Understanding Z-Transforms

The Z-transform of a discrete-time signal x(n) is a power series, written as X(z) = Σ x(n) z⁻ⁿ, where the sum is taken for n from -∞ to ∞.
The variable z is a complex variable.
The direct Z-transform converts a time-domain signal x(n) into its complex-plane representation X(z).
The inverse Z-transform obtains x(n) from X(z).
The Z-transform of a signal x(n) can be denoted as X(z) = Z{x(n)}.
The relationship between x(n) and X(z) is x(n) ↔ X(z).
The Z-transform exists only for values of z where the infinite power series converges.
The region of convergence (ROC) of X(z) is the set of all z values for which X(z) has a finite value.
When citing a Z-transform, its ROC should be indicated.

For the finite-duration signal x₁(n) = {1, 2, 5, 7, 0, 1}, the Z-transform is X₁(z) = 1 + 2z⁻¹ + 5z⁻² + 7z⁻³ + z⁻⁵, with ROC as the entire z-plane except z = 0.
For the signal x₂(n) = {1, 2, 5, 7, 0, 1}, the Z-transform is X₂(z) = z² + 2z + 5 + 7z⁻¹ + z⁻³, and the ROC is the entire z-plane except z = 0 and z = ∞.
The Z-transform of the signal x₃(n) = {0, 0, 1, 2, 5, 7, 0, 1, 0, 0} given an ROC: entire z-plane except z = 0 is X₃(z) = z⁻² + 2z⁻³ + 5z⁻⁴ + 7z⁻⁵ + z⁻⁷.
Z-Transform of x₄(n) = {2, 4, 5, 7, 0, 1} given ROC: entire z-plane except z = 0 and z = ∞ can be expressed as: X₄(z) = 2z² + 4z + 5 + 7z⁻¹ + z⁻³.
If x₅(n) = δ(n), X₅(z) = 1, the ROC is the entire z-plane.
For x₆(n) = δ(n − k) with k > 0, the Z-transform is X₆(z) = z⁻ᵏ, and the ROC is the entire z-plane excluding z = 0.
When x₇(n) = δ(n + k) with k > 0, the Z-transform is X₇(z) = zᵏ, with the ROC being the entire z-plane except z = ∞.
For x(n) = (½)ⁿ u(n), X(z) = 1 / (1 - ½z⁻¹), with |z| > ½.

Z, a complex variable can be expressed in polar form as z = reʲθ, where r = |z| and θ is an angle.
X(z) can be expressed as X(z) = Σ x(n) (reʲθ)⁻ⁿ, with the sum taken from n = -∞ to ∞.
In the ROC of X(z), |X(z)| < ∞, which means |X(z)| = Σ |x(n) r⁻ⁿ|, with the sum taken from n = -∞ to ∞, is finite.
|X(z)| is finite if the sequence x(n) r⁻ⁿ is absolutely summable.
|X(z)| = Σ |x(-n) rⁿ| + Σ |x(n) r⁻ⁿ|, sums taken from n=1 to ∞ and n=0 to ∞, respectively.
For X(z) to converge in a region of the complex plane, both summations must be finite in that region.
Convergence requires values of r such that the product sequence x(-n) rⁿ, where 1 < n < ∞, is absolutely summable.
The ROC for the first sum includes all points within a circle of radius r₁.
Convergence requires values of r such that the product sequence x(n) r⁻ⁿ, where 0 < n < ∞, is absolutely summable.
The ROC for the second sum includes all points outside a circle of radius r₂.
Convergence of X(z) implies that both sums are finite.
The ROC of X(z) is defined by the annular region of the z-plane, r₂ < r < r₁, where both sums are finite.
If r₂ > r₁, there is no common region of convergence, thus X(z) doesn't exist.
Given x(n) = αⁿ u(n), x(n) = αⁿ u(n) ↔ X(z) =1 / (1 - αz⁻¹), |z| > |α|.
Given x(n) = −αⁿ u(−n − 1), the region z is ROC: |z| < |α|.

Causal signal αⁿu(n) and anti-causal signal −αⁿu(−n − 1) have the same closed-form expressions for the z-transform.
Z{αⁿu(n)} = Z{-αⁿu(-n − 1)} = 1 / (1 - αz⁻¹)
A closed-form expression for the z-transform does not uniquely specify the signal in the time domain.
Ambiguity can be resolved only if the ROC is specified along with the closed-form expression.
A discrete-time signal x(n) is uniquely determined by its z-transform X(z) and the region of convergence of X(z).
A causal signal's ROC is the exterior of a circle with some radius r₂, and the ROC of an anti-causal signal is the interior of a circle with some radius r₁.
Given the signal x(n) = αⁿu(n) + bⁿu(-n − 1): when |b| < |α|, there is no X(z) (where |αz⁻¹| < 1 or equivalently |z| > |α|, while the second summation converges if |b⁻¹z| < 1, or equivalently |z| < |b|.
When determining convergence of X(z) and |b| < |α, the ROCs do not overlap, so there are no converging values of z.
In second case, when determining convergence of X(z) and |b| > |α|, there is a ring on the z-plane where power series converge.
With |b| > |α|, the function X(z) can be expressed as X(z) =1/(1 - αz⁻¹) - 1/(1 - bz⁻¹) = (b - α) / (α + b - z - αbz⁻¹), and the ROC is |α| < |z| < |b|.
If there's a ROC for an infinite-duration two-sided signal, it is a ring in the z-plane.

