Inverse Z-transform in Signal Processing - Step-by-Step Execution

Concept Flow - Inverse Z-transform

Start with Z-transform X(z)

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Choose method: Partial Fraction / Power Series / Contour Integral

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Apply chosen method to X(z)

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Calculate coefficients x[n

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Obtain time-domain sequence x[n

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End

The inverse Z-transform converts a function from the Z-domain back to the time domain by applying methods like partial fractions or power series to find sequence coefficients.

Execution Sample

Signal Processing

X_z = "(z^2 + 0.5*z) / (z^2 - 0.8*z + 0.15)"
# Find x[n] using partial fraction expansion
# x[0], x[1], x[2], ...

This code represents a Z-transform function and outlines the process to find its inverse sequence x[n].

Execution Table

Step	Action	Expression	Result/Value
1	Start with given X(z)	X(z) = (z^2 + 0.5z) / (z^2 - 0.8z + 0.15)	Given function
2	Factor denominator	z^2 - 0.8z + 0.15	(z - 0.5)(z - 0.3)
3	Set partial fractions	X(z)/z = A/(z - 0.5) + B/(z - 0.3)	Unknown A, B
4	Multiply both sides by denominator	(z + 0.5) = A(z - 0.3) + B(z - 0.5)	Equation for A, B
5	Plug z=0.5 to solve for A	z=0.5: 0.5 + 0.5 = A(0.5 - 0.3) + B(0.5 - 0.5)	1.0 = 0.2 A => A = 5
6	Plug z=0.3 to solve for B	z=0.3: 0.3 + 0.5 = A(0.3 - 0.3) + B(0.3 - 0.5)	0.8 = B*(-0.2) => B = -4
7	Calculate A and B	A = 5, B = -4	Coefficients found
8	Write inverse Z-transform terms	x[n] = 5*(0.5)^n u[n] - 4*(0.3)^n u[n]	Time domain sequence
9	Calculate first values	x[0] = 5 - 4 = 1.0	x[0] = 1.0
10	Calculate x[1]	5*0.5 - 4*0.3 = 2.5 - 1.2	x[1] = 1.3
11	Calculate x[2]	5*0.5^2 - 4*0.3^2 = 1.25 - 0.36	x[2] = 0.89
12	End	Sequence x[n] found	x[n] = 5*(0.5)^n - 4*(0.3)^n for n>=0

💡 Partial fraction coefficients found and sequence x[n] computed for n>=0

Variable Tracker

Variable	Start	After Step 2	After Step 7	Final
X(z)	(z^2 + 0.5z) / (z^2 - 0.8z + 0.15)	(z^2 + 0.5z) / ((z - 0.5)(z - 0.3))	5/(z - 0.5) - 4/(z - 0.3)	5 z/(z - 0.5) - 4 z/(z - 0.3)
A	unknown	unknown	5	5
B	unknown	unknown	-4	-4
x[n]	unknown	unknown	unknown	5*(0.5)^n - 4*(0.3)^n for n>=0

Key Moments - 3 Insights

Why do we factor the denominator before partial fraction expansion?

Factoring the denominator (Step 2) reveals the poles which are needed to set up the partial fractions correctly (Step 3). Without factoring, we cannot write the terms for A and B.

How do we find coefficients A and B in partial fractions?

We multiply both sides by the denominator (Step 4) and then substitute values of z equal to the poles (Steps 5 and 6) to solve for A and B. This isolates each coefficient.

Why does the inverse Z-transform sequence involve terms like (0.5)^n and (0.3)^n?

These terms come from the poles found in the denominator factors (Step 2). Each pole corresponds to a geometric sequence term in time domain (Step 8).

Visual Quiz - 3 Questions

Look at the execution table, what is the value of coefficient A after Step 7?

A -2.5

B 0.5

C 5

D Unknown

💡 Hint

Check the 'Calculate A and B' row in the execution_table.

At which step do we substitute z=0.3 to solve for coefficient B?

A Step 4

B Step 6

C Step 5

D Step 7

💡 Hint

Look for the step mentioning plugging z=0.3 in the execution_table.

If the denominator had a repeated root, how would the partial fraction setup change?

A Add terms with powers of (z - root) in denominator

B Use only one term per root

C No partial fractions needed

D Change numerator only

💡 Hint

Recall partial fraction rules for repeated roots in the concept_flow.

Concept Snapshot

Inverse Z-transform converts X(z) back to x[n].
Common method: partial fraction expansion.
Steps: factor denominator, set partial fractions, solve coefficients.
Result: sum of terms A*(pole)^n u[n].
Each pole corresponds to geometric sequence in time domain.

Full Transcript

The inverse Z-transform process starts with a given Z-transform function X(z). We factor the denominator to find poles, then express X(z) as partial fractions with unknown coefficients. By multiplying both sides by the denominator and substituting the poles, we solve for these coefficients. Finally, the inverse transform is a sum of geometric sequences based on these poles and coefficients, giving the time-domain sequence x[n]. This step-by-step method helps convert from frequency domain back to time domain signals.