column is just a shifted version of h0h0, which is usually called the impulse response.

Now just to keep notation clean, let h=h0h=h0 denote the impulse response. Can we get a simple formula for the output yy in terms of hh and xx? Observe that we can write

y - 1 y 0 y 1 = h 0 h - 1 h - 2 h 1 h 0 h - 1 h 2 h 1 h 0 x - 1 x 0 x 1 y - 1 y 0 y 1 = h 0 h - 1 h - 2 h 1 h 0 h - 1 h 2 h 1 h 0 x - 1 x 0 x 1
(5)

Each column is just shifted down one. (Each successive row is also shifted right one.) Looking at y-1y-1, y0y0 and y1y1, we can rewrite this formula as

y [ - 1 ] y [ 0 ] y [ 1 ] = + x [ - 1 ] h [ 0 ] h [ 1 ] h [ 2 ] + x [ 0 ] h [ - 1 ] h [ 0 ] h [ 1 ] + x [ 1 ] h [ - 2 ] h [ - 1 ] h [ 0 ] + y [ - 1 ] y [ 0 ] y [ 1 ] = + x [ - 1 ] h [ 0 ] h [ 1 ] h [ 2 ] + x [ 0 ] h [ - 1 ] h [ 0 ] h [ 1 ] + x [ 1 ] h [ - 2 ] h [ - 1 ] h [ 0 ] +
(6)

From this we can observe the general pattern

y [ n ] = + x [ - 1 ] h [ n + 1 ] + x [ 0 ] h [ n + 0 ] + x [ 1 ] h [ n - 1 ] + y [ n ] = + x [ - 1 ] h [ n + 1 ] + x [ 0 ] h [ n + 0 ] + x [ 1 ] h [ n - 1 ] +
(7)

or more concisely

y [ n ] = k = - x [ k ] h [ n - k ] . y [ n ] = k = - x [ k ] h [ n - k ] .
(8)

Does this look familiar? It is simply the formula for the discrete-time convolution of xx and hh, i.e.,

y = x * h . y = x * h .
(9)

Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks