Common Z-transform pairs in Signal Processing - Step-by-Step Execution

Concept Flow - Common Z-transform pairs

Start with time-domain sequence x[n

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Apply Z-transform formula: X(z) = Σ x[n

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Simplify sum to get X(z)

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Identify common pair from table

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Use pair for analysis or inverse transform

The flow shows how a time-domain sequence is converted to its Z-transform using the formula, then matched to common pairs for easy analysis.

Execution Sample

Signal Processing

x[n] = a^n u[n]
X(z) = Σ (a^n) z^(-n) from n=0 to ∞
X(z) = 1 / (1 - a z^{-1}) for |z| > |a|

This code calculates the Z-transform of the sequence a^n times the unit step, resulting in a common transform pair.

Execution Table

Step	Expression	Action	Result	Condition
1	x[n] = a^n u[n]	Define sequence	Sequence ready for transform	n ≥ 0
2	X(z) = Σ x[n] z^{-n}	Apply Z-transform sum	Sum expression formed	n=0 to ∞
3	Σ (a^n) z^{-n}	Substitute x[n]	Σ (a z^{-1})^n	n=0 to ∞
4	Sum of geometric series	Simplify sum	1 / (1 - a z^{-1})	|z| > |a|
5	Identify pair	Match to common Z-transform pair	Confirmed pair	Valid region of convergence
6	-	End	Z-transform found	-

💡 Sum converges only if |z| > |a|, otherwise transform does not exist.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
x[n]	undefined	a^n u[n]	a^n u[n]	a^n u[n]	a^n u[n]
X(z)	undefined	Σ a^n z^{-n}	Σ (a z^{-1})^n	1 / (1 - a z^{-1})	1 / (1 - a z^{-1})

Key Moments - 3 Insights

Why do we need the condition |z| > |a| for the Z-transform to exist?

Because the sum is a geometric series that converges only if the common ratio |a z^{-1}| < 1, which means |z| > |a|. This is shown in step 4 of the execution_table.

Why do we start the sum from n=0 and not from negative n?

Because the sequence x[n] includes the unit step u[n], which is zero for n < 0, so terms before n=0 do not contribute. This is clear in step 1 and step 2.

How do we simplify the sum Σ (a z^{-1})^n to a closed form?

We recognize it as a geometric series with ratio r = a z^{-1}, and use the formula sum = 1/(1-r) when |r|<1, as shown in step 4.

Visual Quiz - 3 Questions

Look at the execution_table at step 4, what is the simplified form of the Z-transform?

A Σ a^n z^n

B a / (1 - z^{-1})

C 1 / (1 - a z^{-1})

D 1 - a z^{-1}

💡 Hint

Refer to the 'Result' column at step 4 in the execution_table.

At which step does the condition |z| > |a| become important?

A Step 4

B Step 1

C Step 2

D Step 5

💡 Hint

Check the 'Condition' column in the execution_table where convergence is discussed.

If the sequence was x[n] = a^n without the unit step u[n], how would the sum limits change?

A Sum from n=0 to ∞

B Sum from n=-∞ to ∞

C Sum from n=1 to ∞

D Sum from n=-∞ to 0

💡 Hint

Without the unit step, the sequence exists for all n, so sum limits cover all integers.

Concept Snapshot

Z-transform converts discrete time sequences x[n] into X(z) using X(z) = Σ x[n] z^{-n}.
Common pairs like x[n] = a^n u[n] transform to X(z) = 1/(1 - a z^{-1}) for |z| > |a|.
The region of convergence ensures the sum converges.
Use these pairs to analyze signals and systems easily.

Full Transcript

The Z-transform takes a sequence x[n] and converts it into a function X(z) by summing x[n] times z to the negative n power. For example, the sequence a^n times the unit step u[n] has a Z-transform of 1 divided by (1 minus a times z to the negative one). This sum converges only if the magnitude of z is greater than the magnitude of a. We start summing from n equals zero because the unit step is zero for negative n. Recognizing the sum as a geometric series helps simplify it to a closed form. These common pairs help us analyze signals and systems in the z-domain efficiently.