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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)
Figure 1
Graph with horizontal axis Re[z] and vertical axis Im[z]. There is a blue circle centered at the origin with its rightmost intersection with the horizontal axis marked α.

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)
Figure 2
Graph with horizontal axis n and vertical axis x_3[n]. There are ten evenly-spaced vertical lines, each with lowest point at the horizontal axis, and each of a different length.

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.

Figure 3
Graph with horizontal axis Re[z] and vertical axis Im[z]. There is a blue circle centered at the origin with its rightmost intersection with the horizontal axis marked 1/2. The graph is titled, ROC.
Figure 4
Graph with horizontal axis Re[z] and vertical axis Im[z]. There is a blue circle centered at the origin with its leftmost intersection with the horizontal axis marked -1/3. The graph is titled, ROC.

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)
Figure 5
Graph with horizontal axis Re[z] and vertical axis Im[z]. There is a blue circle centered at the origin with its rightmost intersection with the horizontal axis marked 1/2 with an X. The spots of vertical value 0 and horizontal value -1/3 and 1/3 are marked with an X and an O respectively. The graph is titled, ROC.

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)
Figure 6
Graph with horizontal axis n and vertical axis x_3[n]. There are ten evenly-spaced vertical lines, each with lowest point at the horizontal axis, and each of a different, decreasing length.

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.

Figure 7
Graph with horizontal axis Re[z] and vertical axis Im[z]. There are red circles spaced out in a circular pattern around the origin, with an X at the origin labeled N-1. The graph is labeled ROC.

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

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