Can two functions have the same frequency domain representation?

Thread starter sanjaysan
Start date Nov 10, 2011
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Domain Frequency Frequency domain Functions Representation

In summary, it is possible for two functions to have the same frequency domain representation, as the Fourier transform only considers the overall distribution of energy in the frequency domain. This can occur if their Fourier transforms have the same value for all frequencies. However, this is not a common occurrence as the Fourier transform takes into account the entire function, including its amplitude, phase, and frequency components. When two functions have the same frequency domain representation, it means they have the same spectral characteristics, even if they are completely different in the time domain.

Nov 10, 2011

sanjaysan

Hi,

I stumbled upon the following two functions which have the same freq domain representation,
1/(j.pi.f)

1.signum(t) = u(t) - u(-t)

2. 2u(t)

what is the reasoning behind them having the same f domain representation?

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Nov 10, 2011

lurflurf

Homework Helper

well
2u(t)-1=signum(t)

Nov 10, 2011

sanjaysan

thanks,that answers my question:)

1. Can two functions have the same frequency domain representation?

Yes, it is possible for two functions to have the same frequency domain representation. This is because the frequency domain representation of a function is determined by the Fourier transform, which converts a function from the time domain to the frequency domain. The Fourier transform is an integral, and it is possible for two different functions to have the same integral value.

2. How can two functions have the same frequency domain representation?

Two functions can have the same frequency domain representation if their Fourier transforms have the same value for all frequencies. This means that the functions have the same distribution of energy in the frequency domain, even if they are different in the time domain.

3. Is it common for two functions to have the same frequency domain representation?

No, it is not common for two functions to have the same frequency domain representation. This is because the Fourier transform takes into account the entire function, including its amplitude, phase, and frequency components. It is unlikely that two functions will have the same combination of these factors.

4. What does it mean when two functions have the same frequency domain representation?

When two functions have the same frequency domain representation, it means that they have the same spectral characteristics. This means that they have the same distribution of energy in the frequency domain, even if they are different in the time domain.

5. Can two functions have the same frequency domain representation but be completely different in the time domain?

Yes, it is possible for two functions to have the same frequency domain representation but be completely different in the time domain. This is because the Fourier transform only takes into account the frequency components of a function, and not its shape or amplitude in the time domain.