Continuous-time vs discrete-time signals in Signal Processing - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to define a continuous-time signal using a sine wave.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 1, 1000)
x = np.sin(2 * np.pi * 5 * [1])
plt.plot(t, x)
plt.title('Continuous-time signal')
plt.show()

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

A5

Bnp.arange(0, 1, 0.01)

Cnp.linspace(0, 1, 10)

Dt

Attempts:

3 left

💡 Hint

Common Mistakes

Using a discrete array like np.arange instead of continuous time vector.

Using a constant number instead of the time variable.

The variable t represents continuous time values, so it should be used inside the sine function to create a continuous-time signal.

2fill in blank

medium

Complete the code to create a discrete-time signal sampled from a sine wave.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

n = np.arange(0, 50)
x = np.sin(2 * np.pi * 0.1 * [1])
plt.stem(n, x)
plt.title('Discrete-time signal')
plt.show()

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

An

Bnp.linspace(0, 1, 50)

Cnp.arange(0, 1, 0.1)

D50

Attempts:

3 left

💡 Hint

Common Mistakes

Using continuous time vectors instead of discrete indices.

Using a constant number instead of the time index.

The variable n represents discrete time indices, so it should be used inside the sine function to create a discrete-time signal.

3fill in blank

hard

Fix the error in the code to correctly plot a discrete-time signal.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

n = np.arange(0, 20)
x = np.sin(2 * np.pi * 0.2 * n)
plt.plot(n, [1])
plt.title('Discrete-time signal')
plt.show()

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

An

Bx

Cnp.linspace(0, 1, 20)

Dnp.sin(n)

Attempts:

3 left

💡 Hint

Common Mistakes

Plotting the time indices instead of the signal values.

Using a different function call argument.

The variable x holds the signal values to plot against the discrete time indices n.

4fill in blank

hard

Fill both blanks to create a dictionary mapping discrete time indices to signal values for samples greater than 0.5.

Signal Processing

n = np.arange(0, 30)
x = np.sin(2 * np.pi * 0.1 * n)
samples = { [1]: [2] for [1] in n if x[[1]] > 0.5 }

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

Ai

Bx[i]

Cn

Dx[n]

Attempts:

3 left

💡 Hint

Common Mistakes

Using the whole array n as a key instead of an index.

Using x[n] which is the whole array instead of a single value.

We use i as the loop variable iterating over n. The dictionary keys are i, and the values are the corresponding signal values x[i] where the condition is met.

5fill in blank

hard

Fill all three blanks to create a dictionary of squared signal values for discrete times where the signal is positive.

Signal Processing

n = np.arange(0, 40)
x = np.sin(2 * np.pi * 0.05 * n)
squared_positive = { [1]: [2]**2 for [1] in n if [3] > 0 }

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

Ak

Bx[k]

Dn

Attempts:

3 left

💡 Hint

Common Mistakes

Using the whole array n as a key.

Checking condition on the wrong variable.

Not squaring the signal value.

We use k as the loop variable. The dictionary keys are k, the values are the squares of x[k], and the condition checks if x[k] is positive.

Practice

1. Which of the following best describes a continuous-time signal?

easy

A. A signal defined only at specific time points

B. A signal that is always zero

C. A signal that only takes integer values

D. A signal defined for every instant of time

Solution

1. Step 1: Understand continuous-time signal definition

   A continuous-time signal is defined for every instant in time, meaning it varies smoothly without gaps.

2. Step 2: Compare with discrete-time signals

   Discrete-time signals are defined only at specific time points, not continuously.

3. Final Answer:

   A signal defined for every instant of time -> Option D

4. Quick Check:

   Continuous-time = defined at all times [OK]

Hint: Continuous means all time points included [OK]

Common Mistakes:

• Confusing continuous with discrete signals
• Thinking discrete signals are continuous
• Assuming continuous signals only take integer values

2. Which syntax correctly represents a discrete-time signal sampled from a continuous signal x(t) at integer times n?

easy

A. x[n] = integral of x(t)

B. x(n) = x(t) for all t

C. x[n] = x(t) at t = n

D. x(t) = x[n] for n in integers

Solution

1. Step 1: Identify discrete-time signal notation

   Discrete-time signals use square brackets [n] to denote samples at integer times.

2. Step 2: Match sampling definition

   Sampling means taking values of x(t) at specific times t = n, so x[n] = x(t) at t = n is correct.

3. Final Answer:

   x[n] = x(t) at t = n -> Option C

4. Quick Check:

   Discrete-time uses square brackets and samples at integers [OK]

Hint: Discrete signals use square brackets for indexing [OK]

Common Mistakes:

• Using parentheses instead of square brackets for discrete signals
• Confusing continuous and discrete notation
• Thinking sampling is an integral operation

3. Given the continuous-time signal x(t) = 2t, what is the discrete-time signal x[n] sampled at integer times?

medium

A. x[n] = 2n

B. x[n] = 2t

C. x[n] = n/2

D. x[n] = 2

Solution

1. Step 1: Understand sampling at integer times

   Sampling means replacing t by integer n in x(t).

2. Step 2: Substitute t = n in x(t) = 2t

   We get x[n] = 2n.

3. Final Answer:

   x[n] = 2n -> Option A

4. Quick Check:

   Replace t with n in formula [OK]

Hint: Replace t with n for discrete samples [OK]

Common Mistakes:

• Leaving t as variable in discrete signal
• Dividing instead of multiplying by 2
• Confusing n and t variables

4. Identify the error in this discrete-time signal definition: x[n] = x(t) at t = n + 0.5

medium

A. The notation x[n] is incorrect

B. Sampling at non-integer times is invalid for discrete-time signals

C. The signal should be continuous, not discrete

D. There is no error; this is a valid discrete-time signal

Solution

1. Step 1: Understand discrete-time signal sampling points

   Discrete-time signals are defined at integer time points n.

2. Step 2: Check sampling time t = n + 0.5

   Sampling at half-integers is not standard discrete-time sampling; it breaks the integer indexing rule.

3. Final Answer:

   Sampling at non-integer times is invalid for discrete-time signals -> Option B

4. Quick Check:

   Discrete-time = integer time samples only [OK]

Hint: Discrete-time signals sample only integer times [OK]

Common Mistakes:

• Thinking discrete signals can sample at any time
• Confusing notation with error
• Assuming half-integers are valid discrete points

5. You have a continuous-time signal x(t) = sin(2πt). You sample it at 1 Hz (once per second). Which discrete-time signal x[n] do you get, and what is a potential problem?

hard

A. x[n] = sin(2πn); signal appears constant at zero

B. x[n] = sin(2πn); signal is perfectly captured

C. x[n] = sin(πn); signal is undersampled

D. x[n] = sin(4πn); signal is oversampled

Solution

1. Step 1: Write discrete-time signal from sampling

   Sampling at 1 Hz means t = n seconds, so x[n] = sin(2πn).

2. Step 2: Evaluate sin(2πn) for integer n

   Since sin(2πn) = 0 for all integers n, the sampled signal is always zero.

3. Step 3: Identify the problem

   This is an example of aliasing where the sampled signal loses information and appears constant.

4. Final Answer:

   x[n] = sin(2πn); signal appears constant at zero -> Option A

5. Quick Check:

   Sampling frequency equals signal frequency causes zero samples [OK]

Hint: Sampling at signal period causes zero values [OK]

Common Mistakes:

• Assuming sampling always captures signal perfectly
• Ignoring aliasing effects
• Confusing frequency and sampling rate