Continuous Time Fourier in LTI system

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SUMMARY

The discussion focuses on solving a problem related to Continuous Time Fourier Transform (CTFT) in Linear Time-Invariant (LTI) systems. Participants emphasize the importance of understanding the power of a signal through its Fourier transform, particularly for sinusoidal inputs. The solution involves using the definition of power and the transfer function to analyze how sinusoids are transformed by the system. Key insights include computing the power from the Fourier transform and utilizing the magnitude of the transfer function |H|.

PREREQUISITES
  • Continuous Time Fourier Transform (CTFT)
  • Linear Time-Invariant (LTI) systems
  • Signal power calculation methods
  • Fourier transform properties of sinusoids
NEXT STEPS
  • Study the definition of power in the context of signals
  • Learn how to compute the Fourier transform of sinusoidal signals
  • Explore the properties of LTI systems and their transfer functions
  • Investigate the relationship between signal spectrum and power
USEFUL FOR

Students and professionals in electrical engineering, particularly those studying signal processing and system analysis, will benefit from this discussion.

beanus
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Homework Statement


Capture.jpg



Homework Equations


Not sure


The Attempt at a Solution


No idea how to even begin.

I don't even know how to start this equation. My textbook has no examples of this type. Do I need to transform x(t)? If someone could simply steer me in the right direction I'm confident I can figure it out.
 
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beanus said:
I don't even know how to start this equation. My textbook has no examples of this type. Do I need to transform x(t)? If someone could simply steer me in the right direction I'm confident I can figure it out.
For Part A. You can solve this several different ways. If you don't know what to do, you should always resort to the definition of power. But in this case you should know that you can also find the power of a signal from its Fourier transform and this particular signal has a really simple Fourier transform. Normally I would just compute the power of the sinusoids from rms values but part A is leading into part B.

For Part B, the transfer function shows how sinusoids are transformed by the system. Your input is the sum of two sinusoids. Again, there is more than one way to do this one but the question is practically begging you to use |H| somehow. From part A you should have the spectrum of X and therefore its magnitude too.
 
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