Noise Analysis

Once we have our device models with the appropriate noise generators, we can determine the effects of these individual noise sources on the behavior of larger circuits. Consider the linear and time invariant (LTI) system with transfer function $H\left(f\right)$ shown in Fig. 1. If we inject noise $S_{i}\left(f\right)$, we would get:

$S_{o}\left(f\right)=\left|H\left(f\right)\right|^{2}\cdot S_{i}\left(f\right)$

(1)

Thus, the output noise spectrum is shaped or "filtered" by the magnitude of the transfer function. Note that the phase is random and cannot be determined. The total integrated noise at the output of the LTI system is:

${\overline {v_{o,T}^{2}}}=\int _{-\infty }^{\infty }\left|H\left(f\right)\right|^{2}\cdot S_{i}\left(f\right)\,df$

(2)

Passive Circuits

For the generalized passive circuit shown in Fig. 2, composed of only "noisy" resistors, plus ideal capacitors and inductors, and with an effective impedance across terminals $A$ and $B$ equal to $Z\left(f\right)$, the equivalent noise is:

${\overline {v_{eq}^{2}}}=4kT\cdot \operatorname {Re} \left[Z\left(f\right)\right]\cdot \Delta f$

(3)

Note that since $Z\left(f\right)$ is frequency dependent, the spectral density of the noise is shaped by the inductors and capacitors, even though only the resistors generate noise. We can then calculate the total integrated noise:

${\overline {v_{eq,T}^{2}}}=\int _{B}4kT\cdot \operatorname {Re} \left[Z\left(f\right)\right]\cdot df$

(4)

Example: An RC Circuit

Given the RC circuit in Fig. 3, we can calculate the equivalent impedance seen at the two terminals as:

$Z={\frac {1}{Y}}={\frac {1}{G+j\omega C}}={\frac {G-j\omega C}{G^{2}+\omega ^{2}C^{2}}}$

(5)

Where $G={\frac {1}{R}}$. Thus, the total integrated noise is:

${\begin{aligned}{\overline {v_{eq,T}^{2}}}&=\int _{B}4kT\cdot \operatorname {Re} \left[Z\left(f\right)\right]\cdot df={\frac {4kT}{2\pi }}\int _{0}^{\infty }{\frac {G}{G^{2}+\omega ^{2}C^{2}}}\cdot d\omega \\&={\frac {kT}{C}}\end{aligned}}$

(6)

Note that the total integrated noise is independent of $R$. This is due to the fact that the thermal noise spectral density is proportional to $R$, but the bandwidth is inversely proportional to $R$, thus cancelling each other out.