The zeros of a z-transform X(z) are the values of z for which X(z) = 0.
The poles of a z-transform are the values of z for which X(z) = ∞.
X(z) = P(z) / Q(z)
Zeros are values of z where P(z) = 0.
Poles are values of z where Q(z) = 0.
Find the poles and zeros of the function expressed as: H(z) = (z + 1) / ((z - ½)(z + ¾)).
To determine the pole-zero plot for signal x(n) = aⁿu(n) given that the z[x(n) = aⁿu(n)] then X(z) = 1 / (1 - az⁻¹), ROC: |z| > |a|.
In standard form if the pole is defined as z = a and the zero is defined as z = 0.

Convolution of two sequences x₁(n) and x₂(n) is x(n) = x₁(n) * x₂(n) = Σ x₁(n − k)x₂(k), with the sum taken from k = 0 to ∞.
- - designates the linear convolution.
In the z-transform domain, X(z) = X₁(z)X₂(z).
Given two sequences, find the z-transform for the convolution of: x₁(n) = 3δ(n) + 2δ(n − 1) and x₂(n) = 2δ(n) – δ(n − 1).
- After applying the z-transform function X₁(z) = 3 + 2z⁻¹ and X₂(z) = 2 − z⁻¹.
- Using X(z) = X₁(z)X₂(z) = x(n) = Z⁻¹(6 + z⁻¹ - 2z⁻²)
When applying the inverse z-transform function: + δ(n − 1) – 2δ(n – 2) = 6δ(n).

Three common methods used to evaluate the inverse Z-transform:
- Direct evaluation of the inverse Z-Transform equation via contour integration.
- Expansion of a given transform into a series of terms, using Power Series Expansion.
- Partial-fraction expansion and table lookup.

The method: express X(z) as the linear combination with the transforms and expressions along with inverse transforms.
Given: X(z) = α₁X₁(z) + α₂X₂(z) + ··· + αₖXₖ(z) .
- Where expressions are X1 with inverse transforms is ₁(n), ..., xₖ(n) available in a table of z-transform pairs; the property is: x(n) = α₁x₁(n) + α₂x₂(n) + ··· + αₖxₖ(n).
For rational functions; it is assumed that α₀ = 1
- X(z) = B(z) / A(z) = (b₀ +b₁z⁻¹ + ··· + bмz⁻ᴹ ) / (1+ α₁z⁻¹ + ··· +αNz⁻ᴺ)
The expression is proper if αₙ ≠ 0 and M < N, and finite zeros should be of less than the number of finite poles.
X(z) = B(z) / A(z) = C₀ + C₁z⁻¹ + ··· + Cм-Nz⁻⁽ᴹ⁻ᴺ⁾ + B₁(z) /A(z).
Eliminating the negative, powers: x (z) =(b₀zᴷ +b₁zᴷ⁻¹ +…+bₘZᴷ⁻ᵐ) / (zᴷ +α₁Zᴷ⁻¹ +…+αₖ)
the method involves distinct and repeating poles.
Distinct poles: X(z) / z = A1/ z - P₁ + A2 / z - P₂ +...+ AN/ z - PN.
Find the values of the unknown coefficients is next.
Perform the partial fraction for expression: X(z) = 1 / (1 – 1.5z⁻¹ + 0.5z⁻²).
- Then multiply for the equation where: X(z) = z² / (z² – 1.5z + 0.5) can be factored .
From: X(z) / z = A1 / z - P₁ + A2 / z - P₂ , the constants is the next step.

For transforms defined as X(z) = A1 /z- P₁ +A2 /z- P₂+…+Ak/z- Pk: Transform can be defined for the form: X(z) = A1 (1-P₁Z⁻¹) +A2/(1-P₂Z⁻¹) …+Ak(1 -Pkz⁻¹). After the transform defined: Z⁻¹[1 / (1- Pkz⁻¹ )] = {(Pk) ⁿµ(µ) (-(Pk) ⁿµ(-µ1)), causal and anticausal respectively.
Causal X: x(n) = [A₁ + A₂ + ··· + Aₖ].
Conditions can be defined for ROC and Pmax.