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.

Practice

1. What is the primary purpose of the Z-transform in signal processing?

easy

A. To convert discrete-time signals into a complex frequency domain for easier analysis

B. To convert continuous signals into discrete signals

C. To amplify the signal strength

D. To filter out noise from signals

Solution

1. Step 1: Understand the role of Z-transform

   The Z-transform changes discrete-time signals into a complex frequency domain representation.

2. Step 2: Compare options with definition

   Only To convert discrete-time signals into a complex frequency domain for easier analysis correctly describes this transformation purpose.

3. Final Answer:

   To convert discrete-time signals into a complex frequency domain for easier analysis -> Option A

4. Quick Check:

   Z-transform = frequency domain conversion [OK]

Hint: Z-transform moves signals to complex frequency domain [OK]

Common Mistakes:

• Confusing Z-transform with filtering
• Thinking it converts continuous to discrete signals
• Assuming it amplifies signals

2. Which of the following is the correct formula for the Z-transform of a discrete sequence x[n]?

easy

A. X(z) = \sum_{n=-\infty}^{\infty} x[n] z^n

B. X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}

C. X(z) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt

D. X(z) = \sum_{n=0}^{N} x[n] z^n

Solution

1. Step 1: Recall the Z-transform formula

   The Z-transform is defined as X(z) = sum from n=0 to infinity of x[n] times z to the power -n.

2. Step 2: Match formula with options

   X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} matches the standard Z-transform formula exactly.

3. Final Answer:

   X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} -> Option B

4. Quick Check:

   Z-transform formula = sum x[n] z^-n [OK]

Hint: Look for sum from 0 to infinity with z to negative n [OK]

Common Mistakes:

• Using positive powers of z instead of negative
• Confusing Z-transform with Fourier transform integral
• Summing over all integers instead of starting at 0

3. Given the sequence x[n] = {1, 2, 3} for n=0,1,2 and zero elsewhere, what is the Z-transform X(z)?

medium

A. X(z) = 1z^{-1} + 2z^{-2} + 3z^{-3}

B. X(z) = 1 + 2z + 3z^{2}

C. X(z) = 1 + 2z^{-1} + 3z^{-2}

D. X(z) = 1z + 2z^{2} + 3z^{3}

Solution

1. Step 1: Apply Z-transform formula to given sequence

   Use X(z) = sum x[n] z^{-n} from n=0 to 2: 1*z^0 + 2*z^{-1} + 3*z^{-2}.

2. Step 2: Write the expression clearly

   This equals 1 + 2z^{-1} + 3z^{-2}, matching X(z) = 1 + 2z^{-1} + 3z^{-2}.

3. Final Answer:

   1 + 2z^{-1} + 3z^{-2} -> Option C

4. Quick Check:

   Sum x[n] z^-n = 1+2z^-1+3z^-2 [OK]

Hint: Use powers of z as negative indices matching n [OK]

Common Mistakes:

• Using positive powers of z
• Starting sum at n=1 instead of 0
• Ignoring zero values outside given n

4. Identify the error in this Z-transform expression: X(z) = \sum_{n=0}^{\infty} x[n] z^{n}

medium

A. The variable z should be real, not complex

B. The summation limits are incorrect

C. The sequence x[n] should be continuous, not discrete

D. The power of z should be negative, not positive

Solution

1. Step 1: Review the Z-transform formula

   The Z-transform uses z raised to the negative power of n, i.e., z^{-n}.

2. Step 2: Compare with given expression

   The expression uses z^{n} (positive power), which is incorrect.

3. Final Answer:

   The power of z should be negative, not positive -> Option D

4. Quick Check:

   Z-transform uses z^-n, not z^n [OK]

Hint: Check sign of exponent on z carefully [OK]

Common Mistakes:

• Using positive powers of z
• Misunderstanding summation limits
• Confusing discrete and continuous signals

5. You have a signal x[n] defined as 1 for n=0 to 3 and 0 elsewhere. How would you express its Z-transform X(z) and what does it represent?

hard

A. X(z) = 1 + z^{-1} + z^{-2} + z^{-3}, representing the signal's frequency components

B. X(z) = z + z^{2} + z^{3} + z^{4}, representing the time-domain signal

C. X(z) = \sum_{n=0}^{\infty} x[n] z^{n}, representing the signal's energy

D. X(z) = 1 + 2z^{-1} + 3z^{-2} + 4z^{-3}, representing the signal's amplitude

Solution

1. Step 1: Write the Z-transform for given x[n]

   Since x[n] = 1 for n=0 to 3, X(z) = sum from 0 to 3 of 1*z^{-n} = 1 + z^{-1} + z^{-2} + z^{-3}.

2. Step 2: Understand what X(z) represents

   X(z) shows the signal's frequency components in the complex z-plane.

3. Final Answer:

   X(z) = 1 + z^{-1} + z^{-2} + z^{-3}, representing the signal's frequency components -> Option A

4. Quick Check:

   Sum of z^-n terms = frequency components [OK]

Hint: Sum z^-n terms for signal duration [OK]

Common Mistakes:

• Using positive powers of z
• Incorrect coefficients in sum
• Misinterpreting what X(z) represents