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Question 1
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Solve for the following signals
The FT of
is:
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The FT of
is:
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The FT of
, where
is the triangular function Choose…
The FT Of
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Question 2
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Consider the following signal:
The value of
is
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The dc-value of the signal
is Choose…
The average power is
The value of
is
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The fundamental Frequency
is
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The FT of
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The FT of
The FT Of
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The value of
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Ask by Matthews Higgins. in Israel
Jan 30,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
Question 1:
- The Fourier Transform of
is . - The Fourier Transform of
is . - The Fourier Transform of
is . - The Fourier Transform of
is .
Question 2:
-
- DC value = 2
- Average power = 10
-
- Fundamental frequency
Hz
Solución
Let’s solve the problems step by step.
Question 1
-
The FT of
:
The Fourier Transform (FT) of the sinc function is a rectangular function. The FT ofis . Here, , so: -
The FT of
:
This is a linear function. The FT ofis given by . Here, and : -
The FT of
:
The triangular function has a known FT. The FT ofis . Here, : -
The FT of
:
The FT ofis . The FT of is :
Question 2
-
The value of
:
The Fourier series coefficientfor can be calculated using the formula: Here,and . The average value of is , so: -
The dc-value of the signal
:
The DC value is the average value over one period:The average value ofis and the average value of is : -
The average power:
The average powercan be calculated as: The average power ofis and for is : -
The value of
:
The Fourier series coefficientcan be calculated similarly. Since and functions do not contribute to negative frequencies, . -
The fundamental frequency
:
The fundamental frequency is given by:The periodis the least common multiple of the periods of the components:
Summary of Answers
- FT of
: - FT of
: - FT of
: - FT of
:
For Question 2:
-
- DC value = 2
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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Beyond the Answer
The Fourier Transform (FT) is a fascinating mathematical tool that transforms signals from the time domain into the frequency domain. It helps us analyze the frequencies contained in a signal, revealing critical insights about its behavior. For instance, the sinc function, prevalent in signal processing, has a rectangular shape in the frequency domain, demonstrating how the FT creates a correspondence between time and frequency!
Real-world applications of the Fourier Transform are everywhere! From audio signal processing, where it helps us manipulate sound waves for music production, to image compression techniques in JPEG files, FT enables efficient data handling. Even engineers use it to analyze vibrations in machinery. Understanding this transforms abstract math into a world of practical wonders!
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