Linear Time Invariant Systems - Graphical Convolution and Convolution to Solve the Zero-State Response of Linear Constant Coefficient Differential Equations (LCCDEs)

Question 1

What is the definition of convolution?

Answer 1

Convolution is the operation that combines an input signal 𝑥(𝑡) with a system’s impulse response ℎ(𝑡) to get the output 𝑦(𝑡)

It can be mathematically expressed as: 𝑦(𝑡) = 𝑥 ∗ ℎ(𝑡) = ∫𝑥(𝜏)ℎ(𝑡−𝜏)d𝜏.

Question 2

What are the ways to compute convolution?

Answer 2

• Analytically
• With simplification/using properties
• Graphically
• Computationally

Question 3

True or False: Convolution involves squaring the functions being convolved.

Answer 3

False

Convolution does not involve squaring.

Question 4

What is the graphical method of convolution often referred to as?

Answer 4

Flip and slide

Question 5

Fill in the blank: The impulse response of a Linear, Constant-Coefficient, Differential Equation (LCCDE) can be found by making 𝑥(𝑡) = ______ and 𝑦(𝑡) = ℎ(𝑡).

Answer 5

𝛿(𝑡)

Question 6

What is the frequency response of a system?

Answer 6

The frequency response is the impulse response but in the frequency domain, denoted as H(ω).

Question 7

How can the output of a system be determined from its input and frequency response?

Answer 7

Y(ω) = H(ω) * X(ω)

Question 8

What is the mathematical expression for the output in terms of convolution in the frequency domain?

Answer 8

Y(ω) = H(ω) * X(ω)

Question 9

What does the convolution integral represent?

Answer 9

The integral of functions after they have been shifted.

Question 10

What is the result of convolving a function with a delta function δ(t)?

Answer 10

The original function shifted by T: x(t) * δ(t - T) = x(t - T)

Question 11

What is the steady-state frequency response?

Answer 11

It is the frequency response of the system when the input is sinusoidal.

Question 12

What is the formula for the output of a convolution in terms of the input and impulse response?

Answer 12

y(t) = ∫x(τ)h(t - τ)dτ

Question 13

Fill in the blank: The frequency response can be determined by the integral H(ω) = ______.

Answer 13

∫h(τ)e^(-jωτ)dτ

Question 14

What is the relationship between the output of a system and its impulse response?

Answer 14

The output is the convolution of the input with the impulse response.

Question 15

What does it mean for a system to be Linear and Time-Invariant (LTI)?

Answer 15

A system is linear if it satisfies the principle of superposition, and time-invariant if its behavior does not change over time.

Question 16

What is the importance of using the frequency domain in system analysis?

Answer 16

It is often easier to work in the frequency domain than in the time domain.

Question 17

How is the impulse response related to the frequency response in LCCDEs?

Answer 17

The impulse response h(t) corresponds to the frequency response H(ω) in the frequency domain.