Signals and Systems

6.18 INVERSE z-TRANSFORM USING PARTIAL FRACTION EXPANSION

In partial fraction expansion method, the function X(z) is expressed as follows:

 

X(z) = α₁X₁(z) + α₂X₂(z) + α₃X₃(z) + ....... + α_mX_m(z)

 

where α₁, α₂, α₃, ............ and α_m are the constants and X₁(z), X₂(z), X₃(z), ........., X_m(z) are the standard z-transforms. The inverse transform of these standard z-transform are known. Using linearity property we can write

 

x(n) = α₁x₁(n) + α₂x₂(n) + α₃x₃(n) + ....... + α_mx_m(n)    (6.72)

 

If the z-transform is rational in nature which can be expressed as the ratio of two polynomials as follows

 

 

Let b0 = 1. If b₀ ≠ 1, the polynomials ...