Common Z-transform pairs in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to compute the Z-transform of a unit step sequence.

Signal Processing

import sympy as sp
z = sp.symbols('z')
n = sp.symbols('n', integer=True, nonnegative=True)
x = sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = X  # Result is [1]

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A1 / (1 - z**-1)

B1 / (z - 1)

Cz / (z - 1)

Dz / (1 - z)

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing z and z^-1 in the formula.

Using z/(z-1) instead of 1/(1-z^-1).

The Z-transform of the unit step sequence u[n] is 1 / (1 - z^-1) for |z| > 1.

2fill in blank

medium

Complete the code to find the Z-transform of the sequence x[n] = a^n u[n].

Signal Processing

a = sp.symbols('a')
x = a**n * sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = X  # Result is [1]

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A1 / (1 - a * z**-1)

B1 / (1 - z**-1 / a)

Cz / (z - a)

Dz / (1 - a)

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping a and z in the denominator.

Forgetting the unit step function u[n].

The Z-transform of x[n] = aⁿ u[n] is 1 / (1 - a z^-1) for |z| > |a|.

3fill in blank

hard

Fix the error in the code to compute the Z-transform of the sequence x[n] = delta[n - k].

Signal Processing

k = sp.symbols('k', integer=True, nonnegative=True)
x = sp.KroneckerDelta(n - k)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = X  # Result should be [1]

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Az**(k-1)

Bz**k

C1 / z**k

Dz**-k

Attempts:

3 left

💡 Hint

Common Mistakes

Using z^k instead of z^-k.

Confusing the sign in the exponent.

The Z-transform of delta[n - k] is z^-k.

4fill in blank

hard

Fill both blanks to compute the Z-transform of x[n] = n * a^n * u[n].

Signal Processing

x = n * a**n * sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = [1] / ([2])**2

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Aa * z**-1

B1 - a * z**-1

Cz - a

D1 - z * a

Attempts:

3 left

💡 Hint

Common Mistakes

Forgetting to square the denominator.

Mixing up numerator and denominator terms.

The Z-transform of n aⁿ u[n] is (a z^-1) / (1 - a z^-1)².

5fill in blank

hard

Fill all three blanks to compute the Z-transform of x[n] = cos(ω₀ n) u[n].

Signal Processing

omega0 = sp.symbols('omega0', real=True)
x = sp.cos(omega0 * n) * sp.Heaviside(n)
X = sp.summation(x * z**(-n), (n, 0, sp.oo))
result = ([1] - [2] * z**-1) / (1 - 2 * [3] * z**-1 + z**-2)

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Az

Bcos(omega0)

Dsin(omega0)

Attempts:

3 left

💡 Hint

Common Mistakes

Using sin(ω₀) instead of cos(ω₀).

Mixing up powers of z in numerator and denominator.

The Z-transform of cos(ω₀ n) u[n] is (z - cos(ω₀) z^-1) / (1 - 2 cos(ω₀) z^-1 + z^-2).