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If G( f) represents the Fourier Transform of a signal g (t) which is real and odd symmetric in time, then G (f) is
- a)Complex
- b)Imaginary
- c)Real
- d)Real and non-negative
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If G( f) represents the Fourier Transform of a signal g (t) which is r...
Explanation: For the real and odd symmetric signal in time domain on the Fourier transform the resulting signal is always imaginary.
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If G( f) represents the Fourier Transform of a signal g (t) which is r...
Introduction:
The Fourier Transform is a mathematical tool used to decompose a complex signal into its constituent frequencies. It represents a signal in terms of its frequency components. The Fourier Transform of a signal g(t) is denoted as G(f), where f represents the frequency.
Real and Odd-Symmetric Signal:
A real and odd-symmetric signal is a signal that satisfies two conditions:
1. Real: The signal is made up of real values, meaning it does not contain any complex components.
2. Odd-Symmetric: The signal is symmetric about the origin (t = 0), meaning g(t) = -g(-t) for all values of t.
Fourier Transform of Real and Odd-Symmetric Signal:
When a real and odd-symmetric signal is Fourier transformed, the resulting spectrum G(f) has certain properties:
1. Imaginary:
The Fourier Transform of a real and odd-symmetric signal is purely imaginary. This means that the real part of G(f) is zero, and only the imaginary part is non-zero. This can be mathematically represented as G(f) = 0 + jB(f), where j represents the imaginary unit and B(f) represents the imaginary part of G(f).
2. Real and Non-Negative:
Furthermore, the imaginary part B(f) of G(f) is always non-negative. This means that the magnitude of the imaginary part is always positive or zero, while the phase can vary. In other words, the spectrum G(f) lies entirely in the upper or lower half of the complex plane.
Conclusion:
In summary, the Fourier Transform of a real and odd-symmetric signal is purely imaginary, with a non-negative imaginary part. Option 'B' correctly states that G(f) is imaginary. This property is a result of the symmetries present in the time-domain signal.
The Fourier Transform is a mathematical tool used to decompose a complex signal into its constituent frequencies. It represents a signal in terms of its frequency components. The Fourier Transform of a signal g(t) is denoted as G(f), where f represents the frequency.
Real and Odd-Symmetric Signal:
A real and odd-symmetric signal is a signal that satisfies two conditions:
1. Real: The signal is made up of real values, meaning it does not contain any complex components.
2. Odd-Symmetric: The signal is symmetric about the origin (t = 0), meaning g(t) = -g(-t) for all values of t.
Fourier Transform of Real and Odd-Symmetric Signal:
When a real and odd-symmetric signal is Fourier transformed, the resulting spectrum G(f) has certain properties:
1. Imaginary:
The Fourier Transform of a real and odd-symmetric signal is purely imaginary. This means that the real part of G(f) is zero, and only the imaginary part is non-zero. This can be mathematically represented as G(f) = 0 + jB(f), where j represents the imaginary unit and B(f) represents the imaginary part of G(f).
2. Real and Non-Negative:
Furthermore, the imaginary part B(f) of G(f) is always non-negative. This means that the magnitude of the imaginary part is always positive or zero, while the phase can vary. In other words, the spectrum G(f) lies entirely in the upper or lower half of the complex plane.
Conclusion:
In summary, the Fourier Transform of a real and odd-symmetric signal is purely imaginary, with a non-negative imaginary part. Option 'B' correctly states that G(f) is imaginary. This property is a result of the symmetries present in the time-domain signal.
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