Practice Sinusoidal Signals (cos(omega0t) And Sin(omega0t)) (4.4.5) - Fourier Transform Analysis of Continuous-Time Aperiodic Signals
Students

Academic Programs

AI-powered learning for grades 8-12, aligned with major curricula

Professional

Professional Courses

Industry-relevant training in Business, Technology, and Design

Games

Interactive Games

Fun games to boost memory, math, typing, and English skills

Sinusoidal Signals (cos(omega0t) and sin(omega0t))

Practice - Sinusoidal Signals (cos(omega0t) and sin(omega0t))

Learning

Practice Questions

Test your understanding with targeted questions

Question 1 Easy

What is the formula for cosine using Euler's identity?

💡 Hint: Hint: Recall how Euler's identity connects exponentials and trigonometric functions.

Question 2 Easy

Name the outputs of the Fourier Transform for cos(omega0t).

💡 Hint: Think about where the frequency content lies for a cosine function.

4 more questions available

Interactive Quizzes

Quick quizzes to reinforce your learning

Question 1

What is the Fourier Transform of cos(omega0t)?

X(jω) = π[δ(ω - ω0) + δ(ω + ω0)]
X(jω) = jπ[δ(ω - ω0) - δ(ω + ω0)]
X(jω) = e^(jω0t)

💡 Hint: Think about how cosines decompose into their frequency components.

Question 2

True or False: The Fourier Transform of sin(omega0t) leads to real-valued impulses.

True
False

💡 Hint: Consider how sine functions behave compared to cosine in Fourier analysis.

Get performance evaluation

Challenge Problems

Push your limits with advanced challenges

Challenge 1 Hard

Discuss how changing the amplitude of a sine wave affects its Fourier Transform. Provide a deeper analysis with mathematical support.

💡 Hint: Consider how changing the amplitude impacts the heights of the impulse responses in the frequency domain.

Challenge 2 Hard

Consider a system with both sine and cosine inputs. Describe how you would analyze them using their Fourier Transforms together. What characterizes their combined behavior?

💡 Hint: Think about how both signals interact and what their combined implications for filtering might be.

Get performance evaluation

Reference links

Supplementary resources to enhance your learning experience.