Inverse Z transform

Function izt calculates the numerical inverse Z-transform of a rational function of z^-1 at specified points.

Function izt calculates the numerical inverse Z-transform of a rational function of z^-1 at specified points using the partial fraction expansion method. It considers the region of convergence (ROC) of the Z-transform.

The Z-transform is assumed to have numerator and denominator coefficients p and q, respectively, arranged in increasing powers of z^-1. The ROC includes the point z0. 'nvec' is the array of input points, which can be a vector, matrix, or tensor. 'tol' specifies the computational error tolerance, with a default value set to 1e-6.

The Z-transform is defined as:

-1 -M

p(1) + p(2) * z + ... + p(M + 1) * z

X(z) = -----------------------------------------

-1 -N

q(1) + q(2) * z + ... + q(N + 1) * z

where M = numel(p) - 1 and N = numel(q) - 1.

Note: If X(z) includes positive powers of z in the numerator or denominator, it can be reformulated to the above structure by dividing the numerator and the denominator by the highest power of z.

Example: Suppose we want to calculate the inverse z transform of

-1 -2 -3

1+ 2z - z +z

X(z)= ---------------------------------, with ROC:1/6<|z|<1/2

-1 -2 -1

(1 - z + 1/4 z )(1 + 1/6 z )

at n = -3, 0, and 3. We have p = [1 2 -1 1] and q = conv([1 -1 1/4], [1 1/6]) = [1.0000 -0.8333 0.0833 0.0417]

To indicate the ROC we chose z0 = 1/4 which falls inside the ROC. We also set nvec = [-3 0 3]. Finally by executing

x = izt(p, q, z0, nvec)

we get

x =

214.5000 7.5625 0.0761

References:

A. V. Oppenheim and R. Schafer, Discrete-Time Signal Processing, Prentice Hall, 3rd Ed. 2010.

A. R. Forouzan, "Region of convergence of derivative of z transform," IET Electronics Letters, vol. 52, no. 8, pp. 617-619.