Common Z-transform pairs in Signal Processing

Introduction

The Z-transform helps us study signals and systems by turning sequences into simpler formulas. Knowing common pairs makes solving problems faster and easier.

When analyzing digital filters in electronics.

When solving difference equations in control systems.

When converting time-based signals into a form easier to work with.

When designing systems that process discrete signals like audio or images.

Syntax

Signal Processing

Z{f[n]} = F(z) = \sum_{n=0}^{\infty} f[n] z^{-n}

This formula means: the Z-transform of a sequence f[n] is a sum of terms f[n] times z to the power of -n.

Here, z is a complex number variable, and n is the index of the sequence starting at 0.

Examples

The Z-transform of a constant sequence 1,1,1,... is a simple fraction.

Signal Processing

Z{1} = 1 / (1 - z^{-1}), |z| > 1

For a geometric sequence where each term is a power of a, the Z-transform is a fraction with a in the denominator.

Signal Processing

Z{a^n} = 1 / (1 - a z^{-1}), |z| > |a|

The Z-transform of the sequence n (0,1,2,3,...) is a fraction with a squared term in the denominator.

Signal Processing

Z{n} = z^{-1} / (1 - z^{-1})^2, |z| > 1

u[n] is the unit step sequence (0 for n<0, 1 for n≥0). Its Z-transform is the same as the constant 1 sequence.

Signal Processing

Z{u[n]} = 1 / (1 - z^{-1}), |z| > 1

Sample Program

This code uses SymPy to calculate the Z-transform of three common sequences: constant 1, geometric a^n, and linear n. It sums the terms and simplifies the results to show the common Z-transform pairs.

Signal Processing

import sympy as sp

z, a = sp.symbols('z a', complex=True)
n = sp.symbols('n', integer=True, nonnegative=True)

# Define sequences
constant_seq = 1
geom_seq = a**n
linear_seq = n

# Define Z-transform variable
Z = lambda f: sp.summation(f * z**(-n), (n, 0, sp.oo))

# Calculate Z-transforms
Z_constant = Z(constant_seq)
Z_geom = Z(geom_seq)
Z_linear = Z(linear_seq)

# Simplify results
Z_constant_simplified = sp.simplify(Z_constant)
Z_geom_simplified = sp.simplify(Z_geom)
Z_linear_simplified = sp.simplify(Z_linear)

print(f"Z-transform of constant sequence 1: {Z_constant_simplified}")
print(f"Z-transform of geometric sequence a^n: {Z_geom_simplified}")
print(f"Z-transform of linear sequence n: {Z_linear_simplified}")

Important Notes

The Z-transform is usually defined for n starting at 0, which fits many digital signal cases.

The region of convergence (ROC) is important to know where the Z-transform formula works.

Common pairs help avoid doing the sum every time and speed up analysis.

Summary

The Z-transform changes sequences into algebraic expressions.

Common pairs include constant, geometric, and linear sequences.

Knowing these pairs helps analyze digital signals and systems easily.

Practice

1. Which of the following is the Z-transform of the sequence x[n] = a^n u[n], where u[n] is the unit step?

easy

A. X(z) = z / (z - a), |z| < |a|

B. X(z) = 1 / (1 - a z^{-1}), |z| > |a|

C. X(z) = (1 - a z^{-1}) / z, |z| > |a|

D. X(z) = a / (1 - z^{-1}), |z| > 1

Solution

1. Step 1: Recall the definition of the Z-transform for geometric sequences

   The Z-transform of a^n u[n] is a standard pair given by the sum of a geometric series: \( \sum_{n=0}^\infty a^n z^{-n} = \frac{1}{1 - a z^{-1}} \) for \( |z| > |a| \).

2. Step 2: Match the formula with the options

   X(z) = 1 / (1 - a z^{-1}), |z| > |a| matches the formula exactly with the correct region of convergence.

3. Final Answer:

   X(z) = 1 / (1 - a z^{-1}), |z| > |a| -> Option B

4. Quick Check:

   Geometric sequence Z-transform = 1/(1 - a z^{-1}) [OK]

Hint: Geometric sequences transform to 1/(1 - a z^{-1}) [OK]

Common Mistakes:

• Confusing region of convergence
• Mixing numerator and denominator terms
• Using wrong inequality for |z|

2. Which of the following is the correct Z-transform expression for the unit impulse sequence δ[n]?

easy

A. X(z) = z / (z - 1)

B. X(z) = z^{-1}

C. X(z) = 1 / (1 - z^{-1})

D. X(z) = 1

Solution

1. Step 1: Understand the unit impulse sequence

   The unit impulse δ[n] is 1 at n=0 and 0 elsewhere.

