# Connexions

You are here: Home » Content » z-Transform Analysis of Discrete-Time Filters

### Recently Viewed

This feature requires Javascript to be enabled.

# z-Transform Analysis of Discrete-Time Filters

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

## zz-transform analysis of discrete-time filters

The zz-transform might seem slightly ugly. We have to worry about the region of convergence, and we haven't even talked about how to invert it yet (it isn't pretty). However, in the end it is worth it because it is extremely useful in analyzing digital filters with feedback. For example, consider the system illustrated below

We can analyze this system via the equations

v [ n ] = b 0 x [ n ] + b 1 x [ n - 1 ] + b 2 x [ n - 2 ] v [ n ] = b 0 x [ n ] + b 1 x [ n - 1 ] + b 2 x [ n - 2 ]
(1)

and

y [ n ] = v [ n ] + a 1 y [ n - 1 ] + a 2 y [ n - 2 ] . y [ n ] = v [ n ] + a 1 y [ n - 1 ] + a 2 y [ n - 2 ] .
(2)

More generally,

v [ n ] = N k = 0 b k x [ n - k ] v [ n ] = N k = 0 b k x [ n - k ]
(3)

and

y [ n ] = M k = 1 a k y [ n - k ] + v [ n ] y [ n ] = M k = 1 a k y [ n - k ] + v [ n ]
(4)

or equivalently

N k = 0 b k x [ n - k ] = y [ n ] - M k = 1 a k y [ n - k ] . N k = 0 b k x [ n - k ] = y [ n ] - M k = 1 a k y [ n - k ] .
(5)

In general, many LSI systems satisfy linear difference equations of the form:

M k = 0 a k y [ n - k ] = N k = 0 b k x [ n - k ] . M k = 0 a k y [ n - k ] = N k = 0 b k x [ n - k ] .
(6)

What does the zz-transform of this relationship look like?

Z k = 0 M a k y [ n - k ] = Z k = 0 M b k x [ n - k ] k = 0 M a k Z { y [ n - k ] } = k = 0 N b k Z { x [ n - k ] } . Z k = 0 M a k y [ n - k ] = Z k = 0 M b k x [ n - k ] k = 0 M a k Z { y [ n - k ] } = k = 0 N b k Z { x [ n - k ] } .
(7)

Note that

Z y [ n - k ] = n = - y [ n - k ] z - n = m = - y [ m ] z - m · z - k = Y ( z ) z - k . Z y [ n - k ] = n = - y [ n - k ] z - n = m = - y [ m ] z - m · z - k = Y ( z ) z - k .
(8)

Thus the relationship above reduces to

k = 0 M a k Y ( z ) z - k = k = 0 N b k X ( z ) z - k Y ( z ) k = 0 M a k z - k = X ( z ) k = 0 N b k z - k Y ( z ) X ( z ) = k = 0 N b k z - k k = 0 M a k z - k k = 0 M a k Y ( z ) z - k = k = 0 N b k X ( z ) z - k Y ( z ) k = 0 M a k z - k = X ( z ) k = 0 N b k z - k Y ( z ) X ( z ) = k = 0 N b k z - k k = 0 M a k z - k
(9)

Hence, given a system like the one above, we can pretty much immediately write down the system's transfer function, and we end up with a rational function, i.e., a ratio of two polynomials in zz. Similarly, given a rational function, it is easy to realize this function in a simple hardware architecture. We will focus exclusively on such rational functions in this course.

## 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.

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