Scaling properties of 1/f Noise don't make sense

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I understand that the Hurst exponent of $frac1f$ (pink) noise is $0$ which means the statistics of $fracx(kt)k^0=x(kt)$ are the same as $x(t) $ assuming $x$ is pink noise. However, I don't see how this aligns with its power spectrum.



It is well known that the power spectrum can be calculated by averaging periodograms i.e. the norm squared of the Fourier transform. So, let $x(t)$ be pink noise, then ...



$frac1omegasimtextPer[x(kt)](omega)=|mathcalF[x(kt)](omega)|^2=|frac1kmathcalF[x(t)](fracomegak)|^2=frac1k^2textPer[x(kt)](fracomegak) sim frac1komega$



I use $sim$ because they are not exactly equivalent but works for these purposes as we are just looking at how these equations scale.



When scaling according to the Hurst exponent, it seems the periodogram actually changes which shouldn't happen. Where is the error in my reasoning?



Note: I am trying to test if my generations of $frac1f$ noise have the right amplitudes.







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  • By the way, the fact that it scales like this with the Hurst exponent I got from the paper hal.in2p3.fr/in2p3-00024797v3/document . It did not explain why though ...
    – Aakash Lakshmanan
    Aug 27 at 16:40














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I understand that the Hurst exponent of $frac1f$ (pink) noise is $0$ which means the statistics of $fracx(kt)k^0=x(kt)$ are the same as $x(t) $ assuming $x$ is pink noise. However, I don't see how this aligns with its power spectrum.



It is well known that the power spectrum can be calculated by averaging periodograms i.e. the norm squared of the Fourier transform. So, let $x(t)$ be pink noise, then ...



$frac1omegasimtextPer[x(kt)](omega)=|mathcalF[x(kt)](omega)|^2=|frac1kmathcalF[x(t)](fracomegak)|^2=frac1k^2textPer[x(kt)](fracomegak) sim frac1komega$



I use $sim$ because they are not exactly equivalent but works for these purposes as we are just looking at how these equations scale.



When scaling according to the Hurst exponent, it seems the periodogram actually changes which shouldn't happen. Where is the error in my reasoning?



Note: I am trying to test if my generations of $frac1f$ noise have the right amplitudes.







share|cite|improve this question




















  • By the way, the fact that it scales like this with the Hurst exponent I got from the paper hal.in2p3.fr/in2p3-00024797v3/document . It did not explain why though ...
    – Aakash Lakshmanan
    Aug 27 at 16:40












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I understand that the Hurst exponent of $frac1f$ (pink) noise is $0$ which means the statistics of $fracx(kt)k^0=x(kt)$ are the same as $x(t) $ assuming $x$ is pink noise. However, I don't see how this aligns with its power spectrum.



It is well known that the power spectrum can be calculated by averaging periodograms i.e. the norm squared of the Fourier transform. So, let $x(t)$ be pink noise, then ...



$frac1omegasimtextPer[x(kt)](omega)=|mathcalF[x(kt)](omega)|^2=|frac1kmathcalF[x(t)](fracomegak)|^2=frac1k^2textPer[x(kt)](fracomegak) sim frac1komega$



I use $sim$ because they are not exactly equivalent but works for these purposes as we are just looking at how these equations scale.



When scaling according to the Hurst exponent, it seems the periodogram actually changes which shouldn't happen. Where is the error in my reasoning?



Note: I am trying to test if my generations of $frac1f$ noise have the right amplitudes.







share|cite|improve this question












I understand that the Hurst exponent of $frac1f$ (pink) noise is $0$ which means the statistics of $fracx(kt)k^0=x(kt)$ are the same as $x(t) $ assuming $x$ is pink noise. However, I don't see how this aligns with its power spectrum.



It is well known that the power spectrum can be calculated by averaging periodograms i.e. the norm squared of the Fourier transform. So, let $x(t)$ be pink noise, then ...



$frac1omegasimtextPer[x(kt)](omega)=|mathcalF[x(kt)](omega)|^2=|frac1kmathcalF[x(t)](fracomegak)|^2=frac1k^2textPer[x(kt)](fracomegak) sim frac1komega$



I use $sim$ because they are not exactly equivalent but works for these purposes as we are just looking at how these equations scale.



When scaling according to the Hurst exponent, it seems the periodogram actually changes which shouldn't happen. Where is the error in my reasoning?



Note: I am trying to test if my generations of $frac1f$ noise have the right amplitudes.









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 27 at 16:39









Aakash Lakshmanan

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  • By the way, the fact that it scales like this with the Hurst exponent I got from the paper hal.in2p3.fr/in2p3-00024797v3/document . It did not explain why though ...
    – Aakash Lakshmanan
    Aug 27 at 16:40
















  • By the way, the fact that it scales like this with the Hurst exponent I got from the paper hal.in2p3.fr/in2p3-00024797v3/document . It did not explain why though ...
    – Aakash Lakshmanan
    Aug 27 at 16:40















By the way, the fact that it scales like this with the Hurst exponent I got from the paper hal.in2p3.fr/in2p3-00024797v3/document . It did not explain why though ...
– Aakash Lakshmanan
Aug 27 at 16:40




By the way, the fact that it scales like this with the Hurst exponent I got from the paper hal.in2p3.fr/in2p3-00024797v3/document . It did not explain why though ...
– Aakash Lakshmanan
Aug 27 at 16:40















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