Noise Analysis

Noise in LTI Systems

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.

Two-Port Noise Analysis

In general, we perform noise analysis in the small signal domain since, for most circuits, the noise signals are relatively small, allowing us to use superposition. Thus, for a circuit with "noisy" elements, we can calculate the noise seen at any port, e.g. the output port, as:

${\overline {v_{o}^{2}}}=\sum _{k}\left|H_{k,o}\right|^{2}\cdot {\overline {v_{n,k}^{2}}}$

(7)

Where ${\overline {v_{n,k}^{2}}}$ is the noise from the $k^{\mathrm {th} }$ noise source, $H_{k,o}$ is the gain from $k^{\mathrm {th} }$ noise source to the port of interest, and we also assume the noise sources are independent of each other.

Input Equivalent Noise Generators

The output noise is a convenient metric to show how "noisy" a circuit is, since it is easy to measure. However, comparing the noise performance of different circuits solely based on the output noise can lead to erroneous results since the output noise also depends on the circuit gain. Thus, we are not sure if a large output noise is due to the inherent noise of the circuit, or possibly due to a relatively large gain. In order to remove this dependency on the circuit gain, we can instead model a "noisy" two-port network using input equivalent noise generators, as shown in Fig. 4.

Thus, we can replace any noisy two-port network with a noiseless two-port network and (1) an equivalent input voltage noise source, ${\overline {v_{n}^{2}}}$, and (2) an equivalent current noise source, ${\overline {i_{n}^{2}}}$. Note that in general, these noise sources are correlated, but let us ignore the correlation for now.