Signals and Systems - Electrical Engineering

544 C H A P T E R 9: The Z-Transform

9.5.2 Partial Fraction Expansion

The basics of partial fraction expansion remain the same for the Z-transform as for the Laplace

transform. A rational function is a ratio of polynomialsN(z)andD(z)inzorz−^1 :

X(z)=

N(z)

D(z)

The poles ofX(z)are the roots ofD(z)=0 and the zeros ofX(z)are the roots of the equationN(z)=0.

Remarks

n The basic characteristic of the partial fraction expansion is that X(z)must be a proper rational function,

or that the degree of the numerator polynomial N(z)must be smaller than the degree of the denominator

polynomial D(z)(assuming both N(z)and D(z)are polynomials in either z−^1 or z). If this condition is

not satisfied, we perform long division until the residue polynomial is of a degree less than that of the

denominator.

n It is more common in the Z-transform than in the Laplace transform to find that the numerator and the

denominator are of the same degree—this is becauseδ[n]is not as unusual as the analog impulse function

δ(t).

n The partial fraction expansion is generated, from the poles of the proper rational function, as a sum of

terms of which the inverse Z-transforms are easily found in a Z-transform table. By plotting the poles and

the zeros of a proper X(z), the location of the poles provides a general form of the inverse within some

constants that are found from the poles and the zeros.

n Given that the numerator and the denominator polynomials of a proper rational function X(z)can be

expressed in terms of positive or negative powers of z, it is possible to do partial fraction expansions in

either z or z−^1. We will see that the partial fraction expansion in negative powers is more like the partial

fraction expansion in the Laplace transform, and as such we will prefer it. Partial fraction expansion in

positive powers of z requires more care.

nExample 9.14

Consider the nonproper rational function

X(z)=

2 +z−^2

1 + 2 z−^1 +z−^2

(the numerator and the denominator are of the same degree in powers ofz−^1 ). Determine how to

obtain an expansion ofX(z)containing a proper rational term to findx[n].

Solution

By division we obtain

X(z)= 1 +

1 − 2 z−^1

1 + 2 z−^1 +z−^2