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

Challenge - 5 Problems

🧠 Conceptual

intermediate

Identify the nature of a signal

Which of the following signals is a discrete-time signal?

A A signal defined for every real number time t, like x(t) = sin(t) for all real t

B A signal defined only at integer time points, like x[n] = sin(n) for n = 0,1,2,...

C A signal that changes continuously over time without gaps

D A signal that is constant and does not change over time

💡 Hint

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

Discrete-time signals have values only at specific time steps (integers), unlike continuous-time signals which have values at every instant.

❓ Predict Output

intermediate

Output of sampling a continuous signal

What is the output array when sampling the continuous-time signal x(t) = t from t=0 to t=4 with a sampling interval of 1?

Signal Processing

import numpy as np

t = np.arange(0, 5, 1)
x = t
print(x)

A [0 2 4 6 8]

B [0 0.5 1 1.5 2]

C [0 1 2 3 4]

D [1 2 3 4 5]

💡 Hint

Sampling with interval 1 means taking values at 0,1,2,3,4.

The continuous signal x(t) = t sampled at t=0,1,2,3,4 gives values equal to those time points.

❓ data_output

advanced

Length of discrete-time signal after downsampling

Given a discrete-time signal x[n] of length 10, what is the length of the signal after downsampling by a factor of 3?

Signal Processing

import numpy as np

x = np.arange(10)
y = x[::3]
print(len(y))

A 4

B 3

C 7

D 10

💡 Hint

Downsampling by 3 means taking every third sample starting at index 0.

From 10 samples, taking every 3rd gives indices 0,3,6,9 → 4 samples total.

❓ visualization

advanced

Plot continuous vs discrete signals

Which plot correctly shows a continuous-time signal and its sampled discrete-time version?

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 2*np.pi, 1000)
x_cont = np.sin(t)

n = np.arange(0, 2*np.pi, np.pi/4)
x_disc = np.sin(n)

plt.figure(figsize=(6,4))
plt.plot(t, x_cont, label='Continuous')
plt.stem(n, x_disc, linefmt='r-', markerfmt='ro', basefmt='k-', label='Discrete')
plt.legend()
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.title('Continuous and Discrete Signals')
plt.show()

A A plot with a smooth blue curve and red dots with stems at sampled points

B A plot with only red dots scattered randomly

C A plot with a jagged blue line and no dots

D A plot with overlapping green and yellow lines

💡 Hint

Discrete signals are shown as dots with stems, continuous as smooth lines.

The continuous signal is a smooth sine wave; the discrete samples are red dots with vertical lines.

🔧 Debug

expert

Identify error in discrete-time signal generation

What error occurs when running this code to generate a discrete-time signal x[n] = n^2 for n=0 to 4?

Signal Processing

n = range(5)
x = [n**2 for n in n]
print(x)

A TypeError because n is used as both list and variable

B SyntaxError due to invalid list comprehension

C NameError because n is not defined

D No error; output is [0, 1, 4, 9, 16]

💡 Hint

Check variable names and list comprehension syntax carefully.

The code correctly uses list comprehension and variable shadowing is allowed; output is squares of 0 to 4.

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