2. Step 2: Calculate the Z-transform

   By definition, \( X(z) = \sum_{n=-\infty}^\infty δ[n] z^{-n} = z^{0} = 1 \).

3. Final Answer:

   X(z) = 1 -> Option D

4. Quick Check:

   Impulse Z-transform = 1 [OK]

Hint: Impulse transforms to 1 always [OK]

Common Mistakes:

• Confusing impulse with step function
• Adding z terms unnecessarily
• Forgetting impulse is zero except at n=0

3. What is the Z-transform of the sequence x[n] = n u[n]?

medium

A. X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1

B. X(z) = 1 / (1 - z^{-1})^2, |z| > 1

C. X(z) = z / (z - 1), |z| > 1

D. X(z) = 1 / (1 - z)^2, |z| > 1

Solution

1. Step 1: Recall the Z-transform of the sequence n u[n]

   The sequence n u[n] has Z-transform \( X(z) = \frac{z^{-1}}{(1 - z^{-1})^2} \) with region of convergence \( |z| > 1 \).

2. Step 2: Match the formula with the options

   X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1 matches the formula exactly including the region of convergence.

3. Final Answer:

   X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1 -> Option A

4. Quick Check:

   Linear sequence Z-transform = z^{-1}/(1 - z^{-1})^2 [OK]

Hint: Linear sequences transform to z^{-1}/(1 - z^{-1})^2 [OK]

Common Mistakes:

• Confusing powers of z
• Using (z - 1) instead of (1 - z^{-1})
• Ignoring region of convergence

4. Identify the error in the following Z-transform expression for x[n] = a^n u[n]: X(z) = 1 / (1 - z a^{-1})

medium

A. The variable z and a are swapped in the denominator

B. The region of convergence is missing

C. The exponent on a should be negative

D. The formula is correct as is

Solution

1. Step 1: Recall the correct Z-transform formula for a^n u[n]

   The correct formula is \( X(z) = \frac{1}{1 - a z^{-1}} \), not \( 1/(1 - z a^{-1}) \).

2. Step 2: Analyze the given expression

   The given expression swaps z and a in the denominator, which changes the meaning and is incorrect.

3. Final Answer:

   The variable z and a are swapped in the denominator -> Option A

4. Quick Check:

   Correct formula uses a z^{-1}, not z a^{-1} [OK]

Hint: Check order of variables in denominator carefully [OK]

Common Mistakes:

• Swapping z and a in denominator
• Ignoring negative powers of z
• Missing region of convergence

5. Given the sequence x[n] = (1/2)^n u[n] + 3 δ[n], what is the combined Z-transform X(z)?

hard

A. X(z) = 3 * z / (z - 1/2), |z| > 1/2

B. X(z) = 3 z^{-1} + 1 / (1 - (1/2) z^{-1}), |z| > 1/2

C. X(z) = 3 + 1 / (1 - (1/2) z^{-1}), |z| > 1/2

D. X(z) = 3 + 1 / (1 - 2 z^{-1}), |z| > 2

Solution

1. Step 1: Find Z-transform of each sequence part

   The Z-transform of (1/2)^n u[n] is \( \frac{1}{1 - (1/2) z^{-1}} \) with \( |z| > 1/2 \). The Z-transform of 3 δ[n] is 3.

2. Step 2: Add the transforms for combined sequence

   Since Z-transform is linear, add both: \( X(z) = 3 + \frac{1}{1 - (1/2) z^{-1}} \).

3. Final Answer:

   X(z) = 3 + 1 / (1 - (1/2) z^{-1}), |z| > 1/2 -> Option C

4. Quick Check:

   Sum of transforms = 3 + 1/(1 - 0.5 z^{-1}) [OK]

Hint: Sum transforms for combined sequences linearly [OK]

Common Mistakes:

• Multiplying instead of adding transforms
• Incorrect region of convergence
• Wrong coefficient placement