# OpenStax-CNX

You are here: Home » Content » Poles and Zeros

### Recently Viewed

This feature requires Javascript to be enabled.

# Poles and Zeros

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

## Poles and zeros

Suppose that X(z)X(z) is a rational function, i.e.,

X ( z ) = P ( z ) Q ( z ) X ( z ) = P ( z ) Q ( z )
(1)

where P(z)P(z) and Q(z)Q(z) are both polynomials in zz. The roots of P(z)P(z) and Q(z)Q(z) are very important.

Definition 1: zero
A zero of X(z)X(z) is a value of zz for which X(z)=0X(z)=0 (or P(z)=0P(z)=0). A pole of X(z)X(z) is a value of zz for which X(z)=X(z)= (or Q(z)=0Q(z)=0).

For finite values of zz, poles are the roots of Q(z)Q(z), but poles can also occur at z=z=. We denote poles in a zz-plane plot by “××” we denote zeros by “”. Note that the ROC clearly cannot contain any poles since by definition the ROC only contains zz for which the zz-transform converges, and it does not converge at poles.

### Example 1

Consider

x1[n]=αnu[n]ZX1(z)=zz-α,z>αx1[n]=αnu[n]ZX1(z)=zz-α,z>α
(2)

and

x2[n]=-αnu[-1-n]ZX2(z)=zz-α,z<αx2[n]=-αnu[-1-n]ZX2(z)=zz-α,z<α
(3)

Note that the poles and zeros of X1(z)X1(z) and X2(z)X2(z) are identical, but with opposite ROCs. Note also that neither ROC contains the point αα.

### Example 2

Consider

x3[n]=12nu[n]+-13nu[n].x3[n]=12nu[n]+-13nu[n].
(4)

We can compute the zz-transform of x3[n]x3[n] by simply adding the zz-transforms of the two different terms in the sum, which are given by

12nu[n]Zzz-12ROC:z>1212nu[n]Zzz-12ROC:z>12
(5)

and

-13nu[n]Zzz+13ROC:z>13.-13nu[n]Zzz+13ROC:z>13.
(6)

The poles and zeros for these zz-transforms are illustrated below.

X3(z)X3(z) is given by

X3(z)=zz-12+zz+13=z(z+13)+z(z-12)(z+13)(z-12)=z(2z-16)(z+13)(z-12)ROC:z>12X3(z)=zz-12+zz+13=z(z+13)+z(z-12)(z+13)(z-12)=z(2z-16)(z+13)(z-12)ROC:z>12
(7)

Note that the poles do not change, but the zeros do, as illustrated above.

### Example 3

Now consider the finite-length sequence

x4[n]=αn0nN-10otherwise.x4[n]=αn0nN-10otherwise.
(8)

The zz-transform for this sequence is

X4(z)=n=0N-1x4[n]z-n=n=0N-1αnz-n=1-(αz)N1-αz=zN-αNzN-1(z-α)ROC:z0X4(z)=n=0N-1x4[n]z-n=n=0N-1αnz-n=1-(αz)N1-αz=zN-αNzN-1(z-α)ROC:z0
(9)

We can immediately see that the zeros of X4(z)X4(z) occur when zN=αNzN=αN. Recalling the “Nth roots of unity”, we see that the zeros are given by

zk=αej2πNk,k=0,1,...,N-1.zk=αej2πNk,k=0,1,...,N-1.
(10)

At first glance, it might appear that there are N-1N-1 poles at zero and 1 pole at αα, but the pole at αα is cancelled by the zero (z0z0) at αα. Thus, X4(z)X4(z) actually has only N-1N-1 poles at zero and N-1N-1 zeros around a circle of radius αα as illustrated below.

So, provided that α<α<, the ROC is the entire zz-plane except for the origin. This actually holds for all finite-length sequences.

## Content actions

### Give feedback:

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