corso di laurea infisica the physicsof energy
TRANSCRIPT
![Page 1: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/1.jpg)
Corso di Laurea in FISICAThe Physics of Energy
Luca Gammaitoni
Corso di Laurea in Fisica, 2020-2021
![Page 2: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/2.jpg)
Random data II
Frequency domain analysis
![Page 3: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/3.jpg)
Fourier transformThe Fourier transform is given by
and the inverse transform is given by
The Fourier transform is also a random variable
![Page 4: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/4.jpg)
Power spectral density
Average value of the squared magnitude of the Fourier transform
![Page 5: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/5.jpg)
Wiener-Khinchin theorem
The autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by
the power spectrum of that process
![Page 6: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/6.jpg)
Parseval’s theorem
The average value of the square of the signal (variance if the signal has zero mean) is equal to the integral of the power spectral density
![Page 7: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/7.jpg)
Examples
![Page 8: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/8.jpg)
![Page 9: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/9.jpg)
![Page 10: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/10.jpg)
![Page 11: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/11.jpg)
![Page 12: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/12.jpg)
Examples
![Page 13: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/13.jpg)
White noise
• White noise is a random signal with a constant power spectral density
• Sequence of serially uncorrelated random variables with zero mean and infinite variance
• If each sample has a normal distribution with zero mean, the signal is said to be Gaussian white noise
![Page 14: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/14.jpg)
Johnson-Nyquist noise
• Relaxation of thermal fluctuation in a resistor
• Small voltage fluctuation associated with thermal motion of electrons
• Example of fluctuation dissipation relationship
![Page 15: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/15.jpg)
Shot noise
• Generated by discrete arrival
• electrons in a wire
• rain on a roof
• Interactions can be ignored
• Arrival independent
• Poisson process
![Page 16: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/16.jpg)
Shot noise
![Page 17: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/17.jpg)
1/f and switching noises• Found in a wide range of transport
processes
• The power spectrum diverges at low frequencies inversely proportional to the frequency
• Scale invariant (look the same at all time scale)
![Page 18: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/18.jpg)
1/f and switching noises
• Many type of defects on a conductor
• lattice vacancies
• dopant atoms
• Different inequivalent types of sites in the material, which have different energies
• There is a probability for a defect to be excited to an higher-energy state and then relaxed to a lower-energy state
![Page 19: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/19.jpg)
Noise in a transistor• Flat if no current flowing: Johnson noise
• 1/f whit current flowing
Characterization of polysilicon bipolar transistors by low-frequency noise and correlation noise measurementsY Mourier, S G-Jarrix, C Delseny, F Pascal, A Pénarier and D Gasquet
![Page 20: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/20.jpg)
Example: physical system pendulumFocus on the pendulum angle
If we come back after a while..
Mass m= 1 Kg, Length l = 1 m, rms motion = 2 10-11 m
![Page 21: Corso di Laurea inFISICA The Physicsof Energy](https://reader034.vdocuments.site/reader034/viewer/2022042902/6269bd3f836fe672633fc644/html5/thumbnails/21.jpg)
To learn more:
Random DataJ. Bendat, A. G. Piersol
Chap. 1 Basic Descriprions and properties