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

Concept Flow - Z-transform definition

Start with discrete signal x[n

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Multiply by z^(-n)

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Sum over all n from -∞ to ∞

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Result: X(z), a function of complex variable z

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Analyze X(z) for system behavior

The Z-transform converts a discrete signal into a complex function by summing the signal multiplied by powers of z inverse.

Execution Sample

Signal Processing

x = [1, 2, 3]
z = 2 + 1j
Xz = sum(x[n] * z**(-n) for n in range(len(x)))

Calculates the Z-transform of a simple discrete signal x at a specific complex value z.

Execution Table

Step	n	x[n]	z^(-n)	x[n]*z^(-n)	Partial Sum X(z)
1	0	1	1	1*1=1	1
2	1	2	0.4-0.2j	(2)*(0.4-0.2j)=0.8-0.4j	1 + 0.8-0.4j = 1.8-0.4j
3	2	3	0.12-0.16j	(3)*(0.12-0.16j)=0.36-0.48j	1.8-0.4j + 0.36-0.48j = 2.16-0.88j

💡 All terms of x[n] summed, final Z-transform value X(z) = 2.16 - 0.88j

Variable Tracker

Variable	Start	After 1	After 2	After 3	Final
n	-	0	1	2	2
x[n]	-	1	2	3	3
z^(-n)	-	1	0.4-0.2j	0.12-0.16j	0.12-0.16j
x[n]*z^(-n)	-	1	0.8-0.4j	0.36-0.48j	0.36-0.48j
Partial Sum X(z)	0	1	1.8-0.4j	2.16-0.88j	2.16-0.88j

Key Moments - 2 Insights

Why do we use z to the power of negative n instead of positive n?

Because the Z-transform formula sums x[n] multiplied by z^(-n), which shifts the signal in the complex plane. This is shown in the execution_table where z^(-n) changes with n.

What does the partial sum represent at each step?

The partial sum accumulates the contributions of each term x[n]*z^(-n). The execution_table shows how the sum builds up step-by-step until the final X(z).

Visual Quiz - 3 Questions

Look at the execution_table at step 2, what is the value of x[1]*z^(-1)?

A 0.12 - 0.24j

B 0.8 - 0.4j

C 1

D 1.4 - 0.2j

💡 Hint

Check the 'x[n]*z^(-n)' column at step 2 in the execution_table.

At which step does the partial sum X(z) become 2.16 - 0.88j?

A Step 3

B Step 2

C Step 1

D Never

💡 Hint

Look at the 'Partial Sum X(z)' column in the execution_table for the final value.

If the signal x[n] had one more term x[3] = 4, how would the execution_table change?

A z^(-n) values would reset to 1

B The partial sum would remain the same

C There would be an additional step 4 with n=3 and x[3]=4

D The sum would stop at step 3

💡 Hint

Adding a term extends the sum, so the execution_table gains a new row for n=3.

Concept Snapshot

Z-transform converts a discrete signal x[n] into X(z) = Σ x[n] * z^(-n) over all n.
It maps time-domain signals into complex frequency domain.
Used to analyze discrete systems and signals.
z is a complex variable; negative powers shift signal components.
Partial sums build X(z) step-by-step.

Full Transcript

The Z-transform takes a discrete signal x[n] and converts it into a complex function X(z) by summing each signal value multiplied by z to the negative n power. This process shifts the signal into the complex frequency domain, helping analyze system behavior. We calculate X(z) by iterating over each n, multiplying x[n] by z^(-n), and adding to a running total. The execution table shows each step's multiplication and partial sum, ending with the final Z-transform value. Key points include understanding why z is raised to negative powers and how partial sums accumulate. Visual quizzes test understanding of these steps and how adding terms affects the sum.