in Electronic Devices and Circuits Noise is any unwanted excitation of a circuit, any input that is not an information-bearing signal. Noise comes from External sources: Unintended coupling with other parts of the physical world. In principle, this kind of noise can be virtually eliminated by careful design. Internal sources: Unpredictable microscopic events that happen in the devices that constitute the circuit. In principle, this kind of noise can be reduced, but never eliminated. Noise is especially important to consider when designing low-power systems because the signal levels (typically voltages or currents) are small.
δx = 1 T The amount of noise in a signal is characterized by its root mean square (RMS) value. X()=X t + δx() t δx = 1 T ( Xt () X)dt = 0 0 T T 0 ( Xt () X) dt X(t) Mean signal level The noise level sets the size of the smallest signal that can be processed meaningfully by a physical system. X(t) X δx RMS noise level t t
δv 1 + δi = δi 1 + δi δv = δv 1 + δv δv δi 1 δi The mean squared levels of (statistically) independent noise sources add: δx()=δx t 1 ()+δx t () t ( δxt ()) = δx 1 () t ( ) + ( δx () t ) +δx 1 t ()δx () t δx =δx 1 +δx +δx 1 δx δx =δx 1 +δx +δx 1 δx δx = δx 1 + δx
The distribution of noise power over the spectrum is called the power spectral density (PSD) of the noise. White noise: Noise power is spread uniformly across the spectrum (cf. white light). log S( f ) S( f ) = C Constant power for a fixed f X(t) t f f log S( f ) S( f ) = C/f Constant power for a fixed f 1/f f Pink noise (a.k.a. flicker or 1/f noise): Noise power is concentrated at lower frequencies (cf. pink light). X(t) t f 1/f f 1/f log f
Usually, the noise in a device is a mixture of white noise and 1/f noise, where the two noise processes are independent. PSD S( f )= C 1 f +C log S( f ) 1/f noise dominant S( f ) C 1/f White noise dominant S( f ) C 1/f noise component white noise component f c log f 1/f noise corner In general, the 1/f noise corner frequency is highly process and bias dependent.
Conventional view: There are two distinct kinds. Shot noise: Variations in the arrival times of discrete charge carriers arising from the unpredictable time that each enters the device. δi = q I f Charge of each carrier Mean current level System bandwidth Requires DC current flow. Noise in diodes, BJTs, and vacuum tubes. Thermal noise: Variations in the arrival times of discrete charge carriers arising from their unpredictable thermal motions within in the device. δi = 4 kt G f Boltzmann constant Absolute temperature Conductance System bandwidth Requires no DC current flow. Noise level is directly proportional to T. Noise in resistors, MOSFETs, and JFETs. Unconventional view: Thermal noise is (two-sided) shot noise arising from diffusion currents.
We model the variation in the arrival times of discrete charge carriers at the terminal of a device as a Poisson process with a mean arrival rate, λ. Pr{carrier arrives in (t, t + dt)} λ dt Knowing how many carriers arrived in some past time interval tells us nothing about how many will arrive in a subsequent time interval. et n be the number of carriers arriving in a given time interval T. For a Poisson process, Average current: n = λ T and δn = λ T I = qn T = qλt T = qλ Fluctuation in current over time scale T ( f ~ 1/T): δi = q δn T = q λt T for every time scale T. = q qλ T ~ qi f
Subthreshold MOS Transistor V B V G V D Gate Bulk Contact p + n + n + Source Channel Drain Depletion Bulk ayer p I f I r Surface Energy ψ s V DS V D I f = q D W Q S δi f = q I f f I r = q D W Q D δi r = q I r f I = I f I r δi = δi f + δi r = q (I f + I r ) f δi = q I sat (1 + e V DS/U T ) f
Subthreshold MOS Transistor Saturation (V DS > 4 U T ) I f Noise is minimum because I r 0. ψ s δi = q I sat f V DS I = I sat V D Ohmic (0 < V DS < 4 U T ) I f I r Noise is intermediate because 0 < I r < I sat. ψ s δi = q I sat (1 + e V DS/U T ) f V DS V D I = I sat (1 e V DS/U T ) inear (V DS = 0) I f I r Noise is maximum because I r = I sat. ψ s V D δi = 4 q I sat f I = 0
White Noise in a Resistor n(x) n I f I r I = I f I r = 0 I f I r 0 x Using the subthreshold MOS transistor with V DS = 0 as a model, we can decompose the uniform concentration of electrons in a shorted resistor into equal forward and reverse components I f = q D n A δi f = q I f f I r = q D n A δi r = q I r f q D n A δi = δi f + δi r = 4 q f q µ n A δi = 4 k T f = 4 k T G f Einstein relation D kt µ = q Nyquist s classic thermal noise result derived from a purely shot noise point of view!
in Resistors and MOS Transistors One formula describes the white noise in all cases: δi = 4 k T µ W Q f Subthreshold MOS Transistor: δi = 4 k T µ W Q S + Q D Average charge per unit area f Above Threshold MOS Transistor: δi = 4 k T µ W 3 Q S + Q S Q D + Q D Q S + Q D f Resistor: δi = 4 k T G f
ψ s V G Traps Assume that there is a uniform density, ρ, of traps in the gate oxide, independent of x. A charge, q, will enter or leave a trap at a time scale set by the tunneling probability: Fluctuation frequency f ~ e α x x dx log f ~ α x df ~ α dx f For a fixed frequency interval df, the total amount of trapped charge Q will fluctuate as A ρ dx δq ~ A ρ dx ~ A df f
ψ s V G Traps x dx We can think of the trapped charge as modulating the transistor s threshold voltage: δq ~ A δv δq T ~ 1 df ~ A C G A f ~ 1 A df f C G ~ A δi ~ g m δv T ~ g m A df f