Z-transform definition in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to define the Z-transform summation formula.

Signal Processing

def z_transform(x, z):
    return sum(x[n] * (z ** (-[1])) for n in range(len(x)))

Drag options to blanks, or click blank then click option'

An

Bz

Cx

D1

Attempts:

3 left

💡 Hint

Common Mistakes

Using z as the exponent instead of n

Using x as the exponent

Using a constant exponent

The Z-transform sums over n, raising z to the power -n.

2fill in blank

medium

Complete the code to calculate the Z-transform for a given sequence x and complex variable z.

Signal Processing

def z_transform(x, z):
    result = 0
    for n in range(len(x)):
        result += x[n] * ([1] ** (-n))
    return result

Drag options to blanks, or click blank then click option'

An

Bz

Cx

Dresult

Attempts:

3 left

💡 Hint

Common Mistakes

Using x or n as the base

Using result as the base

The Z-transform formula uses z raised to the power -n.

3fill in blank

hard

Fix the error in the Z-transform function where the exponent is incorrect.

Signal Processing

def z_transform(x, z):
    return sum(x[n] * (z ** (-[1])) for n in range(len(x)))

Drag options to blanks, or click blank then click option'

A1

Bz

Cx

Dn

Attempts:

3 left

💡 Hint

Common Mistakes

Using z as exponent

Using x as exponent

Using a constant exponent

The exponent must be the index n, not z or other variables.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps each n to its Z-transform term.

Signal Processing

z_terms = {n: x[n] * ([1] ** (-[2])) for n in range(len(x))}

Drag options to blanks, or click blank then click option'

Az

Bn

Cx

Dlen(x)

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping base and exponent

Using x as base or exponent

Each term is x[n] times z to the power -n.

5fill in blank

hard

Fill all three blanks to compute the Z-transform sum using a dictionary comprehension and sum function.

Signal Processing

z_terms = { [1]: x[[2]] * ([3] ** (-[1])) for [1] in range(len(x)) }
result = sum(z_terms.values())

Drag options to blanks, or click blank then click option'

An

Bx

Cz

Dlen(x)

Attempts:

3 left

💡 Hint

Common Mistakes

Using x as index

Using wrong base or exponent

The dictionary keys are n, values are x[n] times z to the -n, then sum all values.

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