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 $Z\left(j\omega \right)$, the equivalent noise is:

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

(3)