Math Problem Statement

A signal \( x(t) = 2\sin(t) \) is passed through an LTI system with impulse response \( h(t) = e^{-t}u(t) \). The maximum value of amplitude at the output of the system is

Solution

To find the maximum value of amplitude at the output of the system, let’s analyze the given information:

  1. Input Signal x(t)x(t): x(t)=2sin(t)x(t) = 2 \sin(t)

  2. Impulse Response h(t)h(t): h(t)=etu(t)h(t) = e^{-t} u(t) where u(t)u(t) is the unit step function, implying that h(t)=0h(t) = 0 for t<0t < 0.

  3. Output y(t)y(t): Since this is a Linear Time-Invariant (LTI) system, the output y(t)y(t) is given by the convolution of x(t)x(t) and h(t)h(t): y(t)=x(t)h(t)y(t) = x(t) * h(t)

To calculate y(t)y(t), let’s examine the convolution:

Step-by-Step Solution:

  1. The Fourier Transform of x(t)=2sin(t)x(t) = 2 \sin(t) is: X(ω)=2j2(δ(ω1)δ(ω+1))=j(δ(ω1)δ(ω+1))X(\omega) = 2 \cdot \frac{j}{2} \left( \delta(\omega - 1) - \delta(\omega + 1) \right) = j \left( \delta(\omega - 1) - \delta(\omega + 1) \right)

  2. The Fourier Transform of h(t)=etu(t)h(t) = e^{-t} u(t) is: H(ω)=11+jωH(\omega) = \frac{1}{1 + j\omega}

  3. The output in the frequency domain is Y(ω)=X(ω)H(ω)Y(\omega) = X(\omega) H(\omega).

  4. Calculating Y(ω)Y(\omega) at ω=1\omega = 1 (the frequency component of x(t)x(t)): Y(1)=j11+j=j1+j1j1j=j12=12ejπ/4Y(1) = j \cdot \frac{1}{1 + j} = \frac{j}{1 + j} \cdot \frac{1 - j}{1 - j} = \frac{j - 1}{2} = \frac{1}{\sqrt{2}} e^{-j\pi/4} The magnitude of this term is 12\frac{1}{\sqrt{2}}.

  5. Since the amplitude of x(t)x(t) is scaled by 12\frac{1}{\sqrt{2}}, the maximum amplitude of the output will be: 2×12=22 \times \frac{1}{\sqrt{2}} = \sqrt{2}

Answer:

The maximum amplitude of the output is 2\sqrt{2}.

Would you like further clarification on any part of this solution?

Here are some related questions for further exploration:

  1. What is the significance of convolution in LTI systems?
  2. How does the frequency response of a system affect the output amplitude?
  3. Can this method be used for any input signal? Why or why not?
  4. How would the output change if h(t)h(t) had a different form?
  5. How does the unit step function u(t)u(t) impact the impulse response?

Tip: When dealing with sinusoidal inputs to LTI systems, analyzing the frequency response can often simplify finding the output amplitude.

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Math Problem Analysis

Mathematical Concepts

Linear Time-Invariant (LTI) Systems
Convolution
Fourier Transform
Amplitude Calculation

Formulas

Output \( y(t) = x(t) * h(t) \)
Fourier Transform of \( x(t) = 2\sin(t) \)
Fourier Transform of \( h(t) = e^{-t}u(t) \)
Magnitude of frequency response: \( |Y(\omega)| \)

Theorems

Frequency Response of LTI Systems

Suitable Grade Level

Undergraduate (Signal Processing or Electrical Engineering Courses)