Sources of noise
White noise:
Other noise:
Shot noise is the noise due to descrete nature of electric charge. Shot noise can be modeled as a Poisson process. A fundamental consideration in electrical and optical devices including photography.
In a Poisson process:
or less that $t$
$$P(t \leq h) = 1 - e^{-\nu h}$$This last expression is the cumulative distribution function. The probably density function is the derivative
$$f(h) = \nu e ^ {-\nu h}$$Frequency domain
A frequency-domain derivation of shot noise
$$i_{noise} = \sqrt{2 q I_{dc}}$$where $q = 1/C$ is the electron charge $1.60\times 10^{-19}$ C.
Signal to Noise Ratio
Signal to noise ratio is normally expressed as the power of the signal divided by the power of the noise.
$$\text{SNR} = \frac{\mu^2}{\sigma^2}$$If the interval is $\delta t$ the current is $I$, rate of charge carriers is $C I h$ where $C$ is Coulomb constant. The SNR is then
$$\text{SNR} = \frac{(C I \delta t)^2}{C I \delta t} = C I \delta t$$Question: Suppose you need 5 sigma accuracy (i.e, $\frac{\mu}{\sigma} > 5$) and are measuring a 1 nA signal, what is the fastest sampling rate you can expect?
Simulated Photon Noise
A sequence of images in which the average number of photons captured per pixel increases by factors of 10x between images. source
Question:
# Demonstration
#
# Simulate shot noise for a 1 pA current
%matplotlib inline
import random
import math
import numpy as np
import matplotlib.pyplot as plt
# constants
C = 6.241509074e18 # number of electrons in a Coulomb
I = 1e-12 # 1 pA of current
h = 1e-6 # 1 microsecond
nu = C*I
print("nu =", nu, "electrons per second")
print("expected number of electrons per step nu*h =", nu*h)
K = 600
# simulate arrival of K charge carriers
dt = np.random.exponential(1/nu, size=(K))
t = np.cumsum(dt)
# plot results
y = np.array([1]*len(t))
fig, ax = plt.subplots(2, 1, figsize=(12, 8))
ax[0].plot(t, y, '.')
ax[1].step(t, np.cumsum(y)/C, where="pre")
ax[1].plot(t, I*t)
ax[1].set_ylabel('coulombs')
ax[1].set_xlabel('seconds')
nu = 6241509.074 electrons per second expected number of electrons per step nu*h = 6.241509074
Text(0.5, 0, 'seconds')
Johnson noise has the same origins as black-body radiation. The average (root mean square) voltage due to thermal noise in a resistor is
$$v_{noise} = \sqrt{4kTRB}$$where $k$ is Boltzmann's constant, T is absolute temperature, R is resistance, and B is bandwidth in Hertz. At 20 deg C
\begin{align*} 4kT & = 1.62 \times 10^{-20} & V^2/Hz-\Omega \\ \sqrt{4ktR} & = 1.27\times 10^{-10}\sqrt{R} & V/Hz^{1/2} \end{align*}The short-circuit current noise is
$$i_{noise} = v_{noise}/R = \sqrt{\frac{4kTB}{R}}$$Question: What is the RMS voltage of 10k ohm resistor?